This is the website for the Caltech Number Theory Seminar organized Matthias Flach, Yu Fu, Elena Mantovan, Liyang Yang. 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
Nov 13 Kimball
| Date | Speaker | Topic | |
|---|---|---|---|
| October 2 | Qiao He (Columbia University) | Intersection of Hecke correspondences and a general conjecture | |
| October 9 | Liubomir Chiriac (Portland State University) | Arithmetic aspects of Hecke traces | |
| October 16 | Aaron Landesman (Harvard) | Malle’s conjecture over function fields | |
| October 23 | Katharine (Katy) Woo (Stanford University) | Manin’s conjecture for Châtelet surfaces | |
| November 6 | Robert Cass (Claremont McKenna College) | ||
| November 13 | Kimball Martin (University of Oklahoma and Osaka Metropolitan University.) | TBD | |
| November 20 | Vacant () | TBD | |
| December 4 | Claus Sorensen (UCSD) | TBD | |
| December 11 | Alex Dunn (Georgia Tech) | TBD |
| … | … | … |
Intersection of Hecke correspondences and a general conjecture
A classical and beautiful result of Gross-Keating relates the intersection of three Hecke correspondences on the integral model of $X_0(1)\times X_0(1)$ with derivative of certain Eisenstein series. Such result can be regarded as an example of arithmetic Siegel-Weil formula and can serve as an ingredient for an arithmetic Gan-Gross-Prasad formula. In this talk, we consider a variant of this formula for the self-product of $X_0(N)$ where $N$ is square-free and the self-product of Shimura curves respectively. Moreover, we explain how these results confirm a general conjecture for GSpin Shimura varieties with vertex levels. This is joint work with Baiqing Zhu.
Arithmetic aspects of Hecke traces
We use a recent implication of Maeda’s Conjecture to motivate several lines of inquiry into the behavior of traces of Hecke operators. Special focus is placed on uniqueness questions and Lehmer-type problems. To support these investigations, we discuss computational techniques drawn from p-adic analysis and Diophantine approximation. These methods aim to provide both conceptual insights and practical tools for further exploration.
Malle’s conjecture over function fields
For $G$ a finite group, Malle’s conjecture predicts the asymptotic growth of the number of $G$ extensions of a fixed global field. In joint work with Ishan Levy, we compute the asymptotic growth of the number of Galois $G$ extensions of $\mathbb F_q(t)$, for $q$ sufficiently large and relatively prime to $|G|$. Time permitting, we may also mention an extension of these methods toward verifying the Poonen-Rains conjectures about average sizes of Selmer groups of elliptic curves in quadratic twist families.
Manin’s conjecture for Châtelet surfaces
We resolve Manin’s conjecture for all Châtelet surfaces over Q (surfaces given by 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 modular forms along polynomial values.
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Based on the simplest-github-page by Christopher Allen