In this talk, we discuss the distributions of class groups of orders in number fields. We explain how studying such distributions is related to counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We introduce two new methods to solve this counting problem, and as an application, we demonstrate how to determine the average size of the 2-torsion in the class groups of cubic orders. Much of this work is joint with Arul Shankar, Artane Siad, and Ila Varma.

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.

In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao established that, for a given number field $K$ with a degree $n\geq 5$, the size of the $2$-torsion is bounded by $h_{2}(K) \ll D^{\frac{1}{2}-\frac{1}{2n}}$, where $D_{K}$ is the discriminant of $K$ over $\mathbb{Q}$. In this presentation, we will introduce new bounds that take into account the geometry of the lattice underlying the ring of integers of $K$. This research is a joint project with Pierre Le Boudec.

The twisted Gan--Gross--Prasad conjectures consider the restriction of representations from GL_n to a unitary group over a quadratic extension E/F. In this talk, I will explain the relative trace formula approach to the global twisted GGP conjecture. In particular, I will discuss how the fundamental lemma that arises can be reduced to the Jacquet--Rallis fundamental lemma, which allows us to obtain the global twisted GGP conjecture under some unramifiedness assumptions and local conditions.

In the first part of the talk I will describe a general context which, in some specific situations, permits us to give a cohomological interpretation to the Langlands-Shahidi theory of L-functions. In the second part of the talk, I will specialize to the context of the general linear group over a totally imaginary base field F, and discuss some recent results of mine on the special values of Rankin-Selberg L-functions for GL(n) x GL(m) over such an F. The talk is based on my preprint: https://arxiv.org/abs/2207.03393

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.

We ask the question, "how does the infinite q-Pochhammer symbol transform under modular transformations?" and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This "Shintani-Faddeev modular cocycle" is an SL_2(Z)-parametrized family of functions generalizing Shintani's double sine function and Faddeev's noncompact quantum dilogarithm. We relate real multiplication values of the Shintani-Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.

Physicists Green, Russo, and Vanhove have discovered solution to differential equations involving automorphic forms appear at the coefficients to the 4-graviton scattering amplitude in type IIB string theory. Specifically, for \Delta the Laplace-Beltrami operator and E_s(g) a Langlands Eisenstein series, solutions f(g) of (\Delta-\lambda) f(g) = E_a(g) E_b(g) for a and b half-integers on certain moduli spaces G(Z)\G(R)/K(R) of real Lie groups appear as coefficients to the analytic expansion of the scattering amplitude. We will briefly discuss different approaches to finding solutions to such equations and focus on a shifted convolution sum of divisor functions which appears as the Fourier modes associated to the homogeneous part of the solution. Initially, it was thought that, when summing over all Fourier modes, the homogeneous solution would vanish but recently we have found an exciting error term. This is joint work with Stephen D. Miller, Danylo Radchenko and Ksenia Fedosova.

Given a reductive group 𝐺 and some auxiliary data, one has the Hitchin fibration associated with the adjoint action of 𝐺 on Lie(𝐺), which is successfully used by B. C. Ngô to prove the endoscopic fundamental lemma for Lie algebras. Following the same idea, there is a group analogue called the multiplicative Hitchin fibration by replacing the Lie algebra with reductive monoids, and one can hope to directly prove the fundamental lemma at group level. This project is almost complete and we report the results so far. There are many new features that are not present in the additive case, among which is a pleasant surprise that there might be some strata in the support theorem that are not explained by endoscopy.

Let $F$ be a number field with ring of adeles $\mathbb{A}_F$. Let $r_1,r_2,r_3$ be a triple of positive integers and let $\pi:=\otimes_{i=1}^3\pi_i$ where the $\pi_i$ are all cuspidal automorphic representations of $\mathrm{GL}_{r_i}(\mathbb{A}_F)$. We denote by $L(s,\pi, \otimes^3)=L(s, \pi_1\times \pi_2 \times \pi_3)$ the corresponding triple product $L$-function. It is the Langlands $L$-function defined by the tensor product representation $\otimes^3:{}^L(\mathrm{GL}_{r_1} \times \mathrm{GL}_{r_2} \times \mathrm{GL}_{r_3}) \to \mathrm{GL}_{r_1r_2r_3}(\mathbb{C})$. In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie.

Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.

The classification of discrete series is one main subject in Langlands program with numerous applications. We first explain the result on the classification of discrete series of odd GSpin groups, generalizing the Mœglin-Tadi ́c classification for classical groups. Note that our approach will give alternate proof for classical groups. This is a joint work with Ivan Mati ́c. We also explain its application on the generic local Langlands correspondence via Langlands-Shahidi method. If time permits, we will explain possible generalization of those to other groups, which is work in progress

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.

A well-known theorem of Waldspurger relates central values of automorphic L-functions for GL(2) to automorphic period integrals over non-split tori. His result was reproved by Jacquet via the comparison of relative trace formulae. Guo-Jacquet�s conjecture aims to generalise Waldspurger�s result as well as Jacquet�s approach to higher dimensions. In this talk, we shall first recall the background of Guo-Jacquet trace formulae. Then we shall focus on an infinitesimal variant of these formulae and try to explain several results on the local comparison of most terms. Our infinitesimal study is expected to be relevant to the study of geometric sides of the original Guo-Jacquet trace formulae.

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.

Orbital integrals for the Lie algebra can be analyzed using the Hitchin fibration. In turn the Hitchin fibration can be analyzed via the morphism g^{reg} ----> g//G from the regular elements of the Lie algebra, to the GIT quotient by the adjoint action. In trying to generalize this story by replacing the action of G on g by the action of G on some sufficiently nice variety M, we must replace the GIT quotient with what we call the regular quotient. This talk will look at the reasons for this, and the difference between the GIT and regular quotients in the case of G acting on G by conjugation (when the derived group of G is not simply connected), G acting on the commuting scheme, and G acting on the Vinberg monoid.

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.

For a smooth projective variety X over an algebraic number field a conjecture of Bloch and Beilinson predicts that the kernel of the Abel-Jacobi map of X is a torsion group. When X is a curve, this follows by the Mordell-Weil theorem. In higher dimensions however there is hardly any evidence for this conjecture. In this talk I will focus on the case when X is a product of smooth projective curves and construct infinitely many nontrivial examples that satisfy a weaker form of the Bloch-Beilinson conjecture. This relies on a recent joint work with Jonathan Love.

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.

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).

The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.