1 Introduction
Recently, Lue Pan gave a new proof of the Fontaine–Mazur conjecture for Hodge–Tate representations of
$\operatorname {\mathrm {Gal}}(\overline {\mathbb Q}/\mathbb {Q})$
with equal Hodge–Tate weights [Reference Pan38]. One of the ingredients in his proof are the Bernstein-centre type of results for the category of mod p representations of
$\operatorname {\mathrm {GL}}_2(\mathbb {Q}_p)$
proved in [Reference Paškūnas39, Reference Paškūnas41]. This caused him to impose some restrictions on the Galois representations at p, when
$p=2$
and
$p=3$
. In this paper we remove these restrictions by proving the required finiteness results in these remaining cases (Theorem 7.3).
Let L be a finite extension of
$\mathbb {Q}_p$
with ring of integers
$\mathscr O$
and residue field k. Let
$G=\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$
and let
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G}(\mathscr O)$
be the category of smooth G-representations on
$\mathscr O$
-torsion modules. Let
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G}(\mathscr O)$
be the full subcategory of
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G}(\mathscr O)$
consisting of representations which are equal to the union of their subrepresentations of finite length. This is equivalent to the requirement that every finitely generated subrepresentation be of finite length, and we call such representations locally finite. We fix a character
$\zeta :Z\rightarrow \mathscr O^{\times }$
of the centre of G, and let
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
be the full subcategory of
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G}(\mathscr O)$
consisting of representations with central character
$\zeta $
. The category
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
is locally finite and by general results of Gabriel [Reference Gabriel31] decomposes as a product
$$ \begin{align*} \operatorname{\mathrm{Mod}}^{\mathrm{l.fin}}_{G, \zeta}(\mathscr O)\cong \prod_{\mathfrak B} \operatorname{\mathrm{Mod}}^{\mathrm{l.fin}}_{G, \zeta}(\mathscr O)_{\mathfrak B} \end{align*} $$
of indecomposable subcategories, called blocks. Moreover, each block is antiequivalent to the category of pseudocompact modules over a pseudocompact ring
$E_{\mathfrak B}$
. The centre of the ring
$E_{\mathfrak B}$
, which we denote by
$Z_{\mathfrak B}$
, is naturally isomorphic to the centre of the category
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G,\zeta }(\mathscr O)_{\mathfrak B}$
, which by definition is the ring of natural transformations of the identity functor. This means that
$Z_{\mathfrak B}$
acts functorially on every object in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
. The finiteness result in the title is an analogue of a result of Bernstein in the theory of smooth representations of p-adic groups on
$\mathbb {C}$
-vector spaces [Reference Bernstein5, Proposition 3.3]:
Theorem 1.1. The ring
$Z_{\mathfrak B}$
is Noetherian, and
$E_{\mathfrak B}$
is a finitely generated
$Z_{\mathfrak B}$
-module.
To prove the theorem we use in an essential way the direct connection between the
$\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p)$
representations and the representations of the absolute Galois group of
$\mathbb {Q}_p$
, which we denote by
$\mathscr G_{\mathbb {Q}_p}$
, discovered by Colmez in [Reference Colmez19], via his celebrated Montreal functor
$\check {\mathbf {V}}$
, which we review in Section 4.4.
For each fixed block
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B}$
there is a finite extension
$L'$
of L with ring of integers
$\mathscr O'$
, such that
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B}\otimes _{\mathscr O} {\mathscr O}'$
decomposes into a finite product of indecomposable subcategories, each of which remains indecomposable after a further extension of scalars. Such absolutely indecomposable blocks have been classified in [Reference Paškūnas40], and they correspond to semisimple representations
$\bar {\rho }: \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2(k)$
, which are either absolutely irreducible or a direct sum of two characters. This bijection realises the semisimple mod p local Langlands correspondence established in a visionary paper of Breuil [Reference Breuil9].
If
$\bar {\rho }$
is absolutely irreducible, then
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B_{\bar {\rho }}}$
contains only one irreducible object
$\pi $
, satisfying
$\check {\mathbf {V}}(\pi ^{\vee })\cong \bar {\rho }$
, where
$\vee $
denotes the Pontryagin dual. Moreover,
$\pi $
is not a subquotient of any parabolically induced representation; such representations are called supersingular.
If
$\bar {\rho }=\chi _1 \oplus \chi _2$
, where
$\chi _1, \chi _2: \mathscr G_{\mathbb {Q}_p}\rightarrow k^{\times }$
are characters, then the irreducible objects in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B_{\bar {\rho }}}$
are the irreducible subquotients of the representation
$$ \begin{align*} (\operatorname{\mathrm{Ind}}^G_B \chi_1\otimes \chi_2 \omega^{-1})_{\mathrm{sm}}\oplus (\operatorname{\mathrm{Ind}}^G_B \chi_2\otimes \chi_1 \omega^{-1})_{\mathrm{sm}}, \end{align*} $$
where we consider
$\chi _1, \chi _2$
as characters of
$\mathbb {Q}_p^{\times }$
via the Artin map
$\operatorname {\mathrm {Art}}_{\mathbb {Q}_p}: \mathbb {Q}_p^{\times } \rightarrow \mathscr G_{\mathbb {Q}_p}^{\mathrm {ab}}$
, and
$\omega (x)= x \lvert x\rvert \pmod {p}$
for all
$x\in \mathbb {Q}_p^{\times }$
corresponds to the cyclotomic character modulo p. (See Section 4.1 for an explicit list.)
All the blocks, except when p is either
$2$
or
$3$
and
$\bar {\rho }=\chi \oplus \chi \omega $
, have been well understood in [Reference Paškūnas39, Reference Paškūnas41]. These exceptional blocks are the main focus of this paper, but our arguments work for all p and all blocks.
The action of
$Z_{\mathfrak B_{\bar {\rho }}}$
induces a functorial ring homomorphism
$$ \begin{align*} c_\tau: Z_{\mathfrak B_{\bar{\rho}}} \rightarrow \operatorname{\mathrm{End}}_{G}(\tau) \end{align*} $$
for every object
$\tau $
in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B_{\bar {\rho }}}$
. Since
$\check {\mathbf {V}}$
is a functor, it induces a ring homomorphism
$$ \begin{align*} \operatorname{\mathrm{End}}_{G}(\tau)\rightarrow \operatorname{\mathrm{End}}^{\mathrm{cont}}_{\mathscr G_{\mathbb{Q}_p}}(\check{\mathbf{V}}(\tau^{\vee}))^{\mathrm{op}},\quad \varphi\mapsto \check{\mathbf{V}}(\varphi). \end{align*} $$
We denote the action of
$\mathscr G_{\mathbb {Q}_p}$
on
$\check {\mathbf {V}} (\tau ^{\vee })$
by
$\rho _{\check {\mathbf {V}} (\tau ^{\vee })}$
.
Let
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}$
be the universal deformation ring parameterising pseudorepresentations lifting
$\operatorname {tr} \bar {\rho }$
with determinant
$\zeta \varepsilon $
, where
$\varepsilon $
is the p-adic cyclotomic character. We may evaluate the universal pseudorepresentation
$T: \mathscr G_{\mathbb {Q}_p} \rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }}$
at
$g\in \mathscr G_{\mathbb {Q}_p}$
to obtain an element
$T(g)\in R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }}$
.
Theorem 1.2. There is an
$\mathscr O$
-algebra homomorphism
$$ \begin{align} \upsilon: R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr} \bar{\rho}}\rightarrow Z_{\mathfrak B_{\bar{\rho}}}, \end{align} $$
satisfying the following compatibility property with the Colmez’s functor: For all
$\tau \in \operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B_{\bar {\rho }}}$
and all
$g\in \mathscr G_{\mathbb {Q}_p}$
, we have
$$ \begin{align*} \check{\mathbf{V}}(c_{\tau}(\upsilon(T(g))))= \rho_{\check{\mathbf{V}}(\tau^{\vee})}(g) + \rho_{\check{\mathbf{V}}(\tau^{\vee})}(g^{-1}) \zeta\varepsilon(g) \end{align*} $$
in
$\operatorname {\mathrm {End}}_{\mathscr G_{\mathbb {Q}_p}}^{\mathrm {cont}} (\check {\mathbf {V}} (\tau ^{\vee }))$
.
The construction of the map (1) is the main point of this paper. Outside the exceptional cases, it has been established in [Reference Paškūnas39, Reference Paškūnas41] using a different argument from ours. Our main result is the following:
Theorem 1.3. The map (1) makes
$Z_{\mathfrak B_{\bar {\rho }}}$
and
$E_{\mathfrak B_{\bar {\rho }}}$
into finitely generated
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }}$
-modules.
Since
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }}$
is known to be Noetherian by the work of Chenevier [Reference Chenevier18], Theorem 1.3 implies Theorem 1.1. Moreover, Theorems 1.2 and 1.3 are sufficient to remove the restrictions in Lue Pan’s paper. Further, we can re-prove most of the results concerning Banach-space representations in [Reference Paškūnas39] (see Section 6.2 and Corollary 6.16).
To give a flavour of the results on Banach-space representations, we will explain a special case. Let
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
be the category of admissible unitary L-Banach-space representations of G with central character
$\zeta $
. This category is abelian [Reference Schneider and Teitelbaum44]. By [Reference Colmez, Dospinescu and Paškūnas21], Colmez’s functor induces a bijection between the equivalence classes of absolutely irreducible nonordinary
$\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
and absolutely irreducible Galois representations
$\rho : \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2(L)$
. We show that there are no extensions between
$\Pi $
and other irreducible representations in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
; hence
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)^{\mathrm {fl}}_{\Pi }$
is a direct summand of
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)^{\mathrm {fl}}$
, where the superscript ‘
$\mathrm {fl}$
’ indicates finite length and the subscript
$\Pi $
indicates that all the irreducible subquotients are isomorphic to
$\Pi $
. We show in Corollary 6.16 that
$\check {\mathbf {V}}$
induces an antiequivalence of categories between
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)^{\mathrm {fl}}_{\Pi }$
and the category of modules of finite length over
$R^{\zeta \varepsilon }_{\rho }$
, the universal deformation ring of
$\rho $
parameterising deformations of
$\rho $
with determinant equal to
$\zeta \varepsilon $
to local Artinian L-algebras. Such results were known before for
$p\ge 5$
[Reference Paškūnas39], and under assumptions on the reduction modulo p of a G-invariant lattice in
$\Pi $
if
$p=2$
or
$p=3$
[Reference Paškūnas41].
Let
be the largest quotient of
such that the Cayley–Hamilton theorem holds for the universal pseudorepresentation with determinant
$\zeta \varepsilon $
lifting
$\operatorname {tr} \bar {\rho }$
. Such algebras have been studied by Bellaïche and Chenevier [Reference Bellaïche and Chenevier3, Reference Chenevier18]; they play a key role in this paper. The subscript ‘
$\mathrm {tf}$
’ will indicate the maximal
$\mathscr O$
-torsion-free quotient. Our second main result asserts the following:
Theorem 1.4. The essential image of
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B_{\bar {\rho }}}$
under
$\check {\mathbf {V}}$
is antiequivalent to the category of pseudocompact
-modules. The map (1) induces an isomorphism
$$ \begin{align} R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr} \bar{\rho}}[1/p]\overset{\cong}{\longrightarrow}Z_{\mathfrak B_{\bar{\rho}}}[1/p]. \end{align} $$
Moreover, if
$p\neq 2$
, then
$Z_{\mathfrak B_{\bar {\rho }}}= (R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}})_{\mathrm {tf}}$
, and if
$p=2$
, then the cokernel of map (1) is killed by
$2$
.
Corollary 1.5.
$Z_{\mathfrak B_{\bar {\rho }}}$
is a complete local Noetherian
$\mathscr O$
-algebra with residue field k. It is
$\mathscr O$
-torsion-free, and
$Z_{\mathfrak B_{\bar {\rho }}} [1/p]$
is normal.
If we are not in the exceptional cases, then Theorem 1.3 is proved in [Reference Paškūnas39, Reference Paškūnas41] essentially by computing first
$E_{\mathfrak B}$
and then its centre
$Z_{\mathfrak B}$
. Moreover, it is proved there that map (1) is an isomorphism and
is
$\mathscr O$
-torsion-free. The argument in this paper, sketched in Section 1.1, is different: We do not compute
$E_{\mathfrak B}$
.
Our original strategy for proving formula (2) in this paper was to use an argument of Gabber in [Reference Kisin34, Appendix]. We showed that
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho}}[1/p]$
is normal,
$Z_{\mathfrak B}[1/p]$
is reduced, and map (1) induces a bijection on maximal spectra
$\operatorname {\mathrm {m-Spec}} Z_{\mathfrak B}[1/p]\rightarrow \operatorname {\mathrm {m-Spec}} R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }} [1/p]$
and an isomorphism of the residue fields. However, this is replaced by a different argument in the final version, which also proves the first part of Theorem 1.4. One important ingredient in the proof is results of Colmez, Dospinescu and the first author [Reference Colmez, Dospinescu and Paškūnas21, Reference Colmez, Dospinescu and Paškūnas22] which imply that the universal framed deformation of
$\bar {\rho }$
with determinant
$\zeta \varepsilon $
lies in the image of
$\check {\mathbf {V}}$
. We show in Appendix A that
acts faithfully on this representation using the theory of Cayley–Hamilton algebras [Reference Procesi43].
The normality of the ring
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}[1/p]$
is proved in Appendix A, where we show that if the generic fibre of the framed deformation ring
$R^{\square }_{\bar {\rho }} [1/p]$
is normal, then
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}[1/p]$
and the corresponding rigid analytic space
$(\operatorname {\mathrm {Spf}} R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }})^{\mathrm {rig}}$
are normal. The same applies to the deformation rings without the fixed-determinant condition. In fact, we prove this statement not only for
$\mathscr G_{\mathbb {Q}_p}$
but for any profinite group satisfying Mazur’s finiteness condition at p. We then show that
$R^{\square }_{\bar {\rho }}[1/p]$
is normal for all
$2$
-dimensional semisimple representations of
$\mathscr G_{\mathbb {Q}_p}$
; the hard cases are precisely those corresponding to the exceptional blocks. If
$p=2$
, then the assertion has been shown in [Reference Colmez, Dospinescu and Paškūnas22], and if
$p=3$
, then we give a proof in Appendix A based on Böckle’s explicit description of the framed deformation ring in [Reference Böckle7]. We note that the argument of [Reference Colmez, Dospinescu and Paškūnas22] has been generalised by Iyengar [Reference Iyengar33], showing that when
$\bar {\rho }$
is the trivial d-dimensional representation of the absolute Galois group of a p-adic field, containing a
$4$
th root of unity if
$p=2$
, then
$R^{\square }_{\bar {\rho }}[1/p]$
is normal, so our results also apply in this setting. We expectFootnote
1
the rings
$R^{\square }_{\bar {\rho }}[1/p]$
to be normal for any d-dimensional representation
$\bar {\rho }$
of
$\mathscr G_F$
, where F is a finite extension of
$\mathbb {Q}_p$
.
1.1 A sketch of the proof
We will now explain the construction of the map (1). To fix ideas we will discuss a special case:
$p=2$
,
$\bar {\rho }=\mathbf 1 \oplus \omega $
and
$\zeta =\mathbf 1$
. Since the cyclotomic character is trivial modulo
$2$
,
$\bar {\rho }=\mathbf 1 \oplus \mathbf 1$
. The corresponding block has two irreducible representations: the trivial
$\mathbf 1$
and the smooth Steinberg representation
$\operatorname {\mathrm {Sp}}$
. Instead of working with representations on
$\mathscr O$
-torsion modules, it is more convenient to use Pontryagin duality and work with representations of G on compact
$\mathscr O$
-modules. We denote by
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
the category antiequivalent to
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)_{\mathfrak B}$
under Pontryagin duality.
Let
$P_{\mathbf 1^{\vee }}$
and
$P_{\operatorname {\mathrm {Sp}}^{\vee }}$
be projective envelopes of
$\mathbf 1^{\vee }$
and
$\operatorname {\mathrm {Sp}}^{\vee }$
in
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, respectively. Then
$P_{\mathfrak B}:= P_{\mathbf 1^{\vee }}\oplus P_{\operatorname {\mathrm {Sp}}^{\vee }}$
is a projective generator of
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, and by results of Gabriel [Reference Gabriel31] the category
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
is equivalent to the category of pseudocompact modules of
$E_{\mathfrak B}:=\operatorname {\mathrm {End}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B})$
. The equivalence is induced by the functors
$$ \begin{align*} N\mapsto \operatorname{\mathrm{Hom}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B}, N),\quad \mathrm m\mapsto \mathrm m \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}} P_{\mathfrak B}. \end{align*} $$
The centre
$Z_{\mathfrak B}$
of the category
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
is naturally isomorphic to the centre of the ring
$E_{\mathfrak B}$
.
Colmez’s functor kills all objects on which
$\operatorname {\mathrm {SL}}_2(\mathbb {Q}_p)$
acts trivially, and these form a thick subcategory. Thus
$\check {\mathbf {V}}$
factors through the quotient category, which we denote by
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
; let
$\mathscr T: {\mathfrak C}(\mathscr O)_{\mathfrak B}\rightarrow \mathfrak Q(\mathscr O)_{\mathfrak B}$
be the quotient functor. Moreover,
$\check {\mathbf {V}}$
induces an equivalence of categories between
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
and its essential image under
$\check {\mathbf {V}}$
. To prove this, one needs to show that
$\check {\mathbf {V}}$
sends nonsplit extensions to nonsplit extensions; such arguments are originally due to Colmez, and in the case
$p=2$
this has been shown by the second author [Reference Tung47].
Since
$\mathscr T (\mathbf 1^{\vee })=0$
, then
$\mathscr T (\operatorname {\mathrm {Sp}}^{\vee })$
is the only irreducible object in
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
up to isomorphism. Moreover, it is shown in [Reference Paškūnas39] that
$\mathscr T (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
is a projective envelope of
$\mathscr T (\operatorname {\mathrm {Sp}}^{\vee })$
, and
$\mathscr T$
induces an isomorphism
$$ \begin{align*} E_{\mathfrak B}':=\operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O)}(P_{\operatorname{\mathrm{Sp}}^{\vee}})\cong\operatorname{\mathrm{End}}_{\mathfrak Q(\mathscr O)}(\mathscr TP_{\operatorname{\mathrm{Sp}}^{\vee}}). \end{align*} $$
Since
$\mathscr T (\operatorname {\mathrm {Sp}}^{\vee })$
is the only irreducible object in
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
, then
$\mathscr T (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
is a projective generator of
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
, and thus
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
is equivalent to the category of pseudocompact
$E_{\mathfrak B}'$
-modules. This implies that
$\mathscr T (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
– and hence, by equivalence of categories,
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
– is flat over
$E_{\mathfrak B}'$
. Since
$\check {\mathbf {V}} (\operatorname {\mathrm {Sp}}^{\vee })$
is a
$1$
-dimensional representation of
$\mathscr G_{\mathbb {Q}_p}$
, in fact the trivial representation with our normalisations, an application of Nakayama’s lemma shows that
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
is a free
$E_{\mathfrak B}'$
-module of rank
$1$
. The action of
$\mathscr G_{\mathbb {Q}_p}$
on
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
commutes with the action of
$E_{\mathfrak B}'$
and so induces a homomorphism
We show that this map is surjective. In general, the argument is carried out in Section 2 in an abstract setting, and then in Proposition 4.18 we verify that the conditions of the abstract setting are satisfied. However, in the special case under consideration, the argument is easier: Since
$\check {\mathbf {V}}$
induces an equivalence of categories between
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
and its essential image, the
$E_{\mathfrak B}'$
-cosocle and
-cosocle of
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }})$
coincide. This implies that there is
$v\in \check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }} )$
, which is a generator of
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }} )$
as both an
$E_{\mathfrak B}'$
- and an
-module. This implies that
$\alpha $
is surjective.
We also show in Section 3 that the natural map
is surjective, where
is the largest quotient of
such that the Cayley–Hamilton theorem holds for the universal pseudorepresentation with determinant
$\zeta \varepsilon $
lifting
$\operatorname {tr} \bar {\rho }$
.
The idea is to show that
$\operatorname {\mathrm {Ker}} \alpha $
contains
$\operatorname {\mathrm {Ker}} \beta $
, since this implies that the action of
$\mathscr G_{\mathbb {Q}_p}$
on
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }} )$
induces surjections:
This is proved using the results of Berger and Breuil [Reference Berger and Breuil4] on universal unitary completions of locally algebraic principal series and density arguments already used in [Reference Colmez, Dospinescu and Paškūnas21] and also in [Reference Tung46]. Morally, the argument should be that
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }} )$
injects into the product of all
$2$
-dimensional crystabelline representations of
$\mathscr G_{\mathbb {Q}_p}$
with mod p reduction isomorphic to
$\bar {\rho }$
and determinant
$\zeta \varepsilon $
; then an element in
$\operatorname {\mathrm {Ker}} \beta $
would kill this product, and hence
$\check {\mathbf {V}} (P_{\operatorname {\mathrm {Sp}}^{\vee }} )$
. In reality the argument is technically a bit more complicated: We also have to consider deformations of such representations to local Artinian L-algebras (see Section 5).
Wang-Erickson proved in [Reference Wang-Erickson48] that
is a finitely generated
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}$
-module. The surjection (3) implies that the image of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}$
is contained in the centre of
$E_{\mathfrak B}'$
, which we denote by
$Z_{\mathfrak B}'$
. Moreover, both
$E_{\mathfrak B}'$
and
$Z_{\mathfrak B}'$
are finite
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}$
-modules. We show that formula (3) induces an isomorphism
by showing that the universal framed deformation of
$\bar {\rho }$
with determinant
$\zeta \varepsilon $
lies in the image of
$\check {\mathbf {V}}$
using [Reference Colmez, Dospinescu and Paškūnas21, Reference Colmez, Dospinescu and Paškūnas22], and
acts faithfully on it (see Proposition A.11). The assertions about the centre in Theorem 1.4, with
$Z_{\mathfrak B}'$
instead of
$Z_{\mathfrak B}$
, are proved by studying the centre of
. This argument is carried out in Appendix A for d-dimensional representations of any profinite group, satisfying Mazur’s finiteness condition at p.
We then transfer this result from
$E_{\mathfrak B}'$
and
$Z_{\mathfrak B}'$
to
$E_{\mathfrak B}$
and
$Z_{\mathfrak B}$
. Let
$M_{\mathfrak B}$
be the kernel of
$P_{\mathfrak B}\rightarrow (P_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
. We show that
$$ \begin{align*} \operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O)}(M_{\mathfrak B})\cong \operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B})=E_{\mathfrak B} \end{align*} $$
by examining various exact sequences and showing that certain
$\operatorname {\mathrm {Ext}}$
-groups vanish. We also show that the cosocle of
$M_{\mathfrak B}$
is a direct sum of finitely many copies of
$\operatorname {\mathrm {Sp}}^{\vee }$
. Thus
$M_{\mathfrak B}$
is a quotient of
$ (P_{\mathfrak B}' )^{\oplus n}$
for some
$n\ge 1$
. This allows us to conclude that
$\operatorname {\mathrm {End}}_{{\mathfrak C}(\mathscr O)}(M_{\mathfrak B})$
and its centre are finitely generated
$Z_{\mathfrak B}'$
-modules, which finishes the proof of Theorem 1.3. The arguments showing the finiteness of
$E_{\mathfrak B}$
and
$Z_{\mathfrak B}$
over
$Z_{\mathfrak B}'$
are carried out in Section 4.3. Then with some more effort we are able to show that
$Z_{\mathfrak B}'=Z_{\mathfrak B}$
(Corollary 6.15).
1.2 What is left to do?
Although we believe that our results will suffice for most number-theoretic applications – for example, [Reference Pan38] – to complete the programme started in [Reference Paškūnas39], one would have to compute the ring
$E_{\mathfrak B}$
in the exceptional cases. This will be harder than [Reference Paškūnas39, Section 10.5], which is already quite involved. We expect that the map (3) induces isomorphisms
This is known to hold for all blocks except for the exceptional ones. Theorem 1.4 implies that to prove this result, it would be enough to show that
is
$\mathscr O$
-torsion-free, and for the second isomorphism in the case
$p=3$
it would be enough to showFootnote
2
that
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }}$
is
$\mathscr O$
-torsion-free.
It seems likely that using the results of this paper, one can remove the restriction on the prime p in Lue Pan’s work [Reference Pan37] on the Fontaine–Mazur conjecture in the residually reducible case, which generalises the work of Skinner and Wiles. We hope to return to these questions in future work.
2 Endomorphism rings
Let E be a pseudocompact
$\mathscr O$
-algebra and let
$\mathrm {PC}(E)$
be the category of left pseudocompact E-modules (see [Reference Brumer12], [Reference Gabriel31, Section IV.3]. Let
$\operatorname {\mathrm {Irr}}(E)$
be the set of equivalence classes of irreducible objects in
$\mathrm {PC}(E)$
.
Let M be in
$\mathrm {PC}(E)$
. We assume that we are given a continuous E-linear action of a profinite group
$\mathscr G$
on M, which makes M into a pseudocompact module over the completed group algebra
. The action induces a homomorphism of
$\mathscr O$
-algebras
. In this section we will study when this map is surjective, as well as its kernel.
If N is a pseudocompact E-module, which is finitely generated as an E-module, then we may present it as
$$ \begin{align*} \prod_{i\in I} E \rightarrow E^{\oplus n} \rightarrow N \rightarrow 0. \end{align*} $$
By applying
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_E(\ast , M)$
we obtain an exact sequence
$$ \begin{align*} 0\rightarrow \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_E(N, M)\rightarrow M^{\oplus n} \rightarrow \oplus_{i\in I} M. \end{align*} $$
We thus may identify
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_E(N, M)$
with a closed submodule of
$M^{\oplus n}$
, which makes
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_E(N, M)$
into a pseudocompact left
-module.
