We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.

In an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

Étant donné un groupe réductif $G$ sur une extension de degré fini de $\mathbb {Q}_p$ on classifie les $G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge$p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble $B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

To every dynamical system $(X,\varphi )$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi )$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi )$, the group $T(\varphi )$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if $(X,\varphi )$ has a dense orbit, then the isomorphism type of the group $T(\varphi )$ is a complete invariant of flow equivalence of the pair $\{\varphi , \varphi ^{-1}\}$.

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Given a manifold $M$ with a submanifold $N$, the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to $M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$, in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$.

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in $G$. This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on $G$. This allows us to demonstrate the conjecture for general semiabelian varieties.

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props

\[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \]

$\mathcal {A}\textit{ssb}_\infty$

$\mathfrak{A}$

${\mathcal {L}} ie_\infty$

${\mathcal {L}} ie_\infty$

$\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$

$\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$

$\odot V$

$V$

$\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$

$\{F_\mathfrak{A}\}$

$GRT=GRT_1\rtimes {\mathbb {K}}^*$

$\{\mathfrak{A}\}$

Let $F$ be a non-Archimedean local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $\unicode[STIX]{x1D712}_{0}$ be a depth-zero character of the maximal compact subgroup $T$ of $T(F)$. This gives by inflation a character $\unicode[STIX]{x1D70C}$ of an Iwahori subgroup $\unicode[STIX]{x2110}\subset T$ of $G(F)$. From Roche [Types and Hecke algebras for principal series representations of split reductive$p$-adic groups, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 361–413], $\unicode[STIX]{x1D712}_{0}$ defines a reductive $F$-split group $\widetilde{G}^{\prime }$ whose connected component $G^{\prime }$ is an endoscopic group of $G$, and there is an isomorphism of $\mathbb{C}$-algebras $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ where $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ is the Hecke algebra of compactly supported $\unicode[STIX]{x1D70C}^{-1}$-spherical functions on $G(F)$ and $\unicode[STIX]{x2110}^{\prime }$ is an Iwahori subgroup of $G^{\prime }(F)$. This isomorphism gives by restriction an injective morphism $\unicode[STIX]{x1D701}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $\unicode[STIX]{x1D701}$ realizes the transfer (matching of strongly $G$-regular semi-simple orbital integrals). If $\operatorname{char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.