1.1 Normed vector spaces

As a preparation for the theory of lattices let us review some properties of normed vector spaces following [Reference Bosch, Güntzer and RemmertBGR84, Ch. 2]. We fix a field $F$ and an absolute value $|\cdot |\colon F \to \mathbf {R}_{\geqslant 0}$ that arises from a nontrivial discrete valuation. Let $R = \{ x \in F, |x| \leqslant 1 \}$ be the corresponding ring of integers. We denote its maximal ideal by $\mathfrak {p}$.

Let $V$ be a finite-dimensional $F$-vector space. Recall that a $\mathfrak {p}$-adic norm on $V$ is a map $\|\cdot \|\colon V \to \mathbf {R}_{\geqslant 0}$ with the following properties:

1. (${\rm V} _{\!1 }$) $\|v\| = 0$ if and only if $v = 0$;

2. (${\rm V} _{\!2 }$) $\|v + v'\| \leqslant \max \{ \|v\|, \|v'\| \}$;

3. (${\rm V} _{\!3 }$) $\|x v\| = |x| \cdot \|v\|$ for all $x \in F$.

We have $\|{-}v\| = \|v\|$ by the homogeneity property (${\rm V}_{\!3}$), so the ultrametric inequality (${\rm V}_{\!2}$) becomes an equality when $\|v\|\ne \|v'\|$.

Let $\pi \in F$ be a uniformizer and set $\varepsilon := |\pi |$. Observe that $\varepsilon \in (0, 1)$.

Lemma 1.1.1 The image of the norm function $\|\cdot \|\colon V \to \mathbf {R}_{\geqslant 0}$ has the form

\[ r_1 \varepsilon^{\mathbf {Z}} \cup \dotsb \cup r_m \varepsilon^{\mathbf {Z}} \cup \{0\}, \]

where $r_1, r_2, \dotsc, r_m \in (\varepsilon,\:1]$ is a sequence of real numbers, $m\leqslant \dim V$.

The ultrametric inequality (${\rm V}_{\!2}$) and the homogeneity property (${\rm V}_{\!3}$) imply that the balls

\[ B(V,\,r) = \{v \in V, \, \|v\| \leqslant r \}, \quad r \in \mathbf {R}_{>0}, \]

are $R$-submodules of $V$ such that:

1. $({\rm B} _{ 1 } )$ $x\,B(V,\,r) = B(V,\,|x|\,r)$ for every scalar $x \in F^\times$;

2. $({\rm B} _{ 2 } )$ $F \otimes _{R} B(V,\,r) = V$.

Each norm endows the vector space $V$ with a topology by means of the fundamental system $\{B(V,\,r)\}_{r>0}$. Lemma 1.1.1 implies that this system is the same as the system of open balls $\{ v \in V, \|v\| < r \}$.

It is important to note that, in general, the norm topology on $V$ can be coarser than the canonical $\mathfrak {p}$-adic topology. An explicit construction of such norms is given in Example 1.1.7. From Lemma 1.1.2 we directly have the following result.

To increase flexibility we will need the notion of a seminorm which is a map $\|\cdot \|\colon V \to \mathbf {R}_{\geqslant 0}$ satisfying the conditions (${\rm V}_{\!2}$) and (${\rm V}_{\!3}$). The following claim is easy to check.

Next we give a criterion for the norm on $V$ to induce the canonical $\mathfrak {p}$-adic topology. Let $\smash {F_{\mathfrak {p}}}$ be the $\mathfrak {p}$-adic completion of the field $F$.

The behaviour of $\mathfrak {p}$-adic norms in Theorem 1.1.5 resembles the behaviour of real-valued quadratic forms on a finite-dimensional $\mathbf {Q}$-vector space $V$. Every such form $q$ extends uniquely to the real vector space $V_\infty = \mathbf {R}\otimes _{\mathbf {Q}} V$. If the form $q$ takes strictly positive values on $V\smallsetminus \{0\}$ then it is positive-semidefinite, and the map $v \mapsto q(v)^{1/2}$ is an archimedean seminorm on $V_\infty$. This seminorm is a norm if and only if the form $q$ is positive-definite. Comparing with Theorem 1.1.5, we conclude that the property of inducing the $\mathfrak {p}$-adic topology can be understood as a form of positive-definiteness for $\mathfrak {p}$-adic norms.

As an offshoot of Theorem 1.1.5 we get a description of all norms that fail to induce the $\mathfrak {p}$-adic topology on $V$.

1.2 The setting

From now on we fix a global function field $F$ and a place $\infty$ of $F$. We will use the following notation:

• – $\kappa \subset F$ is the ring of elements which are regular at all places;

• – $A \subset F$ is the ring of elements which are regular outside $\infty$;

• – $\mathcal {O}_{F,\infty } \subset F$ is the ring of elements which are regular at $\infty$;

• – $F_\infty$ is the $\infty$-adic completion of $F$;

• – $\kappa _\infty$ is the residue field of $F_\infty$ (and of $\mathcal {O}_{F,\infty }$);

• – $c = |\kappa _\infty |$ is the cardinality of this residue field.

Fix an $\infty$-adic absolute value $|\cdot |_\infty \colon F_\infty \to \mathbf {R}_{\geqslant 0}$ such that $|\pi |_\infty ^{-1} = c$ for a uniformizer $\pi \in F_\infty$. Although our theory works with any normalization, this particular choice leads to better-looking formulas.

The ring $\kappa$ is a finite field, called the field of constants of $F$. The ring $A$ is a Dedekind domain of finite type over $\kappa$. We will refer to $A$ as the coefficient ring and it will serve us as an analogue of the ring of integers $\mathbf {Z}$. The local field $F_\infty$ will play the role of the field of real numbers $\mathbf {R}$.

1.3 Lattices

Let $\varLambda$ be a finitely generated projective $A$-module. An $\infty$-adic norm on $\varLambda$ is a map $\|\cdot \|\colon \varLambda \to \mathbf {R}_{\geqslant 0}$ such that:

1. (Λ ₁) $\|\lambda \| = 0$ if and only if $\lambda = 0$;

2. (Λ ₂) $\|\lambda + \lambda '\| \leqslant \max \{ \|\lambda \|, \|\lambda '\| \}$;

3. (Λ ₃) $\|a\lambda \| = |a|_\infty \|\lambda \|$ for all $a \in A$.

Following the construction in [Reference Bosch, Güntzer and RemmertBGR84, § 2.1.3] every such norm extends uniquely to an $\infty$-adic norm on the rational vector space

\[ V = F \otimes_A \varLambda. \]

We will apply the considerations of § 1.1 to the normed vector space $V$. By analogy with § 1.1 we define the following subsets of $\varLambda$:

\[ B(\varLambda,\,r) = \{\lambda\in\varLambda,\:\|\lambda\| \leqslant r\}, \quad r \in \mathbf {R}_{>0}. \]

This time $B(\varLambda,\,r)$ is merely a $\kappa$-vector space.

Proof. By construction, $B(\varLambda,\,s) = \varLambda \cap B(V,\,s)$ for all $s > 0$. We thus have an inclusion of $\kappa$-vector spaces

\[ \frac{B(\varLambda,\,r')}{B(\varLambda,\,r)} \hookrightarrow \frac{B(V,\,r')}{B(V,\,r)}, \]

and the claim follows from Lemma 1.1.2.

We are ready for the main definition of this section.

In other words there is a real number $\varepsilon >0$ such that every nonzero lattice vector $\lambda$ satisfies $\|\lambda \| \geqslant \varepsilon$. By Lemma 1.3.1 this holds if and only if the subsets $\{\lambda \in \varLambda,\:\|\lambda \| \leqslant r\}$ are finite for every $r > 0$.

A norm can induce a non-discrete topology on $\varLambda$. All such norms are described in Example 1.1.7 above.

The discreteness condition of Definition 1.3.2 can be seen as a function field analogue of positive-definiteness. Let $q$ be a real-valued quadratic form on a finitely generated free $\mathbf {Z}$-module $L$. Suppose that the form $q$ takes strictly positive values on $L \smallsetminus \{0\}$. Then $q$ is positive-semidefinite as a quadratic form on the real vector space $\mathbf {R}\otimes _{\mathbf {Z}} L$, and is positive-definite if and only if for each real number $r > 0$ the set $\{v \in L, \,q(v) \leqslant r \}$ is finite, cf. Lemma 9.5 of [Reference SilvermanSil09, VIII.9].

