Schedule for: 22w5178 - Langlands Program: Number Theory and Representation Theory

In this talk we present a partial solution to the basis problem for paramodular forms. Our method involves proving a precise relation between paramodular forms and algebraic modular forms arising from certain positive-definite quinary lattices. The latter can be computed explicitly by the lattice-neighbour method. As an application, we can prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degree two and one. These include some of the type conjectured by Harder at level one and supported by compu- tations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev. This is joint work with Dummigan, Pacetti and Rama.

An important invariant for admissible representations of reductive p-adic groups is the wavefront set, the collection of the maximal nilpotent orbits in the support of the orbital integrals that occur in the Harish-Chandra-Howe local character expansion. We compute the geometric and Okada’s canonical unramified wavefront sets for representations in Lusztig’s category of unipotent reduction for a split group in terms of the Kazhdan-Lusztig parameters. I will emphasise two applications of this calculation: 1) the geometric wave front set of a unipotent supercuspidal representation determines uniquely the nilpotent part of the Langlands parameter; 2) the anti-tempered unipotent Arthur packets are uniquely characterised by the unramified wave front set of their constituents. The talk is based on joint work with Lucas Mason-Brown and Emile Okada.

Thirty years ago, David Vogan conjectured a purely local description of Arthur packets for p-adic groups, closely related to a more developed theory for Real groups by Adams, Barbasch and Vogan. In joint work with Mishty Ray, we have now finished the proof of this conjecture for general linear groups, building on previous work with other members of the ``Voganish Project''. In this talk I'll present the proof for $\mathrm{GL}_n$ and discuss how the strategy seems likely to extend to classical groups, highlighting the work that remains to be done.

Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. In the first lecture, we will recall the definition of stacks of shtukas and their cohomology, the action of the partial Frobenius morphisms and the action of the Hecke algebra. In the second lecture, we will explain the Eichler-Shimura relations, the finiteness property of the cohomology groups, Drinfeld's lemmas and the action of the Weil group (of the function field) on the cohomology groups. In the third lecture, we will explain how this action and the "Zorro lemma" imply the smoothness of the cohomology sheaves.

Kottwitz introduced certain compact ``fake'' unitary group Shimura varieties and determined their local Hasse-Weil zeta functions at primes of good reduction. For primes where the level is arbitrarily deep, the local Hasse-Weil zeta functions were further studied in the case of signature $(1, n − 1)$ (by Xu Shen), and in the case of arbitrary signature but under the assumption that the group at $p$ is a product of Weil restrictions of general linear groups, by Scholze and Shin. In this talk, I will explain joint work with Jingren Chi in which we generalize the work of Scholze-Shin by allowing the group at $p$ to be any inner form of a product of Weil restrictions of general linear groups. New phenomena arise when the group at p is not quasi-split, for example a crucial vanishing property of twisted orbital integrals of the test functions at $p$.

Let $p$ be a prime integer, $F$ be a non-Archimedean local field of residual characteristic $p$ and $G = \mathcal{G}(F)$ be the group of $F$-rational points of a connected reductive group defined over $F$. During the last decade, the study of $p$-modular smooth representations of $G$, i.e. of smooth representations of $G$ with coefficients in a field of characteristic $p$, has been intensively developed for arithmetic reasons (e.g. related to congruences between automorphic forms) but remains really mysterious, even for nice groups as $\mathrm{GL}_{2}(F)$ or $\mathrm{SL}_{2}(F)$, or when one focuses on (admissible) irreducible smooth ones. This talk aims at presenting what is known so far in this setting, then at discussing some joint work with Hauseux, where we prove that understanding $p$-modular representations of $G$ when $\mathcal{G}$ is of $F$-semisimple rank $1$ amounts to understand their restriction to a minimal parabolic subgroup. Doing so, we extend some previous work of Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas (2007) for $\mathrm{GL}_{2}(F)$, but with different methods that give another viewpoint on Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas proofs and suggests that this kind of statement could extend to higher rank groups.

Gross and Reeder showed that for any split reductive group G over a locally compact non-Archimedian field F, one can construct an easy family of cuspidal representations of G(F), called "simple" cuspidals. For G=GL(n), they go back to work of Carayol in the 1970's. In terms of ramification, they come right after the "level 0" cuspidals, which can be obtained from representations of G(k) where k is the residue field of F. But the construction of "simple" cuspidals only essentially involves a non-trivial character of k. I shall decribe the "simple" cuspidals for G=GL(n) or Sp(2n). On the other hand, the Langlands correspondence attaches to a "simple" cuspidal representation π of GL(n) an irreducible degree n representation σ of the Weil group W_F of F, and σ has Swan exponent 1. Describing σ from π is not easy: for example, when n is a power of the residue characteristic p of F, σ is primitive. I shall give part of the recipe (work of Bushnell and I in 2013, Imai and Tsushima in 2015). I shall also examine the more recent case of G=Sp(2n). When F has characteristic 0, J. Arthur attaches to a "simple" cuspidal π of Sp(2n,F) an orthogonal representation σ of W_F of dimension 2n+1. I shall give the description of σ from π, due to Masao Oi (2018) when p is odd, and to Oi and I (2022) when p=2.

