Organizing Committee

Alina Bucur
UCSD

John Conrey
American Institute of Mathematics

David Farmer
AIM

John Jones
Arizona State University

Kiran Kedlaya
University of California, San Diego

Michael Rubinstein
University of Waterloo

Holly Swisher
Oregon State University

John Voight
Dartmouth College

Abstract

In the late 1960s, Robert Langlands discovered a unifying principle in number theory providing a vast generalization of class field theory to include nonabelian extensions of number fields. This principle gives rise to a web of conjectures called the Langlands program which continues to guide research in number theory to the present day. For example, an important first instance of the Langlands program is the modularity theorem for elliptic curves over the rational numbers, an essential ingredient in the proof of Fermat's last theorem.

Despite its many successes, the Langlands program remains vague in many of its predictions, due in part to an absence of data to guide a precise formulation away from a few special cases. In this thematic program, we will experiment with and articulate refined conjectures relating arithmetic-geometric objects to automorphic forms, improve the computational infrastructure underpinning the Langlands program, and assemble additional supporting data. Such data has proven valuable for researchers in number theory, and it will continue to be made available at the L-Functions and Modular Forms Database.

During the semester we will focus on three specific aspects of the Langlands program. First, we will look at elliptic curves over number fields and genus 2 curves over the rationals and will consider their relationship to modular forms. Second, we will consider computational aspects of modular forms in higher rank. Specifically, we will examine K3 surfaces and their connections to modular forms on orthogonal groups. Our third topic concerns analytic aspects of L-functions, building upon and complementing the algebraic, arithmetic, and geometric data.

Computational Aspects of the Langlands Program

Read about the L-functions and Modular Forms Database (LMFDB) created by researchers from this ICERM program.

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Semester Schedule

Wednesday, September 09, 2015

9:00 AM - 1:30 PM EDT

Check-In

Location: 121 S Main St, Providence, RI 02903, USA

1:45 - 2:00 PM EDT

Opening Welcome - Jill Pipher, ICERM Director

Location: 11th Floor Lecture Hall

2:00 - 3:30 PM EDT

Discussion: semester topics and goals - Moderated by Alina Bucur, UCSD

Location: 11th Floor Lecture Hall

3:30 - 4:00 PM EDT

Coffee/Tea Break

Location: 11th Floor Collaborative Space

3:30 - 4:00 PM EDT

Coffee/Tea Break

Location: 11th Floor Collaborative Space

4:00 - 4:30 PM EDT

Discussion: set schedule for the semester

Location: 11th Floor Lecture Hall

4:30 - 5:30 PM EDT

Organizational meeting for seminars and minicourses

Location: 11th Floor Lecture Hall

5:30 - 7:00 PM EDT

Welcome Reception

Location: 11th Floor Collaborative Space

Thursday, September 10, 2015

9:30 - 10:00 AM EDT

Overview of the Langlands program - Kiran S. Kedlaya, UCSD

Location: 11th Floor Lecture Hall

10:00 - 10:15 AM EDT

Coffee/Tea Break

Location: 11th Floor Collaborative Space

10:15 - 11:15 AM EDT

Introduction to the LMFDB - David Farmer, AIM

Location: 11th Floor Lecture Hall

11:30 - 11:35 AM EDT

5-minute Graduate student introductions - Benjamin Breen, Dartmouth College

Location: 11th Floor Lecture Hall

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Lecture Videos

Overview of the Langlands program

Kiran Kedlaya

University of California, San Diego

September 10, 2015 • 12:00 AM

Computational Aspects of the Langlands Program

Sep 9 - Dec 4, 2015

Organizing Committee

Abstract

Semester Schedule

Wednesday, September 09, 2015

Thursday, September 10, 2015

Lecture Videos

Overview of the Langlands program

Introduction to automorphic forms

What are Motives

Introduction to computing complete number theoretic data: number fields

L-functions: the glue that holds the world together

Introduction to the LMFDB

Stable reduction and computation of bad reduction Euler factors - part I

Stable reduction and computation of bad reduction Euler factors - part II

Associated Semester Workshops

Modular Forms and Curves of Low Genus: Computational Aspects

Explicit Methods for Modularity of K3 Surfaces and Other Higher Weig...

Computational Aspects of L-functions

Computational Aspects of the Langlands Program

Sep 9 - Dec 4, 2015

Organizing Committee

Abstract

Confirmed Speakers & Participants

Semester Schedule

Wednesday, September 09, 2015

Thursday, September 10, 2015

Lecture Videos

Overview of the Langlands program

Introduction to automorphic forms

What are Motives

Introduction to computing complete number theoretic data: number fields

L-functions: the glue that holds the world together

Introduction to the LMFDB

Stable reduction and computation of bad reduction Euler factors - part I

Stable reduction and computation of bad reduction Euler factors - part II

Associated Semester Workshops

Modular Forms and Curves of Low Genus: Computational Aspects

Explicit Methods for Modularity of K3 Surfaces and Other Higher Weig...

Computational Aspects of L-functions

Confirmed Speakers & Participants