Automorphic Forms and Arithmetic

Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre-)stabilization of relative trace formulas.

Arithmetic Siegel--Weil formulas predict that derivatives of Eisenstein series Fourier coefficients should encode arithmetic volumes of special cycles on Shimura varieties. Such formulas are arithmetic versions of formulas for complex volumes of moduli spaces in terms of L-functions and Eisenstein series. These "geometrize" the classical Siegel mass and Siegel--Weil formulas, on lattice and lattice vector counting. I will introduce these ideas and overview my proof for arithmetic 1-cycles on unitary Shimura varieties of arbitrarily high dimension. The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic. I will also mention a closely related "near-center arithmetic fundamental lemma (AFL)" relating arithmetic 1-cycles and orbital integrals. This "AFL" is from joint work in progress with Weixiao Lu and Wei Zhang.

Quaternionic modular forms (QMFs) are a class of non-holomorphic automorphic functions that behave similarly to classical Siegel modular forms. They exist on certain real forms of the exceptional groups, and also groups of type B and D with real rank 4, e.g. SO(4,n) with n at least 4. QMFs have semi-classical Fourier expansions and Fourier coefficients. I will explain the proof that, on the exceptional groups, there is a basis of the space of QMFs such that every Fourier coefficient of every element of the basis lies in Q-bar. The main technique is an "automatic convergence" theorem, which says that certain infinite series, which "look like" the Fourier expansion of a QMF, actually converge absolutely and thus are the Fourier expansion of a QMF.

Suppose that $F/\mathbb{Q}_p$ is a finite unramified extension. When $F = \mathbb{Q}_p$, Emerton showed that the mod-p local Langlands correspondence is realized globally in the mod-p cohomology of modular curves. For larger $F$, the mod-p local Langlands correspondence of $\mathrm{GL}_2(F)$ is unknown, but it is interesting to study the subrepresentations of the mod-p cohomology of Shimura curves cut out by a global Galois representation, in analogy with $F = \mathbb{Q}_p$. Under some reasonable hypotheses we prove that these representations of $\mathrm{GL}_2(F)$ are of finite length, as was expected since the work of Breuil-Paskunas from around 2007. This is joint work with C. Breuil, Y. Hu, S. Morra, and B. Schraen.

In 2001, Conrey, Farmer, Keating, Rubinstein, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of L-functions. This heuristic was later extended by Andrade and Keating to include moments and ratios of the family of L-functions associated to hyperelliptic curves of genus g over a fixed finite field.

In joint work with Bergström, Petersen, and Westerland, we related the moment conjecture of Andrade and Keating to the problem of understanding the homology of the braid group with symplectic coefficients. We computed the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations, and showed that the answer matches the number-theoretic predictions. Our results, combined with a recent homological stability theorem of Miller, Patzt, Petersen, and Randal-Williams, imply the conjectured asymptotics for all moments in the function field case, for all large enough odd prime powers $q$.

The decomposition into Bernstein blocks of the category of smooth representations of a $p$-adic group $G$ admits a Galois analogue in terms of enhanced $L$-parameters, in which the role of supercuspidal representations is played by « cuspidal » enhanced $L$-parameters, a notion that originates in the construction by Lusztig of the generalized Springer correspondence.

We will formulate sufficient conditions to be satisfied by the local Langlands correspondence for the supercuspidal representations of the Levi subgroups of G in order to allow to construct a canonical bijection between the associated blocks on both sides.

Next, we will prove that, when these conditions are satisfied, every compound $L$-packet of G contains at least one irreducible representation with non-singular supercuspidal support.

The conditions evacuated above are notably satisfied when $G$ is the exceptional group $G_2$, and when G is a classical group. We will show that, for every irreducible $p$-adic ortho-symplectic dual pair $(G, G')$, the Howe correspondence can be described as a collection of correspondences between simple modules of affine Hecke algebras attached to the Bernstein blocks of enhanced $L$-parameters of $G$ and $G^′$.