1.4 Lattices and vector bundles

We shall relate lattices to vector bundles on the smooth projective curve $X$ over $\operatorname {Spec} \kappa$ which compactifies the affine curve $Y = \operatorname {Spec} A$. By construction we have $X\smallsetminus Y = \{\infty \}$. Let us denote by $\mathcal {O}_X({-}\infty )$ the ideal sheaf of the reduced closed subscheme $\{\infty \} \subset X$.

Pick a lattice $\varLambda$, set $V = F\otimes _A \varLambda$, and let $\mathcal {E}_{\varLambda }$ be the locally free sheaf on the curve $Y$ induced by the $A$-module $\varLambda$. Let $r > 0$ be a real number.

Definition 1.4.1 The quasi-coherent sheaf $\mathcal {E}_{\varLambda, r}$ on the curve $X$ is constructed by gluing the locally free sheaf $\mathcal {E}_{\varLambda }$ to the $\mathcal {O}_{F,\infty }$-module $B(V,\,r)$ via the canonical isomorphism

\[ F\otimes_A \varLambda \xrightarrow{\smash{{{\sim}}}} V \xrightarrow{\smash{{{\sim}}}} F\otimes_{\mathcal{O}_{F,\infty}} B(V,\,r). \]

Combining Theorem 1.4.2 with Lemma 1.1.1 we obtain a correspondence between lattices and systems of vector bundles.

Corollary 1.4.3 There is a one-to-one correspondence between $A$-lattices and pairs which consist of (i) a strictly increasing chain of locally free sheaves

\[ \mathcal{E}_1 \subset \dotsb \subset \mathcal{E}_m \subset \mathcal{E}_1(\infty) \]

on the complete curve $X$ such that $\mathcal {E}_i|_Y = \mathcal {E}_{i+1}|_Y$ for all $i \in \{1, \dotsc, m -1 \}$, and (ii) a sequence of real numbers

\[ \{ r_1 < \dotsb < r_m \} \subset (c^{-1}, 1]. \]

The $A$-module $\varLambda = H^0(Y,\mathcal {E}_i)$ is finitely generated projective and is independent of the choice of $i$. It carries a filtration by $\kappa$-vector spaces $H^0(X,\mathcal {E}_i(j\infty ))$ with $i \in \{1,\dotsc,m\}$ and $j \in \mathbf {Z}$. For each element $\lambda \in \varLambda$ we set

\[ \|\lambda\| = \inf \{r_i c^j \:|\: \lambda \in H^0(X,\,\mathcal{E}_i(j\infty))\}. \]

It is easy to check that the function $\|\cdot \|$ is a norm on $\varLambda$. The normed module $\varLambda$ is a lattice since the vector spaces $H^0(X,\mathcal {E}_i(j\infty ))$ are finite-dimensional.

1.5 The volume

Let $\varLambda$ be a lattice, and consider the vector space $V_\infty = F_\infty \otimes _A \varLambda$ with the induced norm. By Corollary 1.1.6 the norm topology on $V_\infty$ is the canonical $\infty$-adic topology. For each real number $r > 0$ the ball $B(V_\infty,\,r)$ is thus compact and open, and we can make the following definition.

Definition 1.5.1 The $r$-normalized volume of $\varLambda$ is defined by the formula

\[ \operatorname{vol}(\varLambda,\,r) = \mu(V_\infty/\varLambda), \]

where $\mu$ is the unique translation-invariant Haar measure on $V_\infty$ satisfying $\mu (B(V_\infty,\,r)) = 1$. In the case $r = 1$ we will use the simplified notation:

\[ \operatorname{vol}(\varLambda) = \operatorname{vol}(\varLambda,\,1). \]

Denoting by $n$ the rank of the lattice $\varLambda$ we have an equality for every integer $i$:

\[ \operatorname{vol}(\varLambda,\,r c^i) = c^{-ni} \operatorname{vol}(\varLambda,\,r). \]

This results from the fact that $\mu (B(V_\infty,\,r c^i)) = c^{ni} \,\mu (B(V_\infty,\,r))$ for every translation-invariant Haar measure $\mu$.

In the following we denote by $\chi$ the Euler characteristic of coherent sheaves on the curve $X\to \operatorname {Spec}\kappa$.

Proof. The cohomology of the locally free sheaf $\mathcal {E}_{\varLambda, r}$ is computed by a Čech complex

concentrated in degrees $0$ and $1$. We rewrite this complex as follows:

Thus, for every translation-invariant Haar measure $\mu$ on $V_\infty$ we have an equality:

\[ \frac{\mu(V_\infty/\varLambda)}{\mu(B(V_\infty,\,r))} = \frac{|H^1(X,\mathcal{E}_{\varLambda, r})|}{|H^0(X,\mathcal{E}_{\varLambda, r})|}. \]

The claim follows.

As in the case of $\mathbf {Z}$-lattices we can compute the volume of an $A$-lattice via the absolute value of a determinant. In the following formula, we treat the respective wedge products as elements of one-dimensional $F_\infty$-vector space $\det (V_\infty )$. Their quotient is a well-defined element of $F_\infty ^\times$.

Lemma 1.5.3 Pick a basis $v_1, \dotsc, v_n$ of the free module $B(V_\infty,\,r)$ and pick vectors $\lambda _1,\dotsc,\lambda _n$ generating a submodule of finite index $e$ in $\varLambda$. Then

\[ \operatorname{vol}(\varLambda,\,r) = \frac{{|\kappa|}^{(g-1)n}}{e} \biggl|\frac{\lambda_1 \wedge \dotsb \wedge \lambda_n}{v_1 \wedge \dotsb \wedge v_n} \biggr|_\infty, \]

where $g$ is the genus of the coefficient field $F$.

Proof The product $s = \lambda _1 \wedge \dotsb \wedge \lambda _n$ is a rational section of the invertible sheaf $\det (\mathcal {E}_{\varLambda, r})$. Its order at the point $\infty$ equals

\[ [\kappa_\infty:\kappa] v_\infty \biggl(\frac{\lambda_1 \wedge \dotsb \wedge \lambda_n}{v_1 \wedge \dotsb \wedge v_n} \biggr), \]

where $v_\infty \colon F_\infty ^\times \twoheadrightarrow \mathbf {Z}$ is the normalized $\infty$-adic valuation. Hence,

\[ {|\kappa|}^{-\deg(s)} = \frac{1}{e} \cdot \biggl|\frac{\lambda_1 \wedge \dotsb \wedge \lambda_n}{v_1 \wedge \dotsb \wedge v_n} \biggr|_\infty. \]

The claim then follows from Lemma 1.5.2 and Riemann–Roch formula.

Theorem 1.5.4 Let $\varLambda$ be a lattice of rank $n$ and let $r > 0$ be a real number. Then for all integers $i \gg 0$ we have an equality:

\[ |\{\lambda \in \varLambda, \: \|\lambda\| \leqslant r c^i\}| = \frac{c^{ni}}{\operatorname{vol}(\varLambda,\,r)}. \]

Moreover, for all integers $i$ we have a lower bound:

\[ |\{\lambda \in \varLambda, \: \|\lambda\| \leqslant r c^i\}| \geqslant \frac{c^{ni}}{\operatorname{vol}(\varLambda,\,r)}. \]

Proof. Let $i$ be a non-negative integer. Consider a natural short exact sequence of sheaves on the curve $X$:

The coherent sheaf $\mathcal {F}$ is concentrated at the point $\infty$ and has length $n i$ as an $\mathcal {O}_{X,\infty }$-module. We thus have an equality

\[ \frac{|H^0(X,\mathcal{E}_{\varLambda, r}(i\infty))|}{|H^1(X,\mathcal{E}_{\varLambda, r}(i\infty))|} = c^{ni} \frac{|H^0(X,\mathcal{E}_{\varLambda, r})|}{|H^1(X,\mathcal{E}_{\varLambda, r})|}. \]

By construction we have $H^0(X,\:\mathcal {E}_{\varLambda, r}(i\infty )) = \{\lambda \in \varLambda,\:\|\lambda \| \leqslant r c^i\}$. Hence, Lemma 1.5.2 implies that

\[ \frac{|\{\lambda\in\varLambda,\:\|\lambda\| \leqslant r c^i\}|}{|H^1(X,\mathcal{E}_{\varLambda, r}(i\infty))|} = \frac{c^{ni}}{\operatorname{vol}(\varLambda,\,r)}. \]

The cohomology group $H^1(X,\:\mathcal {E}_{\varLambda, r}(i\infty ))$ vanishes for $i \gg 0$ as the invertible sheaf $\mathcal {O}_X(\infty )$ is ample. The first claim of the theorem follows. To prove the second claim note that $c^{ni} \,\operatorname {vol}(\varLambda,\,r)^{-1} = \operatorname {vol}(\varLambda,\,r c^i)^{-1}$ for each integer $i$. It is thus enough to treat the case $i = 0$ which follows from the displayed formula above.

