NTSGrad Spring 2025/Abstracts

This page contains the titles and abstracts for talks scheduled in the Spring 2025 semester. To go back to the main GNTS page for the semester, click here.

1/28

Joey Yu Luo

Mahler integral and L-function

I will define Mahler integrals and L-funciton of elliptic curves, and discuss some conjectural identities between them. They are secretly prediected by the Beilinson-Bloch conjecture, but I will not dig into it.

2/18

Joey Yu Luo

Jacquet-Rallis Fundamental Lemma

Jacquet-Rallis Fundamental Lemma is some mysterious lattice counting equality in the p-adic field. In this talk, I will present the statement of the FL, relate them with orbit integrals, and if time permits, discuss the relating smooth transfer conjecfture(theorem) and arithmetic variants.

3/18

Eiki Norizuki

Arithmetic Statistics of Curves in Positive Characteristic

For a curve in positive characteristic, one can study the a-number, p-ranks and other arithmetic invariants of its Jacobian. A natural question to ask is what the proportion of curves with a given arithmetic invariant in your chosen family of curves is. I will give a flavor of some of the heuristics predicted by Cais, Ellenberg, Zureick-Brown and survey some known results.

4/8

Jiaqi Hou

Maass forms on arithmetic hyperbolic surfaces

Besides holomorphic modular forms, Maass forms are another important class of automorphic forms. I will talk about the definitions and basic properties of Maass forms on hyperbolic surfaces. Then, I will discuss the analytic problem of how to bound Maass forms and present a sup-norm bound by Iwaniec and Sarnak and their proof.

4/15

Alejo Salvatore

Introduction to Arithmetic Topology

Based on results of Artin-Tate, Mumford suggested an analogy between prime ideals and knots. These ideas were developed by Mazur, Kapranov, Reznikov and Morishita. In this talk I will explain how the etale topology of a number field behaves like the cohomology of a 3-manifold and discuss some aspects of the M^2KR dictionary.