Parts of this talk are based on joint works with Ahmed Moussaoui and Maarten Solleveld, and with Yujie Xu.

The local Langlands correspondence (LLC) is a conjectural finite-to-one map from representations of a p-adic reductive group G to the set of L-parameters for G. Recently there have been two major advances in this area: Kaletha's characterization of the LLC in terms of explicit character formulas and harmonic analysis, and then a more geometric construction, due to Fargues-Scholze, involving excursion operators. This talk represents work in progress on what happens when you marry these approaches: you can write down an expected formula for the excursion operator, assuming Kaletha's formulas. Interestingly, the excursion operator is convolution with a function, the "excursion function", which comes entirely out of harmonic analysis.

I will talk about recent work (joint with Boeckle, Iyengar, Manning, Urban) that studies situations when one can prove isomorphism of deformation rings R and Hecke rings T without them being complete intersections. Our work has led to new algebraic invariants which capture the failure of R=T to be isomorphisms of complete intersections. One can try and relate these new invariants to the numerical invariants like Tamagawa factors that appear in the formula conjectured by Block-Kato for the special value L(1,Ad_f) of the adjoint motive associated to a newform f of weight at least 2. One is led to make new guesses about the relation of the higher congruence numbers (defined in work with Iyengar and Manning) associated to Kisin’s local deformation rings at at the augmentation induced by f and the Tamagawa numbers of the adjoint motive of f. The talk will summarize known results and conclude with some more speculative thoughts.

Trianguline representations are a big class of $p$-adic representations that contain all nice enough (cristalline) ones but allow a continuous variation of weights. The trianguline deformation ring is defined as the Zariski closure of such representations in certain ambient spaces. We give a new definition of trianguline deformation rings in 2 dimensional cases as the support of certain modules, so that one does not need to take Zariski closure. It is also integral in nature and can be seen to be compatible with Emerton's Jacquet functor.

Studying the étale cohomology of Shimura varieties with Hecke and Galois actions provides an avenue toward understanding the Langlands correspondence. While the structure of the rational cohomology groups is predicted by conjectures of Kottwitz and Arthur, the integral or mod $\ell$ cohomology groups are more intricate. Although a conjectural formula exists for computing these groups, it relies on an additional conjecture on the existence of moduli spaces of global Galois representations. However, much more can be said if we focus on these cohomology groups with Hecke and local Galois actions. Specifically, I will present an unconditional formula for computing the mod $\ell$ cohomology groups of many Shimura varieties with Iwahori level at $p(\neq \ell$), expressed in terms of coherent sheaves on the moduli space of local Galois representations. This result not only provides evidence supporting the aforementioned conjectures but also has surprising arithmetic applications.

I will first present a formulation of the strong Brumer—Stark conjecture. Equivariant Tamagawa Number Conjecture (ETNC) is a refinement of the strong Brumer—Stark conjecture that gives a formula for the “equivariant” leading term of Artin L-functions of odd characters of totally real fields at s=0. I will sketch a proof of the ETNC. It uses the Euler systems argument and is inspired by the work of Bullach—Burns—Daoud—Seo. This is a joint work with Dasgupta, Silliman and Wang.

(Joint with Ben Bakker) Given an Elliptic curve E with complex multiplication, it is known that E has (potentially) good reduction everywhere. Concretely, this means that the j-invariant of E is an algebraic integer. The generalization of this result to Abelian-Varieties follows from the Neron-Ogg-Shafarevich criterion for good reduction. We generalize this result to Exceptional Shimura varieties S. Concretely, we show that there exists some integral model S_0 of S such that all special points of S extend to (algebraically) integral points of S_0. To prove this we establish a Neron-Ogg-Shafarevich criterion in this setting. Our methods are general and apply, in particular, to arbitrary variations of hodge structures with an immersive Kodaira-Spencer map. We will explain the proof (which is largely in the realm of birational p-adic geometry) and the open questions that remain.