Graduate Instructional Workshop

Arrival: July 24, 2022

Departure: July 30, 2022

Lecture notes (or recordings) will be posted after each talk description below.

General Information

The 5-day graduate instructional workshop will consist of mini-courses on recent developments in the algebraic theory of automorphic forms, L-functions, and related topics. The workshop will help prepare students to be successful researchers by introducing key research techniques and results.

The 5 graduate mini-courses each consists of 4 lectures. The lecturers and topics appear below.

Schedule

All lectures, as well as the panel and registration, will take place in Fenton 110. The door on 13th Street (east side of the building) will be unlocked from 7:30 am until 4:30 pm Monday through Friday. That entrance is wheelchair accessible and leads to the seating area for the lecture hall. There is also a wheelchair accessible entrance to the bottom of the lecture hall (for presenters) to the left of that entrance (west side of the building).
The reception and welcome activity on Monday will be held in the James Common area and Tykeson Lawn, across the street from Fenton Hall.
The panel Tuesday at 4:30 pm is Preparing for the Job Market.

This panel, which is aimed at graduate students, will discuss preparing for the academic job market. After brief introductions, panelists (Cathy Hsu, Jackie Lang, and Sol Friedberg) will respond to participants' questions about preparing for the job market. Topics might include applying to different types of institutions, working abroad, networking, figuring out what kind of positions you would like, two-body problems, and more (depending on participants' questions).

There will be light refreshments available on the second floor of Fenton (inside or just outside the lounge) during morning and afternoon breaks. Participants are encouraged to take refreshments outside to help mitigate COVID risks. Meals will not be provided. There are places to eat on and near campus, as well as near the hotel.
There is a free afternoon on Wednesday. Suggested activities for the free afternoon include a local hike or walk, picking wild blackberries, or visiting one of the three museums on campus (Jordan Schnitzer Museum of Art, Hayward Hall, and the Museum of Natural and Cultural History). Additional possibilities: https://www.eugenecascadescoast.org/things-to-do/ and https://sites.google.com/view/automorphic2021/suggestions-and-fun-stuff
The University of Oregon's COVID-19 regulations and resources are available here: https://coronavirus.uoregon.edu/covid-19-regulations

APAW Week 1 Schedule

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)

Accommodations in shared hotel rooms (at the the Graduate Hotel) will be arranged for funded graduate students for the duration of the workshop. Participants in the collaborative research workshop are strongly encouraged to also attend this graduate instructional workshop. However, due to anticipated space limitations, not all participants in the instructional workshop will necessarily be able to participate in the collaborative research workshop.

Note: As a required part of registration, students requesting funding will need to submit a CV and a short note from their graduate advisor. They will also need to answer a short survey about their mathematical background (to help the speakers set their lectures at an appropriate level).