Automorphic Forms and Arithmetic

The base change fundamental lemma is a statement in harmonic analysis over local fields, which has various applications for Langlands functoriality. It has been established for p-adic fields, but not yet for function fields. I will explain a geometric approach to the base change fundamental lemma over function fields, which is based on analyzing the cohomology of moduli spaces of shtukas.

The arithmetic Gan-Gross-Prasad (AGGP) conjecture, a high dimensional generalization of the Gross–Zagier theorem, relates the height pairing of arithmetic diagonal cycles on certain product Shimura varieties to the first central derivative of Rankin-Selberg tensor product L-functions. The arithmetic fundamental lemma conjecture, proposed about 10 years ago, arises from the relative trace formula approach to the AGGP conjecture. I will recall the statement of the arithmetic fundamental lemma and present (a sketch of) a proof.

The classical Eichler Shimura morphisms identifies Betti cohomology of certain local systems on modular curves with spaces of elliptic modular forms. Faltings has refined such isomorphisms to the p adic context. I will discuss p adic variations of such maps involving Glenn Stevens' modular symbols and Coleman's overconvergent modular forms.

Mazur conjectured, after Faltings's proof of the Mordell conjecture, that the number of rational points on a curve depends only on the genus, the degree of the number field and the Mordell-Weil rank. This conjecture was established in a few cases. In this talk I will explain how to prove this conjecture and some of its generalization. I will focus on how functional transcendence and unlikely intersections on mixed Shimura varieties are applied. This is joint work with Vesselin Dimitrov and Philipp Habegger.

Real-analytic modular forms in the sense of F. Brown are modular objects that have a special structure motivated by problems in mathematical physics and are not, in general, eigenfunctions of the Laplacian. We introduce L-functions for real-analytic modular forms and construct the analogue of a period polynomial in the important subclass of modular iterated integrals of length one. We show that, in terms of their "period polynomials", the behaviour of such real-analytic forms is a cross between the behaviour of holomorphic forms and that of Maass forms. (This is joint work with J. Drewitt)

The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups). In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group. This is joint work with Sug Woo Shin.

The absolute cohomological purity for etale cohomology of Gabber--Thomasson implies that an etale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

Recently, Cai, Friedberg, Ginzburg and Kaplan generalized the doubling method of Piatetski-Shapiro and Rallis. They found global integrals, which represent the standard L-function for pairs of irreducible, automorphic, cuspidal representations \pi - on a (split) classical group G, and \tau - on GL(n). The representation \pi need not have any particular model (such as a Whittaker model, or Bessel model, etc.). These integrals suggest an explicit descent map (an inverse to Langlands functorial lift) from GL(n) to G (appropriate G). I will show that a certain Fourier coefficient applied to a residual Eisenstein series, induced from a Speh representation, corresponding to a self-dual \tau, is equal to the direct sum of irreducible cuspidal representations \sigma \otimes \sigma', on G x G , where \sigma runs over all irreducible cuspidal representations, which lift to \tau (\sigma' is the complex conjugate of an outer conjugation of \sigma). This is a joint work with David Ginzburg.

Together with F. Andreatta we have recently constructed anticyclotomic p-adic L-functions attached to a classical eigenform and a quadratic imaginary field when the prime p is not split in quadratic field. We will present some arithmetic applications to CM elliptic curves in the case p is inert in the quadratic field, analogous to Rubin's formula in the split case.

Bhargava, Kane, Lenstra, Poonen, and Rains proposed heuristics for the distribution of arithmetic data relating to elliptic curves, such as their ranks, Selmer groups, and Tate-Shafarevich groups. As a special case of their heuristics, they obtain Goldfeld's minimalist conjecture, which predicts that 50% of elliptic curves have rank 0 and 50% of elliptic curves have rank 1. After surveying these conjectures, we will explain joint work with Tony Feng and Eric Rains, verifying many of these conjectures over function fields of the form F_q(t), after taking a certain large q limit.

I will explain a new proof of the geometric stabilization theorem for Hitchin fibers, a key ingredients in Ngô's proof of the fundamental lemma. Our approach relies on ideas of Denef-Loeser and Batyrev on p-adic integration and Langlands duality for generic Hitchin fibers. This is joint work with Michael Groechenig and Paul Ziegler.