1.6 A bound on norms of generators

We would like to find a real number $r > 0$ such that the subset $\{\lambda \in \varLambda,\:\|\lambda \| \leqslant r\}$ generates the lattice $\varLambda$. To this end, we will employ the correspondence between lattices and vector bundles. We will use supplementary notation:

• – $g$ is the genus of the smooth projective curve $X\to \operatorname {Spec}\kappa$;

• – $f = [\kappa _\infty :\kappa ]$.

The invertible sheaf $\mathcal {O}_X(\infty )$ has degree $f$. Let us also pick an auxiliary integer $h$ such that the twist $\mathcal {O}_X(h\infty )$ is very ample. We will utilize a classical criterion for global generation of coherent sheaves on curves.

Lemma 1.6.2 Let $\mathcal {E}$ be a locally free sheaf of rank $n$ on the curve $X$. Set

\[ i = \biggl\lceil\frac{-\deg\mathcal{E} + (2g - 2)n + 1}{f}\biggr\rceil + h. \]

Suppose that the sheaf $\mathcal {E}$ is an iterated extension of invertible sheaves of degree at most $2g - 2$. Then the sheaf $\mathcal {E}(i\infty )$ is globally generated.

Proof. Write the degrees of invertible sheaves in the form $(2g - 2) - e_j$ with $e_j \geqslant 0$. We have $\deg \mathcal {E} = (2g-2)n - \sum e_j$ so that

\[ -e_j - \deg \mathcal{E} + (2g-2)n \geqslant 0. \]

Hence, the sheaf $\mathcal {E}' = \mathcal {E}((i-h)\infty )$ is an iterated extension of invertible sheaves of degree at least $2g-1$. Induction on the number of invertible sheaves implies that $H^1(X,\,\mathcal {E}') = 0$ so the sheaf $\mathcal {E}'(h\infty )$ is globally generated by Lemma 1.6.1.

Lemma 1.6.3 Let $\mathcal {E}$ be a locally free sheaf of rank $n$ on the curve $X$. Set

\[ i = \biggl\lceil\frac{-\deg\mathcal{E} + (2g-2)n + 1}{f}\biggr\rceil + h n. \]

Suppose that $H^0(X,\,\mathcal {E}) = 0$. Then the sheaf $\mathcal {E}(i\infty )$ is globally generated.

Proof. Let $\Omega$ be the canonical sheaf of the curve $X\to \operatorname {Spec}\kappa$ and set $\mathcal {F} = \mathcal {E}^* \otimes \Omega$. Serre duality shows that $H^1(X,\,\mathcal {F}) = 0$ so the sheaf $\mathcal {F}(h\infty )$ is globally generated by Lemma 1.6.1. Dualizing and twisting we get an embedding $\mathcal {E}(-h\infty ) \hookrightarrow \Omega ^{\oplus m}$, $m \gg 0$. Hence the sheaf $\mathcal {E}' = \mathcal {E}(-h\infty )$ is an iterated extension of invertible subsheaves $\mathcal {L} \subset \Omega$. Every such subsheaf has degree at most $2g - 2$. Lemma 1.6.2 implies that the sheaf $\mathcal {E}'((i+h)\infty ) = \mathcal {E}(i\infty )$ is globally generated with

\[ i = \biggl\lceil\frac{-\deg\mathcal{E}'+(2g-2)n + 1}{f}\biggr\rceil. \]

The claim follows since $\deg \mathcal {E}' = \deg \mathcal {E} - f h n$.

Theorem 1.6.4 Let $\varLambda$ be a lattice of rank $n$. Suppose that every nonzero vector of $\varLambda$ has norm at least $1$. Then the $A$-module $\varLambda$ is generated by the subset

\[ \{\lambda\in\varLambda,\:\|\lambda\| \leqslant \operatorname{vol}(\varLambda) \cdot C^n\} \]

with $C = {|\kappa |}^{3g + 2f - 1}$.

Proof. Consider the locally free sheaf $\mathcal {E} = \mathcal {E}_{\varLambda, 1}$. We have

\[ H^0(X,\:\mathcal{E}({-}\infty)) = \{\lambda\in\varLambda,\:\|\lambda\| \leqslant c^{-1}\}. \]

Thus $H^0(X,\mathcal {E}({-}\infty )) = 0$ by our assumption on the length of nonzero lattice vectors. Invoking the Riemann–Roch formula, we deduce that

\[ -\deg(\mathcal{E}({-}\infty)) = -\chi(\mathcal{E}) + (-g + 1 + f) n. \]

Hence, by Lemma 1.6.3 the sheaf $\mathcal {E}((i-1)\infty )$ is globally generated when

\[ i = \biggl\lceil\frac{-\chi(\mathcal{E}) + (g - 1 + f + f h)n + 1}{f}\biggr\rceil. \]

To ensure that the invertible sheaf $\mathcal {O}_X(h\infty )$ is very ample it is enough to take any integer $h$ such that $f h\geqslant 2g + 1$. We thus have an estimate $f h\leqslant 2g + f$, and the sheaf $\mathcal {E}(j\infty )$ is globally generated for an integer $j$ such that

\[ j \leqslant \biggl\lceil\frac{-\chi(\mathcal{E}) + (3g + 2f - 1)n + 1}{f}\biggr\rceil - 1. \]

In particular, the $A$-module $\varLambda = H^0(Y,\:\mathcal {E}(j\infty ))$ is generated by the subset

\[ \{\lambda\in\varLambda,\:\|\lambda\| \leqslant c^j\} = H^0(X,\:\mathcal{E}(j\infty)). \]

Since $c = {|\kappa |}^{f}$, we obtain an estimate

\[ c^j \leqslant {|\kappa|}^{-\chi(\mathcal{E})} \cdot {|\kappa|}^{(3g + 2f - 1)n}. \]

The result follows as ${|\kappa |}^{-\chi (\mathcal {E})} = \operatorname {vol}(\varLambda )$ by Lemma 1.5.2.

In the case $A=\kappa [t]$ the estimate of Theorem 1.6.4 is sharp in all ranks. The lattice $A^{\oplus n}$ with the supremum norm has volume ${|\kappa |}^{-n}$, is generated by vectors of norm $1$ and has no nonzero vectors of norm strictly less than $1$.

2.1 Twisted polynomials

Let $K[\tau ]$ be the twisted polynomial ring. This is the ring of polynomials in a formal variable $\tau$ with coefficients in $K$ and with the multiplication subject to the rule $\tau \,\alpha = \alpha ^q\,\tau$ for all $\alpha \in K$. The elements of $K[\tau ]$ will be referred to as $\tau$-polynomials.

The ring $K[\tau ]$ is the endomorphism ring of the $\mathbf {F}_{\hspace {-1.5pt} q}$-module scheme $\mathbf {G}_a$ over $\operatorname {Spec} K$ with the element $\tau$ corresponding to the $q$-Frobenius. This interpretation allows us to evaluate $\tau$-polynomials at points of $\mathbf {G}_a$.

Proof. The determinant of the square matrix in part (2) is the Moore determinant $M(v_1,\dotsc,v_n)$, see [Reference PapikianPap23, Definition 3.1.17]. This determinant is nonzero because the elements $v_1, \dotsc, v_n$ are $\mathbf {F}_{\hspace {-1.5pt} q}$-linearly independent. Hence, the solution vector is uniquely determined.

Consider the polynomial $\varphi = a_0 + a_1 \tau + \dotsb + a_{n-1} \tau ^{n-1}$. By construction, we have $\varphi (v_i) = f(v_i)$ for all $i \in \{1,\dotsc,n\}$ which implies by linearity that $\varphi (v) = f(v)$ for all $v \in V$. It remains to show that the coefficients of $\varphi$ are Galois-invariant and, thus, belong to the subfield $K\subset K^s$.

