# Number Theory

Organizers: Richard Taylor, Brian Conrad, Kannan Soundararajan, and Xinwen Zhu

## Upcoming Events

We care about arithmetic invariants of polynomial equations / motives e.g. conductors or L-functions, which (conjecturally) are often automorphic and related to cycles on Shimura varieties. In this talk, I will focus on L-functions of Asai motives (e.g. Rankin-Selberg motives for GL_n x GL_n)…

A fundamental problem in the arithmetic of varieties over global fields is to determine whether they have a rational point. As a first effective step, one can check that a variety has local points for each place. However, this is not enough, as many classes of varieties are known to…

## Past Events

Abstract: The arithmetic quantum unique ergodicity (AQUE) conjecture predicts that the L^2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”, and…

The general goal of Higher Hida theory is to define and understand the ordinary part of integral coherent cohomology of Shimura varieties. In this talk we will focus on the simplest example of a Shimura variety for a non-split reductive group. We describe the results, notably vanishing…

Automorphy lifting theorems establish situations in which Galois representations over \bar{Q_p} are automorphic if their residual representation has an automorphic lift. In 2018, Allen et. al. proved the first automorphy lifting theorem for n-dimensional Galois representations over a CM field…

I will report on my joint work in progress with Lue Pan which proves that the part of the rational p-adic completed cohomology of a general Shimura variety that is locally analytic with "sufficiently regular" infinitesimal weights is concentrated in the middle degree. I will begin with some…

Abstract

It has long been known how many integers are the sum of two squares, one of which is the square of a prime. However researchershave been frustrated in obtaining a good error term in this seemingly innocuous problem. Recently we discovered the reasons for this difficulty: …

Abstract

Let X be large and H also large but slightly smaller, and consider n ranging from 1 to X. For an arithmetic function f(n) like the k-fold divisor function, what is the best mean square approximation of f(n) by a restricted divisor sum (a function of the sort \sum_{d|n, d < H}…

Scholze has conjectured the existence of the so-called Igusa stacks, which have close relation to Shimura varieties. In my thesis and the joint work in progress with Daniels, van Hoften and Kim, these conjectural stacks are constructed for many interesting classes of Shimura varieties. In this…

Abstract: In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this…

Abstract: In 1989, Zelditch considered the trace of an invariant operator composed with a pseudo-differential operator. The resulting trace formula turned out to be extremely useful in studying the distribution of closed geodesics on hyperbolic surfaces.…

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of…