Lemma 2.1. Let N be a finitely generated projective E-module and let
$\mathrm m$
be a right pseudocompact
-module. Then the natural map
is an isomorphism.
Proof. Since N is finitely generated and projective, we may present it as
$$ \begin{align*} F\overset{e}{\longrightarrow} F \rightarrow N \rightarrow 0, \end{align*} $$
where
$F\cong E^{\oplus n}$
and e is an idempotent. In particular, N is a pseudocompact E-module. The map (4) is induced by a continuous bilinear map
$$ \begin{align*} (v, \phi)\mapsto [ w\mapsto v \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}} \phi(w)]. \end{align*} $$
It is an isomorphism if
$N=F$
. The general case follows by applying the idempotent e to the isomorphism obtained for
$N=F$
.
Lemma 2.2. Let
$\{\rho _i\}_{i\in I}$
be a family of pairwise distinct absolutely irreducible right pseudocompact
-modules. Then the map
is surjective.
Proof. Since
$\mathscr G$
is profinite, each
$\rho _i$
is a finite-dimensional k-vector space. Thus
, given by the action, is continuous for the discrete topology on the target. Since
$\rho _i$
is absolutely irreducible,
$\varphi _i$
is surjective. Moreover,
$\operatorname {\mathrm {Ker}} \varphi _i$
is an open maximal two-sided ideal of
. If
$i\neq j$
, then
$\rho _i\not \cong \rho _j$
, and thus
. This implies that for every finite subset F of I, the map
is surjective. Thus the image of map (5) is dense for the product topology on the target. On the other hand, map (5) is a continuous map between pseudocompact
$\mathscr O$
-modules, and thus its image is closed, which implies surjectivity.
If M is in
$\mathrm {PC}(E)$
, then we let
$\mathfrak r(M)$
be the intersection of open maximal submodules of M. Then
$\mathfrak r(E)$
is a closed two-sided ideal of E and
$\mathfrak r(M)$
is the closure of
$\mathfrak r(E) M$
inside M.
Proposition 2.3. Let us assume that the following hold:
-
1. M is a finitely generated projective E-module.
-
2.
$M/\mathfrak r(M) = \operatorname {\mathrm {cosoc}}_{\mathscr G} M$
. -
3. For all
$S\in \operatorname {\mathrm {Irr}}(E)$
such that
$$ \begin{align*} \rho_S:=\operatorname{\mathrm{Hom}}_{E}^{\mathrm{cont}}(M, S)\neq 0, \end{align*} $$
$\dim _k \rho _S$
is finite and
$\rho _S$
is an absolutely irreducible representation of
$\mathscr G$
.
-
4. If
$S, S'\in \operatorname {\mathrm {Irr}}(E)$
and
$S\not \cong S'$
, then
$\operatorname {\mathrm {Hom}}_{\mathscr G} (\rho _S, \rho _{S'} )=0$
.
Then the map
is surjective.
Proof. For
$S\in \operatorname {\mathrm {Irr}}(E)$
, we let
$\overline {M}_S$
be the smallest quotient of M such that
$$ \begin{align*} \rho_S=\operatorname{\mathrm{Hom}}_E^{\mathrm{cont}}(M, S)=\operatorname{\mathrm{Hom}}_E^{\mathrm{cont}}(\overline{M}_S, S). \end{align*} $$
It follows from assumption (3) that these subspaces are finite-dimensional. Thus
$\overline {M}_S\cong S^{\oplus d}$
such that
$d \cdot \dim _k \operatorname {\mathrm {End}}_E^{\mathrm {cont}}(S)= \dim _k \rho _S$
. Since S is irreducible,
$\operatorname {\mathrm {End}}_E^{\mathrm {cont}}(S)$
is a skew field. It acts on
$\rho _S$
, and this action commutes with the action of
$\mathscr G$
. Since
$\rho _S$
is absolutely irreducible, we conclude that
$\operatorname {\mathrm {End}}_E^{\mathrm {cont}}(S)=k$
and
$\dim _k \rho _S=d$
. Thus
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_E ( \overline {M}_S )\cong M_d(k)$
and thus does not have nontrivial two-sided ideals. Hence, the natural right action of
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_E ( \overline {M}_S )$
on
$\rho _S$
induces an injective ring homomorphism
$$ \begin{align} \operatorname{\mathrm{End}}_E^{\mathrm{cont}}( \overline{M}_S) \rightarrow \operatorname{\mathrm{End}}_k(\rho_S)^{\mathrm{op}}, \end{align} $$
which has to be surjective, as both k-vector spaces have dimension equal to
$d^2$
.
The isomorphism
$M/\mathfrak r(M) \cong \prod _{S\in \operatorname {\mathrm {Irr}}(E)} \overline {M}_S$
induces an isomorphism
$$ \begin{align*} \operatorname{\mathrm{End}}_E^{\mathrm{cont}}( M /\mathfrak r(M)) \cong \prod_{S\in \operatorname{\mathrm{Irr}}(E)} \operatorname{\mathrm{End}}_E^{\mathrm{cont}}(\overline{M}_S). \end{align*} $$
Since
$\rho _S \not \cong \rho _{S'}$
if
$S\not \cong S'$
, it follows from Lemma 2.2 together with isomorphism (6) that the action of
on
$M /\mathfrak r(M)$
induces a surjection
Since M is projective and
$M/\mathfrak r(M)$
is prosemisimple, we have
$$ \begin{align} \operatorname{\mathrm{End}}^{\mathrm{cont}}_E(M) \twoheadrightarrow \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_E(M, M/\mathfrak r(M))\cong \operatorname{\mathrm{End}}_E^{\mathrm{cont}}( M/\mathfrak r(M)). \end{align} $$
If
$\mathrm m$
is an irreducible right pseudocompact
-module and
$\mathfrak a$
is its annihilator, then
is a finite-dimensional simple k-algebra, and thus
is semisimple as a left
-module, and thus
is semisimple as a left
-module. Hence, the surjection
factors through as
Moreover, the maps become isomorphisms after application of
. Thus
as
$M/\mathfrak r(M)= \operatorname {\mathrm {cosoc}}_{\mathscr G} M$
by assumption. Lemma 2.1 together with formula (9) implies that by applying
to formula (8), we obtain isomorphisms
where the
$\mathscr G$
-action on
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_E(M)$
and
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_E(M/\mathfrak r(M))$
is given by
$(g.\varphi )(v):= g (\varphi (v))$
.
If the map
is not surjective, then its cokernel is a nonzero left pseudocompact
-module and thus will have an irreducible quotient
$\mathrm m'$
. If we let
$\mathrm m= \operatorname {\mathrm {Hom}}_k(\mathrm m', k)$
with the right
-action, then
is nonzero, as the evaluation map
is nonzero. By construction, the composition
is the zero map. Thus
cannot be surjective. However, the commutative diagram
implies that the top horizontal arrow is surjective, yielding a contradiction.
We remind the reader that as a consequence of the topological Nakayama’s lemma [Reference Gille, Polo, Artin, Bertin, Gabriel, Raynaud and Serre32, Exposé VIIB, Lemma 0.3.3], the following holds:
Lemma 2.4. Let N be a pseudocompact left E-module. Then N is projective in
$\mathrm {PC}(E)$
if and only if the functor
$\mathrm m \mapsto \mathrm m\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_E N$
from the category of right pseudocompact E-modules to the category of abelian groups is exact. In this case,
$N\twoheadrightarrow N/\mathfrak r(N)$
is a projective envelope of
$N/\mathfrak r(N)$
.
Corollary 2.5. If in addition to the assumptions of Proposition 2.3 we assume that
$M/\mathfrak r(M) \cong E/\mathfrak r(E)$
as E-modules, then M is a free E-module of rank
$1$
and the action of
on M induces a surjection
which is uniquely determined up to a conjugation by
$E^{\times }$
.
We will now give a characterisation of the kernel of
in favourable settings.
Let
$\mathrm m$
be a finite-dimensional L-vector space with continuous
$\mathscr O$
-linear action of E on the right. The image of E in
$\operatorname {\mathrm {End}}_L(\mathrm m)$
is a compact
$\mathscr O$
-module, and thus E stabilises an
$\mathscr O$
-lattice
$\mathrm m^0$
in
$\mathrm m$
. The action of
on M induces a continuous left action of
on
$\mathrm m^0\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_E M$
and hence on
$$ \begin{align*} \mathrm m \otimes_E M= (\mathrm m^0\otimes_E M)[1/p]= (\mathrm m^0\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_E M)[1/p]. \end{align*} $$
Lemma 2.6. Let
$\{\mathrm m_i\}_{i\in I}$
be a family of finite-dimensional L-vector spaces with continuous right
$\mathscr O$
-linear action of E. For each
$i\in I$
, let
$\mathfrak a_i$
be the E-annihilator of
$\mathrm m_i$
and let
$\mathfrak b_i$
be the
-annihilator of
$\mathrm m_i \otimes _E M$
. If M is a free E-module of finite rank and
$\bigcap _{i\in I} \mathfrak a_i=0$
, then
Proof. For each
$i\in I$
, let
$\mathfrak c_i$
be the
$\operatorname {\mathrm {End}}_E^{\mathrm {cont}}(M)$
-annihilator of
$\mathrm m_i \otimes _E M$
. Since
$\mathfrak b_i$
is the preimage of
$\mathfrak c_i$
in
, it is enough to show that
$\bigcap _{i\in I} \mathfrak c_i=0$
.
Let
$w_1, \dotsc , w_n$
be an E-basis of M. Then we may identify
$\operatorname {\mathrm {End}}_E^{\mathrm {cont}}(M)$
with
$M_n(E)$
by mapping
$\varphi $
to the matrix
$ (a_{kj} )$
, given by
$$ \begin{align*} \varphi(w_k)= \sum_{j=1}^n a_{kj} w_j \end{align*} $$
for all
$1\le k\le n$
. If
$v\in \mathrm m_i$
, then
$$ \begin{align*} v \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}} \varphi(w_k)= \sum_{j=1}^n ( v a_{kj})\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}} w_j. \end{align*} $$
Thus
$\varphi $
annihilates
$\mathrm m_i\otimes _E M$
if and only if
$v a_{kj}=0$
for all
$v\in \mathrm m_i$
and all
$1\le k, j\le n$
, which is equivalent to
$\varphi \in M_n(\mathfrak a_i)$
. Since
$\bigcap _{i\in I} \mathfrak a_i=0$
, we have
$\bigcap _{i\in I} M_n(\mathfrak a_i)=0$
and thus
$\bigcap _{i\in I} \mathfrak c_i=0$
.
3 Pseudorepresentations
Let
$\mathscr G$
be a profinite group and let
$\bar {\rho }$
be a continuous semisimple representation of
$\mathscr G$
on a
$2$
-dimensional k-vector space. We fix a continuous group homomorphism
$\psi : \mathscr G\rightarrow \mathscr O^{\times }$
lifting
$\det \bar {\rho }$
. Let
$D^{\mathrm {ps}, \psi }$
be the functor from the category of augmented Artinian
$\mathscr O$
-algebras with residue field k to the category of sets, which maps
$(A, \mathfrak m_A)$
to the set of continuous functions
$t: \mathscr G \rightarrow A$
satisfying the following conditions:
-
•
$t(1)=2$
. -
•
$t(g) \equiv \operatorname {tr} \bar {\rho }(g)\pmod {\mathfrak m_A}, \quad \forall g\in \mathscr G$
. -
•
$t(gh)=t(hg), \quad \forall g, h\in \mathscr G$
. -
•
$\psi (g) t (g^{-1} h )-t(g)t(h) +t(gh)=0, \quad \forall g, h\in \mathscr G$
.
The data
$(t, \psi )$
determines a continuous polynomial law of homogeneous degree
$2$
on
$\mathscr G$
(see [Reference Chenevier18, Lemma 1.9]). This deformation problem is prorepresentable by a local
$\mathscr O$
-algebra
$R^{\mathrm {ps}, \psi }$
with residue field k, complete with respect to profinite topology. We denote by
$T: \mathscr G\rightarrow R^{\mathrm {ps}, \psi }$
the universal deformation. We extend it
$R^{\mathrm {ps}, \psi }$
-linearly to a continuous function
. The homomorphism
$\psi : \mathscr G \rightarrow \mathscr O^{\times }$
induces a continuous
$\mathscr O$
-algebra homomorphism
, which we extend
$R^{\mathrm {ps}, \psi }$
-linearly to a continuous
$R^{\mathrm {ps}, \psi }$
-algebra homomorphism
. Let J be the closed two-sided ideal of
generated by
$a^2- T(a) a + \psi (a)$
for all
.
Proposition 3.1. The ring homomorphism
is surjective.
Proof. Let
$\overline {R}$
and C be the images of
$R^{\mathrm {ps}, \psi }$
and
in
, respectively. Since
$R^{\mathrm {ps}, \psi }$
,
and
are pseudocompact
$\mathscr O$
-modules,
$\overline {R}$
and C are closed subrings of
. It is enough to show that C contains
$\overline {R}$
, since in this case we deduce that C contains the image of
, which is equal to
.
Let
$B= \overline {R}\cap C$
and let
$\mathfrak m_B$
be the intersection of B with the maximal ideal of
$\overline {R}$
. Then B is a closed subring of
$\overline {R}$
. This implies that B is complete for the profinite topology. If
$x\in \mathfrak m_B$
, then
$1+x$
has an inverse in B given by the geometric series. Since B is an
$\mathscr O$
-algebra and the residue field of
$\overline {R}$
is k, we conclude that
$(B, \mathfrak m_B)$
is a local ring with residue field k.
Let
$\overline {T}$
be the specialisation of T along
$R^{\mathrm {ps}, \psi }\twoheadrightarrow \overline {R}$
. The relation
$g^2 - \overline {T}(g) g +\psi (g)=0$
in
implies that
$\overline {T}(g)= g + g^{-1} \psi (g)$
. Thus
$\overline {T}$
takes values in B. The universal property of
$R^{\mathrm {ps}, \psi }$
implies that there is a continuous homomorphism of
$\mathscr O$
-algebras
$\varphi : R^{\mathrm {ps}, \psi } \rightarrow B$
, such that
$\varphi (T(g))= \overline {T}(g)$
for all
$g\in \mathscr G$
. Using the universal property of
$R^{\mathrm {ps}, \psi }$
again, we conclude that if we compose
$\varphi $
with the inclusion
$B \subset \overline {R}$
, we get back the surjection
$R^{\mathrm {ps}, \psi }\twoheadrightarrow \overline {R}$
that we started with. Thus
$B=\overline {R}$
.
Remark 3.2. The proposition does not hold if we do not fix the determinant or consider representations
$\bar {\rho }$
of dimension greater than
$2$
. Counterexamples may be obtained with
$\mathscr G=\mathbb {Z}_p$
and
$\bar {\rho }$
trivial representation of
$\mathscr G$
on an n-dimensional k-vector space, using [Reference Chenevier18, Examples 1.7(i) and 1.11(i)].
4 Representations of
$\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p )$
Let G be a p-adic analytic group and let Z be its centre. We let
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_G(\mathscr O)$
be the category of smooth representations of G on
$\mathscr O$
-torsion modules. Using Pontryagin duality,
$\pi \mapsto \pi ^{\vee }:=\operatorname {\mathrm {Hom}}_{\mathscr O}(\pi , L/\mathscr O)$
equipped with the compact open topology induces an antiequivalence of categories between
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_G(\mathscr O)$
and the category
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_G(\mathscr O)$
of linearly compact
$\mathscr O$
-modules with a continuous G-action [Reference Emerton28, Lemma 2.2.7]. The inverse is given by
$M \mapsto M^{\vee }:=\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{\mathscr O}(M, L/\mathscr O)$
. In particular, if G is compact, then
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_G(\mathscr O)$
is the category of linearly compact
-modules, where
is the completed group algebra. We define
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_G(k)$
and
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_G(k)$
the same way, with
$\mathscr O$
replaced by k. Moreover, for a continuous character
$\zeta : Z \rightarrow \mathscr O^\times $
, adding the subscript
$\zeta $
in any of these categories indicates the corresponding full subcategory of G-representations on which Z acts by
$\zeta $
. Denote by
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
the full subcategory of
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G, \zeta }(\mathscr O)$
consisting of representations in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G, \zeta }(\mathscr O)$
which are equal to the union of their subrepresentations of finite length. We let
$\mathfrak C(\mathscr O)$
be the full subcategory of
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_G(\mathscr O)$
antiequivalent to
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
under the Pontryagin duality.
4.1 Blocks
From now on, we will assume
$G = \operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p )$
. Every irreducible object
$\pi $
of
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_G(\mathscr O)$
is killed by
$\varpi $
and hence is an object of
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_G(k)$
.
Let
$\operatorname {\mathrm {Irr}}_{G, \zeta }$
be the set of irreducible representations in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G, \zeta }(k)$
. We write
$\pi \leftrightarrow \pi '$
if
$\pi \cong \pi '$
or
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }(\pi , \pi ') \neq 0$
or
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }(\pi ', \pi ) \neq 0$
, where
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }(\pi , \pi ')$
is the Yoneda extension group of
$\pi '$
by
$\pi $
in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{G, \zeta }(k)$
. We write
$\pi \sim \pi '$
if there exist
$\pi _1, \dotsc , \pi _n \in \operatorname {\mathrm {Irr}}_{G, \zeta }$
such that
$\pi \cong \pi _1$
,
$\pi ' \cong \pi _n$
and
$\pi _i \leftrightarrow \pi _{i+1}$
for
$1 \leq i \leq n-1$
. The relation
$\sim $
is an equivalence relation on
$\operatorname {\mathrm {Irr}}_{G, \zeta }$
. A block is an equivalence class of
$\sim $
.
Barthel and Livné [Reference Barthel and Livné2] and Breuil [Reference Breuil9] have classified the absolutely irreducible smooth representations
$\pi $
admitting a central character. The blocks containing an absolutely irreducible representation have been determined in [Reference Paškūnas40, Corollary 6.2]. We have the following cases:
-
(i)
$\mathfrak B = \{ \pi \}$
with
$\pi $
supersingular. -
(ii)
$\mathfrak B = \{ (\operatorname {\mathrm {Ind}}_B^G\chi _1 \otimes \chi _2 \omega ^{-1} )_{\mathrm {sm}}, (\operatorname {\mathrm {Ind}}_B^G\chi _2 \otimes \chi _1 \omega ^{-1} )_{\mathrm {sm}} \}$
with
$\chi _2 \chi _1^{-1} \neq \mathbf 1, \omega ^{\pm 1}$
. -
(iii)
$p>2$
and
$\mathfrak B = \{ (\operatorname {\mathrm {Ind}}_B^G \chi \otimes \chi \omega ^{-1} )_{\mathrm {sm}} \}$
. -
(iv)
$p \geq 5$
and
$\mathfrak B = \{ \mathbf 1, \operatorname {\mathrm {Sp}}, (\operatorname {\mathrm {Ind}}^G_B \omega \otimes \omega ^{-1} )_{\mathrm {sm}} \} \otimes \chi \circ \det $
. -
(v)
$p=3$
and
$\mathfrak B = \{ \mathbf 1, \operatorname {\mathrm {Sp}}, \omega \circ \det , \operatorname {\mathrm {Sp}} \otimes \omega \circ \det \} \otimes \chi \circ \det $
. -
(vi)
$p=2$
and
$\mathfrak B = \{ \mathbf 1, \operatorname {\mathrm {Sp}} \} \otimes \chi \circ \det $
.
In these cases,
$\chi , \chi _1, \chi _2: \mathbb {Q}_p^{\times } \rightarrow k^{\times }$
are smooth characters and
$\omega : \mathbb {Q}_p^{\times } \rightarrow k^{\times }$
is the character
$\omega (x) = x \lvert x\rvert \pmod {\varpi }$
, and
$\operatorname {\mathrm {Sp}}$
is the Steinberg representation defined by the exact sequence
$$ \begin{align*} 0 \rightarrow \mathbf 1 \rightarrow (\operatorname{\mathrm{Ind}}_B^G \mathbf 1)_{\mathrm{sm}} \rightarrow \operatorname{\mathrm{Sp}} \rightarrow 0. \end{align*} $$
If
$\pi \in \operatorname {\mathrm {Irr}}_{G, \zeta }$
is not absolutely irreducible then there is a finite extension
$k'$
of k, such that
$\pi \otimes _k k'$
is a finite direct sum of absolutely irreducible representations, see [Reference Paškūnas39, Proposition 5.11], so no information is lost by working with absolutely irreducible representations.
Given a block
$\mathfrak B$
, we denote by
$\pi _{\mathfrak B}$
the direct sum of all representations in
$\mathfrak B$
and let
$P_{\mathfrak B}$
be a projective envelope of
$\pi _{\mathfrak B}^{\vee }$
in
$\mathfrak C(\mathscr O)$
. Then
$E_{\mathfrak B} = \operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B})$
is a pseudocompact
$\mathscr O$
-algebra. We denote the centre of
$E_{\mathfrak B}$
by
$Z_{\mathfrak B}$
.
By [Reference Paškūnas39, Corollary 5.35], the category
$\mathfrak C(\mathscr O)$
decomposes into a direct product of subcategories
$$ \begin{align} \mathfrak C(\mathscr O) \cong \prod_{\mathfrak B \in \operatorname{\mathrm{Irr}}_{G, \zeta} / \sim} \mathfrak C(\mathscr O)_{\mathfrak B}, \end{align} $$
where the objects of
$\mathfrak C(\mathscr O)_{\mathfrak B}$
are those M in
$\mathfrak C(\mathscr O)$
such that for every irreducible subquotient S of M,
$S^{\vee }$
lies in
$\mathfrak B$
. Moreover, the category
$\mathfrak C(\mathscr O)_{\mathfrak B}$
is equivalent to the category of compact right
$E_{\mathfrak B}$
-modules and the centre of
$\mathfrak C(\mathscr O)_{\mathfrak B}$
is isomorphic to
$Z_{\mathfrak B}$
[Reference Paškūnas39, Proposition 5.45].
Lemma 4.1. If
$\mathfrak B$
contains an absolutely irreducible representation, then
$Z_{\mathfrak B}$
is a local pseudocompact
$\mathscr O$
-algebra with residue field k.
Proof. If
$\pi $
is absolutely irreducible, then
$\operatorname {\mathrm {End}}_G(\pi )=k$
, and thus the action of
$Z_{\mathfrak B}$
on
$\pi $
defines a homomorphism of
$\mathscr O$
-algebras
$c_{\pi }: Z_{\mathfrak B}\rightarrow k$
. If
$\pi , \pi '\in \mathfrak B$
are distinct and there is a nonsplit extension
$0\rightarrow \pi \rightarrow \tau \rightarrow \pi '\rightarrow 0$
, then
$\operatorname {\mathrm {End}}_G(\tau )=k$
, and we conclude that
$c_{\pi }=c_{\tau }=c_{\pi '}$
. Using the transitivity property of the relation
$\sim $
on
$\operatorname {\mathrm {Irr}}_{G, \zeta }$
, we conclude that
$c_{\pi }=c_{\pi '}$
for all
$\pi , \pi '\in \mathfrak B$
. It follows from the proof of [Reference Gabriel31, Proposition IV.4.12] that the Jacobson radical of
$Z_{\mathfrak B}$
consists of elements that kill all the irreducible representations. Thus
$\operatorname {\mathrm {Ker}} c_{\pi }$
is the maximal ideal of
$Z_{\mathfrak B}$
with residue field k. The last assertion follows from the fact that
$Z_{\mathfrak B}$
is closed in
$E_{\mathfrak B}$
and [Reference Gabriel31, Proposition IV.4.13].
Lemma 4.2. Let P be projective in
$\mathfrak C(\mathscr O)$
. Then
$P^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
.
Proof. Let J be the Pontryagin dual of P. Then J is injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
and the assertion is equivalent to
$J_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
. Let N be the unipotent subgroup
$ \left(\begin{smallmatrix} 1 & \mathbb {Q}_p \\ 0 & 1 \end{smallmatrix} \right)$
. Then by [Reference Emerton29, Proposition 3.6.2] and [Reference Emerton and Paškūnas30, Corollary 3.12], we have
$J_{N} = 0$
. Thus
$J_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
and the lemma follows. Alternatively, one can deduce the statement from Proposition 4.3.
Proposition 4.3. Let P and M be in
${\mathfrak C}(\mathscr O)$
, such that P is projective and
$M/\varpi M$
is of finite length. Then
$\operatorname {\mathrm {Hom}}_{{\mathfrak C}(\mathscr O)}(M, P)=0$
.
Proof. Let
$K'$
be a compact open pro-p subgroup of
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
such that
$K' \cap Z = \{1\}$
. Then P is projective in
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{K'}(\mathscr O)$
by [Reference Emerton and Paškūnas30, Corollary 3.10], and thus
for some index set I. Thus it is enough to show that
. The topological Nakayama’s lemma for compact
$\mathscr O$
-modules implies that it is enough to show that
. Since
$M/\varpi M$
is of finite length, it is enough to show that
for every
$\pi \in \mathfrak B$
. This follows from [Reference Paškūnas42, Lemma 5.16]. Note that if
$K_n= 1+ M_2 (2 p^n \mathbb {Z}_p )$
and
$K_n'= K_n\cap \operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
, then
$K_n= K_n' ( Z\cap K_n)$
and so
$\pi ^{K_n}= \pi ^{K^{\prime }_n}$
as
$Z\cap K_n$
acts trivially on
$\pi $
; so the argument in [Reference Paškūnas42, Lemma 5.16] carries over to the restriction of
$\pi $
to
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
.