Apply an automorphism $g \in G_K$ to both sides of the matrix equation in part (2). Since the map $f$ is Galois-equivariant, it follows that the polynomial

\[ {}^g \varphi = g(a_0) + g(a_1) \, \tau + \dotsb + g(a_{n-1}) \, \tau^{n-1} \]

satisfies $({}^g \varphi )(g v_i) = f(g v_i)$ for all $i$. The set $g v_1, \dotsc, g v_n$ is yet another basis of $V$ so the linearity implies that $({}^g \varphi )(v) = f(v)$ for all $v \in V$. In particular, $({}^g \varphi )(v_i) = f(v_i)$ for all $i$. Since the matrix equation in part (2) has a unique solution, we deduce that ${}^g \varphi = \varphi$, and the claim follows.

2.2 Artin–Schreier theory

Let $E$ be an $\mathbf {F}_{\hspace {-1.5pt} q}$-module scheme over $\operatorname {Spec} K$ which is isomorphic to $\mathbf {G}_a$. Consider a separable isogeny $\varphi \colon E \to E$, i.e. an $\mathbf {F}_{\hspace {-1.5pt} q}$-linear morphism that induces an isomorphism of the tangent spaces at zero. Let $V$ be the kernel of $\varphi$ viewed as a morphism of sheaves on the small étale site of $\operatorname {Spec} K$. The sheaf $V$ corresponds to the $\mathbf {F}_{\hspace {-1.5pt} q}$-vector space of roots

\[ \{ y \in E(K^s), \:\varphi(y) = 0 \} \]

equipped with the natural Galois action. The short exact sequence of sheaves

induces a long exact sequence of cohomology that terminates in a surjective boundary homomorphism

The boundary homomorphism $\partial _\varphi$ sends $x \in E(K)$ to the class of the cocycle $g \mapsto g y - y$ with $y \in E(K^s)$ any element satisfying $\varphi (y) = x$.

We will refer to $\partial _\varphi$ as the Artin–Schreier homomorphism of the isogeny $\varphi$. We will use the shorthand $\partial _q$ to denote the Artin–Schreier homomorphism of the isogeny $\tau - 1\colon \mathbf {G}_a \to \mathbf {G}_a$ with kernel the constant sheaf $\mathbf {F}_{\hspace {-1.5pt} q}$.

Suppose that the sheaf $V$ is constant or, equivalently, that the $\mathbf {F}_{\hspace {-1.5pt} q}$-vector space $V$ is contained in $E(K)$. Then the cohomology group $H^1(K,\:V)$ coincides with the $\mathbf {F}_{\hspace {-1.5pt} q}$-vector space of continuous homomorphisms $G_K\to V$. In this case, we will often write the Artin–Schreier boundary homomorphism $\partial _\varphi$ in the form of a pairing

\[ [\ ,\ )_{\varphi}\colon E(K) \times G_K \to V, \quad [x,\ )_{\varphi} = \partial_{\varphi}(x). \]

We will use the shorthand $[\ ,\ )_{q}$ for the pairing of the isogeny $\tau -1$.

Lemma 2.2.1 Let $\varphi \colon \mathbf {G}_a \to \mathbf {G}_a$ be a separable isogeny with kernel $V$. Then for every Galois-equivariant epimorphism $f\colon V \twoheadrightarrow \mathbf {F}_{\hspace {-1.5pt} q}$, there is a scaling factor $u \in K$ which makes the following square commutative.

The factor $u$ is computed via Moore determinants:

\[ u = \alpha^{-1} \biggl(\frac{M(w_1,\dotsc,w_n)}{M(w_1,\dotsc,w_n,v)}\biggr)^q.\]

The terms in this formula have the following meaning:

• – $\alpha$ is the top coefficient of the $\tau$-polynomial $\varphi$;

• – $v \in V$ is an element such that $f(v) = 1$;

• – $w_1, \dotsc, w_n \in V$ is a basis of the hyperplane $\ker (f) \subset V$.

The value of $u$ is independent of choices of $v$ and $w_1,\dotsc,w_n$.

The factor $u$ of Lemma 2.2.1 is unique provided that $H^1(K,\,\mathbf {F}_{\hspace {-1.5pt} p}) \ne 0$.

Proof of Lemma 2.2.1 As a first step we shall prolong the morphism $f\colon V \to \mathbf {F}_{\hspace {-1.5pt} q}$ to the following morphism of short exact sequences of sheaves.

By Lemma 2.1.1(1) there is a unique polynomial $\psi _0 \in K[\tau ]$ of degree strictly less than $\dim V = n + 1$ which makes the left square commutative. The composite $(\tau - 1) \psi _0$ vanishes on the $\mathbf {F}_{\hspace {-1.5pt} q}$-vector subspace $V \subset K^s$, and is thus divisible by the polynomial $\varphi$ on the right [Reference PapikianPap23, Corollary 3.1.16]. We get the existence of the polynomial $\psi _1$. By construction, $\deg \psi _0 \leqslant n$ and $\deg \varphi = n+1$ so the relation

\[ \psi_1 \varphi = (\tau - 1)\psi_0 \]

implies that the polynomial $\psi _1 = u$ is constant, and that

\[ u \alpha = \beta^q, \]

where the element $\beta$ is the top coefficient of the polynomial $\psi _0$. This coefficient is computed by Lemma 2.1.1(2) which yields the Moore determinant formula for $u$.

2.3 Drinfeld modules

Fix a global function field $F$ over $\mathbf {F}_{\hspace {-1.5pt} q}$ and a place $\infty$ of $F$. As in § 1.2 this determines a coefficient ring $A$, i.e. the subring of $F$ consisting of elements which are regular outside $\infty$. We will freely use the notation and the terminology of § 1.2 with respect to coefficient rings and related objects.

For us a Drinfeld $A$-module over $\operatorname {Spec} K$ is an $A$-module scheme $E$ such that the underlying $\mathbf {F}_{\hspace {-1.5pt} q}$-module scheme is isomorphic to $\mathbf {G}_a$, and the multiplication morphism $a\colon E \to E$ is an isogeny of degree strictly greater than $1$ for at least one element $a \in A$.

Subject to a choice of an $\mathbf {F}_{\hspace {-1.5pt} q}$-linear isomorphism $E \xrightarrow {\smash {{ {\sim }}}} \mathbf {G}_a$ the data of an $A$-module structure on $E$ amounts to a morphism of $\mathbf {F}_{\hspace {-1.5pt} q}$-algebras $\varphi \colon A \to K[\tau ]$, and one recovers the more common definition of a Drinfeld module.

Let $\operatorname {Lie}_E$ be the tangent space of $E$ at zero. This is a vector space of dimension $1$ over the base field $K$. The action of $A$ on $\operatorname {Lie}_E$ determines a morphism $\iota _E\colon A \to K$, the characteristic homomorphism of the Drinfeld module $E$. The characteristic of $E$ is the ideal $\ker (\iota _E) \subset A$.

Let $\mathfrak {n} \subset A$ be a proper ideal which is not divisible by the characteristic of $E$. The $\mathfrak {n}$-torsion of $E$ is most naturally represented by the $A/\mathfrak {n}$-module

\[ E[\mathfrak{n}] = \operatorname{Hom}_A(A/\mathfrak{n}, E(K^s)). \]

We will use a modified definition which makes it easier to form inverse systems.

The $A/\mathfrak {n}$-module $E[\mathfrak {n}\rangle$ inherits an action of the absolute Galois group $G_K$ from $E(K^s)$. As the $A/\mathfrak {n}$-module $\mathfrak {n}^{-1}/A$ is free of rank $1$ a choice of its generator gives rise to a $G_K$-equivariant isomorphism of $A/\mathfrak {n}$-modules:

\[ E[\mathfrak{n}\rangle \xrightarrow{\smash{{{\sim}}}} E[\mathfrak{n}]. \]

An inclusion of ideals $\mathfrak {n}'\subset \mathfrak {n}$ induces a surjection $E[\mathfrak {n}'\rangle \twoheadrightarrow E[\mathfrak {n}\rangle$, and the resulting morphism $A/\mathfrak {n} \otimes _{A/\mathfrak {n}'} E[\mathfrak {n}'\rangle \xrightarrow {\smash {{ {\sim }}}} E[\mathfrak {n}\rangle$ is an isomorphism.

Lemma 2.3.2 We have a Galois-equivariant short exact sequence of $A$-modules which is natural in the ideal $\mathfrak {n}$ and the Drinfeld module $E$:

The first arrow is given by the composition with the quotient map $\mathfrak {n}^{-1} \twoheadrightarrow \mathfrak {n}^{-1}/A$ and the second arrow is the evaluation at $1 \in \mathfrak {n}^{-1}$.

