ICERM Semester Program on "Computational Aspects of the Langlands Program" (September 9, 2015 - December 4, 2015)
Alina Bucur (UCSD)
Brian Conrey (AIM and University of Bristol)
David Farmer (AIM)
John Jones (Arizona State University)
Kiran Kedlaya (UCSD)
Michael Rubinstein (University of Waterloo)
Holly Swisher (Oregon State University)
John Voight (Dartmouth College)
[Photo Credit: John Voight]
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.