I will propose a way to combine the theory of translation functors with the trace formula to study automorphic representations of connected semisimple anisotropic algebraic groups over the rational numbers whose Archimedean component is a limit of discrete series. I will explain the main ideas of the derivation of a trace formula which, modulo a conjecture on the decomposition of the tensor product of a limit of discrete series with a finite-dimensional representation into basic representations, allows to isolate the non-Archimedean parts of a finite family of C-algebraic automorphic representations containing the ones whose Archimedean component is a given limit of discrete series.

We investigate the $p$-adic monodromy of certain kinds of abelian varieties in $\mathcal{A}_{g}$ and prove a formal density theorem for the locus of deformations with big monodromy. Also, we prove that the small monodromy locus of the deformation space of a supersingular elliptic curve is $p$-adic nowhere dense. The approach is based on a congruence condition of $p$-divisible groups and transform of data between the Rapoport-Zink spaces and deformation spaces.

In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. In this talk we discuss the distributions (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is the more exotic Batman distribution.

Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. Ben-Zvi, Sakellaridis and Venkatesh conjecture that the local unramified part of this theory can be categorified to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. In this talk I will explain the proof of this equivalence and some of its properties. I will also discuss connections to conjectures of Braverman-Kazhdan on constructions of general automorphic L-functions. This is joint work with Tsao-Hsien Chen (in preparation).

Braverman and Kazhdan, L. Lafforgue, Ngo, and Sakellaridis have pursued a set of conjectures asserting that analogues of the Poisson summation formula are valid for all spherical varieties. If proven, these conjectures imply the analytic continuation and functional equations of quite general Langlands L-functions (and thus, by converse theory, much of Langlands functoriality). I will explain techniques for proving the conjectures in special cases that include the first known case where the underlying spherical variety is not a generalized flag variety.

Let F be a non-archimedean local field (such as ℚ_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.

Ribet’s method describes a way to construct a certain extension of fields from the existence of a suitable modular form. To do so, we consider the Galois representation of an appropriate cuspform, which gives rise to a cohomology class that cuts out our desired extension. The process of obtaining a cohomology class from such a representation is usually known as Ribet’s lemma. Several generalizations of this lemma have been stated and proved during the last decades, but the vast majority of them makes the assumption that the representation is residually distinguishable, meaning that the characters of its residual decomposition are non-congruent modulo the maximal ideal. However, recent applications of Ribet’s method, such as for the proof of the 2-part of the Brumer-Stark conjecture, have encountered the challenge that the representation we obtain does not satisfy this assumption. In my talk, I describe the limitations of the residually indistinguishable case and conjecture a new general version of Ribet’s lemma in this context, giving a proof in some particular cases.

In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.

Fourier coefficients of automorphic forms are invariants which encode the analytic and arithmetic properties of automorphic forms. In this talk, we introduce the general notion of Fourier coefficients for automorphic representations of reductive groups and use them to construct explicit endoscopy correspondences, which construct via integral transforms with automorphic kernel functions members in global Arthur packets for classical groups. For instance, we will discuss with some details the recent work joint with Lei Zhang.

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

Recently A. Braverman, D. Kazhdan, and L. Lafforgue have interpreted Langlands' functoriality in terms of a generalized harmonic analysis on reductive groups that requires the existence of new spaces of functions and an associated, generally non-linear, involutive Fourier transform. This talk will discuss some of these objects involved in the local p-adic situation, after introducing some ideas and basic constructions involved. Specifically, the local Fourier transforms have a nice interpretation in terms of their spectral decomposition giving the gamma factors that appear in functional equations of L functions, which, in the standard case allows one to write down the epsilon factors attached to supercuspidal representations as non-abelian Gauss sums. For G=GL(2), we use the local Langlands correspondence to provide L and epsilon factors for odd adjoint power transfers and use this to interpret the Adjoint power Fourier-transform such that its spectral decomposition on supercuspidal representations is given explicitly by certain non-abelian Kloosterman sums, which we use to give a form of the Fourier operator.

Holomorphic modular forms on the Shimura variety S(G) attached to the reductive group G can be interpreted naturally as sections of automorphic vector bundles: locally free sheaves that can be defined analytically by exploiting the structure of a Shimura variety as a quotient of a symmetric space. The construction can also be made algebraic, and in this way one gets a canonical functor from the tensor category of representations of a certain Levi subgroup K of G to the tensor category of vector bundles on S(G), and thus a homomorphism from the representation ring of K to K_0(S(G)). When S(G) is compact we determine how the image of this homomorphism behaves under Chern characters to Deligne cohomology and continuous l-adic cohomology. When S(G) is non-compact and of abelian type, we use perfectoid geometry to define Chern classes in the l-adic cohomology of the minimal compactification of S(G); these are analogous to the topological cohomology classes defined by Goresky and Pardon, using differential geometry. (Joint work with Helene Esnault.)

For a (squarefree) integer d the negative Pell equation is given by: X^2 - d Y^2 = -1. It is easy to see that this equation has no solution over the integers, if d is negative or d is congruent to 3 modulo 4. In this talk we would like to study the asymptotic behavior of integers d such that this equation is solvable. This question is related to the behavior of the class group of the quadratic field generated by a square root of d. The distribution of those class groups is described by the Cohen-Lenstra heuristics.

In the 1990s Moy and Prasad revolutionized p-adic representation theory by showing how to use Bruhat-Tits theory to assign invariants to p-adic representations. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.

