2019 Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting

Organizers:
Jennifer Balakrishnan, Boston University
Noam Elkies, Harvard University
Brendan Hassett, Brown University
Bjorn Poonen, Massachusetts Institute of Technology
Andrew Sutherland, Massachusetts Institute of Technology
John Voight, Dartmouth College

The first annual meeting of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation focused on three main themes: the development and organization of software and databases supporting research in number theory and arithmetic geometry; fundamental research in arithmetic geometry inspired by computation and leading to new algorithms; and explorations of L-functions, modular forms, and Galois representations with elegant and unusual properties.

Collaboration website: https://simonscollab.icerm.brown.edu/

Talks

Frank Calegari
University of Chicago

Fake Abelian Surfaces and Fake Kummer Surfaces

We discuss the problem of finding explicit models for some (geometrically) rational moduli spaces related to fake abelian surfaces and fake Kummer surfaces. On one hand, this is related to recent advances in modularity, and on the other, it raises some intriguing but accessible problems in both explicit number theory and algebraic geometry.

Jennifer Balakrishnan
Boston University

Rational Points on the Cursed Curve

The split Cartan modular curve of level 13, also known as the ‘cursed curve,’ is a genus 3 curve defined over the rationals. By Faltings’ proof of Mordell’s conjecture, we know that it has finitely many rational points. However, Faltings’ proof does not give an algorithm for finding these points.  We will discuss how to determine rational points on this curve using ‘quadratic Chabauty,’ part of Kim’s nonabelian Chabauty program. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman and Jan Vonk.

View Slides (PDF)

Bjorn Poonen
Massachusetts Institute of Technology

Number Field Fragments and Fermat’s Last Theorem

We describe an attempt (unsuccessful so far) to give a new proof of Fermat’s last theorem, involving geometry of numbers and the isotypic components of a Galois number field viewed as a representation of its Galois group.

John Voight
Dartmouth College

The L-functions and Modular Forms DataBase (LMFDB)

The Langlands program is a set of conjectures that lie in deep theories of mathematical symmetry, connecting numerous subfields of mathematics. Recently, it has become feasible to do large-scale computational verification of the predictions of the Langlands program, to test conjectures in higher-dimensional cases and, in particular, to present the results in a way that is widely accessible. To this end, the L-functions and Modular Forms DataBase (LMFDB, http://www.lmfdb.org) was created to connect and organize the work of many mathematicians working in this area. In this talk, we will survey the mathematical underpinnings and the algorithmic infrastructure of the LMFDB in an attempt to navigate and provide compelling visual displays of the Langlands program in action.

View Slides (PDF)

Anthony Várilly-Alvarado
Rice University

Computations on K3 surfaces, Past, Present and Future

Várilly-Alvarado will survey ways in which recent computational advances have allowed us to probe the arithmetic of K3 surfaces and ask computational/experimental questions involving K3 surfaces and elliptic curves that may help shed further light on the subject. I will pay particular attention to Neron-Severi groups, Brauer groups and Galois representations of elliptic curves and K3 surfaces.

View Slides (PDF)

Kiran S. Kedlaya
University of California, San Diego

Computation of Zeta and L-functions: Feasibility and Applications

Thanks in part to improvements in our algorithmic understanding of p-adic cohomology, it has become feasible to compute zeta functions of algebraic varieties over finite fields and L-functions of algebraic varieties over number fields to a far greater extent than ever before. Kedlaya will give a survey of the kinds of computations that are feasible and indicate some problems for which such computations are likely to yield new insights (or have already done so).

View Slides (PDF)

Bianca Viray
University of Washington

Isolated and Sporadic Points on Modular Curves

A corollary of Faltings’s theorem is that infinitely many degree d points on a curve C arise from a parametrization of (a subvariety of) Sym^d C by P^1 or a positive rank abelian variety.  We study so-called isolated degree d points (i.e., those not parametrized by P^1 or a positive rank abelian variety) and show that under a fiber irreducibility condition on a morphism f:C->D, isolated points on C push forward to isolated points on D. We then apply this in the context of modular curves and show that non-CM isolated points map to isolated points on modular curves of bounded level. These results imply analogous results for so-called sporadic points (i.e., points x where there are only finitely many other points of degree at most deg(x)).

Peter Sarnak
Princeton University

An Undetermined Matrix Moment Problem and Its Application to Computing Zeros of L-functions

We examine a basic problem concerning what can be determined efficiently about the locations of the eigenvalues of a matrix in O(2n) given the traces of its first k (<n) powers. We explain how this can be used to compute root numbers and count zeros of L-functions in sub-exponential time (in the conductor). Joint work with M. Rubinstein.

View Slides (PDF)