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/
The first annual meeting of the collaboration drew over 60 people to the Simons Foundation to discuss progress over the previous 15 months and map out directions for future work. The meeting was prefaced by a public lecture from one of the collaboration PIs: “Building Telescopes for Mathematicians” by Andrew Sutherland (MIT). Sutherland described the infrastructure of large-scale cloud-based computations underlying progress in modern number theory. He attracted a diverse audience, mostly from outside the number theory community. Speakers at the meeting played several roles. In addition to the formal talks highlighted below, there were numerous informal conversations over meals and breaks (e.g., on algorithms for computing Picard groups of algebraic surfaces using high-precision numerical solutions to the differential equations governing their periods and Mazur’s phantom abelian varieties associated with rationally connected threefolds—and other cohomologically simple varieties—over number fields).
Formal talks were divided into three categories. First, we asked three collaboration PIs to report on recent research accomplishments. In “Rational Points on the Cursed Curve,” Jennifer Balakrishnan (Boston University) presented work on a question posed by J.P. Serre in 1972, cracking a particularly difficult modular curve known as the ‘cursed curve.’ This research is a tour de force of both computation and theory, relying on recent developments of the quadratic Chabauty technique. “Number Field Fragments and Fermats Last Theorem” by Bjorn Poonen (MIT) proposed an intriguing possible second-generation proof of Fermat’s Last Theorem relying on ideas from algebraic number theory. Much of the work of the collaboration over its first year was improvements to the L-functions and Modular Forms Database, including upgrades to its database infrastructure and classical modular form dataset and interface. This was presented in the talk “The L-Functions and Modular Forms DataBase” by John Voight (Dartmouth).
Second, we sought lectures from researchers working closely with collaboration members but not directly part of the group. “Computation of Zeta and L-functions: Feasibility and Applications” by Kiran S. Kedlaya (UCSD) focused on fundamental results on p-adic approaches to counting points on varieties over finite fields; these underlie algorithms used daily by collaboration members. Anthony V´arilly-Alvarado (Rice University) spoke in his talk “Computations on K3 surfaces, Past, Present and Future” about treating K3 surfaces as a two-dimensional analog of elliptic curves, focusing on level structures and their interpretations via Brauer groups. And “Isolated and Sporadic Points on Modular Curves” by Bianca Viray (University of Washington) presented structural results on isolated and sporadic rational points of fixed degree over modular curves and their images in the j-line, assuming standard conjectures on the uniformity of p-adic representations for elliptic curves.
The third group consisted of leading outside experts that may guide future work of the collaboration. In “Fake Abelian Surfaces and Fake Kummer Surfaces,” Frank Calegari (University of Chicago) sketched explicit constructions among moduli spaces inspired by recent modularity results. Such results are valuable in finding the most accessible path to a given L-function. Finally, “An Undetermined Matrix Moment Problem and Its Application to Computing Zeros of L-functions” by Peter Sarnak (Princeton University/Institute for Advanced Study) developed matrix algebra relevant to computing root numbers and counting zeros of L-functions relatively quickly. Both these presentations may inspire future computational projects.
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.
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.
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.
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.
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.
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).
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)).
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.