Definition 2.3.3 The Kummer boundary homomorphism

\[ \partial_{\mathfrak{n}}\colon E(K) \to H^1(K,\:E[\mathfrak{n}\rangle) \]

is the boundary homomorphism arising from the sequence of Lemma 2.3.2. When the $\mathfrak {n}$-torsion of $E$ is rational we will write it in the form of a pairing:

\[ [\ ,\ )_{\mathfrak{n}}\colon E(K) \times G_K \to E[\mathfrak{n}\rangle. \]

The map $\partial _{\mathfrak {n}}$ can be also called the Artin–Schreier homomorphism but this name is less common in the literature.

Lemma 2.3.5 For each proper principal ideal $(a) = \mathfrak {a}$ of the ring $A$ which is not divisible by the characteristic of $E$ we have a commutative diagram

where $\partial _{a}$ is the Artin–Schreier boundary homomorphism of the separable isogeny $a\colon E \to E$, and $\text {ev}_{a}\colon E[\mathfrak {a}\rangle \xrightarrow {\smash {{ {\sim }}}} E[a]$ is an isomorphism defined by the formula $x\mapsto x(a^{-1})$.

Proof. We have the following commutative diagram.

The claim follows by naturality of boundary homomorphisms.

2.4 Drinfeld modules over local fields

In this section our base field $K$ is assumed to be local. We denote by $\mathcal {O}_K$ the ring of integers of $K$ and by $\mathfrak {p}_K$ the maximal ideal of $\mathcal {O}_K$.

Pick a coefficient ring $A$ over $\mathbf {F}_{\hspace {-1.5pt} q}$. Let $E$ be a Drinfeld $A$-module over $\operatorname {Spec} K$ and let $\iota _E\colon A \to K$ be its characteristic homomorphism.

Definition 2.4.1 The Drinfeld module $E$ has finite residual characteristic when $\iota _E(A) \subset \mathcal {O}_K$. The residual characteristic $\bar {\mathfrak {p}}$ of $E$ is the ideal

\[ \bar{\mathfrak{p}} := \iota_E^{-1}(\mathfrak{p}_K). \]

This ideal is maximal because the residue field $\mathcal {O}_K/\mathfrak {p}_K$ is finite.

The analogous situation for abelian varieties is when the base field $K$ is a non-archimedean local field. The residual characteristic of an abelian variety is just the residual characteristic of the field $K$.

From now on we suppose that the Drinfeld module $E$ has finite residual characteristic. It will be convenient for us to assemble the $\mathfrak {p}$-adic Tate modules $T_\mathfrak {p} E$, $\mathfrak {p}\ne \bar {\mathfrak {p}}$, into a single object.

Definition 2.4.2 The ring of restricted integral adeles is defined by the formula

\[ {A}_{\text{ad},\bar{\mathfrak{p}}} = \varprojlim\nolimits_{\mathfrak{n}} A/\mathfrak{n}, \]

where $\mathfrak {n} \subset A$ runs over proper ideals that are prime to $\bar {\mathfrak {p}}$.

We have a natural isomorphism ${A}_{\text {ad},\bar {\mathfrak {p}}} = \prod _{\mathfrak {p}\ne \bar {\mathfrak {p}}} A_\mathfrak {p}$ where $A_\mathfrak {p}$ is the $\mathfrak {p}$-adic completion of $A$. Compared with the ring of integral adeles of the function field $F$, the ring ${A}_{\text {ad},\bar {\mathfrak {p}}}$ does not include the factors at the places $\bar {\mathfrak {p}}$ and $\infty$.

Definition 2.4.3 The restricted adelic Tate module ${T}_{\text {ad},\bar {\mathfrak {p}}} E$ is defined by the formula

\[ {T}_{\text{ad},\bar{\mathfrak{p}}} E = \varprojlim\nolimits_{\mathfrak{n}} E[\mathfrak{n}\rangle, \]

where $\mathfrak {n} \subset A$ runs over proper ideals that are prime to $\bar {\mathfrak {p}}$ and the transition maps are induced by inclusions of the ideals.

The Tate module ${T}_{\text {ad},\bar {\mathfrak {p}}} E$ is a free module over the ring ${A}_{\text {ad},\bar {\mathfrak {p}}}$ and its rank coincides with the rank of the Drinfeld module $E$. The Galois action on ${T}_{\text {ad},\bar {\mathfrak {p}}} E$ is ${A}_{\text {ad},\bar {\mathfrak {p}}}$-linear. We have a natural ${A}_{\text {ad},\bar {\mathfrak {p}}}$-linear Galois-equivariant isomorphism:

\[ {T}_{\text{ad},\bar{\mathfrak{p}}} E \xrightarrow{\smash{{{\sim}}}} \prod\nolimits_{\mathfrak{p}\ne\bar{\mathfrak{p}}} T_\mathfrak{p} E. \]

Lemma 2.4.4 Consider the inclusion-ordered family $\mathcal {F}$ of proper principal ideals $\mathfrak {a} \subset A$ that are prime to the residual characteristic $\bar {\mathfrak {p}}$. Then the natural homomorphism

\[ {T}_{\text{ad},\bar{\mathfrak{p}}} E \xrightarrow{\smash{{{\sim}}}} \varprojlim\nolimits_{\mathfrak{a} \in \mathcal{F}} E[\mathfrak{a}\rangle \]

is an isomorphism.

3.1 Ramification subgroups

Let $I_K$ be the inertia subgroup of the absolute Galois group $G_K$, and let $P_K$ be the wild inertia subgroup.

Definition 3.1.1 We denote by $J_K$ the maximal quotient of $I_K$ that is abelian of exponent $p$:

\[ J_K := I_K^{\text{ab}}/(I_K^{\text{ab}})^{\times p}. \]

The quotient group $I_K/P_K$ is procyclic of order prime to $p$, so the fact that $P_K$ is a pro-$p$ group implies that the surjection $I_K \twoheadrightarrow I_K/P_K$ is split. As the group $J_K$ is abelian $p$-torsion we conclude that the canonical morphism $P_K \twoheadrightarrow J_K$ is surjective.

The inertia subgroup $I_K$ carries a decreasing separated exhaustive filtration by closed normal subgroups $I_K^{u}$, $u \in \mathbf {Q}_{\geqslant 0}$, the ramification filtration in upper numbering [Reference SerreSer68, Chapitre IV, § 3, Remarque 1, p. 83]. The wild inertia subgroup $P_K$ is the closure of the subgroup $\bigcup _{u > 0} I_K^{u}$.

The ramification filtration of $I_K$ induces a filtration of $J_K$. As the group $J_K$ is abelian the Hasse–Arf theorem implies that the induced filtration is integrally indexed: for each $u\geqslant 0$ the subgroup $J_K^u$ coincides with $J_K^m$, $m = \lceil u \rceil$.

Lemma 3.1.2 For every integer $m \geqslant 0$ such that $m \equiv 0\pmod {p}$ we have

\[ J_K^m = J_K^{m+1}. \]

Proof. Let $L\subset K^s$ be a finite Galois extension of $K$ with Galois group $G$ and upper index ramification filtration $\{ G^u \}_{u\geqslant 0}$. Suppose that the inertia group $G^0$ is abelian $p$-torsion. We need to show that $G^{pi} = G^{pi+1}$ for each $i \geqslant 0$.

We are free to assume that $\smash {G^0} = G$. The group $G$ is abelian, so we have the reciprocity morphism $\omega \colon K^\times \twoheadrightarrow G$ of local class field theory. This morphism transforms the $n$-unit filtration $\smash {\{U_K^{(n)}\}_{n\geqslant 0}}$ to the ramification filtration. For each $i \geqslant 0$ the $p$th power map induces a morphism

\[ U_K^{(i)} \twoheadrightarrow U_K^{(pi)}/U_K^{(pi+1)}. \]

This is surjective since the residue field of $K$ is perfect. Our claim follows.

3.2 Local pairing

Fix a Drinfeld $A$-module ${D}$ over $\operatorname {Spec} \mathcal {O}_K$. By this we mean not only that ${D}$ can be defined by a homomorphism $\varphi \colon A \to \mathcal {O}_K[\tau ]$ but also that the reduction of $\varphi$ modulo $\mathfrak {p}_K$ is a Drinfeld module of the same rank as over $K$.