Let G be the group of rational points of a linear algebraic group over a local field. A representation of G is distinguished by a subgroup H if it admits a non-zero H-invariant linear form. A Galois symmetric pair (G,H) is such that H=Y(F) and G=Y(E) where E/F is a quadratic extension of local fields and Y is a reductive group defined over F. In this talk we show that for a Galois symmetric pair, often the necessary condition for H-distinction of a parabolically induced representation, emerging from the geometric lemma of Berenstein-Zelevinsky, are also sufficient. In particular, we obtain a characterization of H-distinguished representations induced from cuspidal in terms of distinction of the inducing data. We explicate these results further when Y is a classical group and point out some global applications for Galois distinguished automorphic representations of SO(2n+1). This is joint work with Nadir Matringe.

So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture on the number of rational points of bounded anticanonical height on Fano varieties. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In this talk we are interested in subvarieties of biprojective space. There is not much known so far, unless the underlying polynomials are of bidegree (1,1) or (1,2). In this talk we present recent work which combines the circle method with the generalised hyperbola method developed by V. Blomer and J. Bruedern. This allows us to verify Manin's conjecture for certain smooth hypersurfaces in biprojective space of general bidegree.

I will present a formulation of equivariant Tamagawa number conjecture for flat smooth sheaves on separated schemes of finite type over a finite field. After sketching a proof of this I will give application to Chinburg’s conjectures in Galois module theory and tower of fields conjecture. If time permits I will also give an application towards equivariant BSD for abelian varieties defined over global function fields. This is a joint work with David Burns.

We express the bias of global root numbers of Hilbert new forms of cubic level via special values of Dedekind L-functions attached to CM extensions determined by the level. In particular, our formula includes the case when weight 2 appears. We establish the formula by 1) a limit form of Jacquet-Zagier trace formula on PGL_2 associated to certain not necessarily integrable test functions at Archimedean places (when weight 2 occurs), and 2) showing the meromorphic continuation of certain Dirichlet series with coefficients given by special value of Dedekind L-functions via spectral side of the Jacquet-Zagier trace formula. This is a joint work with Q. Pi and H. Wu. arXiv: 2110.08310.

In a 1997 Fields Medalist Maxim Kontsevich suggested that the function F(q) = 1 + (1-q) + (1-q)(1-q^2) + (1-q)(1-q^2)(1-q^3)+…, defined only for q a root of unity, is similar to certain functions arising from the computation of Feynman integrals in quantum field theory. In the last sixteen years this function has been connected to interval orders in decision making theory, ascent sequences and matchings in combinatorics, and Vassiliev invariants in knot theory. Don Zagier related the asymptotic properties of this function to the “half-derivatives” of modular forms and was led to define a notion of “quantum modular form”. In a trilogy of papers, my collaborators (Andrews, Bryson, Ono, Pitman, Zwegers) and I have connected this function to Ramanujan’s mock theta functions and the combinatorics of unimodal sequences. I will tell the story of this function and these many relationships.

Recently, Jiang made a conjecture investigating the connection between two fundamental invariants of an automorphic representation \pi appearing in the discrete spectrum of quasi-split classical groups G(A). The first invariant is the wave front of \pi, WF(\pi), which is the set of maximal unipotent orbits of G, such that \pi admits a non-trivial Fourier coefficients with respect to them. The second invariant is the Arthur parameter \psi of \pi to which one can associate a unipotent orbit \underline{p}(\psi) of the dual group of G. The conjecture says that in any Arthur packet associated to \psi, the Barbasch-Vogan duality of the orbit \underline{p}(\psi) is a sharp upper bound for the wave front of the representations of the packet. This is an important conjecture that vastly generalizes Shahidi's conjecture which claims that in every tempered packet there exists a generic representation. In this talk, I will review this conjecture and present some recent progress towards it. This is a joint work in progress with Dihua Jiang.

Rational maps from the Riemann sphere into itself can be described in terms of poles and principal parts; doing the same for maps into the flag manifold of a loop group gives insight into the topology of moduli of instantons and calorons. (joint work with Michael Murray)

We will give an explicit construction of the regular quotient of Morrissey-Ngô in the case of a symmetric pair. In the case of a quasisplit form (i.e. the regular centralizer group scheme is abelian), we will give a Galois description of the regular centralizer group scheme using parabolic covers. We will then describe how the nonseparated structure of the regular quotient recovers the spectral description of Hitchin fibers given by Schapostnik for U(n,n) Higgs bundles. This work is joint with B. Morrissey.

A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous ``circle problem'' and the other to the equally famous ``divisor problem.'' We discuss these classical unsolved problems. Each identity can be interpreted in three distinct ways. We discuss various methods that have been devised to prove the identities under these different interpretations. Weighted divisor sums naturally arise, and new methods for estimating trigonometric sums need to be developed. Trigonometric analogues and extensions of Ramanujan's identities to Riesz and logarithmic sums are discussed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.

We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of positive definite integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. If time permits, we may discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local-global principle for certain Kleinian sphere packings.

We discuss special value formulas for a p-adic L-function L_p(f, \chi), where f is a fixed newform and \chi varies over the space of Hecke characters of a fixed imaginary quadratic field, as well as some recent applications. These formulas, first found by Bertolini, Darmon, and Prasanna, relate L_p(f, \mathbb{1}), a value which is outside the range of interpolation defining L_p, to arithmetic invariants of cycles on varieties fibered over modular curves.

Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.