Saito--Tunnell theorem is a local version of Waldspurger's formula, relating the existence of E^\times invariant linear forms on representation of GL_2 to local root numbers. I present a generalization of this which relates the existence of GL_n(E) invariant linear forms on GL_2n(F) to local root numbers. The proof relies on a "relative version" of Harish-Chandra' theory of local character expansions.

In this talk, we introduce some Lefschetz-type theorems for Brauer groups of hyperplane sections of smooth projective varieties. This is more or less known when the dimension of the hyperplane section is at least 3, but we will also introduce a version which lowers the dimension from 3 to 2. As a consequence, we reduce the Tate conjecture for divisors on smooth projective varieties from general dimensions to dimension 2, and thus proves a results of Morrow by a different method.

In this talk, I will introduce the functorial descent from cuspidal automorphic representations \pi of GL7(A) with L^S(s, \pi, \wedge^3) having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under mild assumptions on the unramified components of \pi, it is cuspidal and having \pi as a weak functorial lift of each irreducible summand. However, we show that the other descent module supports not only the non-degenerate Whittaker integeral on G2(A), but also every degenerate Whittaker integral. Thus it is generic, but not cuspidal. This is a new phenomenon, compared to the theory of functorial descent for classical and GSpin groups. This work is joint with Joseph Hundley.

For a smooth scheme of finite type over a field its motivic cohomology groups generalize the usual Chow groups and are an important algebraic invariant. In this talk we will explain how, for the special fibers of certain quaternionic Shimura varieties, its motivic cohomology can encode very rich arithmetic information. More precisely we will show that the cycle class map from motivic cohomology to étale cohomology gives a geometric realization of level raising between Hilbert modular forms. The main ingredient for this construction is a form of Ihara's Lemma for Shimura surfaces which we prove by generalizing a method of Diamond-Taylor.

In 1984, Kazhdan and Patterson constructed what are now called the exceptional representations of metaplectic $r$-fold covers of $GL_n$. These representations, which include the Weil representation when $r=n=2$, have seen numerous applications; most spectacularly, when $r=2$ and $n$ is arbitrary, they appear in the integral representation of the symmetric square $L$ function on $GL_n$. Unfortunately they are somewhat difficult to work with in practice, even for $r=2$ and $n>2$. In this talk, I will explain how, for $r=2$ and $n=2q$, these representations can be described in terms of a model space akin to the Schrödinger model of the Weil representation. I will also explain some of my motivations for this problem, and some possible applications of this result.

I will discuss a new construction of Eisenstein cohomology classes on GL_N first introduced by Nori and Sczech. In our approach the corresponding cocycles appear as regularised theta lifts for the dual pair (GL_1,GL_N). This suggests interesting generalisations of this construction by considering more general pairs (GL_k, GL_N), and I will present some calculations regarding this generalisation. Joint work (in progress) with Nicolas Bergeron, Pierre Charollois and Akshay Venkatesh.

The category of mixed Hodge-Tate structures over Q is a mixed Tate category of homological dimension one. By Tannakian formalism, it is equivalent to the category of graded comodules of a commutative graded Hopf algebra. In my recent joint work with A. Goncharov, we give a canonical description A (C) of the Hopf algebra. Such construction can be generalized to A (R) for any dg-algebra R with a Tate line.

We study the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. In the first half of my talk, I will give a lower bound on the number of such characters through a geometric interpretation. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. In the second half of the talk, I will discuss joint work with Ellenberg and Shusterman proving as the size of the constant field grows to infinity, the set of L-functions vanishing at the central point has 0 density.

A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to l-independence of l-adic cohomology of varieties over Henselian valuation fields, possibly of higher rank. This is joint work with Qing Lu.

Deligne--Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. In this talk, we discuss recent progress towards geometric realizations of representations of p-adic groups in three arcs: affine flag varieties, semi-infinite flag varieties, and a finite-ring variant arising from the Moy--Prasad filtration of parahoric subgroups. This is joint work with Alexander Ivanov.

The Ichino-Ikeda conjecture expresses the central value of certain automorphic L-functions in terms of the square of an automorphic period. It is a higher rank generalization of a well-known theorem of Waldspurger on toric periods and is closely related to the global Gan-Gross-Prasad conjecture. In this talk, I am going to explain how explicit Plancherel decompositions of certain symmetric spaces allow to complement previous work of W.Zhang to establish this conjecture for unitary groups under a mild local assumption. Time permitting, I will also explain another application of these Plancherel decompositions to a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of discrete series to adjoint gamma factors.