Samit Dasgupta: Ribet's Method

Abstract: One of the central themes in number theory is the connection between special values of L-functions and certain invariants of associated global algebraic objects (such as regulators). There are many famous concrete conjectures in this direction, such as the conjectures of Stark, Birch-Swinnerton-Dyer, Beilinson, and Bloch-Kato. The main conjectures of Iwasawa theory are another important example. One of the most powerful techniques we have toward understanding some of these conjectures is a method due to Ribet, which creates a link between L-functions and global algebraic objects using modular forms and their associated Galois representations. In this course, we will describe Ribet's method starting with his original proof of the converse to Herbrand's Theorem. This result states that if p is a prime and p divides a certain special value of the Riemann zeta function, then the p-part of an associated component of the class group of Q(mu_p) is non-trivial. We will introduce and describe all of the relevant features of the proof of Ribet's result, including Eisenstein series, congruences with cusp forms, Galois representations, Galois cohomology, class groups, and class field theory. After describing the details of Ribet's proof, we hope to spend some time describing other applications of Ribet's method, just as the Iwasawa main conjecture and the Brumer-Stark conjecture.

Lecture 1 slides Lecture 1 Zoom recording

Lecture 2 slides Lecture 2 Zoom recording

Lecture 3 slides Lecture 3 Zoom recording

Lecture 4 slides Lecture 4 Zoom recording

Lassina Dembélé: An Algorithmic Approach to Hilbert--Siegel Modular Forms and the Paramodularity Conjecture

Abstract: The goal of this mini-course is to explore an algorithmic approach to the paramodularity conjecture of Brumer--Kramer. (Let A be an abelian surface defined over the rationals, with trivial endomorphism ring. This conjecture asserts that there exists a classical Siegel modular form of genus 2 and weight2, with integer coefficients, such that L(A, s) = L(f, s).) First, we will discuss an algorithm for computing Hilbert--Siegel modular forms of low weights. Then, we will study congruences of Hilbert-Siegel modular forms. Combining these with modularity lifting theorems and deformation theory, we will produce examples of abelian surfaces, with trivial conductor and trivial endomorphism ring, which satisfy the paramodularity conjecture.

Sol Friedberg: Automorphic Forms and the Langlands Program: A Brief Introduction

Abstract: This series of lectures will provide a brief introduction to the theory of automorphic forms and representations and to the Langlands Program. This program proposes connections between arithmetic (Galois representations) and analysis (automorphic forms), connections that are mediated by L-functions -- Artin L-functions and automorphic L-functions, resp. The program then uses these connections to make predictions about relations between automorphic forms on different groups, and about the behavior of families of L-functions. In some cases, the special values of these L-functions are of arithmetic interest.

Notes (taken by Pan Yan)

Elena Mantovan: Introduction to Shimura Varieties

Abstract: We shall offer a brief introduction to / overview of the arithmetic theory of Shimura varieties, focusing on those associated with unitary groups. By discussing explicit examples (including the case of modular curves), we will introduce some of the main features of the theory, specifically the realization as moduli spaces of abelian varieties with additional structure, complex uniformization by hermitian symmetric spaces, the construction of CM points and CM-cycles, the geometry in positive characteristic, p-adic uniformization, and most importantly the connection to the algebraic and p-adic theories of automorphic forms.

Notes (taken by Pavel Coupek)

Aaron Pollack: The Rankin-Selberg method

Abstract: L-functions lie at the heart of a large portion of number theory. One tool that we have to better understand L-functions of automorphic forms is the so-called Rankin-Selberg method. Besides being useful for the internal theory of automorphic forms, the Rankin-Selberg method is also useful in applications of L-functions to arithmetic. I will give an introduction to the Rankin-Selberg method.

Notes (taken by Jon Aycock)