2024 Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting

This meeting had five hour-long presentations from experts outside the collaboration, pointing to future developments in databases and quantitative arithmetic geometry. Members of the team presented one hour-long lecture and ten brief “lightning” presentations. We are grateful to Alex Betts (Harvard), a postdoctoral research scientist who delivered a lecture with only a couple days’ notice after one of our confirmed speakers was forced to cancel due to a family illness.

Three of the external talks focused on arithmetic statistics. Akshay Venkatesh (Institute for Advanced Study) spoke on “Quadratic Functions on Class Groups”. He presented joint work with Siad attempting to explain deviations of class group statistics from existing probabilistic heuristics. Their approach involves introducing additional structure on the class group, in the spirit of “biextensions” used to analyze universal families of line bundles. Arul Shankar (Toronto) gave a lecture on “Secondary Terms in the First Moment of the 2-Selmer Groups of Elliptic Curves.” He discussed data for elliptic curves over Q ordered by height — compiled by Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigandt — showing that the empirical average of the Selmer rank is smaller than the proven limit. Shankar presented work with Taniguchi proposing a secondary term compatible with the observed trends. Will Sawin (Princeton) lectured on “Roots of Unity in Arithmetic Statistics.” The distribution of class groups for a given collection of fields may be modeled in probabilistic terms. However, the exact form of the asymptotic hinges on which roots of unity are available. Sawin presented work with Wood on a precise conjecture explaining this dependence.

The talk by Nina Zubrilina (Princeton) was “Root Number Correlation Bias of Fourier Coefficients of Modular Forms.” Murmurations refer to oscillations in the size of reductions of elliptic curves over Q over a range of primes. These are apparent for curves with conductors from suitable intervals and were first observed by He, Lee, Oliver and Pozdnyakov using data from the L-functions and Modular Forms Database (LMFDB). More general patterns have emerged for related classes of objects, including abelian surfaces. Zubrilina discussed unconditional theorems for another collection of objects — modular newforms of weight k. She obtains a precise analytic formula for the oscillation function. Cecília Salgado (Groningen) spoke on “Mordell-Weil Rank Jumps on Families of Elliptic Curves.” For elliptic surfaces over P1 over number fields, the qualitative properties of rational points depend both on the geometry of rational and elliptic multisections and the ranks of the individual fibers. Salgado surveyed results on rational and K3 surfaces, as well as properly elliptic surfaces. The latter need not admit multisections of small genus in general.

Alex Betts (Harvard) gave a lecture “Unexpected Points in Quadratic Chabauty Loci” on joint work with collaboration PI Balakrishnan. The quadratic Chabauty method gives p-adic functions on the Jacobian vanishing at all the rational points. Often the common zeros are completely explained by rational points. This work aims to classify when this fails, in terms of geometric invariants of the underlying curve.

The ten lightning talks focused on early-career researchers affiliated with the collaboration, many of whom are seeking new positions as our project winds down. Topics included the quadratic Chabauty method, Serre’s uniformity conjecture, explicit stable reduction of low-genus curves, constructions techniques for modular forms, enumerations of curves over small fields, and concrete examples.

Alex Betts
Harvard University

Unexpected Points in Quadratic Chabauty Loci

The quadratic Chabauty method attempts to compute the rational or integral points on a curve by constructing a subset of the p-adic points containing them. In some cases, this subset can contain unexpected non-rational points, algebraic over Q. In this talk, Alex Betts will outline results with Jennifer Balakrishnan which give a partial classification of the kinds of unexpected points which can occur on once-punctured elliptic curves of rank 0.
 

Cecília Salgado
University of Groningen

Mordell-Weil Rank Jumps on Families of Elliptic Curves
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Cecília Salgado will discuss recent developments around the variation of the Mordell-Weil rank in 1-dimensional families of elliptic curves, by studying them in the guise of elliptic algebraic surfaces. Salgado will cover recent progress on rational, K3, and surfaces of Kodaira dimension 1.
 

Will Sawin
Columbia University

Roots of Unity in Arithmetic Statistics

Cohen and Martinet considered, for a fixed base field and fixed Galois group, the distribution of the class group of a random extension of the base field with the given Galois group. How is this distribution affected by the roots of unity of the base field? Will Sawin will explain joint work with Melanie Matchett Wood that gives a complete conjectural answer to this question.
 

Arul Shankar
University of Toronto

Secondary Terms in the First Moment of the 2-Selmer Groups of Elliptic Curves

Ranks of elliptic curves are often studied via their 2-Selmer groups. Arul Shankar and collaborator Manjul Bhargava proved that the average size of the 2-Selmer group of elliptic curves is 3, when the family of all elliptic curves is ordered by naive height. On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigand collected and analyzed data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they found a persistently smaller average size of the 2-Selmer group in the data. Thus, it is natural to ask whether there exists a second order main term in the counting function of the 2-Selmer groups of elliptic curves. In this talk, Shankar will discuss joint work with Takashi Taniguchi, in which we prove the existence of such a secondary term.
 