The Drinfeld module ${D}$ has finite residual characteristic, denoted by $\bar {\mathfrak {p}}$ as usual. Let $K^\textit {ur}\subset K^s$ be the maximal unramified extension and let $\mathfrak {n}\subset A$ be a proper ideal not divisible by $\bar {\mathfrak {p}}$. Since ${D}$ is a Drinfeld module over $\operatorname {Spec}\mathcal {O}_K$ the $\mathfrak {n}$-torsion of ${D}$ is contained in ${D}(K^\textit {ur})$. Hence, by § 2.3 we have the Kummer pairing $[\ ,\ )_{\mathfrak {n}}\colon {D}(K^\textit {ur}) \times I_K \to {D}[\mathfrak {n}\rangle$.

Definition 3.2.1 The local restricted adelic Kummer pairing

\[ [\ ,\ )_{\text{ad}}\colon {D}(K^\textit{ur}) \times I_K \to {T}_{\text{ad},\bar{\mathfrak{p}}} {D} \]

is the limit of the Kummer pairings $[\ ,\ )_{\mathfrak {n}}$ over the inclusion-ordered system of proper ideals $\mathfrak {n}$ not divisible by the residual characteristic $\bar {\mathfrak {p}}$.

Since the Tate module ${T}_{\text {ad},\bar {\mathfrak {p}}}{D}$ is $p$-torsion the pairing $[\ ,\ )_{\text {ad}}$ factors through the maximal quotient $J_K = I_K^{\text {ab}}/(I_K^{\text {ab}})^{\times p}$ that is abelian of exponent $p$.

Let $v\colon K \twoheadrightarrow \mathbf {Z} \cup \{\infty \}$ be the normalized valuation of the base field $K$. Every isomorphism of $\mathbf {F}_{\hspace {-1.5pt} q}$-module schemes ${D} \xrightarrow {\smash {{ {\sim }}}} \mathbf {G}_a$ over $\operatorname {Spec} \mathcal {O}_K$ induces a valuation function on ${D}(K)$ by transport of structure from $\mathbf {G}_a(K) = K$. The result is independent of choices because the choices differ by multiplication by an integral unit on the side of $\mathbf {G}_a$. We denote the resulting valuation function by $v$ as well.

Definition 3.2.2 For every $\lambda \in {D}(K)$ we set

\[ \|\lambda\| = \max\{0, \:-v(\lambda)\}. \]

Poonen [Reference PoonenPoo95] introduced the notion of canonical local height on the $A$-module ${D}(K)$. Since the Drinfeld module ${D}$ has good reduction, Proposition 4(4) of [Reference PoonenPoo95, § 3] implies that the map $\|\cdot \|\colon {D}(K) \to \mathbf {Z}_{\geqslant 0}$ is exactly the canonical local height. This map satisfies the ultrametric inequality:

\[ \|\lambda + \lambda'\| \leqslant \max\{ \|\lambda\|, \|\lambda'\| \}. \]

However, $\|\lambda \| = 0$ for every $\lambda \in {D}(\mathcal {O}_K)$ so the height $\|\cdot \|$ is only a seminorm on ${D}(K)$. This seminorm is $\infty$-adic in the following sense.

Lemma 3.2.3 For each $\lambda \in {D}(K)$ and each $a \in A$ we have an equality

\[ \|a\lambda\| = |a|_\infty^s \, \|\lambda\|, \]

where $s$ is the rank of the Drinfeld module ${D}$.

The map $x \mapsto |x|_\infty ^s$ is an $\infty$-adic absolute value on the local field $F_\infty$ albeit its normalization differs from the one of § 1.2 when $s > 1$.

Proof of Lemma 3.2.3 This follows from Proposition 2 and Proposition 4(4) of [Reference PoonenPoo95]. For the reader's convenience let us give a direct argument. The claim is clear when $a = 0$ and holds when $\|\lambda \| = 0$ since ${D}(\mathcal {O}_K)$ is an $A$-submodule of ${D}(K)$. We are thus free to assume that $a \ne 0$ and $\|\lambda \| > 0$.

We identify the $\mathbf {F}_{\hspace {-1.5pt} q}$-module scheme ${D}$ with $\mathbf {G}_a$ over $\operatorname {Spec} \mathcal {O}_K$. The induced $A$-module scheme structure is described by a homomorphism $\varphi \colon A \to \mathcal {O}_K[\tau ]$. Write $\varphi (a) = \alpha _0 + \alpha _1 \tau + \dotsb + \alpha _n \tau ^n$ with $\alpha _n \ne 0$. The top coefficient $\alpha _n$ is a unit since ${D}$ is a Drinfeld module over $\operatorname {Spec} \mathcal {O}_K$. Hence, $v(\alpha _n \lambda ^{q^n}) = q^n v(\lambda )$. Combining this with the estimates $v(\alpha _i \lambda ^{q^i}) \geqslant q^i v(\lambda )$, $i < n$, and the assumption $v(\lambda ) < 0$ we deduce an equality

\[ v(a \, \lambda) = q^n \, v(\lambda). \]

The degree $n$ of the polynomial $\varphi (a)$ is expressed by a formula

\[ n = -s \, [\kappa_\infty:\mathbf {F}_{\hspace{-1.5pt} q}] \, v_\infty(a), \]

where $v_\infty$ is the normalized valuation of the local field $F_\infty$. The absolute value of § 1.2 is given by the formula

\[ |a|_\infty = |\kappa_\infty|^{-v_\infty(a)}. \]

As $|\kappa _\infty | = q^{[\kappa _\infty :\mathbf {F}_{\hspace {-1.5pt} q}]}$ we deduce that $q^n = |a|_\infty ^s$ and the claim follows.

Theorem 3.2.4 Let $\lambda \in {D}(K)$ be an element, and consider the homomorphism

\[ [\lambda,\ )_{\text{ad}}\colon J_K \to {T}_{\text{ad},\bar{\mathfrak{p}}} {D}. \]

Then the following hold.

1. (1) This homomorphism vanishes on the ramification subgroup $J^{\|\lambda \|+1}_K$. In particular, it vanishes altogether when $\|\lambda \| = 0$.

2. (2) If the height $\|\lambda \|$ is prime to $p$, then the homomorphism $[\lambda,\ )_{\text {ad}}$ maps the ramification subgroup $\smash {J^{\|\lambda \|}_K}$ surjectively onto ${T}_{\text {ad},\bar {\mathfrak {p}}} {D}$.

Proof. By Lemmas 2.4.4 and 2.3.5 it is enough to prove the claims for every homomorphism $[\lambda,\ )_{a}\colon J_K \to {D}[a]$ where $a \in A$ runs over non-constant elements that are prime to the residual characteristic $\bar {\mathfrak {p}}$ of ${D}$.

The height $\|\lambda \|$, the group $J_K$ and the homomorphism $[\lambda,\ )_{a}$ remain unchanged when we replace the base field $K$ by an unramified extension. As the Drinfeld module ${D}$ has good reduction and the element $a$ is prime to $\bar {\mathfrak {p}}$ we are free to assume that the torsion module ${D}[a]$ is contained in ${D}(K)$, and consequently in ${D}(\mathcal {O}_K)$.

An $\mathbf {F}_{\hspace {-1.5pt} p}$-vector subspace $V\subset {D}[a]$ is zero if and only if $f(V) = 0$ for each nonzero linear form $f\colon {D}[a] \twoheadrightarrow \mathbf {F}_{\hspace {-1.5pt} p}$. Likewise $V = {D}[a]$ if and only if $f(V) = \mathbf {F}_{\hspace {-1.5pt} p}$ for every such form $f$. It is thus enough to prove our claims after composing the homomorphism $[\lambda,\ )_{a}$ with every $f$.

From now on we identify the $\mathbf {F}_{\hspace {-1.5pt} q}$-module scheme ${D}$ with $\mathbf {G}_a$ over $\operatorname {Spec} \mathcal {O}_K$. The resulting $A$-module scheme structure on $\mathbf {G}_a$ is given by a homomorphism $\varphi \colon A \to \mathcal {O}_K[\tau ]$.