We will denote by
$\operatorname {\mathrm {Ord}}_B$
Emerton’s functor of ordinary parts [Reference Emerton28].
Lemma 4.4. Let
$\pi \hookrightarrow J$
be an injective envelope of
$\pi \in \mathfrak B$
in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
. If
$\pi $
is supersingular, then
$\operatorname {\mathrm {Ord}}_B J=0$
. If
$\pi $
is an irreducible subquotient of
$ (\operatorname {\mathrm {Ind}}_{\overline {B}}^G \chi )_{\mathrm {sm}}$
for some character
$\chi : T\rightarrow k^{\times }$
, where
$\overline {B}$
is the subgroup of lower triangular matrices, then
$\operatorname {\mathrm {Ord}}_B J$
is isomorphic to an injective envelope of
$\chi $
in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{T, \zeta }(\mathscr O)$
, and also in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{T, \zeta }(\mathscr O)$
.
Proof. Since J is injective and
$\operatorname {\mathrm {Ord}}_B$
is adjoint to parabolic induction, which is an exact functor,
$\operatorname {\mathrm {Ord}}_B J$
is injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{T, \zeta }(\mathscr O)$
. Thus it is a direct sum of injective envelopes of characters of T. Moreover,
$$ \begin{align} \operatorname{\mathrm{Hom}}_T(\chi, \operatorname{\mathrm{Ord}}_B J)\cong\operatorname{\mathrm{Hom}}_G( (\operatorname{\mathrm{Ind}}_{\overline{B}}^G \chi)_{\mathrm{sm}}, J). \end{align} $$
Since J is an injective envelope of
$\pi $
, this group is nonzero if and only if
$\pi $
is a subquotient of
$ (\operatorname {\mathrm {Ind}}_{\overline {B}}^G \chi )_{\mathrm {sm}}$
, in which case the dimension of the spaces in formula (12) is equal to the multiplicity with which
$\pi $
occurs as a subquotient of
$ (\operatorname {\mathrm {Ind}}_{\overline {B}}^G \chi )_{\mathrm {sm}}$
. If
$\pi $
is supersingular, then it does not occur as a subquotient of principal series, and thus
$\operatorname {\mathrm {Ord}}_B J=0$
. Otherwise, it follows from [Reference Barthel and Livné2] that there is a unique character
$\chi $
such that the multiplicity is nonzero, in which case it is equal to
$1$
. This proves the first assertion. The same (albeit easier) proof as in [Reference Paškūnas39, Proposition 5.16] implies that every injective object in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{T, \zeta }(\mathscr O)$
is also injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{T, \zeta }(\mathscr O)$
.
Lemma 4.5. Let
$\pi \hookrightarrow J$
be an injective envelope of
$\pi \in \mathfrak B$
in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
. If
$\pi $
is not a character, then
$J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}=0$
; otherwise
$ (J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )^{\vee }$
is nonzero and finitely generated over
$\mathscr O$
.
Proof. If
$J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
is nonzero, then
$\pi \cap J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
is nonzero, as
$\pi \hookrightarrow J$
is essential. Since
$\pi $
is absolutely irreducible, we deduce that it is a character and is the G-socle of
$J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
. Note that the action of G on
$J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
factors through
$$ \begin{align*} G / Z \operatorname{\mathrm{SL}}_2(\mathbb{Q}_p) \cong \mathbb{Q}_p^{\times} / (\mathbb{Q}_p^{\times})^2 \cong \begin{cases} \mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z &\text{if } p>2, \\ \mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z &\text{if } p=2. \end{cases} \end{align*} $$
It follows that
$J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}[\varpi ]$
is a finite-dimensional k-vector space. Dually, this implies that
$ (J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )^{\vee } / \varpi (J^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )^{\vee }$
is finite-dimensional over k, and the lemma follows from Nakayama’s lemma.
Lemma 4.6. Let J be injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
and
$\kappa \in \operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{T, \zeta }(\mathscr O)$
be such that
$\kappa ^{\vee }$
is a finitely generated
$\mathscr O$
-module. Then
$\operatorname {\mathrm {Hom}}_T(\operatorname {\mathrm {Ord}}_B J, \kappa ) = 0$
.
Proof. The following is an analogue of Proposition 4.3 in an easier setting. Let
$T_0$
to be an open torsion-free pro-p subgroup of
$T\cap \operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
. Since
$\operatorname {\mathrm {Ord}}_B J$
is injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{T,\zeta }(\mathscr O)$
by Lemma 4.4, it is injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {sm}}_{T_0}(\mathscr O)$
, as restriction to
$T_0$
is adjoint to
$\operatorname {\mathrm {c-Ind}}_{ZT_0}^{T}$
, which is exact. Thus the Pontryagin dual of
$\operatorname {\mathrm {Ord}}_B J$
is isomorphic to
for some index set I. Hence, it suffices to show that
This is clear, as
is isomorphic to the ring of formal power series in one variable and
$\kappa ^{\vee }$
is a finitely generated
$\mathscr O$
-module.
Lemma 4.7. Let
$\tau $
be a smooth representation of G whose irreducible subquotients consist of characters in
$\mathfrak B$
. Then
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
acts trivially on
$\tau $
and there is an exact sequence
$$ \begin{align} 0 \rightarrow \tau \rightarrow (\operatorname{\mathrm{Ind}}^G_B \tau)_{\mathrm{sm}} \rightarrow Q \rightarrow 0. \end{align} $$
Moreover, the irreducible subquotients of Q are twists of
$\operatorname {\mathrm {Sp}}$
by a character. In particular,
$\operatorname {\mathrm {Hom}}_G(\tau , Q)=0$
.
Proof. Since
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
acts trivially on
$\tau $
by [Reference Tung47, Lemma 1.2.1], the unipotent radical of B acts trivially on
$\tau $
. Thus
$ (\operatorname {\mathrm {Ind}}^G_B \tau \rvert _{B} )_{\mathrm {sm}}$
coincides with the parabolic induction. Since the map
$\tau \rightarrow (\operatorname {\mathrm {Ind}}^G_B \tau )_{\mathrm {sm}}$
defined by
$v \mapsto (g \mapsto gv)$
is G-equivariant and injective, we obtain the first assertion.
To show the second assertion, we choose an increasing and exhaustive filtration
$ \{R^j \}_{j \geq 0}$
of
$\tau $
such that
$R^j / R^{j-1}$
is a character for each j. Then we have
$R^j / R^{j-1} \hookrightarrow (\operatorname {\mathrm {Ind}}^G_B R^j / R^{j-1} )_{\mathrm {sm}}$
with quotient isomorphic to a twist of
$\operatorname {\mathrm {Sp}}$
by a character. Thus the second assertion follows from the exactness of
$ (\operatorname {\mathrm {Ind}}^G_B - )_{\mathrm {sm}}$
.
Lemma 4.8. Let J be injective in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
and let
$\tau $
and Q be as in Lemma 4.7. If
$\tau ^{\vee }$
is finitely generated over
$\mathscr O$
, then
$$ \begin{align*} \operatorname{\mathrm{Hom}}_G(J, \tau) = \operatorname{\mathrm{Hom}}_G(J, (\operatorname{\mathrm{Ind}}^G_B \tau)_{\mathrm{sm}})= \operatorname{\mathrm{Hom}}_G(J, Q) = 0. \end{align*} $$
Proof. Since
$\tau ^{\vee }$
is finitely generated over
$\mathscr O$
, then
$\tau [\varpi ]$
is a finite k-vector space, and thus
$$ \begin{align*} (\operatorname{\mathrm{Ind}}^G_B \tau )_{\mathrm{sm}}^{\vee}/ \varpi \cong (\operatorname{\mathrm{Ind}}^G_B \tau[\varpi])_{\mathrm{sm}}^{\vee} \end{align*} $$
is of finite length in
${\mathfrak C}(\mathscr O)$
and the assertion follows from Proposition 4.3.
Proposition 4.9. Let J and
$\tau $
be as in Lemma 4.8. Then
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }(J, \tau ) = 0$
.
Proof. Consider an exact sequence
$0 \rightarrow \tau \rightarrow I \rightarrow J \rightarrow 0$
. Applying
$\operatorname {\mathrm {Ord}}_B$
to it, we get an exact sequence
$$ \begin{align*} 0 &\rightarrow \operatorname{\mathrm{Ord}}_B \tau \rightarrow \operatorname{\mathrm{Ord}}_B I \rightarrow \operatorname{\mathrm{Ord}}_B J \rightarrow \mathbb R^1 \operatorname{\mathrm{Ord}}_B \tau \rightarrow \mathbb R^1 \operatorname{\mathrm{Ord}}_B I \rightarrow \mathbb R^1 \operatorname{\mathrm{Ord}}_B J \end{align*} $$
of smooth T-representations. It is proved in [Reference Emerton and Paškūnas30] that the functors
$H^i\operatorname {\mathrm {Ord}}_B$
in [Reference Emerton29] coincide with the derived functors
$\mathbb R^i\operatorname {\mathrm {Ord}}_B$
. Moreover,
$H^1\operatorname {\mathrm {Ord}}_B$
coincides with the N-coinvariants twisted by the character
$\alpha ^{-1}$
, where
$\alpha \left( \left( \begin {smallmatrix} a & 0 \\ 0 & d\end {smallmatrix} \right) \right)=\omega (a d^{-1} )$
, by [Reference Emerton29, Proposition 3.6.2].
Since
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
acts trivially on
$\tau $
, we have
$\operatorname {\mathrm {Ord}}_B \tau =0$
and
$\mathbb R^1 \operatorname {\mathrm {Ord}}_B \tau \cong \tau \otimes \alpha ^{-1}$
. Thus the exact sequence reduces to
$$ \begin{align} 0 \rightarrow \operatorname{\mathrm{Ord}}_B I \rightarrow \operatorname{\mathrm{Ord}}_B J \rightarrow \tau \otimes \alpha^{-1}\rightarrow I_N \otimes \alpha^{-1} \rightarrow 0. \end{align} $$
Since the middle map is zero by Lemma 4.6, the map
$\tau \hookrightarrow I$
induces an isomorphism
$\tau \cong I_N$
of T-representations and hence
$$ \begin{align} \operatorname{\mathrm{Hom}}_G(I, (\operatorname{\mathrm{Ind}}^G_B \tau)_{\mathrm{sm}}) \cong \operatorname{\mathrm{Hom}}_G(\tau, (\operatorname{\mathrm{Ind}}^G_B \tau)_{\mathrm{sm}}) \end{align} $$
by the adjunction formula.
It follows from Lemma 4.8 that the first three terms in the long exact sequence obtained by applying
$\operatorname {\mathrm {Hom}}_G(J, -)$
to formula (13) are zero. Thus by applying
$\operatorname {\mathrm {Hom}}_G(I, -)\rightarrow \operatorname {\mathrm {Hom}}_G(\tau ,-)$
to formula (13), we obtain the following commutative diagram with exact columns:
The last part of Lemma 4.7 says that
$\operatorname {\mathrm {Hom}}_G(\tau , Q)=0$
and hence
$\operatorname {\mathrm {Hom}}_G(I, Q)=0$
. Thus all the maps in the top square of the diagram are isomorphisms. The preimage of
$\operatorname {\mathrm {id}}_{\tau }\in \operatorname {\mathrm {Hom}}_G(\tau , \tau )$
in
$\operatorname {\mathrm {Hom}}_G(I, \tau )$
splits the exact sequence
$0 \rightarrow \tau \rightarrow I \rightarrow J \rightarrow 0$
. This proves the proposition.
4.2 Quotient category
Let
$\mathfrak T(\mathscr O)$
be the full subcategory of
$\mathfrak C(\mathscr O)$
whose objects have trivial
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
-action. By [Reference Paškūnas39, Lemma 10.25] (for
$p>2$
) and [Reference Tung47, Lemma 1.2.1] (for
$p=2$
),
$\mathfrak T(\mathscr O)$
is a thick subcategory of
$\mathfrak C(\mathscr O)$
and hence we may consider the quotient category
$\mathfrak Q(\mathscr O) := \mathfrak C(\mathscr O) / \mathfrak T(\mathscr O)$
. Let
$\mathscr T: \mathfrak C(\mathscr O) \rightarrow \mathfrak Q(\mathscr O)$
be the quotient functor; we note that
$\mathscr T$
is the identity on objects. It is shown in [Reference Paškūnas39, Section 10.3] that
$\mathfrak Q(\mathscr O)$
is an abelian category with enough projectives and
$\mathscr T$
is an exact functor.
For a block
$\mathfrak B$
, we denote
$$ \begin{align*} P_{\mathfrak B}' = \bigoplus_{\substack{\pi \in \mathfrak B \\ \pi^{\operatorname{\mathrm{SL}}_2(\mathbb{Q}_p)} = 0}} P_{\pi^{\vee}}, \qquad E_{\mathfrak B}' = \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B}'), \qquad Z_{\mathfrak B}' = Z(E_{\mathfrak B}'), \end{align*} $$
where
$P_{\pi ^{\vee }}$
is a projective envelope of
$\pi ^{\vee }$
in
$\mathfrak C(\mathscr O)$
.
Proposition 4.10. Let
$\mathfrak B$
be a block.
-
1.
$\mathscr T P_{\mathfrak B}'$
is a projective object of
$\mathfrak Q(\mathscr O)$
and
$E_{\mathfrak B}' \cong \operatorname {\mathrm {End}}_{\mathfrak Q(\mathscr O)} (\mathscr T P_{\mathfrak B}' )$
. -
2. The functor
$M \mapsto \operatorname {\mathrm {Hom}}_{\mathfrak Q(\mathscr O)} (\mathscr T P_{\mathfrak B}', M )$
defines an equivalence of categories between
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
and the category of pseudocompact right
$E_{\mathfrak B}'$
-modules, with the inverse given by
$\mathrm m \mapsto \mathrm m \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}'} \mathscr T P_{\mathfrak B}'$
.
Proof. See [Reference Paškūnas39, Lemma 10.27] for the first assertion and [Reference Gabriel31, Section IV.4, Theorem 4, Corollaries 1 and 5] for the second assertion.
4.3 The centre
In this section we will prove key results toward showing that
$E_{\mathfrak B}$
and
$Z_{\mathfrak B}$
are finite over
$Z_{\mathfrak B}'$
. If
$\mathfrak B$
is of type (i), (ii) or (iii), then we have
$P_{\mathfrak B}=P_{\mathfrak B}'$
and thus
$Z_{\mathfrak B}=Z_{\mathfrak B}'$
.
Lemma 4.11. Let E be a ring with centre Z and let
$\mathrm m$
be a finitely generated (right) E-module. Assume that Z is Noetherian and E is (module) finite over Z. Then
$\operatorname {\mathrm {End}}_E(\mathrm m)$
and its centre
$Z(\operatorname {\mathrm {End}}_E(\mathrm m))$
are Noetherian and finite over Z.
Proof. Since
$\mathrm m$
is finitely generated over E, there is a surjection
$E^{\oplus n} \twoheadrightarrow \mathrm m$
for some n. This induces an injection
$\operatorname {\mathrm {End}}_E(\mathrm m) \hookrightarrow \operatorname {\mathrm {Hom}}_E (E^{\oplus n}, \mathrm m )$
and a surjection
$M_n(E^{\mathrm {op}}) \cong \operatorname {\mathrm {End}}_E (E^{\oplus n} ) \twoheadrightarrow \operatorname {\mathrm {Hom}}_E (E^{\oplus n}, \mathrm m )$
of Z-modules. Since
$M_n(E^{\mathrm {op}})$
is finite over
$E^{\mathrm {op}}$
, it is finite and Noetherian over Z. It follows that
$\operatorname {\mathrm {Hom}}_E (E^{\oplus n}, \mathrm m )$
and thus
$\operatorname {\mathrm {End}}_E(\mathrm m)$
, which can be identified with a Z-submodule of
$\operatorname {\mathrm {Hom}}_E (E^{\oplus n}, \mathrm m )$
, are finite and Noetherian over Z. This proves the lemma.
Lemma 4.12. Let
$M_{\mathfrak B} = \ker (P_{\mathfrak B} \rightarrow (P_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )$
. Then the following hold:
-
1.
$(M_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
. -
2. For all
$\pi \in \mathfrak B$
,
$\operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)} (M_{\mathfrak B}, \pi ^{\vee } )$
is finite-dimensional over k.
Proof. It suffices to consider the block
$\mathfrak B$
of type (iv), (v) or (vi); otherwise
$M_{\mathfrak B} = P_{\mathfrak B}$
and the assertion is trivial. Let
$J_{\mathfrak B}$
be the Pontryagin dual of
$P_{\mathfrak B}$
. Then we have an exact sequence
$0 \rightarrow J_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} \rightarrow J_{\mathfrak B} \rightarrow M_{\mathfrak B}^{\vee } \rightarrow 0$
. By applying
$\operatorname {\mathrm {Hom}}_{\operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p )}(\pi , -)$
, we get a long exact sequence
$$ \begin{align*} 0 &\rightarrow \operatorname{\mathrm{Hom}}_{G}(\pi, J_{\mathfrak B}^{\operatorname{\mathrm{SL}}_2(\mathbb{Q}_p)}) \rightarrow \operatorname{\mathrm{Hom}}_{G}(\pi, J_{\mathfrak B}) \rightarrow \operatorname{\mathrm{Hom}}_{G}(\pi, M_{\mathfrak B}^{\vee}) \\ &\rightarrow \operatorname{\mathrm{Ext}}^1_{G, \zeta}(\pi, J_{\mathfrak B}^{\operatorname{\mathrm{SL}}_2(\mathbb{Q}_p)}) \rightarrow \operatorname{\mathrm{Ext}}^1_{G, \zeta}(\pi, J_{\mathfrak B}) = 0, \end{align*} $$
where
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }$
is the extension group in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
.
If
$\pi \in \mathfrak B$
is a character, then both
$\operatorname {\mathrm {Hom}}_{G} (\pi , J_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )$
and
$\operatorname {\mathrm {Hom}}_{G}(\pi , J_{\mathfrak B})$
are
$1$
-dimensional over k, and
$\operatorname {\mathrm {Ext}}^1_{G, \zeta } (\pi , J_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} ) = 0$
by [Reference Tung47, Lemma 1.2.1]. Thus
$\operatorname {\mathrm {Hom}}_G (\pi , M_{\mathfrak B}^{\vee } ) = 0$
and the first assertion follows (see [Reference Colmez and Dospinescu20, Lemma III.40] for another proof).
If
$\pi \in \mathfrak B$
is not a character, then
$\operatorname {\mathrm {Hom}}_{G} (\pi , J_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} ) = 0$
and
$\operatorname {\mathrm {Hom}}_{G}(\pi , J_{\mathfrak B})$
is
$1$
-dimensional over k. Since
$\pi $
is killed by
$\varpi $
, for the second assertion it is enough to show that
$\operatorname {\mathrm {Ext}}^1_{G, \zeta } (\pi , (J_{\mathfrak B}[\varpi ])^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} )$
is finite-dimensional over k. This holds, because
$(J_{\mathfrak B}[\varpi ])^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
is of finite length by Lemma 4.5, and if
$\chi $
is a character, then
$\operatorname {\mathrm {Ext}}^1_{G, \zeta }(\pi , \chi \circ \det )$
is finite-dimensional.
Corollary 4.13. There is a surjection
$ (P_{\mathfrak B}' )^{\oplus n} \twoheadrightarrow M_{\mathfrak B}$
for some
$n \in \mathbb {N}$
.
Proof. This follows from Lemma 4.12, which shows that the cosocle of
$M_{\mathfrak B}$
contains no characters and each irreducible representation in
$\mathfrak B$
appears with finite multiplicity.
Lemma 4.14. Both
$\operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B})$
and
$\operatorname {\mathrm {Ext}}^1_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B})$
are equal to zero.
Proof. The first assertion follows because
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
acts trivially on
$P_{\mathfrak B} / M_{\mathfrak B}$
and
$P_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
(see Lemma 4.2), and the second assertion follows from Proposition 4.9 applied to
$J=P_{\mathfrak B}^{\vee }$
and
$\tau = J_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
. It follows from Lemma 4.5 that
$\tau ^{\vee }$
is a finitely generated
$\mathscr O$
-module.
Proposition 4.15. There is a natural isomorphism
$E_{\mathfrak B} \cong \operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B})$
. In particular,
$Z_{\mathfrak B} \cong Z(\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}))$
.
Proof. Set
$\phi \in E_{\mathfrak B} = \operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B})$
. Then the composition
$M_{\mathfrak B} \xrightarrow {\phi } P_{\mathfrak B} \twoheadrightarrow P_{\mathfrak B} / M_{\mathfrak B}$
is the zero map, since
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
acts trivially on
$P_{\mathfrak B} / M_{\mathfrak B} \cong (P_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}$
and
$(M_{\mathfrak B})_{\operatorname {\mathrm {SL}} (\mathbb {Q}_p )}= 0$
by Lemma 4.12. Thus
$\phi $
maps
$M_{\mathfrak B}$
to
$M_{\mathfrak B}$
, and restriction to
$M_{\mathfrak B}$
induces a ring homomorphism
$E_{\mathfrak B} \rightarrow \operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B})$
.
Applying the functor
$\operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, -)$
to the exact sequence
$0 \rightarrow M_{\mathfrak B} \rightarrow P_{\mathfrak B} \rightarrow P_{\mathfrak B}/M_{\mathfrak B} \rightarrow 0$
, we get the exact sequence
$$ \begin{align*} 0 \rightarrow \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) \rightarrow \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, P_{\mathfrak B}) \rightarrow \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, P_{\mathfrak B} / M_{\mathfrak B}). \end{align*} $$
Since the last term is equal to zero by Lemma 4.12, we obtain
$\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) \cong \operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, P_{\mathfrak B})$
. On the other hand, by applying the functor
$\operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(-, P_{\mathfrak B})$
to the same short exact sequence, we get the exact sequence
$$ \begin{align*} 0 &\rightarrow \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B}) \rightarrow \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B}) \rightarrow \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, P_{\mathfrak B}) \\ &\rightarrow \operatorname{\mathrm{Ext}}^1_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B}). \end{align*} $$
Since
$\operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B}) = 0$
and
$\operatorname {\mathrm {Ext}}^1_{\mathfrak C(\mathscr O)}(P_{\mathfrak B} / M_{\mathfrak B}, P_{\mathfrak B}) = 0$
by Lemma 4.14, we deduce
$\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B}) \cong \operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}, P_{\mathfrak B})$
and the proposition follows.
Corollary 4.16. There is a natural surjective homomorphism
$$ \begin{align*}Z_{\mathfrak B}'\rightarrow Z(\operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}))\cong Z_{\mathfrak B}.\end{align*} $$
Proof. Note that we have
$(M_{\mathfrak B})^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
by Lemma 4.2 and
$(M_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
by Lemma 4.12. It follows from [Reference Paškūnas39, Lemma 10.26] that the functor
$\mathscr T$
induces an isomorphism
$$ \begin{align*} \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) &\cong \operatorname{\mathrm{End}}_{\mathfrak Q(\mathscr O)}(\mathscr T M_{\mathfrak B}), \quad \varphi \mapsto \mathscr T\varphi. \end{align*} $$
Since
$Z_{\mathfrak B}'$
is the centre of
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
, it acts on
$\mathscr T M_{\mathfrak B}$
, and this action induces a homomorphism
$Z_{\mathfrak B}'\rightarrow Z (\operatorname {\mathrm {End}}_{\mathfrak Q(\mathscr O)}(\mathscr T M_{\mathfrak B}) )$
, which we may compose with the previous isomorphism to obtain a homomorphism
$Z_{\mathfrak B}'\rightarrow Z (\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) )$
. Since
$Z_{\mathfrak B}$
is the centre of
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, a similar argument shows that
$Z_{\mathfrak B}'$
is a
$Z_{\mathfrak B}$
-algebra and the surjection
$ (P_{\mathfrak B}' )^{\oplus n} \twoheadrightarrow M_{\mathfrak B}$
in Corollary 4.13 is
$Z_{\mathfrak B}$
-equivariant. It induces a
$Z_{\mathfrak B}$
-equivariant surjection
$ (\mathscr T P_{\mathfrak B}' )^{\oplus n} \twoheadrightarrow \mathscr T M_{\mathfrak B}$
. We deduce that the map
$$ \begin{align*} Z_{\mathfrak B}'\rightarrow Z(\operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B})) \end{align*} $$
is a homomorphism of
$Z_{\mathfrak B}$
-algebras. Proposition 4.15 implies that the composition
$$ \begin{align*} Z_{\mathfrak B}\rightarrow Z_{\mathfrak B}' \rightarrow Z(\operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}))\cong Z_{\mathfrak B} \end{align*} $$
is the identity map, which implies that the homomorphism is surjective.
4.4 Colmez’s Montreal functor
Let
$\mathscr G_{\mathbb {Q}_p}$
be the absolute Galois group of
$\mathbb {Q}_p$
. We will consider
$\zeta $
as a character of
$\mathscr G_{\mathbb {Q}_p}$
via local class field theory, normalised so that the uniformisers correspond to geometric Frobenius. Let
$\varepsilon : \mathscr G_{\mathbb {Q}_p} \rightarrow \mathbb {Z}_p^{\times }$
be the p-adic cyclotomic character.
