Number Theory

I will outline the proof of the (unramified, global) geometric Langlands conjecture, emphasizing the main intermediate results used along the way. The overall project is joint with Gaitsgory, Arinkin, Beraldo, Campbell, Chen, Faergeman, Lin, and Rozenblyum. I will also describe recent work on…

Clozel, Harris and Taylor proposed a generalized Ihara's lemma for definite unitary groups. In this talk, we prove some cases of their conjecture under the assumption of banal coefficients. The proof relies on the recent work of Hemo and Zhu on unipotent categorical local Langlands…

We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by the 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 estimating sums of Fourier coefficients of…

The study of coherent cohomology on (the special fiber of) Shimura varieties has various applications to arithmetic problems, such as congruences of automorphic forms, weight part of Serre's conjecture, and liftability of mod p automorphic forms. One of the basic problems is to prove certain…

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.…

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…