Let $f\colon {D}[a] \twoheadrightarrow \mathbf {F}_{\hspace {-1.5pt} p}$ be a nonzero linear form. According to Lemma 2.2.1 there is an element $u \in \mathbf {G}_a(K)$ such that

\[ f \circ [\lambda,\ )_{\varphi(a)} = [u\lambda,\ )_{p}. \]

This element is given by a formula with Moore determinants $M(-)$:

\[ u = \alpha^{-1}\biggl( \frac{M(w_1,\dotsc,w_n)}{M(w_1,\dotsc,w_n,v)}\biggr)^p. \]

Here $\alpha$ is the top coefficient of the polynomial $\varphi (a)$ and $w_1, \dotsc,w_n,v$ is a suitable $\mathbf {F}_{\hspace {-1.5pt} p}$-basis of ${D}[a]$.

By our assumption ${D}$ is a Drinfeld module over $\operatorname {Spec}\mathcal {O}_K$. This implies that the top coefficient $\alpha$ is an integral unit. Since $a$ is prime to the residual characteristic the torsion module ${D}[a] \subset {D}(\mathcal {O}_K)$ maps injectively to ${D}(k)$ where $k$ is the residue field of $\mathcal {O}_K$. In particular, the image of the basis $w_1,\dotsc,w_n,v$ in ${D}(k)$ remains $\mathbf {F}_{\hspace {-1.5pt} p}$-linearly independent. Hence, the corresponding Moore determinants are nonzero. We conclude that the element $u$ is an integral unit. It follows that $v(u\lambda ) = v(\lambda )$.

Consider the Artin–Schreier polynomial $X^p - X = u\lambda$. The Galois group $G$ of its splitting field is the image of the absolute Galois group under the homomorphism $[u\lambda,\ )_{p}$. The upper index ramification filtration of $G$ is given by images of absolute ramification subgroups:

\[ G^{i} = [u\lambda, I_K^{i})_{p}. \]

This filtration was calculated by Hasse (see Thomas [Reference ThomasTho05], Proposition 2.1 and the following paragraph). Set $b = \max \{0,\,-v(u\lambda )\}$. Then the subgroups $G^{i}$ vanish for $i > b$, and if the integer $b$ is prime to $p$, then $G^{b} = G = \mathbf {F}_{\hspace {-1.5pt} p}$. Our theorem follows since $v(u\lambda ) = v(\lambda )$ as shown previously.

3.3 Open image theorem

The following arguments involve passage from the base field $K$ to a finite separable extension $L$. On account of this we have to fix a few conventions. First, it will be more convenient for us to view the homomorphism $[\lambda,\ )_{\text {ad}}$ as a morphism from the inertia subgroup $I_K$ rather than its quotient $J_K$.

Second, we need to be careful with the canonical local height on the $A$-module ${D}(L)$. This involves the normalized valuation of $L$ and so differs from the height on the submodule ${D}(K)$ by an integer factor, the ramification index of $L/K$. We will denote the height on ${D}(L)$ by $\|\cdot \|_{L}$.

Lemma 3.3.1 Let $\lambda \in {D}(K) \smallsetminus {D}(\mathcal {O}_K)$ be an element, and suppose that the homomorphism $[\lambda,\ )_{\text {ad}}\colon I_K \to {T}_{\text {ad},\bar {\mathfrak {p}}} {D}$ is not surjective. Then there are:

• – a non-constant element $a \in A$ prime to the residual characteristic $\bar {\mathfrak {p}}$ of ${D}$;

• – a finite separable extension $L/K$; and

• – an element $\lambda ' \in {D}(L)\smallsetminus {D}(\mathcal {O}_L)$;

such that

\[ a \lambda' = \lambda, \quad v_p(\|\lambda'\|_{L}) < v_p(\|\lambda\|_{K}). \]

Here $v_p\colon \mathbf {Q} \twoheadrightarrow \mathbf {Z}\cup \{\infty \}$ denotes the normalized $p$-adic valuation.

Proof. Lemmas 2.4.4 and 2.3.5 imply that there is a non-constant element $a \not \in \bar {\mathfrak {p}}$ such that the homomorphism $[\lambda,\ )_{a}\colon I_K \to {D}[a]$ is not surjective.

The height $\|\lambda \|$, the inertia group $I_K$ and the homomorphism $[\lambda,\ )_{a}$ remain unchanged when we replace the base field $K$ by an unramified extension. We are thus free to assume that the torsion scheme ${D}[a]$ is constant or, in other words, that all the $K^s$-points of ${D}[a]$ are defined over the field $K$. Under this assumption the homomorphism $[\lambda,\ )_{a}$ is defined on the absolute Galois group $G_K$, see § 2.2. Its kernel determines a finite abelian $p$-torsion extension $L/K$.

Let $I \subset {D}[a]$ be the image of the inertia subgroup $I_K$. The ramification index $e$ of the extension $L/K$ equals the cardinality of $I$. By our choice of the absolute value $|\cdot |_\infty$ in § 1.2 we have an equality $|A/(a)| = |a|_\infty$. Thus, the torsion module ${D}[a]$ has cardinality $|a|_\infty ^s$, and the assumption $I \ne {D}[a]$ implies that $e < |a|_\infty ^s$.

By construction there is an element $\lambda ' \in {D}(L)$ such that $a \lambda ' = \lambda$. We have $\|\lambda \|_{L} = e \, \|\lambda \|_{K}$ so Lemma 3.2.3 shows that

\[ |a|_\infty^s \, \|\lambda'\|_{L} = e \, \|\lambda\|_{K}. \]

The integers $e$ and $|a|_\infty$ are powers of $p$ so the strict inequality $e < |a|_\infty ^s$ implies a strict inequality of $p$-adic valuations $v_p(\|\lambda '\|_{L}) < v_p(\|\lambda \|_{K})$ as claimed.

Proof. Suppose that the homomorphism is not surjective. Invoking Lemma 3.3.1 we obtain a non-constant element $a \in A\smallsetminus \bar {\mathfrak {p}}$, a finite separable extension $L/K$ and an element $\lambda ' \in {D}(L)$ such that $a \lambda ' = \lambda$. As the pairing $[\ ,\ )_{\text {ad}}$ is $A$-linear in the first argument we deduce that

\[ a [\lambda', I_L)_{\text{ad}} = [\lambda, I_L)_{\text{ad}}. \]

The right-hand side has finite index in $[\lambda, I_K)_{\text {ad}}$. Hence, it is enough to prove the theorem with $\lambda$ replaced by $\lambda '$ and $K$ replaced by $L$.

Lemma 3.3.1 also shows that we have a strict inequality of $p$-adic valuations:

\[ v_p(\|\lambda'\|_{L}) < v_p(\|\lambda\|_{K}). \]

The height $\|\cdot \|_{L}$ takes only integer values, so the valuation $v_p(\|\lambda '\|_{L})$ is bounded below by zero. Hence, after repeating our argument finitely many times we arrive at a situation where the homomorphism $[\lambda ',\ )_{\text {ad}}$ is onto.

4.1 The conductor of a Drinfeld module

Let $A$ be a coefficient ring over $\mathbf {F}_{\hspace {-1.5pt} q}$ and let $E$ be a Drinfeld $A$-module of stable reduction over $\operatorname {Spec} K$. Drinfeld's theory of Tate uniformization [Reference DrinfeldDri74, § 7] represents $E$ as an analytic quotient of a Drinfeld module ${D}$ over $\operatorname {Spec}\mathcal {O}_K$ by a Galois-invariant lattice $\varLambda \subset {D}(K^s)$, a finitely generated projective $A$-submodule which is discrete with respect to the adic topology. We thus have an analytic morphism $e_\varLambda \colon {D} \to E$ extending to a short exact sequence:

This sequence is determined functorially by the Drinfeld module $E$. We will refer to $\varLambda$ as the period lattice of $E$ and to $e_\varLambda$ as the exponential map of $\varLambda$. A detailed exposition of Tate uniformization theory can be found in a recent book by Papikian [Reference PapikianPap23, § 6.2].

Throughout this section we make an additional assumption: the lattice $\varLambda$ is defined over $K$, which is to say, $\varLambda \subset {D}(K)$. We will study the action of inertia on the Tate modules of $E$.

Lemma 4.1.2 For each proper ideal $\mathfrak {n}\subset A$ that is not divisible by the characteristic of $E$ we have a Galois-equivariant short exact sequence of $A/\mathfrak {n}$-modules:

The homomorphism $\delta _{\mathfrak {n}}$ has the following description.

1. (1) For each $x \in E[\mathfrak {n}\rangle$ there is an $A$-module morphism $\tilde {x}\colon \mathfrak {n}^{-1} \to {D}(K^s)$ which lifts $x$ in the sense that $e_\varLambda \circ \tilde {x}$ equals the composite of $x$ and the quotient morphism $\mathfrak {n}^{-1} \twoheadrightarrow \mathfrak {n}^{-1}/A$.

