2023 Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting

Simons Foundation Collaboration on Arithmetic Geometry, Number Theory, and Computation
Annual Meeting: January 11–12, 2023

This meeting was the first held in-person at the Simons Foundation since 2020. Our goals were to highlight work by early-career researchers supported by the project and to promote connections between our efforts and exciting developments around the world.

Two current collaboration researchers and one recent alumna gave hour-long lectures about their work. Wanlin Li (Washington University in St. Louis) gave a masterful lecture on the reduction types of abelian varieties over number fields, focusing on the density of primes with ordinary reduction and the infinitude of primes with basic reduction; this extends results of collaboration PI Noam Elkies. Edgar Costa (MIT) focused on new algorithms for enumerating isogeny classes of principally polarized abelian surfaces (without extra endomorphisms) over the rational numbers. David Roe (MIT) presented two major recent developments in the L-functions and modular forms database (LMFDB). One is expanded databases of finite groups of moderate size, including subgroup relations and complex characters. The second enumerates more than eight million modular curves of level ≤ 70, building on the groups database, but also providing information on ranks of Jacobians, gonality, defining equations and rational points.

Five current collaboration members gave brief talks about their current research. Alexander Betts (Harvard) presented p-adic approaches to heights that should permit the quadratic Chabauty method to be used in more examples. Juanita Duque-Rosero (Dartmouth) enumerated triangular modular curves of small genus. Avinash Kulkarni (Dartmouth) discussed average numbers of solutions to enumerative problems over the p-adic numbers. Grant Molnar (Dartmouth) counted elliptic curves with 7-isogeny, finding formulas similar to those for counting rational points on stacks. Oana Padurariu (Boston University) analyzed rational points on Atkin–Lehner quotients of Shimura curves.

Collaboration director Brendan Hassett (Brown) drew connections between constructions of torsors for abelian varieties relevant to rationality questions and invariant-theoretic approaches to moduli spaces in arithmetic statistics.

The distinguished external speakers presented important advances highly relevant to future research initiatives of the collaboration. Wei Ho (Michigan/Princeton/IAS) gave a broad overview of recent breakthroughs in arithmetic statistics, focusing on the conceptual components of the counting arguments. Robert Lemke Oliver (Tufts) discussed efforts to bound the number of extensions of the rational numbers, of fixed degree, with absolute discriminant less than X. He obtained results for large classes of Galois groups, e.g., solvable groups, sporadic groups, exceptional groups and classical groups of bounded rank. David Zureick-Brown (Emory) gave an update on Mazur’s program to classify images of £-adic representations for elliptic curves over the rationals, including fast algorithms to compute these images.

Further benchmarks of collaboration progress
The high level of LMFDB usage seen during the pandemic continued in 2022, when the database was used in 176 countries and all 50 states in the U.S.:

Alexander Betts
Harvard University

Computing Local Heights for Quadratic Chabauty
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The quadratic Chabauty method uses the theory of p-adic heights to try to compute rational points on curves, especially modular curves. With a very small number of exceptions, the method has only been applied successfully to curves with potentially good reduction, which ensures that the local contributions to the height at places away from p are all zero. In this talk, Alexander Betts will report on an ongoing project with Juanita Duque-Rosero, Sachi Hashimoto and Pim Spelier in which we develop algorithms to compute these local height contributions, with an eye towards using them in quadratic Chabauty computations in new regimes.
 

Edgar Costa
Massachusetts Institute of Technology

Computing Isogeny Classes of Principally Polarized Abelian Surfaces Over the Rationals
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Edgar Costa will describe a practical algorithm, given a principally polarized abelian surface (PPAS) over QQ, to compute all the other PPASs in its isogeny class with a trivial endomorphism ring. This is joint work in progress with Raymond van Bommel, Shiva Chidambaram and Jean Kieffer.
 

Juanita Duque-Rosero
Dartmouth College

Triangular Modular Curves
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Triangular modular curves are generalizations of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. The talk will focus on describing an explicit enumeration of triangular modular curves of low genus. This is work in progress with John Voight.
 

Brendan Hassett
Brown University

Rationality and Arithmetic

Consider a class of smooth projective varieties that are geometrically rational. When are they rational over a given ground field? Brendan Hassett will discuss arithmetic aspects of rationality criteria and relations with constructions used in arithmetic statistics.
 

