Joint Berkeley-Stanford Algebraic Number Theory

 By Faltings's Theorem, formerly known as the Mordell
Conjecture, a smooth projective curve of genus at least 2 that is
defined over a number field K has at most finitely many K-rational
points. Votja later gave a second proof. Many authors, including
Bombieri, de Diego,…

In the last several years, it has been realized that the local Langlands correspondence has a categorical formulation, with variants given independently by Emerton and Helm, Fargues and Scholze, Hellmann, and Zhu. All of these approaches essentially postulates an equivalence of categories, with…

(Joint work with Frank Calegari and Vesselin Dimitrov.) The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some…

Interesting moduli spaces don't have many integral points.  More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than H^{\…

We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group.  We present a proof of a subconvex bound in the t-aspect.  More generally, we address the spectral aspect in the case of uniform parameter growth.

These results are…

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis)
concerning the bias of cubic Gauss sums.
This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by…

Investigating the p-adic integration map constructed by J.-M. Fontaine during the 90's, which is the main tool for proving the Hodge--Tate decomposition of the Tate module of an abelian variety over a p-adic field, we realized that the group of p-adic points of the above-named abelian variety,…

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$, discriminant bounded by $X$, and Galois closure $S_n$. For $C$ a fixed curve given by an affine equation $y^m = f(x)$ where $m \geq 2…

Abstract: Igusa varieties are smooth varieties in characteristic p arising naturally as étale covers of certain subvarieties (central leaves) of Shimura varieties, for example of the ordinary locus of the modular curve. Igusa varieties over the (mu)-ordinary locus of a Shimura variety are used…

The absolute trace of a totally positive algebraic integer is defined to be the average of its conjugate elements in $\mathbb{R}$. We show that there are infinitely many totally positive algebraic integers of absolute trace at most $1.81$ but only finitely many of absolute trace at most $1.80$.…