This is the website for the Caltech Number Theory Seminar organized Matthias Flach, Yu Fu, Elena Mantovan, and Roy Zhao (on leave). This webpage is maintained by Yu Fu.
The seminar is scheduled to meet at 4:00-5:00 PM on Thursdays in 387 Linde Hall
Titles and abstracts are also available on the Caltech Calendar
Anticyclotomic Iwasawa theory for newforms at Eisenstein primes
Anticyclotomic Iwasawa theory for elliptic curves at Eisenstein case was first studied in the recent work of Castella–Grossi–Lee–Skinner. We will talk about generalizations of their results to newforms of arbitrary weights, meanwhile removing their techinical assumptions. Using Hida arguments, one could also deduce weight 2 results for multiplicative reduction. The applications include new cases of p-part BSD formula, p-converse theorems and improvements in arithmetic statistics. This is joint work with Timo Keller.
Relative Trace Formula and Uniform Non-vanishing of Hilbert Modular L-values
Anticyclotomic Iwasawa theory for elliptic curves at Eisenstein case was first studied in the recent work of Castella–Grossi–Lee–Skinner. We will talk about generalizations of their results to newforms of arbitrary weights, meanwhile removing their techinical assumptions. Using Hida arguments, one could also deduce weight 2 results for multiplicative reduction. The applications include new cases of p-part BSD formula, p-converse theorems and improvements in arithmetic statistics. This is joint work with Timo Keller.
Towards an explicit Bezrukavnikov’s equivalence in mixed characteristic
Let G be a complex reductive group. A celebrated theorem of Kazhdan-Lusztig establishes an isomorphism between the extended affine Hecke algebra of G and certain equivariant K-group of the Steinberg variety of the Langlands dual group of G. This isomorphism plays a crucial role in Kazhdan-Lusztig’s proof of the Deligen-Langlands conjecture. In the equal characteristic setting, Bezrukavnikov studied the categorification of this isomorphism and proved an equivalence between two geometric realizations of the affine Hecke algebra, which can be seen as the tamely ramified local geometric Langlands correspondence. In mixed characteristics, Bando and Yun-Zhu independently used implicit approaches to derive Bezrukavnikov’s equivalence from the equal characteristic setting. In this talk, I will discuss an explicit approach to establish Bezrukavnikov’s equivalence building on a previous joint work with Anschütz, Lourenço, and Wu. This talk is based on an ongoing project with Bando, Gleason, and Lourenço.
Computing crystalline deformation rings via the Taylor-Wiles-Kisin patching method
A celebrate Crystalline deformation rings play an important role in Kisin’s proof of the Fontaine-Mazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.
Degree $d$ points on curves
Given a plane curve $C$ defined over $\mathbb{Q}$, when the genus of the curve is greater than one, Faltings’ theorem tells us that the set of rational points on the curve is finite. It is then natural to consider higher degree points, that is, points on this curve defined over fields of degree $d$ over $\mathbb{Q}$. We ask for which natural numbers $d$ are there points on the curve in a field of degree $d$. For positive proportions of certain families of curves, we give results about which degrees of points do not occur. This talk is based on joint work with Andrew Granville.
Dieudonné theory via cohomology of classifying stacks
Classically, Dieudonné theory offers a linear algebraic classification of finite group schemes and p-divisible groups over a perfect field of characteristic p>0. In this talk, I will discuss generalizations of this story from the perspective of p-adic cohomology theory (such as crystalline cohomology, and the newly developed prismatic cohomology due to Bhatt–Scholze) of classifying stacks. Time permitting, I will discuss some applications.
Diophantine problems arising from tetrahedra
From the humble right triangle stem two rich, classical Diophantine problems: Pythagorean triples and the Congruent Number Problem. Both of these involve the study of rational points on algebraic curves. I’ll discuss a higher-dimensional version of these problems leading to certain algebraic surfaces and their rational points. This is joint work with Dinakar Ramakrishnan.
Based on the simplest-github-page by Christopher Allen