Seminars Sorted by Series

I will begin this talk by reviewing the Eichler-Shimura isomorphism between the space of cusp forms of weight k and level N, and a certain cohomology group. Shimura called this cohomology group as parabolic cohomology. In the context of automorphic...

In this talk we present explicit bounds for the Weil height and the number of integral points on classical surfaces first studied by Clebsch (1871) and Klein (1873). Building on Hirzebruch's work in which he related these surfaces to a Hilbert...

The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation $\rho:Gal_Q\rightarrow GL_2(\overline{Q}_p)$ is modular if it is unramified outside finitely many places and de Rham...

Weissauer proved using the theory of endoscopy that the Galois representations associated to classical modular forms of weight two appear in the middle cohomology of both a modular curve and a Siegel modular threefold. Correspondingly, there are...

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams.

Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L...

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...

In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula...

I will first review what we know about the toroidal and minimal compactifications of Shimura varieties and their integral models, and the well-positioned subschemes of these integral models.  Then I will explain some p-adic analogues of Harris and...

The Tate conjecture predicts that in many instances, Langlands functoriality should be given by algebraic cycle classes. In previous joint work with Ichino, we showed that the Jacquet-Langlands correspondence for cohomological modular forms on GL(2)...

When an algebraic variety over the rational numbers contains infinitely many rational points, we may study their distribution. In particular, for Fano varieties, the asymptotic behavior of the number of rational points of bounded height is predicted...

I will explain how to factor the system of Beilinson-Kato elements as a product of two modular symbols (an algebraic avatar of the Rankin-Selberg formula).  This is joint work with Shanwen Wang.

We discover a surface related to the pair correlation of zeros of the Riemann zeta function. We make a conjecture on the shape of the surface and present partial results and numerical evidence towards the conjecture. This is joint work with Debmalya...

Given a compact Riemann surface, a classical theorem of of Narasimhan and Seshadri characterize vector bundles arising from unitary representations of the fundamental group as the polystable vector bundles of degree 0. Given a projective curve with...

In the early 2000’s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(Z)...

I will talk about several results on Hecke algebras attached to Bernstein blocks of (arbitrary) reductive p-adic groups, where we construct a local Langlands correspondence for these Bernstein blocks. Our techniques draw inspirations from the...

We count squarefree numbers in short intervals [X, X+H] for H > X^{1/5 - $\delta$}, where $\delta$ > 0 is some absolute constant. This improves on the exponent 1/5 shown by Filaseta and Trifonov in 1992. 

In improving bounds on the number of...

In 2001, Conrey, Farmer, Keating, Rubinstein, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of L-functions. This heuristic was later extended by Andrade and Keating to include...

Let π be a cuspidal automorphic representation of Sp_2n over Q which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is...

In my recent work on a geometric proof of the endoscopic fundamental lemma for spherical Hecke algebras, there are many new features not present in its Lie algebra analogue originally proved by B.C.~Ng\^o. One of such new features is an asymptotic...

I will introduce Apollonian circle packings, and describe the local-global conjecture, which predicts the set of curvatures of circles occurring in a packing. I will then describe reciprocity obstructions, a phenomenon rooted in reciprocity laws...

In joint work with Jacob Tsimerman we study when the primitive of a given algebraic function can be constructed using primitives from some given finite set of algebraic functions, their inverses, algebraic functions, and composition. When the given...

Motivated by work of Cantat-Dujardin, we study orbits by translations in K3 surfaces with two elliptic fibrations. We prove in particular that all orbits are infinite away from a proper Zariski-closed subset.

Among the tools, beyond the Pila-Wilkie...

We will describe emerging understanding of the structures related to the arithmetic of Zeta and Multizeta values for function fields through various results and conjectures.

In a recent joint work with S. Iyengar, C. Khare and J. Manning, we use their notion of congruence modules in higher codimension to give a new construction of the bottom class of the rank d=[F:\Q] Euler system attached to nearly ordinary Hilbert...

Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural...

It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight one, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a "derived"...

In this talk, I will report a recent joint work with Shouwu Zhang about a generic positivity of the Beilinson-Bloch height for the Gross-Schoen and Ceresa cycles of curves of genus at least 3. We also construct a Zariski open dense subset U of the...

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive.  

The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to...

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...

I will first discuss the notion of automorphic representations on a unitary group that are $P$-ordinary (at $p$), where $P$ is some parabolic subgroup. In the “ordinary” setting (i.e. when $P$ is minimal), such a representation $\pi$ has a...

In the 1940s Mahler initiated the program of determining the bass note spectrum $$\mathrm{Spec}(P):=\left\{\inf_{\underline{x} \in\Lambda \setminus \underline{0}}\left\vert P(\underline{x})\right\vert, \Lambda \subset \mathbb{R}^k \text{ a...

The Cohen-Lenstra heuristics are influential conjectures in arithmetic statistics from 1984 which predict the average number of p-torsion elements in class groups of quadratic fields, for p an odd prime. So far, this average number has only been...

We prove an effective equidistribution theorem for semisimple
closed orbits on compact adelic quotients. The obtained error depends
polynomially on the minimal complexity of intermediate orbits and the
complexity of the ambient space. As an application...

Are there infintely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative? Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests...

In the 1960s, Kirillov’s orbit method provided a striking correspondence between irreducible representations of a Lie group $G$ and certain geometric objects called coadjoint orbits. In 2021, Nelson and Venkatesh profitably adapted this method to...

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several...

We resolve Manin's conjecture for all Châtelet surfaces over $Q$ (surfaces given by equations of the form $x^2 + ay^2 = f(z)$) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is...

Quaternionic modular forms (QMFs) are a type of non-holomorphic automorphic function that exist on certain forms of the exceptional groups, and on orthogonal groups SO(4,n) with n at least 3.  They have a robust notion of Fourier coefficients...

We show that for every quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. By work of Shlapentokh, this implies that Hilbert's tenth problem over the ring of...

I will discuss recent results for algebraicity of critical values of Spin L-functions for 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...

I will discuss joint work in progress with Will Sawin on function field multiple Dirichlet series constructed based on a set of axioms from algebraic geometry. These series have applications to moments of Dirichlet L-functions for characters of...

A construction of Thurston assigns a hyperbolic 3-manifold to any polyhedron; a natural question is: which such are arithmetic? We report on ongoing work aiming to answer this question. 

I will talk about a unification of two bounded height results around abelian varieties. One is due to Silverman from 1983, which states, for an abelian scheme A/C with no fixed part over a curve C, that the set of points on C where the generic...

How many algebraic integers of bounded height have a minimal polynomial with a given Galois group? One approach to this problem is via Malle's conjecture. In this talk we will discuss an alternative approach using a construction with the Galois...

In recent work with Hajir, Larsen and Maire, we have proved that a large class of finitely generated pro-p groups G can be realized as tamely ramified extensions of a number field K, though ramification at an infinite number of primes is required...

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit...

Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. In this lecture I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis)...

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...

Central predictions of arithmetic quantum chaos such as the Quantum Unique Ergodicity conjecture and the sup-norm problem ask about the mass distribution of automorphic forms, most classically in terms of their weight or Laplace eigenvalue (for...

I will give an overview of recent progress in random multiplicative functions (random models for multiplicative functions) and their connection to the study of the Fyodorov-Hiary-Keating conjecture and Polya's question on nonnegative character sums...