Workshop project descriptions

Netan Dogra and Jan Vonk

Project: Non-abelian Chabauty 

Description: This project group is concerned with the explicit determination of the (finite) set of rational points on curves, following the non-abelian method of Chabauty introduced by Minhyong Kim. We will apply this to examples of Shimura curves, especially those relevant to Mazur's program B. The aim is to build on existing algorithms and also to explore new approaches to the algorithmic resolution of (modular) Diophantine equations.


John Jones and Jen Paulhus

Project: Groups in the LMFDB

Description: The LMFDB contains not only number theoretic objects directly connected to L-functions, but also related mathematical objects such as transitive permutation groups and Sato-Tate groups.  This project will focus on building a section for abstract finite groups which arise in other areas of the LMFDB such as automorphism groups, images of Galois representations, and inertia groups for field extensions.  Building on initial work on finite groups for the LMFDB, we will work on computational questions such as the best ways to represent group elements and subgroups, and how to most efficiently add new groups to the database.  We also hope to complete the coding necessary for these pages to be inserted into the LMFDB.


Kiran Kedlaya and Dave Roberts

Project: Hypergeometric Motives in the LMFDB

Description: The beta version of the LMFDB contains a partially working section on hypergeometric motives over Q (beta.lmfdb.org/Motive/Hypergeometric/Q/). The project is to improve this to a fully working hypergeometric motives section, properly documented and linked to other parts of the LMFDB. Among the issues that need to be addressed are computing complete L-functions corresponding to specialization points of the currently listed families.


Sally Koutsoliotas and Ralf Schmidt

Project: New examples of transcendental L-functions

Description: Higher degree transcendental L-functions are difficult to study experimentally because there is no known method for constructing the underlying automorphic objects.  Recently, some new examples of these L-functions have been found.  We will examine what new phenomena appear in these examples (individually, as well as collectively); develop more robust ways of computing them; and incorporate these data into the LMFDB.


Nicolas Mascot and Jeroen Sijsling

Project: Explicit arithmetic of Jacobians

Let A be an abelian variety over a number field F. In many contexts, such as modularity, it is important to study the endomorphism ring End(A) of A, as well as the structure of its ell-torsion subgroups A[ell] for prime numbers ell. In the case where A = Jax(X) is the Jacobian of an algebraic curve X over F, calculating these invariants is possible. Heuristically, the endomorphism ring End(A) can be calculated by considering the period matrix of X, and recent developments have allowed the verification of these heuristic results. Similarly, there has been recent progress on calculating the Galois-module structure on A[ell] in this case.

The goal of this working group is to improve the available machinery on these topics and to broaden their applications, in particular (1) the optimization of the endomorphism verification for superelliptic and plane curves and their applications to canonical lifting of endomorphisms and (2) the construction of curves, either individually or in families, whose ell-torsion defines exotic Galois representations, such as those needed for solving the Gross problem at p <= 7.


Celine Maistret and Vladimir Dokchitser

Project: Arithmetic of hyperelliptic curves over local fields

Description: This project is based on the “cluster picture” approach developed in [3] to study hyperelliptic curves over local fields. The cluster picture of a curve given by $y^2=f(x)$ is an elementary combinatory object that describes the $p$-adic distances between the roots of $f(x)$. It can be used to recover the curve’s Galois representation, conductor, special fibre of the minimal regular model, minimal discriminant, deficiency, Tamagawa number of its Jacobian, local root numbers and a basis of integral differentials ([1], [2], [3], [4], [5]).

The aims of this project are 1) to create a user’s guide for non-specialists, 2) implement the results into a computer algebra, 3) theoretically extend the cluster approach e.g. to non-semistable cases and to other invariants such as local solubility. 

[1] A. Betts, On the computation of Tamagawa numbers and Neron component groups of Jacobians of semistable hyperelliptic curves, arxiv:1808.05479
[2] M. Bisatt, Clusters, inertia and root numbers, arXiv:1902.08981 
[3] T. Dokchitser, V. Dokchitser, C. Maistret, A. Morgan, Arithmetic of Hyperelliptic curves over local fields, arXiv:1808.02936 
[4] O. Faraggi, S. Nowell, Models of Hyperelliptic Curves with Tame Potentially Semistable Reduction, arXiv:1906.06258.
[5] S. Kunzweiler, Differential Forms on Hyperelliptic Curves with Semistable Reduction, arXiv:1902.07784.


Everett Howe and Rachel Pries

Project: Cyclic covers with isogenous Jacobians

Description: The goal of this project is to classify pairs of cyclic covers of the projective line, defined over a finite field, which are isogenous. This will build on work of Mestre (Couples de Jacobiennes isogenes de courbes hyperelliptiques de genre arbitraire) and recent work of Howe, Sutherland, and Voloch.  We will start by focusing on the zeta functions of the curves in the 20 special families found by Moonen. For some of the families, we will also determine when certain unramified extensions of the curves have the same zeta function. If possible, we will add information about the abelian varieties over finite fields that occur in the Moonen families to the LMFDB.


Padmavathi Srinivasan and Isabel Vogt

Project: Galois representations of abelian surfaces

Description: The goal of this project is to gather data on the Galois representations associated to abelian surfaces defined over Q, and expand the data on genus 2 curves in the LMFDB.  One direction is to compute mod ell images for small primes ell, building on work of Drew Sutherland.  Another direction would be to write code to determine the set of all primes ell for which the mod ell representation is reducible.

References:
J. Cullinan, Symplectic stabilizers with applications to abelian varieties, International Journal of Number Theory
8, 2012.
P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press,
1990.
E. Larson and D. Vaintrob, Determinants of subquotients of Galois representations associated with abelian
varieties, Journal of the Institute of Mathematics of Jussieu 13, 2014.
D. Sutherland, Computing images of Galois representations attached to elliptic curves, Forum of Mathematics, Sigma
4, 2016.



Bianca Viray and David Zureick-Brown

Project: Determining Isolated Points on Rank 0 Low Level Modular Curves

Description: The goal of the project is to compute all isolated points on a number of low level modular curves.  The first focus will be on those modular curves whose Jacobian has rank 0, as in that case the problem reduces to determining all P^1-isolated points.   Given a complete description of the isolated points on a modular curve, a secondary goal will be to see how this information can be leveraged to understand more of the arithmetic of this curve.