We give an overview of a general trend of results that say that the distribution of the number of F_q-points of certain families of curves of genus g is asymptotically given by a sum of q+1 independent, identically distributed random variables as g goes to infinity. In particular, we discuss the distribution of the number of F_q-points for cyclic l-covers of genus g. (This is joint work with Bucur, David, Feigon, Kaplan, Ozman, Wood.) This work generalizes previous results in which only connected components of the moduli space were considered.

I will begin with an introduction to Galois deformation theory and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss Kisin's work on flat deformations and explain how to generalize to Galois representations valued in an arbitrary reductive group. Kisin's techniques led to the successful determination of the connected components of the flat deformation ring in the 2-dimensional case. If time permits, I will touch on difficulties of going beyond GL_2.

In this talk, I will introduce the famous result by Gangl-Kaneko-Zagier about a family of period polynomial relations among double zeta value of even weight. Then I will generalize their result in various ways, from which we can see the appearance of modular forms in low levels. At the end, I will give a generalization of the Eichler-Shimura-Manin correspondence to the case of the space of newforms of level 2 and 3 and a certain period polynomial space.

Colmez conjectured a formula relating periods of abelian varieties with complex multiplication to derivatives of Artin L-functions. IÂ’ll explain how to prove an averaged version of ColmezÂ’s conjectural formula, using the arithmetic of integral models of orthogonal Shimura varieties. This is joint work with F. Andreatta, E. Goren, and K. Madapusi Pera.

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.

Rankin-Selberg convolution L-functions are important functions in number theory. Their properties play a central role in many of deepest works on the Ramanujan-Petersson Conjecture. In a recent paper, Hoffstein and Hulse defined generalizations of these L-functions, the so-called "shifted-convolution" L-functions. They obtained the meromorphic continuation of the functions in many cases. Here we consider symmetrizations of these L-functions, and we exactly evaluate their special values at diagonal weights for all shifts. This is joint work with Michael Mertens.

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.

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest.

I will speak about recent progress on the enumeration of number fields, with particular attention to joint work with Taniguchi, which proved the existence of a negative secondary term in the counting function for cubic fields by discriminant. Among other results, we also found surprising biases in arithmetic progressions -- e.g., cubic field discriminants are more likely to be 5 (mod 7) than 3 (mod 7). Our work applies the analytic theory of the Shintani zeta function, which I will describe briefly. I will also discuss other approaches to related questions (and in particular an independent, and different, proof of the secondary term due to Bhargava, Shankar, and Tsimerman), using approaches as diverse as the geometry of numbers, algebraic geometry, and class field theory.

The prime numbers are the multiplicative building blocks of the integers, and much thought has been given towards understanding their behavior. In this talk, we will examine prime numbers from two points of view. We will first consider questions on the distribution of the primes. In particular, we will illustrate how the breakthrough work of Maynard and Tao on bounded gaps between primes settles an old problem of Erdos and Turan. Secondly, we will explore the relationship between prime numbers and zeros of the Riemann zeta-function, as a way to motivate the study of the moments of the Riemann zeta function and more general L-functions. In particular, we consider arbitrary products of L-functions attached to irreducible cuspidal automorphic representations of GL(m) over the rationals. The Langlands program suggests essentially all L-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive automorphic L-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive automorphic L-functions twisted by quadratic Dirichlet characters.

By studying universal motivic unipotent representations of fundamental group of varieties and comparing their different realizations, we combine Kim's recent method in Diophantine geometry with Deligne-Goncharov's theory of motivic fundamental groups to develop a machinery for approaching Diophantine problems concerning integral points.

Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

Higher theta functions are the residues of Eisenstein series on covers of the adelic points of classical groups. On the one hand, they generalize the Jacobi theta function. On the other, their Whittaker-Fourier coefficients are not understood, even for covers of $GL_2$. In this talk I explain how, using methods of descent, one may establish a series of relations between the coefficients of theta functions on different groups. In the first instance, this allows us to prove a version of Patterson's famous conjecture relating the Fourier coefficient of the biquadratic theta function to quartic Gauss sums. This is based on joint work with David Ginzburg.

Shnirelman discovered in the 1970s that the eigenfunctions of the Laplacian on a compact Riemannian manifold whose flow is ergodic with respect to Liouville measure exhibit an analogue of classical ergodicity at the quantum level. This phenomenon became known as "Quantum Ergodicity" and Schnirelman's proof was completed by Zelditch and Colin de Verdiere in the 1980s. Following a brief introduction to the subject, I will show that Quantum Ergodicity can also hold in systems which are essentially integrable, provided they have some arithmetic structure. Finally, in the absence of such an arithmetic structure, a very different phenomenon occurs: scarring. This talk is based on joint work with Par Kurlberg, KTH Stockholm.

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.

In the late 80Â’s Henniart used the then recently introduced Laumon l-adic local Fourier transform to prove the numerical local Langlands correspondence for GL(n). More recently, Bloch-Esnault and independently Lopez have developed a complex version of this transform. I will explain the fascinating picture that emerges when HenniartÂ’s strategy of proof is translated to this setting of local geometric Langlands parameters.

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.

For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.

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 will describe new algorithms to compute an explicit finite simplicial model for compact, congruence locally symmetric spaces and Hecke actions thereon. Joint work with Aurel Page.

The Malle-Bhargava heuristics give asymptotic predictions for the density of number fields of bounded discriminant with a given Galois group G, in terms of the number of G-extensions of p-adic fields Q_p. These heuristics can also be applied when the discriminant is replaced by any of a wide variety of other “counting functions”. I’ll discuss how some of these alternate counting functions are built, the idea of global mass formulas, and some cases in which the heuristic predictions can be compared to known results.

I will build a commutative ring that satisfies "standard conjectures", starting from a finite field. What is this ring? What does it say about the finite field? This talk will be elementary: No background beyond the first year graduate algebra will be necessary. Joint with Mats Boij, Bill Huang, and Greg Smith.

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.