Number Theory

How do the finite/infinite dichotomy of the Killing–Cartan classification of simple Lie groups & algebras appear in arithmetic geometry? I will explain how this Lie-theoretic dichotomy is realized in the finiteness or infinitude of positive integer solutions to certain Diophantine equations…

We present a new structure on the first Galois cohomology of families of symplectic self-dual p-adic representations of $G_Qp$ of rank two. This is a functorial decomposition into free rank one Lagrangian submodules encoding Bloch-Kato subgroups and epsilon factors, mirroring an underlying…

We present results on the maximal dimension of compact subvarieties of the moduli space of abelian varieties and of moduli of complex curves of compact type. Equivalently, this is the maximal dimension of a compact complex parameter space for a maximally varying family of abelian varieties/…

In characteristic zero, Castelnuovo proved that a unirational surface is rational. In positive characteristic, this fails. We discuss the plethora of non-rational, often general-type, surfaces that are unirational in positive characteristic. In 1977, Shioda conjectured that these…

Let K/Qp be a finite unramified extension and r a crystalline representation of G_K. The problem of  finding all possibilities for the mod p reduction of r goes back to Serre. I will report on recent work with Bhargav Bhatt and Toby Gee, where we use prismatization to obtain some new…

 Cosine polynomials of the form f(x) = cos(a_1 x) + cos(a_2 x) + … + cos(a_n x)  appear extensively in number theory and combinatorics. An old problem of Ankeny and Chowla asks: if a_1 … a_n are distinct positive integers, how small must the minimum value of f(…

The celebrated abc conjecture asserts that every solution to the equation a+b=c in triples of coprime integers (a,b,c) must satisfy rad(abc) >= c^{1-\epsilon}, with finitely many exceptions. We prove a power-saving bound on the exceptional set of such triples.…

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit models and their ranks will be found using a 2-…

Abstract:   In this talk, we investigate rational solutions to systems of forms in many variables of different degrees. We begin by providing a history related to rational solutions to homogeneous polynomial equations. We then outline the motivation behind our work and highlight a…

A longstanding question in the theory of Shimura varieties concerns their perfectoidness at infinite level—a property that would reveal deep connections between étale and coherent cohomology. In this talk, we establish a criterion for perfectoidness via Sen theory, building on a new development…