Colmez [Reference Colmez19] has defined an exact and covariant functor
$\mathbf V$
from the category of smooth, finite-length representations of G on
$\mathscr O$
-torsion modules with a central character to the category of continuous finite-length representations of
$\mathscr G_{\mathbb {Q}_p}$
on
$\mathscr O$
-torsion modules. This functor is modified in [Reference Paškūnas39, Section 5.7] to an exact covariant functor
$$ \begin{align*} \check{\mathbf{V}}: \mathfrak C(\mathscr O) \rightarrow \operatorname{\mathrm{Mod}}^{\mathrm{pro}}_{\mathscr G_{\mathbb{Q}_p}}(\mathscr O) \end{align*} $$
as follows. Let M be in
$\mathfrak C(\mathscr O)$
; if it is of finite length, we define
$\check {\mathbf {V}}(M) := \mathbf V (M^\vee )^\vee (\zeta \varepsilon )$
, where
$\vee $
denotes the Pontryagin dual. For general
$M \in \mathfrak C(\mathscr O)$
, we may write
$M \cong \varprojlim M_i$
, with
$M_i$
of finite length in
$\mathfrak C(\mathscr O)$
, and define
$\check {\mathbf {V}}(M) := \varprojlim \check {\mathbf {V}}(M_i)$
. With this normalisation, we have the following:
-
•
$\check {\mathbf {V}} (\pi ^{\vee } ) = 0$
if
$\pi \cong \chi \circ \det $
. -
•
$\check {\mathbf {V}} (\pi ^{\vee } )= \chi _1$
if
$\pi \cong (\operatorname {\mathrm {Ind}}^G_B \chi _1 \otimes \chi _2 )_{\mathrm {sm}}$
. -
•
$\check {\mathbf {V}} (\pi ^{\vee } ) = \chi $
if
$\pi \cong \operatorname {\mathrm {Sp}} \otimes \chi \circ \det $
. -
•
$\check {\mathbf {V}} (\pi ^{\vee } ) = \mathbf V(\pi )$
is a
$2$
-dimensional absolutely irreducible Galois representation if
$\pi $
is supersingular.
The functor
$\check {\mathbf {V}}$
induces a bijection
$\mathfrak B\mapsto \bar {\rho }_{\mathfrak B}$
between blocks containing an absolutely irreducible representation and equivalence classes of semisimple representations
$\bar {\rho }: \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2(k)$
such that all irreducible summands of
$\bar {\rho }$
are absolutely irreducible. The representation
$\bar {\rho }_{\mathfrak B}$
can be described explicitly according to the classification of blocks given in Section 4.1: in case (i),
$\bar {\rho }_{\mathfrak B}=\check {\mathbf {V}} (\pi ^{\vee } )$
is absolutely irreducible; in case (ii),
$\bar {\rho }_{\mathfrak B}=\chi _1\oplus \chi _2$
; in cases (iii) and (vi),
$\bar {\rho }_{\mathfrak B}=\chi \oplus \chi $
; and in cases (iv) and (v),
$\bar {\rho }_{\mathfrak B}= \chi \oplus \chi \omega $
.
Since the functor
$\check {\mathbf {V}}: \mathfrak C(\mathscr O) \rightarrow \operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{\mathscr G_{\mathbb {Q}_p}}(\mathscr O)$
kills characters and hence every object in
$\mathfrak T(\mathscr O)$
, it factors through
$\mathscr T: \mathfrak C(\mathscr O) \rightarrow \mathfrak Q(\mathscr O)$
. We denote
$\check {\mathbf {V}}: \mathfrak Q(\mathscr O) \rightarrow \operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{\mathscr G_{\mathbb {Q}_p}}(\mathscr O)$
by the same letter.
Proposition 4.17. For each block
$\mathfrak B$
, the functor
$\check {\mathbf {V}}$
induces an equivalence of categories between
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
and its essential image in
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{\mathscr G_{\mathbb {Q}_p}}(\mathscr O)$
.
Proof. This is due to [Reference Paškūnas39] for cases (i)–(iv), [Reference Tung46, Proposition 2.8] for case (v) and [Reference Tung47, Proposition 1.3.2] for case (iv).
Proposition 4.18. The map
is surjective. Moreover, if
$\mathfrak B$
is supersingular, then
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )\cong M_2 ( (E_{\mathfrak B}' )^{\mathrm {op}} )$
; otherwise,
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )\cong (E_{\mathfrak B}' )^{\mathrm {op}}$
.
Proof. It suffices to show that
$M=\check {\mathbf {V}} (P_{\mathfrak B}' )$
satisfies all four conditions in Proposition 2.3. We note that the functor
$\mathrm m \mapsto \mathrm m \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}'} \check {\mathbf {V}} (P_{\mathfrak B}' )$
is exact by the equivalence of categories in Proposition 4.10(2), the exactness of
$\check {\mathbf {V}}$
and the isomorphism
$$ \begin{align*} \check{\mathbf{V}}(\mathrm m \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}'} P_{\mathfrak B}') \cong \mathrm m \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}'} \check{\mathbf{V}}(P_{\mathfrak B}') \end{align*} $$
(see the proof of [Reference Paškūnas39, Lemma 5.53]). Thus by Lemma 2.4,
$\check {\mathbf {V}} (P_{\mathfrak B}' )$
is a projective
$E_{\mathfrak B}'$
-module. Let
$\mathfrak r$
be the Jacobson radical of
$E_{\mathfrak B}'$
. Since
$$ \begin{align*} \left(E_{\mathfrak B}'\middle/{\mathfrak r}\right)\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}'}\check{\mathbf{V}}\left(P_{\mathfrak B}'\right)\cong \check{\mathbf{V}}\big(\left(E_{\mathfrak B}'\middle/\mathfrak r\right) \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}'} P_{\mathfrak B}'\big) \end{align*} $$
is either a
$1$
- or a
$2$
-dimensional k-vector space, the topological Nakayama’s lemma implies that
$\check {\mathbf {V}} (P_{\mathfrak B}' )$
is a finitely generated
$E_{\mathfrak B}'$
-module, and thus Proposition 2.3(1) holds. Proposition 4.17 implies that Proposition 2.3(2) holds.
If
$\mathrm m$
is a right pseudocompact
$E_{\mathfrak B}'$
-module then [Reference Brumer12, Lemma 2.4] implies that
$$ \begin{align} \operatorname{\mathrm{Hom}}_{\mathscr O}^{\mathrm{cont}}( \mathrm m \operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{E_{\mathfrak B}'} \check{\mathbf{V}}(P_{\mathfrak B}') , k) \cong \operatorname{\mathrm{Hom}}_{E_{\mathfrak B}'}^{\mathrm{cont}}( \check{\mathbf{V}}(P_{\mathfrak B}'), \operatorname{\mathrm{Hom}}_{\mathscr O}^{\mathrm{cont}}(\mathrm m, k)). \end{align} $$
Since in our situation
$E_{\mathfrak B}'$
is a compact
$\mathscr O$
-algebra, the irreducible (left or right)
$E_{\mathfrak B}'$
-modules are finite-dimensional vector spaces with discrete topology, and thus the map
$\mathrm m\mapsto \mathrm m^*:=\operatorname {\mathrm {Hom}}_k(\mathrm m, k)$
induces a bijection between irreducible left and irreducible right
$E_{\mathfrak B}'$
-modules. Moreover, if
$\mathrm m$
is an irreducible right
$E_{\mathfrak B}'$
-module, then it follows from formula (16) that
$\rho _{\mathrm m^*}\cong (\mathrm m \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}'} \check {\mathbf {V}} (P_{\mathfrak B}' ) )^*$
; thus Proposition 2.3(3) and (4) follow from Proposition 4.17.
If
$\mathfrak B$
is supersingular, then it contains only one irreducible
$\pi $
, which is not a character. Thus
$P_{\mathfrak B}=P_{\mathfrak B}'$
,
$E_{\mathfrak B}=E_{\mathfrak B}'$
is a local ring and
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}} P_{\mathfrak B}= \pi ^{\vee }$
. Thus
$ k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}} \check {\mathbf {V}}(P_{\mathfrak B})\cong \check {\mathbf {V}} (\pi ^{\vee } )\cong \bar {\rho }_{\mathfrak B}$
is an absolutely irreducible
$2$
-dimensional representation. Since
$E_{\mathfrak B}$
is a local ring, we deduce that
$\check {\mathbf {V}}(P_{\mathfrak B})$
is a free
$E_{\mathfrak B}$
-module of rank
$2$
. Thus
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_{E_{\mathfrak B}} (\check {\mathbf {V}}(P_{\mathfrak B}) )\cong M_2 ( (E_{\mathfrak B}' )^{\mathrm {op}} )$
.
If
$\mathfrak B$
is of type (iii) or (vi), then the block in the quotient category contains only one irreducible object, and Colmez’s functor maps it to a
$1$
-dimensional
$\mathscr G_{\mathbb {Q}_p}$
-representation. The same argument as in the supersingular case shows that
$E_{\mathfrak B}'$
is a local ring and
$\check {\mathbf {V}} (P_{\mathfrak B}' )$
is a free
$E_{\mathfrak B}'$
-module of rank
$1$
, and thus
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )= (E_{\mathfrak B}' )^{\mathrm {op}}$
.
If
$\mathfrak B$
is of type (ii), (iv) or (v), then
$\mathfrak Q(\mathscr O)_{\mathfrak B}$
contains exactly two irreducible objects, and Colmez’s functor sends them to distinct
$1$
-dimensional
$\mathscr G_{\mathbb {Q}_p}$
-representations
$\chi _1, \chi _2$
. It follows from Corollary 2.5 that
$\check {\mathbf {V}}(P_{\mathfrak B}')$
is a free
$E_{\mathfrak B}'$
-module of rank
$1$
, and thus
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )= (E_{\mathfrak B}' )^{\mathrm {op}}$
.
4.5 Banach-space representations
Let
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
be the category of admissible unitary L-Banach-space representations [Reference Schneider and Teitelbaum44, Section 3] on which Z acts by the character
$\zeta $
. We note that
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
is an abelian category [Reference Schneider and Teitelbaum44, Theorem 3.5]. Any
$\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
has an open, bounded and G-invariant lattice
$\Theta $
, and
$\Theta \otimes _{\mathscr O} k$
is an admissible smooth k-representation of G. Let
$\Theta ^d = \operatorname {\mathrm {Hom}}_{\mathscr O}(\Theta , {\mathscr O})$
be the Schikhof dual of
$\Theta $
endowed with the topology of pointwise convergence. Then
$\Theta ^d$
is an object of
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_G(\mathscr O)$
[Reference Paškūnas39, Lemma 4.4]. If
$\Theta ^d$
is in
$\mathfrak C(\mathscr O)$
, then
$\Xi ^d$
is in
$\mathfrak C(\mathscr O)$
for every open bounded G-invariant lattice
$\Xi $
in
$\Pi $
, since
$\Theta $
and
$\Xi $
are commensurable and
$\mathfrak C(\mathscr O)$
is closed under subquotients [Reference Paškūnas39, Lemma 4.6].
If
$\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
, then we let
$$ \begin{align*} \check{\mathbf{V}}(\Pi) = \check{\mathbf{V}}(\Theta^d) \otimes_{\mathscr O} L, \end{align*} $$
where
$\Theta $
is any open bounded G-invariant lattice in
$\Pi $
. Then
$\check {\mathbf {V}}$
is exact and contravariant on
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
.
5 Density
5.1 Capture
Let
$G = \operatorname {\mathrm {GL}}_2 (\mathbb {Q}_p )$
and
$K = \operatorname {\mathrm {GL}}_2 (\mathbb {Z}_p )$
. Write Z for the centre of G and
$Z(K)$
for the centre of K. Let
$\psi : Z(K) \rightarrow \mathscr O^{\times }$
be a continuous character. We identify Z with
$\mathbb {Q}_p^{\times }$
and
$Z(K)$
with
$\mathbb {Z}_p^{\times }$
via the map
$ \left(\begin{smallmatrix} x & 0 \\ 0 & x \end{smallmatrix} \right) \mapsto x$
.
Lemma 5.1. Let
$\{V_i\}_{i \in I}$
be a family of continuous representations of K on finite-dimensional L-vector spaces with central character
$\psi $
, and let
$M \in \operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{K, \psi }(\mathscr O)$
be
$\mathscr O$
-torsion-free. The following conditions are equivalent:
-
1. For all
$i \in I$
, the smallest quotient
$M \twoheadrightarrow Q$
such that
is equal to M. -
2. The intersection of the kernels of all
for each
$i \in I$
is equal to zero. -
3. The image of the evaluation map
is a dense subspace, where
$$ \begin{align*} \bigoplus_{i \in I} \operatorname{\mathrm{Hom}}_K(V_i, \Pi(M)) \otimes_L V_i \rightarrow \Pi(M) \end{align*} $$
$\Pi (M) := \operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{\mathscr O}(M, L)$
is an L-Banach space equipped with the supremum norm.
Proof. See [Reference Colmez, Dospinescu and Paškūnas21, Lemmas 2.7 and 2.10].
Definition 5.2. We say that
$\{V_i\}_{i \in I}$
captures M if it satisfies one of the equivalent conditions in Lemma 5.1.
Since
$1+p\mathbb {Z}_p$
(resp.,
$1+4\mathbb Z_2$
) is a free pro-p group of rank 1 if
$p>2$
(resp.,
$p=2$
), there are a smooth nontrivial character
$\chi : \mathbb {Z}_p^\times \rightarrow L^\times $
and a continuous character
$\eta _0 : \mathbb {Z}_p^{\times } \rightarrow L^\times $
such that
$\psi = \chi \eta _0^2$
. Let e be the smallest integer such that
$\chi $
is trivial on
$1+p^e\mathbb {Z}_p$
. Let
$$ \begin{align*} J = \left(\begin{smallmatrix} \mathbb{Z}_p^\times & \mathbb{Z}_p \\ p^e \mathbb{Z}_p & \mathbb{Z}_p^\times \end{smallmatrix}\right) \end{align*} $$
and let
$\chi \otimes \mathbf 1: J \rightarrow L^\times $
be the character which sends
$ \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$
to
$\chi (a)$
. Then the representation
$\tau =\operatorname {\mathrm {Ind}}^{K}_{J} (\chi \otimes \mathbf 1)$
is a principal series type. That is, for an irreducible smooth
$\overline {L}$
-representation
$\pi $
of G, we have
$\operatorname {\mathrm {Hom}}_{L[K]}(\tau , \pi ) \neq 0$
if and only if
$\pi \cong (\operatorname {\mathrm {Ind}}^G_B \psi _1 \otimes \psi _2 )_{\mathrm {sm}}$
, where B is a Borel subgroup and
$\psi _1\rvert _{\mathbb {Z}_p^\times } = \chi $
and
$\psi _2\rvert _{\mathbb {Z}_p^\times } = \mathbf 1$
[Reference Breuil and Mézard11, Section A2.2].
Proposition 5.3. The family
$$ \begin{align*} \{\operatorname{\mathrm{Ind}}^{K}_{J} (\chi \otimes \mathbf 1) \otimes \operatorname{\mathrm{Sym}}^{2a} L^2 \otimes (\det)^{-a} \otimes \eta \eta_0 \circ \det \}_{a \in \mathbb{N} , \eta}, \end{align*} $$
where
$\eta $
runs over all the characters with
$\eta ^2 = \mathbf 1$
, captures every projective object in
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{K, \psi ^{-1}}(\mathscr O)$
.
Proof. See [Reference Paškūnas41, Proposition 2.7].
We will denote the family of representations in this proposition by
$\{V_i\}_{i\in I}$
. We note that each
$V_i$
is a twist of a locally algebraic representation by a unitary character
$\eta _0$
, which might not be locally algebraic. However, twisting by its inverse will get us to a locally algebraic situation, which is sufficient for all arguments that follow.
5.2 Locally algebraic vectors in
$\Pi (P)$
Let
$\zeta : Z \rightarrow \mathscr O^{\times }$
be a continuous character and
$\psi = \zeta \rvert _{K}$
. Let P be a projective object in
$\mathfrak C(\mathscr O)$
and
$E = \operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(P)$
. In particular, P is a torsion-free compact linear-topological
$\mathscr O$
-module. Define
$$ \begin{align*} \Pi(P) := \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_{\mathscr O}(P, L) \end{align*} $$
with the topology induced by the supremum norm. Then we have
$E [1/p ] \cong \operatorname {\mathrm {End}}^{\mathrm {cont}}_{G}(\Pi (P))$
.
If V is a continuous representation of K on a finite-dimensional L-vector space, then
Since P is projective in
$\operatorname {\mathrm {Mod}}^{\mathrm {pro}}_{K, \psi ^{-1}}(\mathscr O)$
by [Reference Emerton and Paškūnas30, Corollary 3.10], the family of finite-dimensional K-representations associated to
$\psi $
in Proposition 5.3, which we denote by
$\{V_i\}_{i \in I}$
, captures P. We view
$V_i$
as a representation of
$KZ$
by letting
$ \left(\begin{smallmatrix} \varpi & 0 \\ 0 & \varpi \end{smallmatrix} \right)$
act by
$\zeta (\varpi )$
.
Proposition 5.4. For each
$i \in I$
, we define
$A_i := \operatorname {\mathrm {End}}_G (\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i )$
. Then
-
1.
$A_i$
is isomorphic to
$L[T]$
and -
2.
$\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
is flat over
$A_i$
.
Proof. We may write
$V_i= \operatorname {\mathrm {Ind}}^{K}_{J} (\chi \otimes \mathbf 1)\otimes _L W_i$
, where the action of
$KZ$
on
$W_i$
extends to an action of G (see Proposition 5.3). Then
$$ \begin{align*} \operatorname{\mathrm{c-Ind}}^G_{KZ} V_i \cong \operatorname{\mathrm{c-Ind}}^G_{KZ}(\operatorname{\mathrm{Ind}}^{K}_{J} (\chi \otimes \mathbf 1))\otimes_L W_i. \end{align*} $$
Here we view
$\operatorname {\mathrm {Ind}}^{K}_{J} (\chi \otimes \mathbf 1)$
as a representation of
$KZ$
by letting
$ \left(\begin{smallmatrix} \varpi & 0 \\ 0 & \varpi \end{smallmatrix} \right)$
act on
$\operatorname {\mathrm {Ind}}^{K}_{J} (\chi \otimes \mathbf 1)$
by
$\zeta (\varpi ) \zeta ^{-1}_{W_i}( \varpi )$
, where
$\zeta _{W_i}$
is the central character of
$W_i$
. Since the restriction of
$W_i$
to any compact open subgroup of G remains absolutely irreducible, the isomorphism induces an isomorphism of L-algebras
$$ \begin{align*} \operatorname{\mathrm{End}}_G(\operatorname{\mathrm{c-Ind}}^G_{KZ} V_i) \cong \operatorname{\mathrm{End}}_G(\operatorname{\mathrm{c-Ind}}^G_{KZ}(\operatorname{\mathrm{Ind}}^{K}_{J} (\chi \otimes \mathbf 1))). \end{align*} $$
Thus we may assume that
$W_i$
is the trivial representation.
By [Reference Bushnell and Kutzko14], the K-type
$\operatorname {\mathrm {Ind}}^{K}_{J} (\chi \otimes \mathbf 1)$
is a G-cover of the
$K_M$
-type
$\chi \otimes \mathbf 1$
, where
$M = \mathbb {Q}_p^{\times } \times \mathbb {Q}_p^{\times }$
and
$K_M = \mathbb {Z}_p^{\times } \times \mathbb {Z}_p^{\times }$
. Thus there is an algebra isomorphism
$$ \begin{align*} j_M: \operatorname{\mathrm{End}}_{M}(\operatorname{\mathrm{c-Ind}}^M_{K_M} (\chi \otimes \mathbf 1)) \xrightarrow{\sim} \operatorname{\mathrm{End}}_G(\operatorname{\mathrm{c-Ind}}^G_J(\chi \otimes \mathbf 1)) \end{align*} $$
such that for each
$f \in \operatorname {\mathrm {End}}_{M} (\operatorname {\mathrm {c-Ind}}^M_{K_M} \chi \otimes \mathbf 1 )$
, we have
$\operatorname {\mathrm {Supp}}(j_M f) = J \operatorname {\mathrm {Supp}}(f) J$
. It follows that
$$ \begin{align*} L[T] \xrightarrow{\sim} \operatorname{\mathrm{End}}_{M}(\operatorname{\mathrm{c-Ind}}^{M}_{K_MZ} (\chi \otimes \mathbf 1)) \xrightarrow[j_M]{\sim} A_i, \end{align*} $$
where T maps to an element in
$A_i$
supported at
$JZ \left(\begin{smallmatrix} p & 0 \\ 0 & 1 \end{smallmatrix} \right) JZ$
. Here we view
$\chi \otimes \mathbf 1$
as a representation of
$K_M Z$
by letting
$ \left(\begin{smallmatrix} \varpi & 0 \\ 0 & \varpi \end{smallmatrix} \right)$
act by
$\zeta (\varpi )$
. This shows the first assertion.
To prove the second assertion, it suffices to show that
$\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
is torsion-free, since
$A_i$
is a principal ideal domain. After tensoring with
$\overline {L}$
, this is equivalent to
$\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
having no
$T - \lambda $
torsion, which is easily seen using the fact that the functions in
$\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
are compactly supported.
In particular, Frobenius reciprocity gives
$$ \begin{align} \operatorname{\mathrm{Hom}}_K(V_i, \Pi(P)) \cong \operatorname{\mathrm{Hom}}_G(\operatorname{\mathrm{c-Ind}}^G_{KZ} V_i, \Pi(P)). \end{align} $$
Hence
$\operatorname {\mathrm {Hom}}_K(V_i, \Pi (P))$
is naturally an
$A_i$
-module and we may transport the action of
$A_i$
onto
via formula (17).
If V is a continuous representation of K on a finite-dimensional L-vector space and if
$\Theta $
is an open, bounded K-invariant lattice in V, let
$\lvert \cdot \rvert $
be the norm on
$V^*$
given by
$\lvert \ell \rvert := \mathop {\mathrm {sup}}_{v \in \Theta } \lvert \ell (v)\rvert $
, so that
$\Theta ^d = \operatorname {\mathrm {Hom}}_{\mathscr O}(\Theta , \mathscr O)$
is the unit ball in
$V^*$
with respect to
$\lvert \cdot \rvert $
. The topology on
is given by the norm
$\lVert \phi \rVert := \mathop {\mathrm {sup}}_{v \in P} \lvert \phi (v)\rvert $
, and
is the unit ball in this Banach space.
Proposition 5.5. For all
$i \in I$
, the submodule
is dense in
, where
$\ell _{A_i}(A_i \phi )$
is the length of
$A_i \phi $
as an
$A_i$
-module.
Proof. See [Reference Colmez, Dospinescu and Paškūnas21, Proposition 2.19].
Proposition 5.6. Let
$\mathfrak m$
be a maximal ideal of
$A_i$
and let
$\Pi $
be a completion of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
with respect to a G-invariant norm. Then
$\Pi $
is the universal unitary completion of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
. Moreover, the action of
$A_i$
on
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
extends to a continuous action of
$A_i$
on
$\Pi $
.
Proof. Let
$\Pi ^{\circ }$
be a G-invariant
$\mathscr O$
-lattice of
$\Pi $
. Then
$$ \begin{align*} \Theta := \Pi^{\circ} \cap (A_i / \mathfrak m^n \otimes_{A_i} \operatorname{\mathrm{c-Ind}}^G_{KZ} V_i) \end{align*} $$
is a G-invariant
$\mathscr O$
-lattice of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
. By [Reference Emerton27, Proposition 1.17], it suffices to show that
$\Theta $
is of finite type over
$\mathscr O[G]$
.
By Proposition 5.4(i), we have that
$\kappa (\mathfrak m) := A_i / \mathfrak m \cong L[T] / f(T)$
– where
$f(T) \in L[T]$
is an irreducible polynomial – is a finite extension of L. Define a finite, increasing, exhaustive filtration
$ \{ R^j \}_{n \geq j \geq 0}$
of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
by G-invariant
$A_i$
-submodules
$R^j = \mathfrak m^{n-j} / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
. Then we have
$$ \begin{align*} R^j / R^{j-1} \cong \kappa(\mathfrak m) \otimes_{A_i} \operatorname{\mathrm{c-Ind}}^G_{KZ} V_i \end{align*} $$
for each j and
$ \{ \Theta ^j := \Theta \cap R^j \}_{n \geq j \geq 0}$
is a finite, increasing, exhaustive filtration of
$\Theta $
such that
$\Theta ^j$
is a G-invariant
$\mathscr O$
-lattice of
$R^j$
for each j. Moreover,
$\Theta ^j / \Theta ^{j-1}$
gives rise to a G-invariant
$\mathscr O$
-lattice of
$R^j / R^{j-1} \cong A_i / \mathfrak m \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
, and thus is finitely generated over
$\mathscr O[G]$
by the proof of [Reference Berger and Breuil4, Theorem 4.3.1]. This implies that
$\Theta $
is finitely generated over
$\mathscr O[G]$
, and the first assertion follows.
If
$\phi \in A_i$
, then
$\phi (\Theta ) \subset \varpi ^n \Theta $
for some
$n\in \mathbb Z$
, as
$\Theta $
is finitely generated over
$\mathscr O[G]$
. This implies the second assertion.
We denote the universal unitary completion of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
by
$\Pi _{i,\mathfrak m, n}$
. If
$n=1$
, then
$\Pi _{i,\mathfrak m, 1}$
is the universal unitary completion of
$\kappa (\mathfrak m) \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
studied in [Reference Berger and Breuil4, Theorem 4.3.1] and [Reference Breuil and Emerton10, Proposition 2.2.1].
Corollary 5.7. If
is such that
$A_i \phi \cong A_i / \mathfrak m^n$
for a maximal ideal
$\mathfrak m$
of
$A_i$
, then
$\phi $
induces an injection
$\Pi _{i,\mathfrak m, n} \hookrightarrow \Pi (P)$
. Moreover,
$\Pi _{i,\mathfrak m, n}$
admits a filtration of length n such that each graded piece is isomorphic to the universal unitary completion
$\Pi _{i,\mathfrak m, 1}$
of
$\kappa (\mathfrak m) \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
.