2. (2) The residue class $\delta _{\mathfrak {n}}(x)$ is represented by the element $\tilde {x}(1) \in \varLambda$ where $\tilde {x}$ is a lift of $x$.

3. (3) Given a representative $\lambda$ of the residue class $\delta _{\mathfrak {n}}(x)$ one can pick a lift $\tilde {x}$ such that $\tilde {x}(1) = \lambda$.

Proof. Consider a commutative diagram

with the vertical arrows given by $A$-module multiplication. The bottom row of this diagram is exact by construction and the top row is exact since the $A$-module $\mathfrak {n}$ is flat.

Consider the natural isomorphism $\mathfrak {n}\otimes _A M \xrightarrow {\smash {{ {\sim }}}} \operatorname {Hom}_A(\mathfrak {n}^{-1},\:M)$. Under this isomorphism the multiplication map $\mathfrak {n} \otimes _A M \to M$ is transformed to the evaluation map $f \mapsto f(1)$. Hence, the middle vertical arrow in the diagram above is surjective by Lemma 2.3.2, and our claims follow from the snake lemma.

Lemma 4.1.1 implies that Drinfeld modules ${D}$ and $E$ have the same residual characteristic. As usual, we denote this characteristic by $\bar {\mathfrak {p}}$. Consider the restricted adelic Tate module ${T}_{\text {ad},\bar {\mathfrak {p}}} E$ and let $\operatorname {GL}({T}_{\text {ad},\bar {\mathfrak {p}}} E)$ be the group of its automorphisms as an ${A}_{\text {ad},\bar {\mathfrak {p}}}$-module. The action of the inertia subgroup $I_K$ gives rise to the (local) restricted adelic monodromy representation

\[ \rho\colon I_K \to \operatorname{GL}({T}_{\text{ad},\bar{\mathfrak{p}}} E). \]

Similarly, for each prime $\mathfrak {p}\ne \bar {\mathfrak {p}}$ we have the $\mathfrak {p}$-adic monodromy representation $\rho _{\mathfrak {p}}\colon I_K \to \operatorname {GL}(T_\mathfrak {p} E)$. The representation $\rho _{\mathfrak {p}}$ is the composite of $\rho$ and the projection $\operatorname {GL}({T}_{\text {ad},\bar {\mathfrak {p}}} E) \to \operatorname {GL}(T_\mathfrak {p} E)$.

Taking the limit of sequences of Lemma 4.1.2 we obtain a short exact sequence of Galois representations:

As the Drinfeld module ${D}$ has good reduction we have at our disposal the local adelic Kummer pairing

\[ [\ ,\ )_{\text{ad}}\colon {D}(K) \times I_K \to {T}_{\text{ad},\bar{\mathfrak{p}}}{D}. \]

Lemma 4.1.3 The monodromy representation $\rho$ factors as a composition

where the second arrow is given by the formula $f \mapsto 1 + {T}_{\text {ad},\bar {\mathfrak {p}}}(e_\varLambda ) \circ f \circ \delta _{\text {ad}}$.

Proof. Let $\mathfrak {n} \subset A$ be a proper ideal not divisible by $\bar {\mathfrak {p}}$ and let $x \in E[\mathfrak {n}\rangle$ be an element. Pick a representative $\lambda$ of the residue class $\delta _{\mathfrak {n}}(x)$. We need to show that for each automorphism $g \in I_K$ one has

\[ g \circ x = x + e_\varLambda \circ [\lambda, g)_{\mathfrak{n}}. \]

Let $\tilde {x}\colon \mathfrak {n}^{-1} \to {D}(K^s)$ be a lift of $x$. By Lemma 4.1.2(3) we are free to assume that $\tilde {x}(1) = \lambda$. Lemma 2.3.4 shows that

\[ [\lambda, g)_{\mathfrak{n}} = g \circ \tilde{x} - \tilde{x}. \]

Hence, $e_\varLambda \circ [\lambda, g)_{\mathfrak {n}} = g \circ x - x$, and we are done.

Let us return to the setting of Drinfeld modules. Following § 3.2 we consider Poonen's canonical local height $\|\cdot \|$ on the $A$-module ${D}(K)$. We have seen that with a suitable normalization of the absolute value this height is an $\infty$-adic seminorm. Remarkably, this restricts to a proper norm on every period lattice $\varLambda$.

This fits to our definition of a normed lattice (Definition 1.3.2) with a caveat: the homogeneity property (3) holds for a differently normalized $\infty$-adic absolute value. When necessary we will correct for this discrepancy by considering the norm $\|\cdot \|^{{1}/{s}}$.

Theorem 4.1.7 Let $E$ be a Drinfeld module of stable reduction over $\operatorname {Spec} K$ with period lattice $\varLambda$ defined over $K$. Then there is an integer $m\geqslant 0$ such that the monodromy representation $\rho \colon J_K \to \operatorname {GL}({T}_{\text {ad},\bar {\mathfrak {p}}} E)$ maps the ramification subgroup $\smash {J^{m}_K}$ to $1$. Explicitly, if the period lattice $\varLambda$ is generated by elements $\lambda _1, \dotsc, \lambda _n$, then one can take

\[ m = \max \|\lambda_i\| + 1. \]

In view of Theorem 4.1.7 the following definition makes sense.

The invariant $\operatorname {\mathfrak {f}}(E/K)$ relates to conductors of torsion point extensions in the following way. For each proper ideal $\mathfrak {n}$ that is prime to $\bar {\mathfrak {p}}$ consider the extension $K_\mathfrak {n}\subset K^s$ generated by the $\mathfrak {n}$-torsion points of $E$, which is to say, the group $\operatorname {Gal}(K^s/K_\mathfrak {n})$ is the kernel of the torsion points representation $G_K \to \operatorname {GL}(E[\mathfrak {n}])$. Let ${K_\mathfrak {n}}^{\kern -.9ex\circ }\subset K_\mathfrak {n}$ be the maximal unramified subextension.

Lemma 4.1.10 The extensions $K_\mathfrak {n}/{K_\mathfrak {n}}^{\kern -.9ex\circ }$ are abelian and their conductors are related to $\operatorname {\mathfrak {f}}(E/K)$ as follows. If $\operatorname {\mathfrak {f}}(E/K) \ne 0$, then we have an equality

\[ \operatorname{\mathfrak{f}}(E/K) + 1 = \max_\mathfrak{n}\,\operatorname{\mathfrak{f}}(K_\mathfrak{n}/{K_\mathfrak{n}}^{\kern-.9ex\circ}). \]

If $\operatorname {\mathfrak {f}}(E/K) = 0$, then $\operatorname {\mathfrak {f}}(K_\mathfrak {n}/{K_\mathfrak {n}}^{\kern -.9ex\circ }) = 0$ for all $\mathfrak {n}$.

Proof. The Galois representation $E[\mathfrak {n}]$ and $E[\mathfrak {n}\rangle$ are non-canonically isomorphic, cf. § 2.3. Thus, the Galois group of the extension $K_\mathfrak {n}/{K_\mathfrak {n}}^{\kern -.9ex\circ }$ is the image of the absolute inertia subgroup $I_K$ under the torsion points representation $G_K \to \operatorname {GL}(E[\mathfrak {n}\rangle )$. The latter is obtained from the adelic Tate module representation by reduction modulo the ideal $\mathfrak {n}{A}_{\text {ad},\bar {\mathfrak {p}}}$. Hence, the extension $K_\mathfrak {n}/{K_\mathfrak {n}}^{\kern -.9ex\circ }$ is abelian by Corollary 4.1.4. When this extension is nontrivial its conductor and its highest break in upper numbering $b(K_\mathfrak {n}/{K_\mathfrak {n}}^{\kern -.9ex\circ })$ are related in the following way:

\[ \operatorname{\mathfrak{f}}(K_\mathfrak{n}/{K_\mathfrak{n}}^{\kern-.9ex\circ}) = b(K_\mathfrak{n}/{K_\mathfrak{n}}^{\kern-.9ex\circ}) + 1; \]

see [Reference SerreSer67, § 4.2, Proposition 1]. In view of this formula our claim is immediate.

The normalization of the conductor $\operatorname {\mathfrak {f}}(E/K)$ thus differs by $1$ from the one of local class field theory. In our case this choice leads to better-looking formulas.