Automorphic Forms and Arithmetic

We will present a new notion of isogeny between ‘p-divisible groups with additional structure’ that employs the cohomological stacks of Drinfeld and Bhatt-Lurie—-in particular the theory of apertures developed in prior work with Gardner—-and combines it with some invariant theoretic tools familiar to the geometric Langlands and representation theory community, namely the Vinberg monoid and the wonderful compactification. This gives a uniform construction of p-Hecke correspondeces and Rapoport-Zink spaces associated with unramified local Shimura data. In particular, we give the first general construction of RZ spaces associated with exceptional groups. This work is joint with Si Ying Lee.

In the quantum theory of angular momentum, the Racah--Wigner coefficient, often known as the 6-j symbol, is a numerical invariant assigned to a tetrahedron with half-integer edge-lengths. The 6 edge-lengths may be viewed as representations of SU(2) satisfying certain multiplicity-one conditions. One important property of the 6j symbol is its hidden symmetry outside the tetrahedral ones, originally discovered by Regge. In this talk, we explore a generalized construction, dubbed tetrahedral symbol, in the context of rank-1 semisimple groups over local fields, and explain how the extra symmetries may be explained by relative Langlands duality. Joint work with Akshay Venkatesh.

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin--Selberg, with applications to the Bloch--Kato conjecture and to Iwasawa theory. For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Bloch--Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.

Since the celebrated work of Wiles, Taylor--Wiles on the connection between modular forms and elliptic curves through Galois representations, studying the link between these objects has been a central subject in number theory. In such a correspondence, the weight of modular forms is matched with the Hodge--Tate weight of Galois representation. Similarly, Serre weight conjectures explain how weights of mod p automorphic forms are related to Galois representations. In this talk, I will introduce Serre weight conjectures, and we discuss a proof of the conjecture for $\mathrm{GSp}_4$ under technical assumptions using a novel geometric result on Galois deformation rings. If time permits, we also explain an application to modularity lifting for $\mathrm{GSp}_4$. This is based on a joint work with Daniel Le and Bao Viet Le Hung.

I will discuss recent results for algebraicity of critical values of Spin L-functions for $\mathrm{GSp}_6$. I will also discuss ongoing work toward the construction of p-adic L-functions interpolating these values. I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. This is joint work with Giovanni Rosso and Shrenik Shah. All who are curious about this topic are welcome at this talk, even without prior experience with Spin L-functions.

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on "everything" related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring - the stable Bernstein center - which should be intimately related with the local Langlands correspondence. However, while it's easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue's theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I'll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G.

a conjecture of Fargues and Scholze, inspired by the geometric Langlands program, upgrades the local Langlands correspondence for a reductive group G over p-adic field F to an equivalence of categories between a certain category of l-adic sheaves on the stack $\mathrm{Bun}_G$ (generalizing the category of smooth representations of G) and a category of coherent sheaves on the stack of L-parameters. In this talk I will explain how to compute and match the K-theory of the categories involved in the case of $\mathrm{GL}_n$. This includes the computation of the K-theory of the fibers of the stack of L-parameters over its coarse moduli space in a way that is inspired by the Berstein-Zelevinsky classification of irreducible $\mathrm{GL}_n(F)$ representations.

In his tensor factorization of the completed cohomology of the modular tower (with Z_p-coefficients), Emerton observed that the local factors corresponding to primes \ell different from p could be uniquely characterized, up to isomorphism, by a short list of properties relating such a factor to the associated local Galois representation (in particular, such factors satisfied a form of Ihara's lemma and agreed generically with the representations of GL_2 associated to the local Galois representation via local Langlands.) In later joint work Emerton and I proved that, given an n-dimensional local Galois representation, there was at most one family of representations of GL_n over the same base that satisfied a similar set of desiderata; such families were later shown to always exist by joint work of mine with Gil Moss. We show that an analogous construction can be made for any quasi-split group over a local field F, and explain a (conjectural) connection between this construction and the categorical local Langlands correspondence. This is joint work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

Fargues conjectured the existence of eigensheaves for discrete L-parameters. I will explain an extension of this conjecture to A-parameters, based on the constructon in the specral side of the categorical local Langlands. I will also mention a related notion of cuspidal coherent sheaves. I will recall the categroical conjecture and discuss the example of GL(2) in detail.

The Kronecker limit formula is an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subheading terms of certain Artin L-functions. In this talk we will formulate a “non-Artinian” generalization of (averaged) Colmez conjecture, relating the following two quantities:

(1) the Faltings height of certain arithmetic Chow cycles on unitary Shimura varieties, and

(2) the sub-leading terms of the adjoint L-functions of (cohomological) automorphic representations of unitary groups U(n).

The $n=1$ case of our conjecture recovers the averaged Colmez conjecture. We are able to prove our conjecture when $n=2$ using a relative trace formula approach that can be formulated for the general $n$.

The “arithmetic relative Langlands” philosophy suggests that there are a lot of other similar (conjectural) phenomena involving subleading terms of L-functions and Faltings-like heights.

Joint work with Ryan Chen and Weixiao Lu.

The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will first review the basic theory of local models for Iwahori level, concentrating on the general linear and general symplectic group cases. The main goal will be to explain what can be said about local models when the level structure is $\Gamma_1(p)$, which is slightly deeper than Iwahori level. For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order $p$, and then I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.

In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of $L$-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of $L$-functions at certain integers. Of these, Stark’s Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde and other collaborators on these topics. The first is a proof of the Brumer-Stark conjecture. The second is the proof of an exact formula for Brumer-Stark units that had been conjecturally developed over the past 20 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a $p$-adic solution to the question of explicit class field theory for these fields.

All number theorists learn that the sum $1+2q+2q^4+2q^9+...$ is a modular form of weight 1/2. The usual proof that is taught uses the Poisson summation formula; but can one prove it in a purely algebraic way?

I will discuss this a little bit to put us in the right frame of mind. Then I will describe an algebraic theory of indefinite theta functions, i.e., theta functions attached to indefinite quadratic forms, which uses related ideas in its development.

This is all joint work with Kenz Kallal (Princeton).

In this talk, we will describe a generating series or arithmetic CM cycles in unitary Shimura varies, which is expected to be modular. Next, we describe a possible `arithmetic relative trace formula’ with one side being the arithmetic intersection between our generating series and a known arithmetic theta series of arithmetic divisors, and the other side is a derivative of `theta integrals’ which is known to be modular. So the identity gives some evidence of the modularity conjecture. Finally proof of the conjectured identity is reduced to (arithmetic) fundamental lemma and transfers, a lot of which was known to Wei Zhang and others. We propose a fundamental lemma (and prove it too) and an arithmetic fundamental lemma at infinity. This is a preliminary joint work with A. Mihatsch and S. Sankaran.

Gauss’s celebrated conjecture on the class number one problem for imaginary quadratic fields was finally settled by Heegner, Baker and Stark in the middle of the 20th century. An important idea of Goldfeld, combined with the formula of Gross and Zagier forc an elliptic curve of rank $3$, then led to an effective method for determining all imaginary quadratic fields of small class number. It is believed that there are infinitely many real quadratic fields of class number one, but analogues of the class number one problem have been formulated for subfamilies with small regulators (like the real quadratic fields of Richaud Degert type). I will describe a conditional approach to this class number one problem based on the theory of Stark-Heegner points, which exploits a conjectural variant of the Gross Zagier formula for certain elliptic curves of rank $\ge 2$.