NTS ABSTRACTSpring2024

Back to the number theory seminar main webpage: Main page

Jan 25

Jason Kountouridis

The monodromy of simple surface singularities in mixed characteristic

Given a smooth proper surface X over a $p$-adic field, it is a generally open problem in arithmetic geometry to relate the mod $p$ reduction of X with the monodromy action of inertia on the $\ell$-adic cohomology of X, the latter viewed as a Galois representation ($\ell \neq p$). In this talk, I will focus on the case of X degenerating to a surface with simple singularities, also known as rational double points. This class of singularities is linked to simple simply-laced Lie algebras, and in turn this allows for a concrete description of the associated local monodromy. Along the way we will see how Springer theory enters the picture, and we will discuss a mixed-characteristic version of some classical results of Artin, Brieskorn and Slodowy on simultaneous resolutions.

Feb 01

Brian Lawrence

Conditional algorithmic Mordell

Conditionally on the Fontaine-Mazur, Hodge, and Tate conjectures, there is an algorithm that finds all rational points on any curve of genus at least 2 over a number field. The algorithm uses Faltings's original proof of the finiteness of rational points, along with an analytic bound on the degree of an isogeny due to Masser and Wüstholz. (Work in progress with Levent Alpoge.)

Feb 08

Haoyang Guo

Frobenius height of cohomology in mixed characteristic geometry

In complex geometry, it is known that the i-th cohomology of a variation of Hodge structures on a smooth projective complex variety has the weight increased by at most i. In this talk, we consider the mixed and the positive characteristic analogues of this fact. We recall the notion of prismatic cohomology and prismatic crystals introduced by Bhatt and Scholze, and show that Frobenius height of the i-th prismatic cohomology of a prismatic F-crystal behaves the same. This is a joint work with Shizhang Li.

Feb 15

Haoyang Guo

p-adic Gross--Zagier and rational points on modular curves

Faltings' theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings' theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross--Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross--Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross--Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points.