Proof. The assumption
$A_i \phi \cong A_i / \mathfrak m^n$
implies that
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
injects into
$\Pi (P)$
. The first assertion follows immediately from Proposition 5.6. To show the second assertion, we let
$ \{R^j \}_{n \geq j \geq 0}$
be the filtration of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
defined in Proposition 5.6. Let
$\Pi ^j$
be the closure of
$R^j$
in
$\Pi _{i,\mathfrak m, n}$
; then
$\Pi ^n= \Pi _{i,\mathfrak m,n}$
. Since
$\mathfrak m R^j = R^{j-1}$
, we have
$\mathfrak m \Pi ^j = \Pi ^{j-1}$
and hence
$\Pi ^j= \mathfrak m^{n-j} \Pi _{i,\mathfrak m,n}$
. If
$\Pi ^j= \Pi ^{j-1}$
, then
$\Pi ^j=\mathfrak m \Pi ^j$
and hence
$\Pi ^j= \mathfrak m^j \Pi ^j= \mathfrak m^n \Pi _{i,\mathfrak m,n}=0$
. Since
$\Pi ^j\neq 0$
for
$1\le j\le n$
, we conclude that
$\Pi ^j\neq \Pi ^{j-1}$
for
$1\le j\le n$
. Moreover,
$\Pi ^j / \Pi ^{j-1}$
contains
$R^j / R^{j-1} \cong A_i / \mathfrak m \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
as a dense subspace, and thus is isomorphic to
$\Pi _{i,\mathfrak m, 1}$
. This proves the corollary.
Note that the image of any
$\phi \in \operatorname {\mathrm {Hom}}_G (A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i, \Pi (P) )$
is isomorphic to
$A_i / \mathfrak m^k \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
for some
$0 \leq k \leq n$
. Hence it induces an injection
$\Pi _{i, \mathfrak m, k} \hookrightarrow \Pi (P)$
by Corollary 5.7.
Proposition 5.8. Let P be a projective object in
$\mathfrak C(\mathscr O)$
. Then the image of the evaluation map
$$ \begin{align*} \bigoplus_{i \in I} \bigoplus_{\mathfrak m, n} \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_{G}(\Pi_{i,\mathfrak m, n}, \Pi(P)) \otimes_L \Pi_{i,\mathfrak m, n} \rightarrow \Pi(P), \end{align*} $$
where
$\mathfrak m$
runs through maximal ideals of
$A_i$
and
$n \in \mathbb {N}$
is a dense subspace.
Proof. Let
$\Pi $
be the closure of the image of the evaluation map and M the image of P under
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{L}(\Pi (P), L) \twoheadrightarrow \operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{L}(\Pi , L)$
. Then we have
Here the first isomorphism is due to the fact that any module M over a commutative ring A such that every finitely generated submodule is of finite length admits a decomposition
$M \cong \oplus _{\mathfrak m} M[\mathfrak m^{\infty }]$
with
$\mathfrak m$
running through maximal ideals of A. The second isomorphism is given by formula (17), and the last isomorphism is due to Corollary 5.7. Similarly, we have
Thus by Proposition 5.5, we have
for each
$i \in I$
. Combining this with Lemma 5.1, we deduce the proposition.
Corollary 5.9. Define
$\mathrm m_{i,\mathfrak m, n} := \operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)} (P, \Theta ^d ) \otimes _{\mathscr O} L$
, where
$\Theta $
is an open bounded G-invariant lattice in
$\Pi _{i,\mathfrak m, n}$
. Then
$$ \begin{align*} \bigcap_{i \in I} \bigcap_{\mathfrak m, n} \mathfrak{a}_{i,\mathfrak m, n} = 0, \end{align*} $$
where
$\mathfrak {a}_{i,\mathfrak m, n} := \operatorname {\mathrm {ann}}_E (\mathrm m_{i,\mathfrak m, n} )$
.
Proof. For
$\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
with a G-invariant
$\mathscr O$
-lattice
$\Theta $
, we have
$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(P, \Theta^d) \otimes_{\mathscr O} L \cong \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_{G}(\Pi, \Pi(P)). \end{align*} $$
Thus the evaluation map in Proposition 5.8 induces an
$E [1/p ]$
-homomorphism
$$ \begin{align*} \bigoplus_{i \in I} \bigoplus_{\mathfrak m, n} \mathrm m_{i,\mathfrak m, n} \otimes_L \Pi_{i,\mathfrak m, n} \rightarrow \Pi(P) \end{align*} $$
with a dense image. Since
$E [1/p ]$
acts faithfully on the right-hand side of the map, it acts faithfully on the left-hand side as well. This proves the corollary, since the E-action on the left-hand side factors through the quotient
$E / \bigcap _{i \in I} \bigcap _{\mathfrak m, n} \mathfrak {a}_{i,\mathfrak m, n}$
.
6 Main results
Given a block
$\mathfrak B$
, we have defined
$\pi _{\mathfrak B}$
,
$P_{\mathfrak B}$
and
$E_{\mathfrak B}$
in Section 4.1 and
$P_{\mathfrak B}'$
and
$E_{\mathfrak B}'$
in Section 4.2. We assume that all irreducibles in
$\mathfrak B$
are absolutely irreducible. This can be achieved by replacing k with a finite extension. Let
$\bar {\rho }_{\mathfrak B}$
be the
$2$
-dimensional semisimple Galois representation of
$\mathscr G_{\mathbb {Q}_p}$
over k defined by
$\check {\mathbf {V}} (\pi _{\mathfrak B}^{\vee } )$
in cases (i), (ii), (iv) and (v), and by a direct sum of two copies of
$\check {\mathbf {V}} (\pi _{\mathfrak B}^{\vee } )$
in cases (iii) and (vi); see Section 4.4 for an explicit description. We write
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
for the universal pseudodeformation ring of
$\operatorname {tr} \bar {\rho }_{\mathfrak B}$
with a fixed determinant
$\zeta \varepsilon $
. This ring is Noetherian by [Reference Chenevier18, Proposition F]. We let
$T: \mathscr G_{\mathbb {Q}_p}\rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
be the universal object (see Section 3).
6.1 Finiteness
Let
$\{V_i \}_{i \in I}$
be a family of K-representations defined in Proposition 5.3 and let
$A_i = \operatorname {\mathrm {End}}_G (\operatorname {\mathrm {c-Ind}}^G_{KZ} V_i )$
. For each
$i \in I$
, a maximal ideal
$\mathfrak m$
of
$A_i$
and
$n \in \mathbb {N}$
, we write
$\Pi _{i,\mathfrak m, n}$
for the universal unitary completion of
$A_i / \mathfrak m^n \otimes _{A_i} \operatorname {\mathrm {c-Ind}}^G_{KZ} V_i$
.
Lemma 6.1. Assume
$\Pi _{i,\mathfrak m, n}$
is a subrepresentation of
$\Pi (P_{\mathfrak B}' )$
. Then
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is a finite free
$A_{i} / \mathfrak m^n$
-module of rank equal to
$\dim _{A_i / \mathfrak m} (\check {\mathbf {V}} (\Pi _{i,\mathfrak m, 1} ) ) \leq 2$
. Moreover,
-
(i) if
$\mathop {\mathrm {rank}}\nolimits _{A_i / \mathfrak m^n} \check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} ) = 2$
, then
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is a deformation to
$A_{i} / \mathfrak m^n$
of the absolutely irreducible
$2$
-dimensional L-representation
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, 1} )$
of
$\mathscr G_{\mathbb {Q}_p}$
; and -
(ii) if
$\mathop {\mathrm {rank}}\nolimits _{A_{i} / \mathfrak m^n} \check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} ) = 1$
, then the action of
$\mathscr G_{\mathbb {Q}_p}$
on
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is given by an
$(A_{i} / \mathfrak m^n)^{\times }$
-valued character lifting
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, 1} )$
.
Proof. Since
$V_i$
is a principal series type, the
$\mathscr G_{\mathbb {Q}_p}$
-module
$$ \begin{align*} \check{\mathbf{V}}(\Pi_{i,\mathfrak m, n}) / \mathfrak m \check{\mathbf{V}}(\Pi_{i,\mathfrak m, n}) \xrightarrow{\sim} \check{\mathbf{V}}(\Pi_{i,\mathfrak m, 1}) \end{align*} $$
has dimension
$r \leq 2$
over
$\kappa (\mathfrak m) := A_i / \mathfrak m$
by [Reference Berger and Breuil4, Theorem 4.3.1] and [Reference Breuil and Emerton10, Proposition 2.2.1]. Nakayama’s lemma implies that we have a surjection
$$ \begin{align*} (A_i / \mathfrak m^n)^{\oplus r} \twoheadrightarrow \check{\mathbf{V}}(\Pi_{i,\mathfrak m, n}). \end{align*} $$
Proposition 5.7 and the exactness of
$\check {\mathbf {V}}$
imply that
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
has length
$nr$
as an
$A_i / \mathfrak m^n$
-module. Hence the surjection is an isomorphism and the lemma follows.
Lemma 6.2. Under the same assumptions as in Lemma 6.1, there is a natural map
$\theta _{i,\mathfrak m, n}: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} \rightarrow A_i / \mathfrak m^n$
, which induces a map
Proof. If
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is of rank
$2$
over
$A_i/\mathfrak m^n$
, it follows from Lemma 6.1(i) that
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is a deformation of the
$2$
-dimensional Galois representation
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, 1} )$
to
$A_i/{\mathfrak m}^n$
with determinant
$\zeta \varepsilon $
. It follows from [Reference Chenevier18, Section 4.1, Theorem 3.17] that there is an
$\mathscr O$
-algebra map
$\theta _{i,\mathfrak m, n}: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} \rightarrow A_i / \mathfrak m^n$
such that the specialisation of T along
$\theta _{i,\mathfrak m, n}$
is equal to
$\operatorname {tr}_{A_i/{\mathfrak m}^n} \check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
. This map induces a homomorphism of
$\mathscr O$
-algebras
and Cayley–Hamilton for
$M_2(A_i / \mathfrak m^n)$
implies that J lies in the kernel of this map.
If
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
is given by a character
$\chi _{i,\mathfrak m, n}: \mathscr G_{\mathbb {Q}_p} \rightarrow (A_i / \mathfrak m^n)^{\times }$
, then the same argument applies to the representation
$\chi _{i,\mathfrak m, n} \oplus \chi _{i,\mathfrak m,n}^{-1} \zeta \varepsilon $
, so that we get a map
Its image commutes with the idempotent which projects onto the direct summand
$\chi _{i,\mathfrak m, n}$
. Hence, the image is contained in
$\operatorname {\mathrm {End}}_{A_i / \mathfrak m^n} (\chi _{i,\mathfrak m, n} )\times \operatorname {\mathrm {End}}_{A_i / \mathfrak m^n} (\chi _{i,\mathfrak m,n}^{-1}\zeta \varepsilon )$
, and we may project to
$\operatorname {\mathrm {End}}_{A_i / \mathfrak m^n} (\chi _{i,\mathfrak m, n} )$
to obtain the required homomorphism.
In Propositions 3.1 and 4.18, we established surjections
Theorem 6.3. The maps just given induce a surjection
In particular,
$E_{\mathfrak B}'$
and
$Z_{\mathfrak B}'$
are finite over
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
and hence Noetherian.
Proof. For the first part we have to show that
$\operatorname {\mathrm {Ker}} \beta \subset \operatorname {\mathrm {Ker}} \alpha $
. Let
$M = \check {\mathbf {V}} (P_{\mathfrak B}' )$
and define
$\mathrm m_{i,\mathfrak m, n} := \mathrm m (\Pi _{i,\mathfrak m, n} )$
, with
$i,\mathfrak m, n$
as in Corollary 5.9. Then the assumptions in Lemma 2.6 are satisfied by Corollary 5.9. It follows that the kernel of
$\alpha $
is given by
$\cap _{i \in I} \cap _{\mathfrak m, n} \mathfrak b_{i,\mathfrak m, n}$
, where
$\mathfrak b_{i,\mathfrak m, n}$
is the
-annihilator of
$$ \begin{align*} \mathrm m_{i,\mathfrak m, n} \otimes_{E_{\mathfrak B}'} \check{\mathbf{V}}(P_{\mathfrak B}') \cong \check{\mathbf{V}}(\mathrm m_{i,\mathfrak m, n} \otimes_{E_{\mathfrak B}'} P_{\mathfrak B}') \cong \check{\mathbf{V}}(\Pi_{i,\mathfrak m, n}). \end{align*} $$
Since the action of
on
$\check {\mathbf {V}} (\Pi _{i,\mathfrak m, n} )$
factors through
by Lemma 6.2,
$\mathfrak b_{i,\mathfrak m, n}$
contains the kernel of
$\beta $
and hence
$\operatorname {\mathrm {Ker}} \beta \subset \operatorname {\mathrm {Ker}} \alpha $
.
The second assertion is a consequence of the first assertion and the finiteness of
over
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
[Reference Wang-Erickson48, Proposition 3.6].
Corollary 6.4.
$E_{\mathfrak B}$
and
$Z_{\mathfrak B}$
are finite over
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
and hence Noetherian.
Proof. By [Reference Paškūnas39, Lemma 10.26], we have
$$ \begin{align*} \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) &\cong \operatorname{\mathrm{End}}_{\mathfrak Q(\mathscr O)}(\mathscr T M_{\mathfrak B}), \\ \operatorname{\mathrm{Hom}}_{\mathfrak C(\mathscr O)}(P_{\mathfrak B}', M_{\mathfrak B}) &\cong \operatorname{\mathrm{Hom}}_{\mathfrak Q(\mathscr O)}(\mathscr T P_{\mathfrak B}', \mathscr T M_{\mathfrak B}), \end{align*} $$
since
$M_{\mathfrak B}^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = (M_{\mathfrak B})_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = (P_{\mathfrak B}' )_{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )} = 0$
. Define
$\mathrm m := \operatorname {\mathrm {Hom}}_{\mathfrak C(\mathscr O)} (P_{\mathfrak B}', M_{\mathfrak B} )$
, which is a finitely generated right
$E_{\mathfrak B}'$
-module by Corollary 4.13. Moreover, we have
$$ \begin{align*} \operatorname{\mathrm{End}}_{E_{\mathfrak B}'}(\mathrm m) \cong \operatorname{\mathrm{End}}_{\mathfrak Q(\mathscr O)}(\mathscr T M_{\mathfrak B})\cong \operatorname{\mathrm{End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) \end{align*} $$
by Proposition 4.10(2) and the isomorphism above. Theorem 6.3 implies that the conditions of Lemma 4.11 are satisfied with
$E=E_{\mathfrak B}'$
and
$Z=Z_{\mathfrak B}'$
. Thus
$\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B})$
and its centre are finite over
$Z_{\mathfrak B}'$
, and thus finite over
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
by Theorem 6.3, and hence Noetherian. This implies the corollary, since
$\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) \cong E_{\mathfrak B}$
and
$Z (\operatorname {\mathrm {End}}_{\mathfrak C(\mathscr O)}(M_{\mathfrak B}) ) \cong Z_{\mathfrak B}$
by Proposition 4.15.
Remark 6.5. Since
$Z_{\mathfrak B}$
is Noetherian, the
$\mathfrak m$
-adic topology coincides with the linearly compact topology in Lemma 4.1.
Let us spell out the properties of the map
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow Z_{\mathfrak B}$
constructed in Corollary 6.4. Since
$Z_{\mathfrak B}$
acts functorially on every object in
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, the homomorphism
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow Z_{\mathfrak B}$
induces a functorial ring homomorphism
$$ \begin{align*} c_M: R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}} \rightarrow \operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O)}(M) \end{align*} $$
for every object M in
${\mathfrak C}(\mathscr O)$
. Since
$\check {\mathbf {V}}$
is a functor, it induces a ring homomorphism
$$ \begin{align*} \operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O)}(M)\rightarrow \operatorname{\mathrm{End}}^{\mathrm{cont}}_{\mathscr G_{\mathbb{Q}_p}}(\check{\mathbf{V}}(M)),\quad \varphi\mapsto \check{\mathbf{V}}(\varphi). \end{align*} $$
We denote the action of
$\mathscr G_{\mathbb {Q}_p}$
on
$\check {\mathbf {V}}(M)$
by
$\rho _{\check {\mathbf {V}}(M)}$
. Finally, for all
$g\in \mathscr G_{\mathbb {Q}_p}$
we may evaluate the universal pseudorepresentation
$T: \mathscr G_{\mathbb {Q}_p} \rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
at
$g\in \mathscr G_{\mathbb {Q}_p}$
to obtain an element
$T(g)\in R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
.
Proposition 6.6. For each
$M\in {\mathfrak C}(\mathscr O)_{\mathfrak B}$
and each
$g\in \mathscr G_{\mathbb {Q}_p}$
,
$$ \begin{align*} \check{\mathbf{V}}(c_M(T(g)))= \rho_{\check{\mathbf{V}}(M)}(g) + \rho_{\check{\mathbf{V}}(M)}(g^{-1}) \zeta\varepsilon(g) \end{align*} $$
in
$\operatorname {\mathrm {End}}_{\mathscr G_{\mathbb {Q}_p}}^{\mathrm {cont}} (\check {\mathbf {V}}(M) )$
.
Proof. Since
$g^2 -T(g)g + \zeta \varepsilon (g)=0$
in
, the equality
$T(g)\operatorname {\mathrm {id}} = g + \zeta \varepsilon (g) g^{-1}$
holds in that ring. The rest is just unravelling the definitions.
Since
$P_{\mathfrak B}$
is a projective generator for
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, the functor
$$ \begin{align*} N \mapsto \mathrm m(N):= \operatorname{\mathrm{Hom}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B}, N) \end{align*} $$
induces an equivalence of categories between
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
and the category of right pseudocompact
$E_{\mathfrak B}$
-modules. The inverse functor is given by
$\mathrm m \mapsto \mathrm m \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}} P_{\mathfrak B}$
.
Corollary 6.7. For N in
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
the following assertions are equivalent:
-
1. There is a surjection
$P_{\mathfrak B}^{\oplus n} \twoheadrightarrow N$
for some
$n\ge 1$
. -
2.
$\mathrm m(N)$
is a finitely generated
$E_{\mathfrak B}$
-module. -
3.
$\mathrm m(N)$
is a finitely generated
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
-module. -
4.
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}} N$
is of finite length in
${\mathfrak C}(\mathscr O)$
. -
5. The cosocle of N in
${\mathfrak C}(\mathscr O)$
is of finite length.
The equivalent conditions hold if N is finitely generated over
for a compact open subgroup H of G.
Proof. (1) implies (2), since
$\mathrm m$
is exact. (2) implies (3) by Corollary 6.4. Since
$$ \begin{align*} k\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}}\mathrm m(N)\cong \mathrm m( k\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}}N), \end{align*} $$
and the functor
$\mathrm m$
is an antiequivalence, (3) implies (4). Let
$N \twoheadrightarrow \operatorname {\mathrm {cosoc}}(N)$
be the cosocle of N. Since the maximal ideal of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
acts trivially on every semisimple object, the surjection factors through
$$ \begin{align*} k\operatorname{\mathrm{\mathbin{\widehat{\otimes}}}}_{R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}} N \twoheadrightarrow \operatorname{\mathrm{cosoc}}(N), \end{align*} $$
and so (4) implies (5). If
$\operatorname {\mathrm {cosoc}}(N)$
is of finite length, then there is a surjection
$\pi _{\mathfrak B}^{\oplus n} \twoheadrightarrow \operatorname {\mathrm {cosoc}}(N)$
for some
$n\ge 1$
. Since
$P_{\mathfrak B}$
is projective, there is a map
$\varphi : P_{\mathfrak B}^{\oplus n} \rightarrow N$
lifting
$(P_{\mathfrak B})^{\oplus n} \twoheadrightarrow \pi _{\mathfrak B}^{\oplus n} \twoheadrightarrow \operatorname {\mathrm {cosoc}}(N)$
. The cokernel of
$\varphi $
will have zero cosocle and hence
$\varphi $
is surjective, so that (5) implies (1).
If N is finitely generated over
, which we may assume to be pro-p, then
$ ((N/\varpi N)^{\vee } )^H$
is a finite-dimensional k-vector space, and hence the G-socle of
$N^{\vee }$
is of finite length, which dually implies that (5) holds.
Since every irreducible in
$\mathfrak B$
is admissible, its Pontryagin dual is finitely generated over
for any compact open subgroup H of G. It follows from Corollary 6.7(4) that
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}}P_{\mathfrak B}$
is also a finitely generated
-module. This implies that the assumptions made in [Reference Paškūnas39, Section 4] are satisfied with the category
${\mathfrak C}(\mathscr O)$
there equal to
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, and we will record some consequences.
6.2 Banach-space representations
The category
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
decomposes into a direct sum of categories [Reference Paškūnas39, Proposition 5.36]:
$$ \begin{align} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L) \cong \bigoplus_{\mathfrak B \in \operatorname{\mathrm{Irr}}_{G, \zeta} / \sim} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L)_{\mathfrak B}, \end{align} $$
where the objects of
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B}$
are those
$\Pi $
in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
such that for every open bounded G-invariant lattice
$\Theta $
in
$\Pi $
, the irreducible subquotients of
$\Theta \otimes _{\mathscr O} k$
lie in
$\mathfrak B$
. This condition is equivalent to requiring that
$\Theta ^d$
be an object of
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
.
As in the previous subsection, we fix a block
$\mathfrak B$
consisting of absolutely irreducible representations. Let
$\operatorname {\mathrm {Mod}}^{\mathrm {fg}}_{E_{\mathfrak B} [1/p ]}$
be the category of finitely generated right
$E_{\mathfrak B} [1/p ]$
-modules. The functor
$$ \begin{align*} \mathrm m: \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L)_{\mathfrak B} \rightarrow \operatorname{\mathrm{Mod}}^{\mathrm{fg}}_{E_{\mathfrak B}[1/p]}, \quad \Pi \mapsto \mathrm m(\Pi):= \operatorname{\mathrm{Hom}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B},\Theta^d)\otimes_{\mathscr O} L, \end{align*} $$
where
$\Theta $
is any open bounded G-invariant lattice in
$\Pi $
and is exact, contravariant and fully faithful by [Reference Paškūnas39, Lemma 4.45]. Moreover, it induces an antiequivalence of categories
$$ \begin{align} \mathrm m: \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L)_{\mathfrak B}^{\mathrm{fl}} \overset{\cong}{\longrightarrow} \operatorname{\mathrm{Mod}}^{\mathrm{fl}}_{E_{\mathfrak B}[1/p]}, \end{align} $$
where the superscript ‘
$\mathrm {fl}$
’ indicates the subcategories of objects of finite length in the respective categories [Reference Paškūnas39, Theorem 4.34]. We write antiequivalence instead of equivalence to indicate that
$\mathrm m$
is contravariant.
If
$\mathfrak m$
is a maximal ideal of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} [1/p ]$
, then we let
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B, \mathfrak m}^{\mathrm {fl}}$
be the full subcategory of
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
consisting of finite-length Banach-space representations, which are killed by some power of
$\mathfrak m$
. The functor
$\mathrm m$
induces an antiequivalence between this category and the category of
$E_{\mathfrak B} [1/p ]$
-modules of finite length, which are killed by a power of
$\mathfrak m$
. The Chinese remainder theorem [Reference Paškūnas39, Theorem 4.36] implies that we have an equivalence of categories
$$ \begin{align} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L)_{\mathfrak B}^{\mathrm{fl}}\cong \bigoplus_{\mathfrak m \in \operatorname{\mathrm{m-Spec}} R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}[1/p]} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{G, \zeta}(L)_{\mathfrak B, \mathfrak m}^{\mathrm{fl}}. \end{align} $$
Corollary 6.8. If
$\Pi _1\in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B, \mathfrak m_1}^{\mathrm {fl}}$
and
$\Pi _2\in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B, \mathfrak m_2}^{\mathrm {fl}}$
for distinct maximal ideals
$\mathfrak m_1$
and
$\mathfrak m_2$
of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} [1/p ]$
, then the Yoneda
$\operatorname {\mathrm {Ext}}^i(\Pi _1, \Pi _2)$
computed in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
vanish for all
$i\ge 0$
.
Proof. It follows from formula (21) that the assertion holds for the Yoneda
$\operatorname {\mathrm {Ext}}$
groups computed in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B}^{\mathrm {fl}}$
. It follows from [Reference Paškūnas39, Proposition 4.46, Corollary 4.48] that these coincide with Yoneda
$\operatorname {\mathrm {Ext}}$
groups computed in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B}$
, which is a direct summand of
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
(see formaul (19)).
We will determine the set of isomorphism classes
$\operatorname {\mathrm {Irr}}(\mathfrak m, L')$
of irreducible objects in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')_{\mathfrak B, \mathfrak m}^{\mathrm {fl}}$
for a sufficiently large finite extension
$L'$
of L. Recall that
$\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
is absolutely irreducible if
$\Pi \otimes _L L'$
is irreducible in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')$
for all finite extensions
$L'$
of L. It follows from formula (20) that for such
$\Pi $
, Schur’s lemma holds, so that
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_G(\Pi )=L$
. This result is also proved in [Reference Dospinescu and Schraen26] in a much more general setting. It follows from formula (20) that irreducibles in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)_{\mathfrak B, \mathfrak m}$
correspond to irreducible modules of the algebra
$E_{\mathfrak B} \otimes _{R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}} \kappa (\mathfrak m)$
. Corollary 6.4 implies that this algebra is finite-dimensional over
$\kappa (\mathfrak m)$
, and thus
$\operatorname {\mathrm {Irr}}(\mathfrak m, L')$
is finite for every finite extension
$L'$
of L, and there is a finite extension
$L'$
of L such that all
$\Pi $
in
$\operatorname {\mathrm {Irr}}(\mathfrak m, L')$
are absolutely irreducible.
