Automorphic Forms and Arithmetic

I will talk about my recent rationality results on the ratios of critical values for Rankin-Selberg L-functions for GL(n) x GL(m) over a totally imaginary base field. In contrast to a totally real base field, when the base field is totally imaginary, some delicate signatures enter the reciprocity laws for these special values. These signatures depend on whether or not the totally imaginary base field contains a CM subfield. This is a generalization of my work with Günter Harder on rank-one Eisenstein cohomology for GL(N), where N = n + m. The rationality result comes from interpreting Langlands's constant term theorem in terms of an arithmetically defined intertwining operator between Hecke summands in the cohomology of the Borel-Serre boundary of a locally symmetric space for GL(N). The signatures arise from Galois action on certain local systems that intervene in boundary cohomology.

Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is 'r-modular' for r=2,3.... To define this generalization, the modular curve gets replaced with Drinfeld's concept of a 'shtuka space'. The r-modularity of E is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2- and 3-modular.

In this talk I will present a project in progress with Joaquín Rodrigues Jacinto concerning the study of locally analytic representations of p-adic Lie groups and its relation with p-adic D-modules of rigid spaces à la Ardakov. I will sketch how both theories are essentially two different looks of the same kind of objects and how they can be interpreted in terms of sheaves in suitable stacks on analytic rings. If time permits I will mention two possible applications in the cohomology of Shimura varieties.

The classical Satake isomorphism relates the spherical Hecke algebra of a reductive group $G$ over a local field $F$ to representations of the Langlands dual group. When $F$ is of mixed characteristic $(0,p)$ and the Hecke algebra has characteristic prime to $p$, the Satake isomorphism has been geometrized by X. Zhu, J. Yu, Fargues-Scholze, and Richarz-Scholbach using techniques from p-adic geometry.
In this talk, we consider the case where the Hecke algebra has characteristic p. I will speak on my recent joint work with Yujie Xu, where we geometrize and obtain explicit formulas for the mod p Satake isomorphism of Herzig and Henniart-Vignéras using mod p étale sheaves on Witt vector affine flag varieties. Our methods involve the constant term functors inspired from the geometric Langlands program, especially the geometry of certain generalized Mirković-Vilonen cycles. The situation is quite different from l-adic sheaves ($l \neq p$) because only three of the six functors preserve constructible sheaves.

I will talk about my recent joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of reductive $p$-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties. In particular, we obtain "mixed" L-packets containing F-singular supercuspidals and non-supercuspidals.

How many automorphic representations of level n have a specified local factor at the infinite places? When this local factor is a discrete series representation, this questions is asymptotically well-understood as n grows. Non-tempered local factors, on the other hand, violate the Ramanujan conjecture and should be very rare. We use the endoscopic classification for representations to quantify this rarity in the case of cohomological representations of unitary groups, and discuss some applications to the growth of cohomology of Shimura varieties.

Let $(G, X)$ be a Shimura datum, and let $K$ be a compact open subgroup of $G(\mathbb{A}_f)$. One hopes that under mild assumptions on $G$ and $K$, the points of the Shimura variety $Sh_K(G, X)$ form a family of motives; in abelian type this is well-understood, but in non-abelian type it is completely mysterious. I will discuss joint work with Christian Klevdal showing that for non-abelian type Shimura varieties the points (over number fields, say) at least yield compatible systems of l-adic representations (to be precise, after projection to the adjoint group of G). These should be the l-adic realizations of the conjectural motives.

To a connected reductive group $G$ over a local field $F$ we define a compact topological group $\tilde\pi_1(G)$ and an extension $G(F)_\infty$ of $G(F)$ by $\tilde\pi_1(G)$. From any character $x$ of $\tilde\pi_1(G)$ of order n we obtain an n-fold cover $G(F)_x$ of the topological group $G(F)$. We also define an L-group for $G(F)_x$, which is a usually non-split extension of the Galois group by the dual group of $G$, and deduce from the linear case a refined local Langlands correspondence between genuine representations of $G(F)_x$ and L-parameters valued in this L-group.
This construction is motivated by Langlands functoriality. We show that a subgroup of the L-group of $G$ of a certain kind naturally lead to a smaller quasi-split group $H$ and a double cover of $H(F)$. Genuine representations of this double cover are expected to be in functorial relationship with representations of $G(F)$. We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the theory of endoscopy.

I will report on a joint work with Klingler and Ullmo. Given a polarizable variation of Hodge structures on a smooth complex quasi-projective variety S (e.g. the one associated to a family of pure motives over S), Cattani, Deligne and Kaplan proved that its Hodge locus (the locus of closed points of S where exceptional Hodge tensors appear) is a *countable* union of closed algebraic subvarieties of S. In this talk I will discuss when this Hodge locus is actually algebraic.
The first part of the talk will introduce the Hodge theoretic formalism and highlight differences and similarities with the world of Shimura varieties. If time permits I will present some applications of such a viewpoint to either the Lawrence-Venkatesh method or to the existence of genus four curves of "Mumford's type".

I will first recall the general expectations of Shimura, Langlands, and Kottwtiz on the shape of the zeta function of a Shimura variety, or more generally its étale cohomology. I will then report on some recent progress which partially fulfills these expectations, for Shimura varieties of unitary groups and special orthogonal groups. Finally, I will give a preview of some foreseeable developments in the near future.

For a Hecke eigenform $f$, we state an adjoint L-value formula relative to each quaternion algebra $D$ over $\mathbb{Q}$ with discriminant $\partial$ and reduced norm $N$. A key to prove the formula is the theta correspondence for the quadratic $\mathbb{Q}$-space $(D,N)$. Under the $R=\mathbb{T}$-theorem, the $p$-part of the Bloch-Kato conjecture is known; so, the formula is an adjoint Selmer class number formula. We also describe how to relate the formula to a consequence of the Tate conjecture for quaternionic Shimura varieties.

The first half of this talk will be devoted to describing the structural properties of the set of local field valued points of a certain class of algebraic stacks. I will then describe two applications in the second half, one joint with Wyss and Ziegler, and the other one with Esnault. The first application relates the p-adic volume of certain moduli spaces to BPS invariants and the second application is an elementary proof of the existence of a Fontaine-Laffaille structure for rigid flat connections.