Wei Ho
Institute for Advanced Study

Recent Progress in Arithmetic Statistics

There has been significant progress in arithmetic statistics in the last few years by many different people. Wei Ho will discuss a range of recent developments in counting invariants related to number fields and elliptic curves, and present some applications.
 

Avinash Kulkarni
Dartmouth College

Integral Geometry in Non-Archimedean Spaces
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In this lightning talk, Avinash Kulkarni will discuss a non-archimedean integral geometry formula for the action of a compact K-analytic group on a homogeneous space. This formula is analogous to a result over the reals obtained by Howard. Some applications will be discussed. Joint work with Antonio Lerario and Peter Burgisser.
 

Robert Lemke-Oliver
Tufts University

Uniform Exponent Bounds on The Number of Primitive Extensions of Number Fields
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A folklore conjecture asserts the existence of a positive constant \(c_n\) such that the number of degree \(n\) extensions \(K/Q\) with discriminant bounded by \(X\) is asymptotic to \(c_n\) \(X\). This conjecture is known if \(n\) is at most 5, but even the weaker conjecture that there exists an absolute constant \(C>1\) such that the number of such fields is at most \(O(X^C)\) remains unknown and apparently out of reach.

Robert Lemke-Oliver will discuss progress on this weaker conjecture (which is termed the “uniform exponent conjecture”) in two ways. First, Lemke-Oliver will reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second, for almost all such groups, Lemke-Oliver will prove the strongest known upper bound on the number of such extensions. These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups and classical groups of bounded rank.
 

Wanlin Li
Washington University in St. Louis

Ordinary and Basic Reductions of Abelian Varieties

Given an abelian variety \(A\) defined over a number field, a conjecture attributed to Serre states that the set of primes at which \(A\) admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fité 2021, etc.).

In this talk, Wanlin Li will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including the case where \(A\) has almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfán, Mantovan, Pries and Tang.

Apart from ordinary reduction, Li will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries and Tang.
 

Grant Molnar
Dartmouth College

Counting Elliptic Curves with a 7-isogeny
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In this talk, Grant Molnar will present new asymptotics for the number of elliptic curves height up to X which admit a (cyclic) 7-isogeny, and discuss directions for future work. This research is joint with John Voight.
 

Oana Padurariu
Boston University

Rational Points on Atkin-Lehner Quotients of Geometrically Hyperelliptic Shimura Curves
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Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X0(D,N). In this talk, Oana Padurariu will describe how we created a database containing all their Atkin-Lehner quotients and how we computed their sets of Q-rational points when these sets are finite. Oana Padurariu will also determine which rational points are CM for many of these curves. This is joint work with Ciaran Schembri.
 

David Roe
Massachusetts Institute of Technology

Modular Curves and Finite Groups: Building Connections Via Computation
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The study of rational points on modular curves has a long history in number theory. Mazur’s 1970s papers that describe the possible torsion subgroups and isogeny degrees for rational elliptic curves rest on a computation of the rational points on X0(N) and X1(N), and a large body of work since then continues this tradition. Modular curves are parameterized by open subgroups H of GL(2, Zhat), and correspondingly parameterize elliptic curves E whose adelic Galois representation lim E[n] is contained in H. For general H, the story of when X_H has non-cuspidal rational or low degree points (and thus when there exist elliptic curves with the corresponding level structure) becomes quite complicated, and one of the best approaches we have for understanding its large-scale computation. David Roe will describe a new database of modular curves, including rational points, explicit models and maps between models, along with some of the mathematical challenges faced along the way.

The close connection between modular curves and finite groups also arises in other areas of number theory and arithmetic geometry. Most well-known are Galois groups associated to field extensions, but one attaches automorphism groups to algebraic varieties and Sato-Tate groups to motives. Building on existing tables of groups, Roe and collaborators have added a new finite groups section to the L-functions and modular forms database, which they hope will prove useful both to number theorists and to others who are using and studying finite groups.
 

David Zureick-Brown
Emory University

l-adic Images of Galois for Elliptic Curves over Q
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David Zureick-Brown will discuss recent joint work with Jeremy Rouse and Drew Sutherland on Mazur’s “Program B” — the classification of the possible “images of Galois” associated an elliptic curve (equivalently, classification of all rational points on certain modular curves XH). The main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.

Zureick-Brown will also discuss the framework and various applications (for example: a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.