Proposition 6.9. Let
$L'$
be a finite extension of L and let
$\Pi $
be absolutely irreducible in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')_{\mathfrak B}$
. Let
$c_{\Pi }: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow L'$
be the composition
$$ \begin{align*} c_{\Pi}: R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}\rightarrow Z_{\mathfrak B}\rightarrow \operatorname{\mathrm{End}}^{\mathrm{cont}}_G(\Pi)=L'. \end{align*} $$
Then one of the following holds:
-
1. If
$\Pi $
is a subquotient of
$ (\operatorname {\mathrm {Ind}}_B^G \psi _1\otimes \psi _2 \varepsilon ^{-1} )_{\mathrm {cont}}$
for some unitary characters
$\psi _1, \psi _2: \mathbb {Q}_p^{\times } \rightarrow (L')^{\times }$
, then
$T_{c_{\Pi }}= \psi _1+\psi _2$
; -
2. otherwise,
$\check {\mathbf {V}}(\Pi )$
is a
$2$
-dimensional absolutely irreducible
$L'$
-representation of
$\mathscr G_{\mathbb {Q}_p}$
,
$\det \check {\mathbf {V}}(\Pi )= \zeta \varepsilon $
and
$T_{c_{\Pi }}= \operatorname {tr} \check {\mathbf {V}}(\Pi )$
,
where
$T_{c_{\Pi }}$
is the specialisation of the universal pseudorepresentation
$T: \mathscr G_{\mathbb {Q}_p}\rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
along
$c_{\Pi }$
.
Proof. Let
$\Psi $
be the unitary principal series representation in (1). If
$\psi _1\neq \psi _2\varepsilon ^{-1}$
, then
$\Psi ^{\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )}=0$
, and by looking at its reduction modulo p one may conclude that
$\Psi $
is absolutely irreducible. If
$\psi _1= \psi _2\varepsilon $
, then
$\Psi $
is a nonsplit extension
$$ \begin{align*} 0 \rightarrow \psi_1\circ \det \rightarrow \Psi \rightarrow \widehat{\operatorname{\mathrm{Sp}}} \otimes \psi_1\circ\det \rightarrow 0, \end{align*} $$
where
$\widehat {\operatorname {\mathrm {Sp}}}$
is the universal unitary completion of the smooth Steinberg representation. This representation is absolutely irreducible, since its mod p reduction is. In both cases,
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_G(\Psi )=L'$
, and thus
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
acts on all irreducible subquotients of
$\Psi $
via the same homomorphism
$c_{\Psi }$
. Since
$\check {\mathbf {V}}(\Psi )= \psi _2$
, regarded as a representation of
$\mathscr G_{\mathbb {Q}_p}$
via the class field theory,
$g + \varepsilon \zeta (g) g^{-1}$
acts on it via the scalar
$\psi _2(g)+ \psi _2 (g^{-1} )\varepsilon \zeta (g) = \psi _2(g) + \psi _1(g)$
for all
$g\in \mathscr G_{\mathbb {Q}_p}$
. Proposition 6.6 implies that the specialisation of T at
$c_{\Psi }$
is the character
$\psi _1+\psi _2$
.
If we are not in part (1), then [Reference Colmez, Dospinescu and Paškūnas21, Corollary 1.2, Theorem 1.9] imply that
$\check {\mathbf {V}}(\Pi )$
is absolutely irreducible
$2$
-dimensional and
$\det \check {\mathbf {V}}(\Pi )= \zeta \varepsilon $
. A calculation with
$2\times 2$
matrices implies that
$g+ \zeta \varepsilon (g) g^{-1}$
acts on
$\check {\mathbf {V}}(\Pi )$
by a scalar
$ (\operatorname {tr} \check {\mathbf {V}}(\Pi ) )(g)$
. Proposition 6.6 implies that the specialisation of T at
$c_{\Psi }$
is the character
$\operatorname {tr} \check {\mathbf {V}}(\Pi )$
.
Corollary 6.10. Let
$L'$
be a finite extension of L and let
$x: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow L'$
be an
$\mathscr O$
-algebra homomorphism. If
$T_x= \psi _1+ \psi _2$
for characters
$\psi _1, \psi _2:\mathscr G_{\mathbb {Q}_p} \rightarrow (L')^{\times }$
, then one of the following holds:
-
1. If
$\psi _1\psi _2^{-1}=\mathbf 1$
, then
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')= \{ (\operatorname {\mathrm {Ind}}_B^G \mathbf 1\otimes \varepsilon ^{-1} )_{\mathrm {cont}}\otimes \psi _1\circ \det \}$
. -
2. If
$\psi _1 \psi _2^{-1}= \varepsilon ^{\pm 1}$
, then
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')= \{ \mathbf 1, \widehat {\operatorname {\mathrm {Sp}}}, (\operatorname {\mathrm {Ind}}_B^G \varepsilon \otimes \varepsilon ^{-1} )_{\mathrm {cont}} \} \otimes \psi \circ \det $
. -
3. If
$\psi _1 \psi _2^{-1}\neq \varepsilon ^{\pm 1}, \mathbf 1$
, then
$$ \begin{align*} \operatorname{\mathrm{Irr}}(\mathfrak m_x, L')= \{ (\operatorname{\mathrm{Ind}}_B^G \psi_1\otimes \psi_2 \varepsilon^{-1})_{\mathrm{cont}}, (\operatorname{\mathrm{Ind}}_B^G \psi_2\otimes \psi_1 \varepsilon^{-1})_{\mathrm{cont}}\}, \end{align*} $$
where we consider
$\psi _1$
and
$\psi _2$
as unitary characters of
$\mathbb {Q}_p^{\times }$
via the class field theory and
$\psi $
in (2) is either
$\psi _1$
or
$\psi _2$
.
Proof. We have explained in the course of the proof of Proposition 6.9 that the representations listed in this corollary are absolutely irreducible and are contained in
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')$
. Moreover, using the functor of ordinary parts one may show that they are pairwise nonisomorphic.
We will show that the list is exhaustive. We may enlarge
$L'$
so that all
$\Pi \in \operatorname {\mathrm {Irr}}(\mathfrak m_x, L')$
are absolutely irreducible. Since
$c_{\Pi }=x$
, we cannot be in part (2) of Proposition 6.9, and thus we must be in part (1), and hence
$\Pi $
is already in our list.
Proposition 6.11. Let
$L'$
be a finite extension of L and let
$x: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow L'$
be an
$\mathscr O$
-algebra homomorphism. If
$T_x =\operatorname {tr} \rho $
, where
$\rho : \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2(L')$
is absolutely irreducible, then
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')=\{\Pi \}$
, with
$\Pi $
absolutely irreducible nonordinary and
$\check {\mathbf {V}}(\Pi )\cong \rho $
.
Proof. It follows from [Reference Colmez, Dospinescu and Paškūnas21, Theorem 1.1] that such
$\Pi $
exists. We may enlarge
$L'$
so that all
$\Pi ' \in \operatorname {\mathrm {Irr}}(\mathfrak m_x, L')$
are absolutely irreducible. Since
$c_{\Pi '}=x$
, we cannot be in part (1) of Proposition 6.9; thus we must be in part (2), and
$\operatorname {tr} \check {\mathbf {V}}(\Pi )=\operatorname {tr} \check {\mathbf {V}}(\Pi ')$
. Since both
$\check {\mathbf {V}}(\Pi ')$
and
$\check {\mathbf {V}}(\Pi )$
are absolutely irreducible, we deduce that
$\check {\mathbf {V}}(\Pi )\cong \check {\mathbf {V}}(\Pi ')$
, and [Reference Colmez, Dospinescu and Paškūnas21, Theorem 1.8] implies that
$\Pi \cong \Pi '$
.
6.3 The centre
We fix a block
$\mathfrak B$
as in the previous section and explore the relation between
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
and
$Z_{\mathfrak B}$
. So far we have constructed a finite map
$$ \begin{align} R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}\rightarrow Z_{\mathfrak B}'\twoheadrightarrow Z_{\mathfrak B} \end{align} $$
(Theorem 6.3 and Corollary 4.16). We show in Corollary A.14 that
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} [1/p ]$
is normal, and we know by [Reference Chenevier17, Theorem 2.1] that it is equidimensional, and the locus corresponding to absolutely irreducible pseudorepresentations is Zariski dense in
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} [1/p ]$
.
Proposition 6.12. Let
$R^{\square , \zeta \varepsilon }_{\bar {\rho }_{\mathfrak B}}$
be the universal framed deformation ring of
$\bar {\rho }_{\mathfrak B}$
with fixed determinant
$\zeta \varepsilon $
, let S be its maximal
$\mathscr O$
-torsion-free quotient and let
$V_S$
be a free S-module of rank
$2$
with
$\mathscr G_{\mathbb {Q}_p}$
-action induced by the universal deformation
$\mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2 ( R^{\square , \zeta \varepsilon }_{\bar {\rho }_{\mathfrak B}} )\twoheadrightarrow \operatorname {\mathrm {GL}}_2(S)$
. There is N in
${\mathfrak C}(\mathscr O)$
with a continuous action of S, which commutes with the action of G, such that we have an isomorphism of
$S [\mathscr G_{\mathbb {Q}_p} ]$
-modules
$\check {\mathbf {V}}(N)\cong V_{S}$
.
Proof. If
$x\in \operatorname {\mathrm {m-Spec}} S [1/p ]$
, then the specialisation of
$V_S$
at x lies in the image of
$\check {\mathbf {V}}$
by [Reference Colmez, Dospinescu and Paškūnas22, Theorem 10.1]. Since
$S [1/p ]$
is reduced (Propositions A.9 and A.13) and Jacobson, such points will be dense, and the existence of such N follows from [Reference Colmez19, Theorem II.3.3].
The subscript ‘
$\mathrm {tf}$
’ will indicate the maximal
$\mathscr O$
-torsion-free quotient.
Theorem 6.13. The surjection
in Theorem 6.3 identifies
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )$
with
. In particular, map (22) induces an isomorphism
$$ \begin{align} R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}[1/p]\overset{\cong}{\longrightarrow} Z_{\mathfrak B}'[1/p]. \end{align} $$
Moreover, if
$p\neq 2$
, then
$Z_{\mathfrak B}'= (R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} )_{\mathrm {tf}}$
, and if
$p=2$
, then the cokernel of map (22) is killed by
$2$
.
Proof. As already explained in the proof of Proposition 4.3, projective objects in
${\mathfrak C}(\mathscr O)$
are also projective in the category of compact
-modules, where
$K'$
is an open pro-p subgroup of
$\operatorname {\mathrm {SL}}_2 (\mathbb {Q}_p )$
intersecting Z trivially, and thus are
$\mathscr O$
-torsion-free. Hence,
$P_{\mathfrak B}'$
is
$\mathscr O$
-torsion-free. Since
$E_{\mathfrak B}'$
and
$Z_{\mathfrak B}'$
act faithfully on
$P_{\mathfrak B}'$
, we deduce that both rings are
$\mathscr O$
-torsion-free. Since
$\operatorname {\mathrm {End}}_{E_{\mathfrak B}'}^{\mathrm {cont}} (\check {\mathbf {V}} (P_{\mathfrak B}' ) )$
is either
$ (E_{\mathfrak B}' )^{\mathrm {op}}$
or
$M_2 (E_{\mathfrak B}' )$
by Proposition 4.18, we deduce that the map in Theorem 6.3 factors through
If a lies in the kernel of this map, then it will kill
$\check {\mathbf {V}} (P_{\mathfrak B}' )$
and hence
$\mathrm m\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}'} \check {\mathbf {V}} (P_{\mathfrak B}' )$
for all compact right
$E_{\mathfrak B}'$
-modules. Thus a will kill all objects in the essential image of
$\check {\mathbf {V}}$
, and it will therefore also kill the representation
$V_S$
defined in Proposition 6.12.
It follows from Propositions A.9 and A.13 that the ring
$R^{\square , \zeta \varepsilon }_{\bar {\rho }_{\mathfrak B}} [1/p ]$
is normal and the absolutely irreducible locus is dense in
$\operatorname {\mathrm {Spec}} R^{\square , \zeta \varepsilon }_{\bar {\rho }_{\mathfrak B}} [1/p ]$
. Corollary A.7 implies that
acts faithfully on
$V_S$
, hence
$a=0$
and formula (24) is injective.
The assertions about the centre follow from Proposition A.11.
We immediately obtain the following:
Corollary 6.14.
$Z_{\mathfrak B}'$
is a complete local Noetherian
$\mathscr O$
-algebra with residue field k. It is
$\mathscr O$
-torsion-free and
$Z_{\mathfrak B}' [1/p ]$
is normal.
Corollary 6.15.
$Z_{\mathfrak B}= Z_{\mathfrak B}'$
.
Proof. Since
$Z_{\mathfrak B}$
acts faithfully on
$P_{\mathfrak B}$
it is
$\mathscr O$
-torsion-free. Thus it is enough to show that the surjection
$Z_{\mathfrak B}'\twoheadrightarrow Z_{\mathfrak B}$
(see Corollary 4.16) induces an isomorphism after inversion of p. Since
$Z_{\mathfrak B}' [1/p ]$
is reduced by Corollary 6.14, it is enough to show that
$\operatorname {\mathrm {m-Spec}} Z_{\mathfrak B} [1/p ]$
contains a subset
$\Sigma $
of
$\operatorname {\mathrm {m-Spec}} Z_{\mathfrak B}' [1/p ]$
, which is dense in
$\operatorname {\mathrm {Spec}} Z_{\mathfrak B}' [1/p ]$
. We may take
$\Sigma $
to be the absolutely irreducible locus in
$\operatorname {\mathrm {m-Spec}} R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }_{\mathfrak B}} [1/p ]$
, as it is dense in
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr} \bar {\rho }_{\mathfrak B}} [1/p ]$
by [Reference Chenevier17, Theorem 2.1] and lies in
$\operatorname {\mathrm {m-Spec}} Z_{\mathfrak B} [1/p ]$
by the main result of [Reference Colmez, Dospinescu and Paškūnas21].
Corollary 6.16. Let
$L'$
be a finite extension of L and let
$x: R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}} \rightarrow L'$
be an
$\mathscr O$
-algebra homomorphism. If the specialisation of the universal pseudodeformation
$T:\mathscr G_{\mathbb {Q}_p}\rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
at x is not of the form
$\psi + \psi \varepsilon $
, then
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')^{\mathrm {fl}}_{\mathfrak B, \mathfrak m_x}$
is equivalent to the category of modules of finite length over the completion of
at
$\mathfrak m_x$
.
Moreover, if
$T_x =\operatorname {tr} \rho $
, where
$\rho : \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_2(L')$
is absolutely irreducible, then
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')^{\mathrm {fl}}_{\mathfrak B, \mathfrak m_x}$
is equivalent to the category of modules of finite length over the deformation ring
$R^{\zeta \varepsilon }_{\rho }$
, which parameterises the deformations of
$\rho $
with determinant
$\zeta \varepsilon $
to local Artinian
$L'$
-algebras.
In particular, if
$\Pi '\in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')$
is killed by
$\mathfrak m_x$
then
$\Pi '$
is isomorphic to a direct sum of finitely many copies of
$\Pi $
in Proposition 6.11.
Proof. After extending scalars, we may assume that
$L=L'$
. If
$T_x\neq \psi +\psi \varepsilon $
for any character
$\psi $
, then it follows from Corollaries 6.10 and 6.11 that
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')$
does not contain characters. We may apply [Reference Paškūnas39, Theorem 4.36] to deduce that
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')^{\mathrm {fl}}_{\mathfrak B, \mathfrak m_x}$
is antiequivalent to the category of
$E_{\mathfrak B}'\otimes _R \widehat {R}_{\mathfrak m_x}$
-modules of finite length, where
$R=R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
. Theorem 6.13 implies that this ring coincides with the completion of
at
$\mathfrak m_x$
.
Let us assume that
$T_x= \operatorname {tr} \rho $
with
$\rho $
absolutely irreducible. Then
$\operatorname {\mathrm {Irr}}(\mathfrak m_x, L')=\{\Pi \}$
, with
$\Pi $
absolutely irreducible by Proposition 6.11 and
$\check {\mathbf {V}}(\Pi )\cong \rho $
. It follows from [Reference Chenevier18, Sections 4.1 and 4.2] that
is an Azumaya algebra over
$\widehat {R}_{\mathfrak m_x}$
. Since
$\rho $
is an absolutely irreducible
$2$
-dimensional module of
, we conclude that
, and thus
is isomorphic to the ring of
$2\times 2$
matrices over
$\widehat {R}_{\mathfrak m_x}$
. Since
$M_2 (\widehat {R}_{\mathfrak m_x} )$
is Morita equivalent to
$\widehat {R}_{\mathfrak m_x}$
, which is isomorphic to
$R^{\zeta \varepsilon }_{\rho }$
by [Reference Kisin35, Lemma 2.3.3, Proposition 2.3.5], we obtain the first assertion.
In particular, the full subcategory of
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L')^{\mathrm {fl}}_{\mathfrak B, \mathfrak m_x}$
consisting of representations killed by
$\mathfrak m_x$
is equivalent to the category of finite-dimensional vector spaces over
$L'$
, and hence the last assertion follows.
6.4 Complements
We will prove Theorem 1.1. Let
$\mathfrak B$
be an arbitrary block, so that we do not assume that it contains an absolutely irreducible representation.
If
$\pi _1, \pi _2\in \operatorname {\mathrm {Irr}}_{G, \zeta }(k)$
, then it follows from [Reference Paškūnas39, Proposition 5.11] that there is a finite extension
$k'$
of k such that
$\pi _1\otimes _k k'$
is a finite direct sum of absolutely irreducible representations; then
$\pi _2\otimes _k k'$
is a finite direct sum of irreducible representations, each of them occurring with multiplicity
$1$
. It is implied by [Reference Paškūnas39, Proposition 5.33] that
$$ \begin{align} \operatorname{\mathrm{Ext}}^1_{k[G], \zeta}(\pi_1, \pi_2)\otimes_k k'\cong \operatorname{\mathrm{Ext}}^1_{k'[G],\zeta}(\pi_1\otimes_k k', \pi_2\otimes_k k'). \end{align} $$
If
$\operatorname {\mathrm {Ext}}^1_{k[G], \zeta }(\pi _1, \pi _2)\neq 0$
, then it follows from this formula that there are irreducible summands
$\pi _1'$
of
$\pi _1\otimes _k k'$
and
$\pi _2'$
of
$\pi _2 \otimes _k k'$
such that
$\operatorname {\mathrm {Ext}}^1_{k'[G], \zeta } (\pi _1', \pi _2' )\neq 0$
. Since
$\pi _1'$
is absolutely irreducible, we conclude by inspecting the list of blocks in Section 4.1 that
$\pi _2'$
is absolutely irreducible, and thus if
$\mathfrak B$
is the block containing
$\pi _1$
, then
$\pi \otimes _k k'$
is a finite direct sum of absolutely irreducible representations for all
$\pi \in \mathfrak B$
.
Let
$L'$
be a finite extension of L with ring of integers
$\mathscr O'$
and residue field
$k'$
. If
$\pi ^{\prime }_1, \pi _2'\in \operatorname {\mathrm {Irr}}_{G,\zeta }(k')$
are absolutely irreducible, then it follows from [Reference Paškūnas39, Proposition 5.11] that there exist unique
$\pi _1, \pi _2\in \operatorname {\mathrm {Irr}}_{G, \zeta }(k)$
such that
$\pi _1'$
is a direct summand of
$\pi _1\otimes _k k'$
and
$\pi _2'$
is a direct summand of
$\pi _2\otimes _k k'$
. It follows from formula (25) that if
$\pi _1'$
and
$\pi _2'$
lie in the same block in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O')$
, then
$\pi _1$
and
$\pi _2$
lie in the same block in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O)$
.
Thus if
$\pi _1\in \mathfrak B$
and we let
$\mathfrak B_1, \dotsc , \mathfrak B_r$
be the blocks of irreducible subquotients of
$\pi _1\otimes _k k'$
in
$\operatorname {\mathrm {Mod}}^{\mathrm {l.fin}}_{G, \zeta }(\mathscr O')$
and
$\mathfrak B\otimes _k k'$
be the set of isomorphism classes of irreducible subquotients of
$\pi \otimes _k k'$
for all
$\pi \in \mathfrak B$
, then
$$ \begin{align*} \mathfrak B\otimes_k k'= \bigcup_{i=1}^r \mathfrak B_i. \end{align*} $$
It follows from [Reference Paškūnas39, Corollary 5.40] that
$P_{\mathfrak B}\otimes _{\mathscr O} \mathscr O'\cong \prod _{i=1}^r P_{\mathfrak B_i}$
and
$$ \begin{align*} E_{\mathfrak B}\otimes_{\mathscr O} \mathscr O'\cong \operatorname{\mathrm{End}}_{{\mathfrak C}(\mathscr O')}(P_{\mathfrak B}\otimes_{\mathscr O} \mathscr O')\cong \prod_{i=1}^r E_{\mathfrak B_i}. \end{align*} $$
Since the blocks
$\mathfrak B_i$
contain only absolutely irreducible representations, it follows from Corollary 6.4 that
$E_{\mathfrak B}\otimes _{\mathscr O} \mathscr O'$
is a finite module over its centre
$Z(E_{\mathfrak B}\otimes _{\mathscr O} \mathscr O')$
and
$$ \begin{align*} Z(E_{\mathfrak B})\otimes_{\mathscr O} \mathscr O'\cong Z(E_{\mathfrak B}\otimes_{\mathscr O} \mathscr O')\cong \prod_{i=1}^r Z_{\mathfrak B_i} \end{align*} $$
is Noetherian (see the argument in the proof of [Reference Dospinescu, Paškūnas and Schraen25, Lemma 4.14] for the first isomorphism). Since
$\mathscr O'$
is a finite free
$\mathscr O$
-module, this implies that
$Z(E_{\mathfrak B})$
is Noetherian, and
$E_{\mathfrak B}$
is a finitely generated
$Z_{\mathfrak B}$
-module, which finishes the proof of Theorem 1.1.
7 Application to Hecke eigenspaces
Let R be a linearly compact local
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
-algebra with residue field k; we do not assume that R is Noetherian. If
$x:R\rightarrow \overline {\mathbb {Q}}_p$
is an
$\mathscr O$
-algebra homomorphism, then we denote by
$T_x$
the specialisation of the universal pseudorepresentation
$T: \mathscr G_{\mathbb {Q}_p} \rightarrow R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
along
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}\rightarrow R \overset {x}{\rightarrow } \overline {\mathbb {Q}}_p$
.
Let M be an object of
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, which we assume to be
$\mathscr O$
-torsion-free. Then
$\Pi (M):=\operatorname {\mathrm {Hom}}_{\mathscr O}^{\mathrm {cont}}(M, L)$
is a unitary L-Banach-space representation of G.
We assume that we are given a continuous action of R on M, which commutes with the action of G, such that the following hold:
-
• the action of R on M is faithful;
-
• the two actions of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
on M induced by the maps coincide; and
$$ \begin{align*} R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}\rightarrow R, \qquad R^{\mathrm{ps}, \zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak B}}\rightarrow Z_{\mathfrak B} \end{align*} $$
-
• M is a finitely generated
-module.
Theorem 7.1. Let
$x: R\rightarrow \overline {\mathbb {Q}}_p$
be an
$\mathscr O$
-algebra homomorphism and let
$\Pi (M)[\mathfrak m_x]$
be the subspace of
$\Pi (M)$
annihilated by the kernel of x. Then under the foregoing assumptions,
$\Pi (M)[\mathfrak m_x]$
is nonzero and is of finite length in
$\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{G, \zeta }(L)$
. Moreover,
-
• if
$T_x$
is the trace of an absolutely irreducible Galois representation defined over
$\kappa (x)$
, then for some multiplicity
$$ \begin{align*} \Pi(M)[\mathfrak m_x]\cong \Pi^{\oplus m} \end{align*} $$
$m>0$
, where
$\Pi $
is an absolutely irreducible nonordinary
$\kappa (x)$
-Banach-space representation of G satisfying
$\operatorname {tr} \check {\mathbf {V}}(\Pi )= T_x$
; and
-
• if
$T_x$
is the trace of a reducible Galois representation, then (after a possible extension of scalars) all the irreducible subquotients of
$\Pi (M)[\mathfrak m_x]$
occur as subquotients of a direct sum of unitary parabolic induction where
$$ \begin{align*} (\operatorname{\mathrm{Ind}}_B^G \psi_1 \otimes \psi_2 \varepsilon^{-1})_{\mathrm{cont}} \oplus (\operatorname{\mathrm{Ind}}_B^G \psi_2 \otimes \psi_1 \varepsilon^{-1})_{\mathrm{cont}}, \end{align*} $$
$\psi _1, \psi _2:\mathscr G_{\mathbb {Q}_p} \rightarrow \kappa (x)^{\times }$
are characters such that
$T_x= \psi _1 + \psi _2$
.
Proof. Since
$P_{\mathfrak B}$
is a projective generator for
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
, the functor
$N \mapsto \mathrm m(N):= \operatorname {\mathrm {Hom}}_{{\mathfrak C}(\mathscr O)}(P_{\mathfrak B}, N)$
induces an equivalence of categories between
${\mathfrak C}(\mathscr O)_{\mathfrak B}$
and the category of right pseudocompact
$E_{\mathfrak B}$
-modules. The inverse functor is given by
$\mathrm m \mapsto \mathrm m \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{E_{\mathfrak B}} P_{\mathfrak B}$
. In particular, the assumption that R acts faithfully on M implies that R acts faithfully on
$\mathrm m(M)$
.
