Number Theory

Abstract:   I will describe recent work joint with Olga Balkanova and Dmitry Frolenkov on a restricted divisor function and its associated divisor problem. This problem shares properties with the usual Dirichlet divisor problem  and the Hardy-Littlewood problem to count lattice points…

Abstract: Suppose that n is 0 or 4 mod 6. We show that there are infinitely many primes of the form p^2 + nq^2 with both p and q prime, and obtain an asymptotic for their number. In particular, when n = 4 we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec.

Joint w. Ben…

Given an algebraic curve defined over a number field, one can define the Néron-Tate height on the Jacobian and prove its positivity. This height pairing and its positivity play important roles in the proof of the Mordell-Weil theorem, in Vojta's proof of the Mordell conjecture, and in the…

The fastest known deterministic algorithms for factorising polynomials in F_p[x] have a worst-case runtime that is exponential in log p. We will discuss a new deterministic algorithm which factorises an integer polynomial modulo many primes simultaneously with amortised runtime that is…

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…