In this talk I will explain the spectral decomposition of the space of automorphic forms (and more generally of cohomology groups of stacks of shtukas), indexed by Langlands paremeters, a refined decomposition obtained with Xinwen Zhu (after an idea of Drinfeld) and I will mention the link with the l-adic geometric Langlands program by trace of Frobenius constructions (due to Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum, Varshavsky).

Let H and G be reductive groups over Q such that H is contained in G. In this talk I will explain the connection between the non-vanishing of periods integrals for (G, H) and the local geometry of Eigenvarieties of G. Furthermore, we will use this connection in the construction of p-adic L-functions. The talk will be based on the following two situations: 1) H= GL(n)xGL(n) and G= GL(2n) (joint work with M. Dimitrov, A. Graham, A. Jorza and C. Williams). 2) H= GL(n) and G= GL(n) over an imaginary quadratic field (work in progress with P-H. Lee and C. Williams)

The classical Shimura correspondence lifts automorphic representations on the double cover of $\mathrm{SL}_2$ (corresponding to classical half-integral weight forms) to automorphic representations on $\mathrm{PGL}_2$. Though efforts have been made for many years to generalize this map to higher rank groups and higher degree covers, our knowledge is limited. In this talk I present joint work with Omer Offen that points to a new Shimura lift for automorphic representations on the triple cover of $\mathrm{SL}_3$ -- we establish the Fundamental Lemma for a relative trace formula. Further work is on-going. We expect that this project will both prove the existence of a global lift of automorphic representations and characterize the image of the lift by means of a period involving a theta function on $\mathrm{SO}_8$, thereby confirming a 2001 conjecture of Bump, Friedberg and Ginzburg.

This is a report on several joint projects analyzing the parametrization of V. Lafforgue and C. Xue of automorphic representations of reductive groups over function fields, and the corresponding local parametrizations, due to Genestier-Lafforgue and Fargues-Scholze. With Gan and Sawin we show that pure supercuspidal representations usually have ramified Galois parameters, and attach Weil-Deligne parameters unambiguously to all discrete series representations. With Böckle, Feng, Khare, and Thorne, we construct cyclic base change of a cuspidal automorphic representation of any sufficiently large prime degree, and apply this to base change for local representations. Finally, with Ciubotaru, we show that the unramified components of any cuspidal automorphic representation of a simple group (with a few surprising exceptions) are tempered and generic, provided at least one component is tempered and another is unramified generic.

We outline a general strategy based on types and covers to describe the $L$-packets of representations of classical groups over a non-archimedean local field of odd residual characteristic : L-packets are parametrized by Jordan sets, and the Jordan set of a given cuspidal representation can be computed using types and covers that transform the problem into a computation of defining relations for Hecke algebras. We explain the results obtained in the symplectic case in a joint work with Guy Henniart and Shaun Stevens : the inertial Jordan sets of cuspidal representations are fully determined. We explain the ambiguity that may remain and a possible way to solve it, with examples from joint works with Laure Blasco in $\mathrm{Sp}(4)$ and Geo Kam-Fai Tam in ramified unitary groups and work in progress in $\mathrm{Sp}(4)$ with Henniart and Stevens.

Let $K$ be a CM field with maximal totally real subfield $F=K^+$, let $\pi$ be a conjugate self-dual cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_K)$, and let $P$ be a fixed prime of $F$. As $\chi$ ranges over the set $X_K(P)$ of primitive ring class characters of $K$ of $P$-power conductor, the root number $\epsilon(1/2, \pi \otimes \chi)$ of the twisted standard L-function $L(s, \pi \otimes \chi)$ of $\pi$ is generically independent of the choice of $\chi$. That is, for one of $k \in {0, 1}$, we know that $\epsilon(1/2, \pi \otimes \chi) = (-1)^k$ for all but finitely many $\chi$ in $X_K(P)$. In the case of $k=0$, we study the central values $L(1/2, \pi \otimes \chi)$ as $\chi$ varies in $X_K(P)$ using “toric period” integral presentations, namely (1) Eulerian integral presentations and (2) presentations implied by the Ichino-Ikeda Gan-Gross-Prasad conjecture for unitary groups. In particular, we present a generalization of “Mazur’s conjecture”, and explain how this can be deduced from a certain equidistribution criterion. We then describe an approach to reducing this criterion to the celebrated theorems of Ratner and Margulis-Tomanov on p-adic unipotent flows (which would generalize the well-known theorems of Vatsal and Cornut-Vatsal for ${\rm GL}_2$).

In the last years, the study of the images of the Galois representations associated to regular algebraic cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, via global Langlands correspondence, has been an effective strategy to address the inverse Galois problem for finite groups of Lie type. In this talk we will explain how, by combining this strategy with Langlands functoriality and globalization of supercuspidal representations, we can construct residual Galois representations with controlled image and obtain new families of finite groups of type $B_m$, $C_m$ and $D_m$ arising as Galois groups over $\mathbb{Q}$.