We claim that
$\mathrm m(M)$
is a finitely generated R-module. The topological Nakayama’s lemma implies that it is enough to show that
$k \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R \mathrm m(M)$
is a finite-dimensional k-vector space. Since
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R \mathrm m(M)\cong \mathrm m (k \operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R M )$
, it is enough to show that
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R M$
is of finite length in
${\mathfrak C}(\mathscr O)$
. Since by assumption M is a finitely generated
-module,
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R M$
is a finitely generated
-module. Since by assumption the actions of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
on M induced by
$Z_{\mathfrak B}$
and by R, coincide we deduce that the maximal ideal of
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
annihilates
$k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R M$
. Corollary 6.7(4) applied to
$N=k\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_R M$
implies the claim.
Since
$\mathrm m(M)$
is a finitely generated and faithful R-module, its localisation
$\mathrm m(M)_{\mathfrak m_x}$
is a finitely generated faithful
$R_{\mathfrak m_x}$
-module. If
$\mathrm m(M)\otimes _R \kappa (x)=0$
, then
$\mathrm m(M)_{\mathfrak m_x}=0$
by Nakayama’s lemma, and since
$R_{\mathfrak m_x}$
acts faithfully,
$R_{\mathfrak m_x}=0$
and hence
$\kappa (x)=0$
, giving a contradiction. In particular,
$\mathrm m(M)\otimes _{R}\kappa (x)$
is a nonzero, finite-dimensional
$\kappa (x)$
-vector space. Since R is a compact
$\mathscr O$
-module and
$\kappa (x)$
is a subfield of
$\overline {\mathbb {Q}}_p$
, we have that
$\kappa (x)$
is a finite extension of L and the image of R is contained in the ring of integers of
$\kappa (x)$
.
Let Q be the maximal
$\mathscr O$
-torsion-free Hausdorff quotient of
$M/\mathfrak m_x M$
. It follows from [Reference Schneider and Teitelbaum44, Proposition 1.3] that
$\Pi (Q)$
is a closed subspace of
$\Pi (M)$
, which then implies that
$\Pi (Q)=\Pi (M)[\mathfrak m_x]$
. It follows from the equivalence of categories already explained that
$\mathrm m(Q)$
is isomorphic to the image of
$\mathrm m(M)$
in
$\mathrm m(M)\otimes _{R}\kappa (x)$
. In particular, Q and thus
$\Pi (Q)$
are nonzero.
The last two assertions follow from the antiequivalence (20) and Corollaries 6.10 and 6.16.
Remark 7.2. If M is finitely generated as a
-module, then the argument in the proof of Theorem 7.1 shows that
$\mathrm m(M)$
is a finitely generated
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
-module, and since R acts faithfully on
$\mathrm m(M)$
, then R is a finitely generated
$R^{\mathrm {ps}, \zeta \varepsilon }_{\operatorname {tr}\bar {\rho }_{\mathfrak B}}$
-module and hence is Noetherian.
The result allows us to remove the restrictions imposed on the Galois representation
$\bar {\rho }_{\mathfrak m, p}$
in [Reference Pan38, Corollary 6.3.6], by taking M to be the Pontryagin dual of the representation denoted by
$\tilde {H}^1(K^p, E/\mathscr O)_{\mathfrak m, \zeta '}$
in [Reference Pan38, Theorem 6.3.5] and taking R to be the closure of the subring generated by the Hecke operators in
$\operatorname {\mathrm {End}}^{\mathrm {cont}}_{\mathscr O}(M)$
. Since [Reference Pan38, Corollary 6.3.6] is the only place where the restriction on p is used, the proof of [Reference Pan38, Theorem 6.4.7] goes through without a change to give the following result:
Theorem 7.3 Lue Pan +
$\varepsilon $
Let
$\rho : \operatorname {\mathrm {Gal}} (\overline {\mathbb {Q}}/\mathbb {Q} )\rightarrow \operatorname {\mathrm {GL}}_2(L)$
be promodular and absolutely irreducible. If
$\rho $
is unramified outside finitely many places and
$\rho \rvert _{\mathscr G_{\mathbb {Q}_p}}$
is Hodge–Tate with weights
$0,0$
, then
$\rho $
is associated to a weight
$1$
modular form.
The promodular condition means that the Hecke eigenvalues associated to
$\rho $
appear in completed cohomology; see [Reference Pan38, Definition 6.1.2] for the precise definition. The original theorem in Lue Pan’s paper had to additionally assume that if p is
$2$
or
$3$
, then
$ (\bar {\rho }\rvert _{\mathscr G_{\mathbb {Q}_p}} )^{\mathrm {ss}}$
is not isomorphic to
$\chi \oplus \chi \omega $
for any character
$\chi : \mathscr G_{\mathbb {Q}_p}\rightarrow k^{\times }$
.
A Normality of
$R^{\mathrm {ps}} [1/p ]$
Let
$\mathscr G$
be a profinite group satisfying Mazur’s finiteness condition at p: The group of continuous group homomorphisms
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{\mathrm {grp}} (\mathscr G', \mathbb F_p )$
is finite for every open subgroup
$\mathscr G'$
of
$\mathscr G$
. Let
$\bar {\rho }: \mathscr G \rightarrow \operatorname {\mathrm {GL}}_d(k)$
be a continuous semisimple representation such that all the irreducible summands of
$\bar {\rho }$
are absolutely irreducible. Let
$\psi : \mathscr G\rightarrow \mathscr O^{\times }$
be a character lifting
$\det \bar {\rho }$
. Let
$\bar {D}: k[\mathscr G]\rightarrow k$
be the pseudorepresentation associated to
$\bar {\rho }$
in [Reference Chenevier18], so that
$\bar {D}(1+t g)= \det (1 + t \bar {\rho }(g))$
for all
$g\in \mathscr G$
. We may consider the framed deformation ring
$R^{\square }_{\bar {\rho }}$
, its quotient
$R^{\square , \psi }_{\bar {\rho }}$
parameterising framed deformations of
$\bar {\rho }$
with determinant equal to
$\psi $
, the universal deformation ring
$R^{\mathrm {ps}}$
of
$\bar {D}$
, and its quotient
$R^{\mathrm {ps}, \psi }$
parameterising deformations of
$\bar {D}$
with determinant
$\psi $
. This last ring is constructed as follows: If
$D^u:\mathscr G\rightarrow R^{\mathrm {ps}}$
is the universal deformation of
$\bar {D}$
, then for each
$g\in \mathscr G$
, we have that
$D^u(1 +t g)= a_0(g)+\dotsb +a_d(g)t^d$
, with
$a_i(g)\in R^{\mathrm {ps}}$
, and
$R^{\mathrm {ps}, \psi }$
is the quotient of
$R^{\mathrm {ps}}$
by the ideal generated
$\psi (g) a_d(g)-1$
for all
$g\in \mathscr G$
. The finiteness condition on
$\mathscr G$
ensures that all these rings are Noetherian. The characteristic polynomial of the universal framed deformations of
$\bar {\rho }$
induces maps
$R^{\mathrm {ps}}\rightarrow R^{\square }_{\bar {\rho }}$
and
$R^{\mathrm {ps}, \psi }\rightarrow R^{\square , \psi }_{\bar {\rho }}$
.
Theorem A.1. If
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal, then both
$R^{\mathrm {ps}} [1/p ]$
and the associated rigid space
$(\operatorname {\mathrm {Spf}} R^{\mathrm {ps}})^{\mathrm {rig}}$
are normal.
We also prove a version of the theorem with a fixed determinant. We apply this theorem to
$\mathscr G=\mathscr G_{\mathbb {Q}_p}$
to prove that the rings
$R^{\mathrm {ps}, \psi }, R^{\mathrm {ps}}$
and associated rigid analytic spaces are normal for all
$2$
-dimensional
$\bar {\rho }$
. There is essentially one case that we need to handle, namely
$\bar {\rho }= \mathbf 1 \oplus \omega $
, where
$\omega $
is the cyclotomic character modulo p; in the other cases, all the rings are regular. The trickiest cases are when
$p=2$
and
$p=3$
. The case
$p=2$
is treated in [Reference Colmez, Dospinescu and Paškūnas22]. We deal with the case
$p=3$
using the work of Böckle [Reference Böckle7].
The argument of [Reference Colmez, Dospinescu and Paškūnas22] has been extended by Iyengar in [Reference Iyengar33], to the case when
$\bar {\rho }$
is the trivial d-dimensional representation of a Galois group of a p-adic field F, under the assumption that F contains a primitive
$4$
th root of unity if
$p=2$
. Thus our theorem applies in that setting.Footnote
3
We will split the proof into several steps. We start with commutative algebra lemmas and recall that all excellent rings are G-rings [45, Tag 07QS].
Lemma A.2. Let A be a G-ring and set
$\mathfrak p\in \operatorname {\mathrm {Spec}} A$
. Then
$A_{\mathfrak p}$
satisfies Serre’s condition
$(R_i)$
(resp.,
$(S_i)$
) if and only if the completion
$A_{\mathfrak p}$
at
$\mathfrak p$
does.
Proof. Let
$B=A_{\mathfrak p}$
and let
$\hat {B}$
be the completion of
$A_{\mathfrak p}$
at
$\mathfrak p$
. Since A is a G-ring, the fibre rings
$\kappa (\mathfrak q)\otimes _B \hat {B}$
are regular for all
$\mathfrak q\in \operatorname {\mathrm {Spec}} B$
. The assertion follows from [Reference Matsumura and Reid36, Theorem 23.9].
Lemma A.3. Let A be a complete local Noetherian
$\mathscr O$
-algebra with residue field k and
, set
$\mathfrak q \in \operatorname {\mathrm {Spec}} B [1/p ]$
, and let
$\mathfrak p$
be the image of
$\mathfrak q$
in
$\operatorname {\mathrm {Spec}} A [1/p ]$
. Then
$A_{\mathfrak p}$
satisfies Serre’s condition
$(R_i)$
(resp.,
$(S_i)$
) if and only if
$B_{\mathfrak q}$
does. In particular,
$A [1/p ]$
is normal if and only if
$B [1/p ]$
is normal.
Proof. The proof is a variation on [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin15, Appendix A]. We may assume that A and hence B are
$\mathscr O$
-torsion-free. Set
$\mathfrak p'\in \operatorname {\mathrm {Spec}} A_{\mathfrak p} \subset \operatorname {\mathrm {Spec}} A$
. We claim that the ring
$\kappa (\mathfrak p')\otimes _A B$
is regular. By Cohen’s structure theorem, there is a subring
$C\subset A/\mathfrak p'$
such that C is formally smooth over
$\mathscr O$
and
$A/\mathfrak p'$
is finite over C. Then
$$ \begin{align*} \kappa(\mathfrak p')\otimes_A B\cong \kappa(\mathfrak p') \otimes_{A/\mathfrak p'} B/\mathfrak p'B. \end{align*} $$
Since
$A/\mathfrak p'$
is finite over C, we have
Thus
where
$Q(C)$
is the quotient field of C. Since C is formally smooth over
$\mathscr O$
, the ring
is isomorphic to a ring of formal power series over
$\mathscr O$
, and thus is regular. Tensoring with
$Q(C)$
over C is just localisation with respect to the multiplicative set
$C\setminus \{0\}$
, and thus
is regular. Since
$Q(C)$
is of characteristic
$0$
, the extension
$\kappa (\mathfrak p)/ Q(C)$
is separable, and it follows from [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin15, Lemma A.3] that
is regular. We deduce that
$\kappa (\mathfrak p')\otimes _{A_{\mathfrak p}} B_{\mathfrak q}$
is regular, since it is a localisation of
$\kappa (\mathfrak p')\otimes _A B$
at
$\mathfrak q$
.
It follows from [Reference Matsumura and Reid36, Theorem 23.9] that
$A_{\mathfrak p}$
satisfies
$(R_i)$
(resp.,
$(S_i)$
) if and only if
$B_{\mathfrak q}$
does. To conclude that
$A [1/p ]$
is normal if and only if
$B [1/p ]$
is, we only have to show that the map
$\operatorname {\mathrm {Spec}} B [1/p ]\rightarrow \operatorname {\mathrm {Spec}} A [1/p ]$
is surjective, and this is clear because
$(\mathfrak p, x_1, \dotsc , x_r)$
maps to
$\mathfrak p$
.
Lemma A.4. Let
$A\rightarrow B$
be a finite étale map of local rings. Then A satisfies Serre’s condition
$(R_i)$
(resp.,
$(S_i)$
) if and only if B does.
Proof. If
$\mathfrak p\in \operatorname {\mathrm {Spec}} A$
, then the fibre ring
$\kappa (\mathfrak p)\otimes _A B$
is a finite étale
$\kappa (\mathfrak p)$
-algebra and hence a product of fields, and thus is regular. The assertion follows from [Reference Matsumura and Reid36, Theorem 23.9].
Proposition A.5. Let
$L'$
be a finite extension of L and let
$\rho : \mathscr G \rightarrow \operatorname {\mathrm {GL}}_n(L')$
be a continuous representation with mod p semisimplification isomorphic to
$\bar {\rho }$
. If
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal, then the ring
$R^{\square }_{\rho }$
, representing the framed deformations of
$\rho $
to Artinian
$L'$
-algebras, is also normal.
Proof. We may choose a finite extension
$L''$
of
$L'$
with the ring of integers
$\mathscr O''$
and residue field
$k''$
such that
$\rho \otimes _{L'} L''$
has a
$\mathscr G$
-invariant
$\mathscr O''$
-lattice
$\Theta $
with
$\Theta \otimes _{\mathscr O''} k''\cong \bar {\rho }\otimes _k k''$
(see the proof of [Reference Colmez, Dospinescu and Paškūnas22, Lemma 9.5]). Thus
$\Theta $
is a deformation of
$\bar {\rho }\otimes _k k''$
to
$\mathscr O''$
.
It follows from Lemma A.4 that
$R^{\square }_{\rho }$
is normal if and only if
$L''\otimes _{L'} R^{\square }_{\rho }$
is normal. The same argument shows that
$L''\otimes _{L} R^{\square }_{\bar {\rho }} [1/p ]$
is normal. Moreover, we may identify
$L''\otimes _{L'} R^{\square }_{\rho }$
with the framed deformation ring of
$\rho \otimes _{L'} L''$
to local Artinian
$L''$
-algebras, and
$\mathscr O''\otimes _{\mathscr O} R^{\square }_{\bar {\rho }}$
with the framed deformation ring of
$\bar {\rho }\otimes _k k''$
to local Artinian
$\mathscr O''$
-algebras. After these identifications, we may assume that
$L=L'=L''$
, and so
$\Theta $
is a deformation of
$\bar {\rho }$
to
$\mathscr O$
and hence induces an
$\mathscr O$
-algebra homomorphism
$x: R^{\square }_{\bar {\rho }} \rightarrow \mathscr O$
.
It follows from [Reference Kisin35, Lemma 2.3.3, Proposition 2.3.5] that
$R^{\square }_{\rho }$
is isomorphic to the completion of
$ (R^{\square }_{\bar {\rho }} )_{\mathfrak p}$
at
$\mathfrak p=\operatorname {\mathrm {Ker}} x$
. Since
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal,
$ (R^{\square }_{\bar {\rho }} )_{\mathfrak p}$
will satisfy
$(R_1)$
and
$(S_2)$
. Lemma A.2 implies that the same holds for the completion. Thus
$R^{\square }_{\rho }$
is normal by Serre’s criterion for normality [Reference Matsumura and Reid36, Theorem 23.8].
Lemma A.6. Let A be a normal Noetherian ring and let G be a group acting on A by ring automorphisms. Then the subring of G-invariants
$A^G$
is normal.
Proof. If A is a domain, then the assertion is proved in [Reference Bruns and Herzog13, Proposition 6.4.1]. The same proof works in our setting, as we will explain for the lack of a reference. Since A is Noetherian and normal, it is a finite product of normal domains. Let
$\operatorname {\mathrm {Frac}}(A)$
denote its total ring of fractions. Then
$\operatorname {\mathrm {Frac}}(A)$
is a finite product of fields. The group G acts on
$\operatorname {\mathrm {Frac}}(A)$
and we have
$$ \begin{align} A^G= A\cap \operatorname{\mathrm{Frac}}(A)^G. \end{align} $$
We claim that
$\operatorname {\mathrm {Frac}}(A)^G$
is a finite product of fields. The claim implies that
$\operatorname {\mathrm {Frac}}(A)^G$
is its own ring of fractions. Since A is normal, equation (26) implies that
$A^G$
is reduced and integrally closed in its ring of fractions and has only finitely many minimal prime ideals – and hence is normal by [45, Tag 037B, Lemma 10.37.16].
To prove the claim, we note that
$\operatorname {\mathrm {Spec}} \operatorname {\mathrm {Frac}}(A)$
consists of finitely many primes and is in bijection with the set
$\mathscr E$
of idempotents
$e\in \operatorname {\mathrm {Frac}}(A)$
, such that
$e \operatorname {\mathrm {Frac}}(A) e$
is a field. We have
$1=\sum _{e\in \mathscr E} e$
and
$e e'=0$
if
$e\neq e'$
. If
$e\in \operatorname {\mathrm {Frac}}(A)$
is a G-invariant idempotent, then
$$ \begin{align*} \operatorname{\mathrm{Frac}}(A)^G= (e \operatorname{\mathrm{Frac}}(A) e)^G \times ((1-e) \operatorname{\mathrm{Frac}}(A)(1-e))^G, \end{align*} $$
and thus we may assume that the action of G on
$\mathscr E$
is transitive. If
$x\in \operatorname {\mathrm {Frac}}(A)^G$
and
$e x=0$
for some
$e\in \mathscr E$
, then using the transitivity of the action we obtain that
$ex=0$
for all
$e\in \mathscr E$
, and so
$x=\sum _{e\in \mathscr E} ex=0$
. Hence, if
$x\in \operatorname {\mathrm {Frac}}(A)^G$
is nonzero, then
$ex$
is nonzero, and we denote its inverse in the field
$e \operatorname {\mathrm {Frac}}(A) e$
by
$(xe)^{-1}$
. If we let
$y= \sum _{e\in \mathscr E} (xe)^{-1} e \in \operatorname {\mathrm {Frac}}(A)$
, then
$xy=1$
. Thus x is a unit in
$\operatorname {\mathrm {Frac}}(A)$
and its inverse y is unique. Uniqueness implies that y is G-invariant. Thus if G acts transitively on
$\mathscr E$
, then
$\operatorname {\mathrm {Frac}}(A)^G$
is a field.
Proof of Theorem A.1. Let
$D^u: \mathscr G\rightarrow R^{\mathrm {ps}}$
be the universal pseudorepresentation lifting
$\overline {D}$
. Let
$\mathrm {CH}(D^u)$
be the closed two-sided ideal of
defined in [Reference Chenevier18, Section 1.17], so that
is the largest quotient of
where the Cayley–Hamilton theorem for
$D^u$
holds. Following [Reference Chenevier18, Section 1.17], we will call such an algebra a Cayley–Hamilton
$R^{\mathrm {ps}}$
-algebra of degree d. Then E is a finitely generated
$R^{\mathrm {ps}}$
-module [Reference Wang-Erickson48, Proposition 3.6]. If
$f: E\rightarrow M_d(B)$
is a homomorphism of
$R^{\mathrm {ps}}$
-algebras for a commutative
$R^{\mathrm {ps}}$
-algebra B, then we say f is a homomorphism of Cayley–Hamilton algebras if
$\det \circ f: E\rightarrow B$
is equal to the specialisation of
$D^u$
along
$R^{\mathrm {ps}}\rightarrow B$
.
There is a commutative
$R^{\mathrm {ps}}$
-algebra
$A^{\mathrm {gen}}$
together with a homomorphism of
$R^{\mathrm {ps}}$
-algebras
$j: E\rightarrow M_d(A^{\mathrm {gen}})$
satisfying the following universal property: If
$f: E\rightarrow M_d(B)$
is a map of Cayley–Hamilton
$R^{\mathrm {ps}}$
-algebras for a commutative
$R^{\mathrm {ps}}$
-algebra B, then there is a unique map
$\tilde {f}: A^{\mathrm {gen}}\rightarrow B$
of
$R^{\mathrm {ps}}$
-algebras such that
$f= M_d (\tilde {f} )\circ j$
(see, for example, [Reference Wang-Erickson48, Theorem 3.8] or [Reference Böckle, Iyengar and Paškūnas8, Lemma 3.1]). Since E is finitely generated as an
$R^{\mathrm {ps}}$
-module,
$A^{\mathrm {gen}}$
is of finite type over
$R^{\mathrm {ps}}$
.
Let
$\Lambda _i: E\rightarrow R^{\mathrm {ps}}$
,
$0\le i\le d$
, be the coefficients of the characteristic polynomial of
$D^u$
; these are homogeneous polynomial laws satisfying
$D^{u}(t-a)= \sum _{i=0}^n (-1)^i \Lambda _i(a) t^{d-i}$
in
$R^{\mathrm {ps}}[t]$
for all
$a\in E$
[Reference Chenevier18, Section 1.10]. Now
$E [1/p ]$
is a
$\mathbb Q$
-algebra and the pair
$ (E [1/p ], \Lambda _1 )$
is a trace algebra satisfying the d-dimensional Cayley–Hamilton identity in the sense of [Reference Procesi43, Definition 2.6] (see [Reference Chenevier16, Footnote 10]). Moreover, for
$\mathbb Q$
-algebras, the homomorphisms of Cayley–Hamilton algebras coincide with the notion of maps of algebras with traces in [Reference Procesi43, Section 2.5]. Thus
$j: E [1/p ] \rightarrow M_d (A^{\mathrm {gen}} [1/p ] )$
is injective and its image is equal to the
$\operatorname {\mathrm {GL}}_d$
-invariants [Reference Procesi43, Theorem 2.6]. Moreover,
$R^{\mathrm {ps}} [1/p ]= (A^{\mathrm {gen}} [1/p ] )^{\operatorname {\mathrm {GL}}_d}$
(see [Reference Chenevier16, Proposition 2.3] and [Reference Wang-Erickson48, Theorem 2.20]). By Lemma A.6 it is enough to show that
$A^{\mathrm {gen}} [1/p ]$
is normal. Further, it is enough to show that the localisation of
$A^{\mathrm {gen}} [1/p ]$
at every maximal ideal is normal (see [45, Tag 037B, Lemma 10.37.10]). (The superscript ‘gen’ in
$A^{\mathrm {gen}}$
stands for generic matrices in [Reference Procesi43, Section 1.1].)
Let
$\mathfrak m$
be a maximal ideal of
$A^{\mathrm {gen}} [1/p ]$
. Its residue field
$\kappa (\mathfrak m)$
is a finite extension of L, as
$A^{\mathrm {gen}} [1/p ]$
is finitely generated over
$R^{\mathrm {ps}} [1/p ]$
. By specialising j at
$\mathfrak m$
we obtain a continuous representation
$\rho :\mathscr G\rightarrow \operatorname {\mathrm {GL}}_d(\kappa (\mathfrak m))$
such that
$$ \begin{align*} \det(1+ t \rho(g))=D^u\otimes_{R^{\mathrm{ps}}} \kappa(\mathfrak m)( 1 + tg),\quad \forall g\in \mathscr G. \end{align*} $$
This implies that if we choose a
$\mathscr G$
-invariant
$\mathscr O_{\kappa (\mathfrak m)}$
-lattice
$\Theta $
in
$\rho $
, then the semisimplification of
$\Theta /\varpi _{\kappa (\mathfrak m)}\Theta $
is isomorphic to
$\bar {\rho }$
, so that we are in the setup of Proposition A.5. The universal property of
$A^{\mathrm {gen}}$
implies that the completion of
$A^{\mathrm {gen}} [1/p ]$
at
$\mathfrak m$
is the universal framed deformation ring
$R^{\square }_{\rho }$
, which is normal by Proposition A.5. Since
$A^{\mathrm {gen}}$
is finitely generated over
$R^{\mathrm {ps}}$
, which is a complete local Noetherian ring,
$A^{\mathrm {gen}}$
and hence its localisation
$ (A^{\mathrm {gen}} [1/p ] )_{\mathfrak m}$
are excellent, and thus a G-ring. Lemma A.2 implies that
$ (A^{\mathrm {gen}} [1/p ] )_{\mathfrak m}$
satisfies
$(R_1)$
and
$(S_2)$
and hence is normal.
Let
$\tilde {R}$
be the normalisation of
$R^{\mathrm {ps}}$
in
$R^{\mathrm {ps}} [1/p ]$
. Then
$(\operatorname {\mathrm {Spf}} R^{\mathrm {ps}})^{\mathrm {rig}}= (\operatorname {\mathrm {Spf}} \tilde {R} )^{\mathrm {rig}}$
[Reference de Jong24, Lemma 7.2.2]. Since
$R^{\mathrm {ps}} [1/p ]$
is normal, so is
$\tilde {R}$
, and thus
$ (\operatorname {\mathrm {Spf}} \tilde {R} )^{\mathrm {rig}}$
is normal [Reference de Jong24, Proposition 7.2.4(c)]. Alternatively, one could use the fact that the local rings of
$(\operatorname {\mathrm {Spf}} R^{\mathrm {ps}})^{\mathrm {rig}}$
are excellent [Reference Conrad23, Theorem 1.1.3] and [Reference de Jong24, Lemma 7.1.9] together with Lemma A.2.
The following is a corollary to the proof; it does not require the assumption that
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal:
Corollary A.7. Let V be a free
$R^{\square }_{\bar {\rho }} [1/p ]$
-module of rank d with
$\mathscr G$
-action given by
$\rho ^{\square }: \mathscr G \rightarrow \operatorname {\mathrm {GL}}_d (R^{\square }_{\bar {\rho }} )$
. Then
acts faithfully on V. The same holds with the fixed determinant.
