# Number Theory

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

## Upcoming Events

Igusa stacks are p-adic geometric objects that roughly parametrize abelian varieties up to isogeny. In a joint work with Daniels, van Hoften, and Zhang, we constructed Igusa stacks for Hodge type Shimura data, and discussed how its cohomology relates to the cohomology of Shimura varieties.…

## Past Events

Let k be a number field. We provide an asymptotic formula for the number of Galois extensions of k with absolute discriminant bounded by some X, as X tends to infinity. We also provide an asymptotic formula for the closely related count of extensions of k whose normal closure has…

Computer theorem provers (which know the axioms of mathematics and can check proofs) have existed for decades, but it's only recently that they have been noticed by mainstream mathematicians. Modern work of Tao, Scholze and others has now been taught to Lean (one of these systems), and (…

We shall explain how to generalize theorems of Polya and Andre in both a qualitative and (more importantly) quantitative way. As a consequence of our methods, we prove new irrationality results, including various products of logarithms as well as the Dirichlet L-value L(2,chi_{-3}). This is (…

In 1978, Apery shocked the mathematical world with an elementary but ingenious proof that zeta(3) is irrational. Despite a great deal of effort and expectation, these methods have not yielded the irrationality of any further Dirichlet L-value. I will revisit the proof and then explain why some…

Mordell (1922) proved that the rational points of an elliptic curve $E / {\bf Q}$ form a finitely-generated abelian group. It is still not known which finitely-generated abelian groups can occur as $E({\bf Q})$. Mazur (1977) proved that the possible torsion subgroups $T$ are the cyclic groups of…

We introduce a mod p analogue of the Mumford—Tate conjecture, which governs the p-adic monodromy of families of mod p abelian varieties. It turns out that the conjecture is closely related to a notion of formal linearity of mod p Shimura varieties. Surprisingly, the conjecture…

For large x and coprime a and q, the arithmetic progression n = a mod q contains approximately pi(x)/phi(q) primes up to x. For which moduli q is this a good approximation? In this talk, I will focus on results for smooth moduli, which were a key ingredient in Zhang's proof of bounded gaps…

I'll speak about joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance set if the distance between any pair of points in the…

Given an étale Z_p-local system of rank n on an algebraic variety X, continuous cohomology classes of the group GL_n(Z_p) give rise to classes in (absolute) étale cohomology of the variety. These characteristic classes can be thought of as p-adic analogs of Chern-Simons characteristic classes of…

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…