Akshay Venkatesh
Institute for Advanced Study

Quadratic Functions on Class Groups
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Hecke proved that the different of a number field is always a square in the ideal class group. In work in progress with collaborator Artane Siad, Akshay Venkatesh will show that a choice of a square root induces an interesting algebraic structure, a quadratic function on the class group, which clarifies the origin of some peculiar class group statistics.
 

Nina Zubrilina
Princeton University

Root Number Correlation Bias of Fourier Coefficients of Modular Forms
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In a recent machine learning based study, He, Lee, Oliver and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a broader class of arithmetic L-functions when split by root number.

Nina Zubrilina will discuss this root number correlation bias when the average is taken over all weight k modular newforms. Zubrilina will point to a source of this phenomenon in this case and compute the correlation function exactly.

Eran Assaf
Dartmouth College

Algebraic Modular Forms: From Computation to Insights
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Algebraic modular forms allow us to effectively compute systems of Hecke eigenvalues. In this talk, Eran Assaf will give a brief survey of how, in different settings, these computations have provided insights regarding modularity and transfer in the Langlands program.
 

Raymond van Bommel
MIT

Reduction of Plane Quartics and Cayley Octads
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In this joint work with Jordan Docking, Vladimir Dokchitser, Reynald Lercier, and Elisa Lorenzo García, Raymond van Bommek will give a conjectural characterization of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of “cluster pictures”
for hyperelliptic curves.
 

Shiva Chidambaram
MIT

Non-Maximal Primes for Galois Representations of Picard Curves
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Serre’s uniformity conjecture predicts that the ell-torsion Galois representation of a non-CM elliptic curve over Q has maximal image for every prime ell > 37. For typical genus 2 curves, the largest known non-maximal prime is currently 31, which was obtained by implementing an algorithm due to Dieulefait. We introduce an algorithm to compute non-maximal primes for Picard curves y^3 = f_4(x), which are genus 3 curves having an automorphism of order 3. After running it on several large datasets, the largest non-maximal prime we obtain is 13, hinting that they are rarer. This is joint work with Pip Goodman.
 

Edgar Costa
MIT

Eichler-Shimura Construction for Hilbert Modular Forms
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Edgar Costa will demonstrate how to reconstruct an abelian variety associated with a Hilbert modular form. Joint work with Raymond van Bommel, Noam Elkies, Maarten Derickx, Timo Keller, Samuel Schiavone, and John Voight.
 

Juanita Duque-Rosero
Boston University

Computing Local Heights on Hyperelliptic Curves for Quadratic Chabauty
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Local heights are arithmetic invariants used in the quadratic Chabauty method for determining rational points on curves. In this talk, Juanita Duque-Rosero will present an algorithm to compute these local heights on hyperelliptic curves at odd primes v not equal to p. This is joint work with Alexander Betts, Sachi Hashimoto, and Pim Spelier.
 

Aashraya Jha
Boston University

Quadratic Chabauty over Number Fields
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Aashraya Jha will discuss the method of quadratic Chabauty for curves defined over non-trivial extensions of the rationals, and how it differs from quadratic Chabauty over the rationals. Jha will give some examples of computing points with this method to demonstrate its utility.
 

Jun Lau
Boston University

Census of genus 6 curves over F_2
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In joint work with Kiran Kedlaya and Steve Huang, Jun Lau will present algorithms to determine genus 6 curves defined over F_2, up to isomorphism over F_2. Furthermore, Lau will compute the number of curves over F_2 weighted by their F_2-automorphism groups.
 

Adam Logan
Brown University

Computing Shadows of Ceresa Cycles
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The Ceresa cycle on the Jacobian of a curve was introduced by Ceresa in his 1982 dissertation to illuminate the distinction between algebraic and homological equivalence on a variety. He proved
by a nonconstructive Hodge-theoretic argument that for a very general curve the Ceresa cycle is not torsion in the group of cycles modulo algebraic equivalence.

In recent years there has been a lot of interest in proving that the Ceresa cycle is or is not torsion for
explicit curves. One type of obstruction to the cycle being torsion lies in the Jacobian of the reduction of the curve to a finite field. Adam Logan will describe this obstruction, some extensions of it, and calculations of these for a set of plane quartics. This is joint work with J. Ellenberg, P. Srinivasan, and A. Venkatesh.
 

Ciaran Schembri
Dartmouth College

Torsion and Moduli of Surfaces with Quaternionic Multiplication
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In a celebrated work Mazur classified which torsion subgroups can occur for elliptic curves defined over the rationals. A natural analogue is to consider surfaces with geometric endomorphisms by a
quaternion order (PQM surfaces) since the associated moduli space is 1-dimensional. In this talk Ciaran Schembri will discuss progress towards classifying which torsion subgroups are possible for these surfaces. Schembri will also give a description of the moduli problem for PQM surfaces. This is joint work with Eran Assaf, Jef Laga, Freddy Saia, Ari Shnidman, Jacob Swenberg and John Voight.
 

Sam Schiavone
MIT

Reconstructing Genus 4 Curves
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Sam Schiavone will describe how to recover the canonical model of a genus 4 curve from its period lattice. As an application, Schiavone will show how this can be used to glue genus 2 curves along their torsion.