Proof. We use the notation of the proof of Theorem A.1, so that
and there is a map
$j: E\rightarrow M_d(A^{\mathrm {gen}})$
satisfying a universal property. This map is an injection after inverting p. Let
$V^{\mathrm {gen}}$
be a free
$A^{\mathrm {gen}}$
-module of rank d, with E-action given by j. Thus
$E [1/p ]$
acts faithfully on
$V^{\mathrm {gen}} [1/p ]$
.
Suppose that
$a\in E [1/p ]$
kills off V. Since
$E [1/p ]$
acts faithfully on
$V^{\mathrm {gen}} [1/p ]$
, there is a maximal ideal
$\mathfrak m$
of
$A^{\mathrm {gen}} [1/p ]$
such that a acts nontrivially on
$V^{\mathrm {gen}}\otimes _{A^{\mathrm {gen}}} A^{\mathrm {gen}}_{\mathfrak m}$
. Let
$\hat {A}^{\mathrm {gen}}_{\mathfrak m}$
be the completion of
$A^{\mathrm {gen}}_{\mathfrak m}$
with respect to the maximal ideal. Since
$\hat {A}^{\mathrm {gen}}_{\mathfrak m}$
is faithfully flat over
$A^{\mathrm {gen}}_{\mathfrak m}$
, a acts nontrivially on the completion of
$V^{\mathrm {gen}} [1/p ]$
at
$\mathfrak m$
, which we denote by
$\hat {V}^{\mathrm {gen}}_{\mathfrak m}$
. However, as explained in the proof of Theorem A.1,
$\hat {V}^{\mathrm {gen}}_{\mathfrak m}$
is isomorphic as an E-module to the completion of V at a maximal ideal of
$R^{\square }_{\bar {\rho }} [1/p ]$
. Since a annihilates V, it will also annihilate the completion, giving a contradiction.
Define
$E^{\psi }:= E\otimes _{R^{\mathrm {ps}}}R^{\mathrm {ps}, \psi }$
and
$A^{\mathrm {gen}, \psi }:=A^{\mathrm {gen}}\otimes _{R^{\mathrm {ps}}}R^{\mathrm {ps}, \psi }$
. Then
$j: E\rightarrow M_d(A^{\mathrm {gen}})$
induces a map
$j: E^{\psi }\rightarrow M_d (A^{\mathrm {gen}, \psi } )$
which satisfies the same universal property as j. Then the same proof works with
$A^{\mathrm {gen}, \psi }$
instead of
$A^{\mathrm {gen}}$
.
Lemma A.8. Let R be a complete local Noetherian
$\mathscr O$
-algebra with residue field k, and let
$\rho : \mathscr G\rightarrow \operatorname {\mathrm {GL}}_d(R)$
be a continuous representation. Assume that R is
$\mathscr O$
-torsion-free and reduced, and the set of
$x\in \operatorname {\mathrm {m-Spec}} R [1/p ]$
such that
$\rho _x$
is absolutely irreducible is dense in
$\operatorname {\mathrm {Spec}} R [1/p ]$
. Then
$$ \begin{align*} \mathscr C:=\{ X\in M_d(R): X \rho(g)= \rho(g) X, \quad \forall g\in \mathscr G\} \end{align*} $$
consists of scalar matrices.
Proof. Set
$X\in \mathscr C$
with matrix entries
$x_{ij}$
. Let
$\mathfrak p_1, \dotsc , \mathfrak p_n$
be the minimal primes of R. Since R is reduced, it embeds into
$\prod _{s=1}^n \kappa (\mathfrak p_s)$
. It is enough to show that the image of X in
$M_d(\kappa (\mathfrak p_s))$
for
$1\le s \le n$
is scalar, since if the images of
$x_{ij}$
and
$x_{ii}-x_{jj}$
for
$i\neq j$
are zero in
$\kappa (\mathfrak p_s)$
for
$1\le s\le n$
, then they are zero in A and so X is scalar. If
$x\in \operatorname {\mathrm {m-Spec}} R [1/p ]$
is such that
$\rho _x: \mathscr G\rightarrow \operatorname {\mathrm {GL}}_d(\kappa (x))$
is absolutely irreducible, then
$\mathscr C\otimes _R \kappa (x)$
is
$1$
-dimensional. Since
$\mathfrak q\mapsto \dim _{\kappa (\mathfrak q )} \mathscr C\otimes _R \kappa (\mathfrak q)$
is upper semicontinuous and such x are dense, we deduce that
$\dim _{\kappa (\mathfrak p_s )} \mathscr C\otimes _R \kappa (\mathfrak p_s)=1$
for all minimal primes
$\mathfrak p_s$
. This implies that
$\mathscr C\otimes _R \kappa (\mathfrak p_s)$
consists of scalar matrices.
Proposition A.9. Assume that
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
is nonzero. Then
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal (resp., reduced) if and only if
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
is normal (resp., reduced).
Proof. Let
$\Gamma $
be the pro-p completion of the abelianisation of
$\mathscr G$
. Because
$\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{\mathrm {grp}} (\mathscr G, \mathbb F_p )$
is finite,
$\Gamma \cong \Delta \times \mathbb {Z}_p^r$
, where
$\Delta $
is a finite p-group. The map
$\mathscr G\rightarrow (R^{\square }_{\bar {\rho }} )^{\times }$
,
$g\mapsto \psi (g)^{-1} \det \rho ^{\square }(g)$
, factors through
$\Gamma $
and thus induces an
$\mathscr O$
-algebra homomorphism
; and
$R^{\square , \psi }_{\bar {\rho }}$
is equal to the quotient of
$R^{\square }_{\bar {\rho }}$
by the augmentation ideal in
.
Let
$\mathscr X(\Gamma )$
be the functor which sends a local Artinian
$\mathscr O$
-algebra
$(A, \mathfrak m_A)$
to the group of continuous characters
$\chi : \Gamma \rightarrow 1+\mathfrak m_A$
. This functor is represented by
. For such
$(A, \mathfrak m_A)$
, the group
$\mathscr X(\Gamma )(A)$
acts on
$D^{\square }(A)$
by twisting. The action induces a homomorphism of local
$\mathscr O$
-algebras
. Let
$R^{\mathrm {inv}}= \{ a\in R^{\square }_{\bar {\rho }}: \gamma (a)= a\otimes 1 \}$
be the subring of
$\mathscr X(\Gamma )$
-invariants in
$R^{\square }_{\bar {\rho }}$
. Analogously,
$\mathscr X(\Delta )$
acts on
$D^{\square }$
, the action induces the map
$\delta : R^{\square }_{\bar {\rho }}\rightarrow R^{\square }_{\bar {\rho }}\operatorname {\mathrm {\mathbin {\widehat {\otimes }}}}_{\mathscr O} \mathscr O[\Delta ]$
and we let
$R^{\mathrm {inv.t}}= \{ a\in R^{\square }_{\bar {\rho }}: \delta (a)= a\otimes 1 \}$
be the subring of
$\mathscr X(\Delta )$
-invariants in
$R^{\square }_{\bar {\rho }}$
.
The action of
$\mathscr X(\Gamma )$
and
$\mathscr X(\Delta )$
on
$D^{\square }$
is free, since if
$\rho _A: \mathscr G_{\mathbb {Q}_p}\rightarrow \operatorname {\mathrm {GL}}_d(A)$
is a framed deformation of
$\bar {\rho }$
, then for each
$g\in \mathscr G_{\mathbb {Q}_p}$
at least one matrix entry of
$\rho _A(g)$
will not lie in
$\mathfrak m_A$
and thus is a unit. Hence,
$\rho _A(g)=\rho _A(g) \chi _A(g)$
for all
$g\in G$
implies that
$\chi _A$
is the trivial character.
The map
$R^{\mathrm {inv.t}} \rightarrow R^{\square }_{\bar {\rho }}$
is finite and becomes étale after inversion of p by [Reference Allen1, Proposition 1.1.11(2)]. Thus
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal if and only if
$R^{\mathrm {inv.t}} [1/p ]$
is normal, by Lemma A.4. Since
$R^{\mathrm {inv}}$
is the subring of
$\mathscr X(\Gamma /\Delta )$
-invariants in
$R^{\mathrm {inv.t}}$
and
$\Gamma /\Delta \cong \mathbb {Z}_p^r$
, we have
by [Reference Allen1, Proposition 1.1.11(2)]. Thus
$R^{\mathrm {inv.t}} [1/p ]$
is normal if and only if
$R^{\mathrm {inv}} [1/p ]$
is normal, by Lemma A.3. The map
$R^{\mathrm {inv}}\rightarrow R^{\square , \psi }_{\bar {\rho }}$
is finite and becomes étale after inversion of p by [Reference Allen1, Proposition 1.1.11(3)]. Lemma A.4 implies that
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
is normal if and only if
$R^{\mathrm {inv}} [1/p ]$
is normal. Putting all the equivalences together proves the assertion.
Since reducedness is equivalent to (
$R_0$
) and (
$S_1$
), the same proof works.
Corollary A.10. If
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal, then
$R^{\mathrm {ps}, \psi } [1/p ]$
and the associated rigid analytic space
$ (\operatorname {\mathrm {Spf}} R^{\mathrm {ps}, \psi } )^{\mathrm {rig}}$
are normal.
Proof. Proposition A.9 implies that
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
is normal. The proof of Theorem A.1, with
$R^{\square }_{\bar {\rho }} [1/p ]$
replaced by
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
, implies the assertion.
Proposition A.11. Let
,
$E_{\mathrm {tf}}$
the maximal
$\mathscr O$
-torsion-free quotient of E,
$Z(E_{\mathrm {tf}})$
the centre of
$E_{\mathrm {tf}}$
and
$R^{\mathrm {ps}}_{\mathrm {tf}}$
the maximal
$\mathscr O$
-torsion-free quotient of
$R^{\mathrm {ps}}$
. Then
$R^{\mathrm {ps}}_{\mathrm {tf}}$
is a subring of
$Z(E_{\mathrm {tf}})$
.
If
$R^{\square }_{\bar {\rho }} [1/p ]$
is reduced and the set
$x\in \operatorname {\mathrm {m-Spec}} R^{\square }_{\bar {\rho }} [1/p ]$
, such that
$\rho ^{\square }_x$
is absolutely irreducible, is dense in
$\operatorname {\mathrm {Spec}} R^{\square }_{\bar {\rho }} [1/p ]$
, then
$d \cdot Z(E_{\mathrm {tf}}) \subset R^{\mathrm {ps}}_{\mathrm {tf}}$
. In particular, if
$p\nmid d$
, then
$R^{\mathrm {ps}}_{\mathrm {tf}}=Z(E_{\mathrm {tf}})$
. Moreover, the same holds for rings with fixed determinant.
Proof. As in the proof of Theorem A.1, there is an injection
$$ \begin{align*} j: E[1/p] \hookrightarrow M_d(A^{\mathrm{gen}}[1/p]). \end{align*} $$
Moreover,
$\operatorname {tr} \circ j$
induces a surjection
$E [1/p ]\twoheadrightarrow R^{\mathrm {ps}} [1/p ]$
. Thus
$R^{\mathrm {ps}}_{\mathrm {tf}}$
is a subring of
$Z(E_{\mathrm {tf}})$
. Corollary A.7 gives us an injection
$E_{\mathrm {tf}} \hookrightarrow M_d (R^{\square }_{\bar {\rho }} [1/p ] )$
. If
$a\in E_{\mathrm {tf}}$
, then the characteristic polynomial of
$j(a)$
has coefficients in
$R^{\mathrm {ps}}_{\mathrm {tf}}$
. Moreover,
$Z(E_{\mathrm {tf}})$
is contained in the centraliser of
$\rho ^{\square }(\mathscr G)$
in
$M_d (R^{\square }_{\bar {\rho }} [1/p ] )$
. According to Lemma A.8, the centraliser is equal to scalar matrices. Since
$j(z)$
is a scalar matrix, we deduce that
$d z \in R^{\mathrm {ps}}_{\mathrm {tf}}$
.
It follows from Proposition A.9 that
$R^{\square , \psi }_{\bar {\rho }} [1/p ]$
is reduced. Since twisting by characters does not change the property of being absolutely irreducible, the proof of Proposition A.9 shows that the absolutely irreducible locus is dense in
$\operatorname {\mathrm {Spec}} R^{\square , \psi }_{\bar {\rho }} [1/p ]$
. Then the same proof goes through.
Proposition A.12. If
$p=3$
,
$\mathscr G=\mathscr G_{\mathbb Q_3}$
and
$\bar {\rho }= \mathbf 1\oplus \omega $
, then
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal.Footnote
4
Proof. We will first relate the framed deformation ring
$R^{\square }_{\bar {\rho }}$
to the ring studied in [Reference Böckle7]. Let
$\mu _3$
be the group of
$3$
rd roots of unity in
$\overline {\mathbb {Q}}_3$
, let
$E=\mathbb Q_3(\mu _3)$
and let
$E(3)$
be the compositum of all extensions
$E \subset E' \subset \overline {\mathbb {Q}}_3$
such that
$[E':E]$
is a power of
$3$
. Then the Galois group
$\operatorname {\mathrm {Gal}}(E(3)/E)$
is the maximal pro-
$3$
quotient of
$\operatorname {\mathrm {Gal}} (\overline {\mathbb {Q}}_3/E )$
, and thus the map
$\rho ^{\square }: \mathscr G_{\mathbb Q_3} \rightarrow \operatorname {\mathrm {GL}}_2 (R^{\square }_{\bar {\rho }} )$
factors through the surjection
$\mathscr G_{\mathbb Q_3}\twoheadrightarrow \operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)$
. Since
$\operatorname {\mathrm {Gal}}(E/ \mathbb Q_3)$
has order
$2$
, Schur–Zassenhaus implies that the surjection
$\operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)\twoheadrightarrow \operatorname {\mathrm {Gal}}(E/ \mathbb Q_3)$
has a splitting, which gives us an isomorphism
$$ \begin{align*} \operatorname{\mathrm{Gal}}(E(3)/\mathbb Q_3)\cong \operatorname{\mathrm{Gal}}(E(3)/E) \rtimes G, \end{align*} $$
where
$G=\{1, \sigma \}$
is a subgroup of
$\operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)$
. One may define a closed subfunctor, denoted by
$\mathrm {EH}_1$
in [Reference Böckle7], of the framed deformation functor
$D^{\square }$
such that
$\mathrm {EH}_1(A)$
consists of pairs
$(V_A, \beta _A)$
, where
$V_A$
is a deformation of
$\omega \oplus 1$
to A and
$\beta _A=(v_1, v_2)$
is an A-basis of
$V_A$
lifting a fixed basis
$\beta _k$
of
$\omega \oplus 1$
, such that
$\sigma $
acts by
$-1$
on
$v_1$
and by
$1$
on
$v_2$
. It follows from the Iwahori decomposition for the group
$1+M_2(\mathfrak m_A)$
that a framed deformation
$(V_A, \beta _A)\in D^{\square }(A)$
can be conjugated to a framed deformation in
$\mathrm {EH}_1(A)$
by a unique element of the form
$ \left( \begin {smallmatrix} 1 & b \\ 0 & 1 \end {smallmatrix} \right) \left( \begin {smallmatrix} 1 & 0\\ c & 1 \end {smallmatrix} \right)$
, with
$b, c \in \mathfrak m_A$
. Hence if
$\mathrm {EH}_1$
is represented by R, then
Now R is a complete intersection by [Reference Böckle7, Theorem 1.1], thus so is
$R^{\square }_{\bar {\rho }}$
; and to show the normality of
$R^{\square }_{\bar {\rho }} [1/p ]$
, it is enough to show that the singular locus in
$R [1/p ]$
has codimension at least
$2$
.
Let
$\rho : \operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)\rightarrow \operatorname {\mathrm {GL}}_2(R)$
be the representation obtained for the action of the Galois group on
$V_R$
with respect to the basis
$\beta _{R}$
such that
$\rho (\sigma )= \left( \begin {smallmatrix} -1 & 0 \\ 0 & 1\end {smallmatrix} \right)$
. It follows from the argument of [Reference Colmez, Dospinescu and Paškūnas22, Lemma 4.1] that
$x\in \operatorname {\mathrm {m-Spec}} R [1/p ]$
is singular if and only if there is an exact sequence
$0\rightarrow \delta \rightarrow \rho _x \rightarrow \delta \varepsilon \rightarrow 0$
for some character
$\delta : \operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)\rightarrow \kappa (x)^{\times }$
. Thus the singular locus is contained in the reducible locus, and it is enough to show that it has positive codimension inside the reducible locus: We know that R is a domain by [Reference Böckle7, Theorem 1.1], and there are absolutely irreducible lifts of
$\bar {\rho }$
, so that the reducible locus has codimension
$1$
inside
$\operatorname {\mathrm {Spec}} R$
.
We will now describe the ring R as computed in [Reference Böckle7] and compute the reducible locus. We know from [Reference Böckle7, Lemma 3.2] that the representation
$\rho : \operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)\rightarrow \operatorname {\mathrm {GL}}_2(R)$
factors through a quotient
$\operatorname {\mathrm {Gal}}(E(3)/\mathbb Q_3)\twoheadrightarrow P \rtimes G$
, where P is a pro-p group with generators
$x_1, x_2, x_3, x_4$
and one relation
$$ \begin{align*} r= x_1^3 [x_1, x_2][x_3, x_4][x_4, x_3^{-1}] [x_2, x_1^{-1}] x_1^3, \end{align*} $$
where
$[g, h]=g hg^{-1} h^{-1}$
. The action of
$\sigma \in G$
on the generators is given by
$$ \begin{align*} \sigma(x_1)= x_1^{-1}, \qquad \sigma(x_2)= x_2, \qquad \sigma(x_3)=x_3^{-1}, \qquad \sigma(x_4)=x_4. \end{align*} $$
Let
and let
$A_i\in \operatorname {\mathrm {GL}}_2(S)$
be the matrices
$$ \begin{align*} A_1&=\left(\begin{smallmatrix} \sqrt{1+bc} & b \\ c & \sqrt{1+bc}\end{smallmatrix}\right), & A_2&= \sqrt{1+a}\left(\begin{smallmatrix} \sqrt{1+d} & 0 \\ 0 & \sqrt{1+d}^{-1}\end{smallmatrix}\right), \\ A_3&=\left(\begin{smallmatrix} \sqrt{1+b'c'} & b' \\ c' & \sqrt{1+b'c'}\end{smallmatrix}\right), & A_4&= \sqrt{1+a'}\left(\begin{smallmatrix} \sqrt{1+d'} & 0 \\ 0 & \sqrt{1+d'}^{-1}\end{smallmatrix}\right), \end{align*} $$
and let
$$ \begin{align*} B= A_1^3 [A_1, A_2][A_3, A_4][A_4, A_3^{-1}] [A_2, A_1^{-1}] A_1^3. \end{align*} $$
Then [Reference Böckle7, Theorem 4.1(c)] asserts that
$R= S/I$
, where I is the ideal of S generated by the matrix entries of
$B - \left( \begin {smallmatrix}1 & 0 \\ 0 & 1\end {smallmatrix} \right)$
and
$\rho : P\rtimes G\rightarrow \operatorname {\mathrm {GL}}_2(R)$
is obtained by mapping
$\sigma \mapsto \left( \begin {smallmatrix} -1 & 0 \\ 0 & 1\end {smallmatrix} \right)$
and
$x_i\mapsto A_i$
for
$1\le i\le 4$
.
It follows from this description that the locus in
$\operatorname {\mathrm {Spec}} R$
parameterising reducible representations, where
$\sigma $
acts on the rank
$1$
subrepresentation by
$-1$
(resp.,
$1$
), is equal to
$V( c, c')$
(resp.,
$V(b, b'))$
.
The images of
$A_1$
and
$A_3$
in
$\operatorname {\mathrm {GL}}_2(R/( c, c'))$
are unipotent upper-triangular matrices. It is easy to compute the commutator of a unipotent upper-triangular matrix with a diagonal matrix. One obtains that the image of B in
$\operatorname {\mathrm {GL}}_2(S/( c, c'))$
is the matrix
$ \left( \begin {smallmatrix} 1 & 6b- 2bd -2 b'd' \\ 0 & 1\end {smallmatrix} \right)$
. Thus
$$ \begin{align*} R/( c, c')= S/(c, c', 3b -bd -b'd') \end{align*} $$
is an integral domain, as
is factorial and
$3b -bd -b'd'$
is an irreducible element in
$S/(c, c')$
.
Let
$X^{\mathrm {sing}}$
be the singular locus in
$\operatorname {\mathrm {Spec}} R [1/p ]$
. The point
$x\in \operatorname {\mathrm {Spec}} R/(c,c')$
corresponding to the representation
$ \big( \begin {smallmatrix} \varepsilon ^3 & 0 \\ 0 & 1 \end {smallmatrix} \big)$
will not lie in
$X^{\mathrm {sing}}$
, since this representation is not an extension of
$\delta \varepsilon $
by
$\delta $
. Thus
$X^{\mathrm {sing}} \cap \operatorname {\mathrm {Spec}} R/(c,c') [1/p ]$
is of codimension at least
$1$
. In the same way, we obtain that
$X^{\mathrm {sing}} \cap \operatorname {\mathrm {Spec}} R/(b,b') [1/p ]$
is of codimension at least
$1$
in
$\operatorname {\mathrm {Spec}} R/(b,b') [1/p ]$
. Thus
$X^{\mathrm {sing}}$
is of codimension at least
$1$
in the reducible locus in
$\operatorname {\mathrm {Spec}} R [1/p ]$
and of codimension at least
$2$
in
$\operatorname {\mathrm {Spec}} R [1/p ]$
.
Proposition A.13. If
$\mathscr G=\mathscr G_{\mathbb {Q}_p}$
, then
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal and the absolutely irreducible locus is dense in
$\operatorname {\mathrm {Spec}} R^{\square }_{\bar {\rho }} [1/p ]$
for all semisimple
$2$
-dimensional
$\bar {\rho }$
.
Proof. Since
$R^{\square }_{\bar {\rho }} [1/p ]$
is excellent, the singular locus is closed in
$R^{\square }_{\bar {\rho }} [1/p ]$
. If it is nonempty, then it will contain a maximal ideal x such that
$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\mathscr G_{\mathbb{Q}_p}}(\rho^{\square}_x, \rho^{\square}_x(1))\neq 0 \end{align*} $$
(see [Reference Colmez, Dospinescu and Paškūnas22, Lemma 4.1]). Thus
$\bar {\rho }$
is of the form
$\bar {\chi } \oplus \bar {\chi } \omega $
. After twisting by a character, we may assume that
$\bar {\rho }= \mathbf 1 \oplus \omega $
. If
$p=2$
or
$p=3$
, then
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal by [Reference Colmez, Dospinescu and Paškūnas22, Proposition 4.3] or Proposition A.12, respectively. If
$p\ge 5$
, it follows from the proof of [Reference Paškūnas39, Proposition B2, Theorem B.3], based on the work of Böckle [Reference Böckle6], that
$R^{\square }_{\bar {\rho }}$
is formally smooth over
. (The only change is that because in our setting
$\bar {\rho }$
is split, the generator
$x_{p-2}$
maps to the matrix
$ \left( \begin {smallmatrix} 1 & x \\ 0 & 1\end {smallmatrix} \right)$
instead of
$ \left( \begin {smallmatrix} 1 & 1 \\ 0 & 1\end {smallmatrix} \right)$
. This adds an extra variable but does not change the relation coming from [Reference Paškūnas39, equation (261)].) Thus
$R^{\square }_{\bar {\rho }} [1/p ]$
is normal.
Hence,
$R^{\square }_{\bar {\rho }} [1/p ]$
is a product of normal domains, and if the absolutely irreducible locus were not dense, there would be a component without absolutely irreducible points. (Let I be the ideal of
$R^{\square }_{\bar {\rho }}$
generated by the matrix entries of
$ (\rho ^{\square }(gh)-\rho ^{\square }(hg) )^2$
for all
$g, h\in \mathscr G_{\mathbb {Q}_p}$
. Then a specialisation of
$\rho ^{\square }$
at
$x\in \operatorname {\mathrm {m-Spec}} R^{\square }_{\bar {\rho }} [1/p ]$
is absolutely irreducible over
$\kappa (x)$
if and only if
$x\not \in V(I)$
. Thus if an irreducible component of
$R^{\square }_{\bar {\rho }} [1/p ]$
contains an absolutely irreducible point, then such points are dense in the component.) In the course of the proof of Proposition A.11, we have shown that
$R^{\mathrm {ps}} [1/p ]$
is a subring of
$R^{\square }_{\bar {\rho }} [1/p ]$
. Thus there would exist an irreducible component of
$R^{\mathrm {ps}} [1/p ]$
without absolutely irreducible points. This would contradict [Reference Chenevier17, Theorem 2.1].
Corollary A.14. If
$\mathscr G=\mathscr G_{\mathbb {Q}_p}$
, then
$R^{\mathrm {ps}, \psi } [1/p ]$
,
$R^{\mathrm {ps}} [1/p ]$
and the corresponding rigid analytic spaces are normal for all semisimple
$2$
-dimensional
$\bar {\rho }$
.
Acknowledgements
The first author would like to thank Toby Gee for organising a seminar on Lue Pan’s paper [Reference Pan38] and for the invitation to participate. Lue Pan’s paper was a major source of motivation to deal with the exceptional cases. The first author would also like to thank Patrick Allen and Gebhard Böckle for stimulating correspondence. Parts of the paper were written when the second author was visiting Academia Sinica. The second author would like to thank Carl Wang-Erickson for stimulating correspondence. The authors would like to thank Gabriel Dospinescu and Toby Gee for their detailed comments on an earlier draft. The authors would also like to thank the anonymous referee for a quick and thorough reading of the paper. We acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen.
Conflict of Interest
None.
Open access
