Netan Dogra and Jan Vonk
Project: Nonabelian Chabauty
Description: This project group is concerned with the explicit determination of the (finite) set of rational points on curves, following the nonabelian 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 Lfunctions, but also related mathematical objects such as transitive permutation groups and SatoTate 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 Lfunctions corresponding to specialization points of the currently listed families. Sally Koutsoliotas
Project: New examples of transcendental Lfunctions
Description: Higher degree transcendental Lfunctions are difficult to study experimentally because there is no known method for constructing the underlying automorphic objects. Recently, some new examples of these Lfunctions 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.
Adam Logan and Matthias Schuett
Project: Explicit modularity for K3 surfaces and CalabiYau threefolds
Description: It is known that K3 surfaces of Picard number $20$ and rigid CalabiYau threefolds over $\mathbb Q$ are modular. In addition, there are some results that establish the modularity of certain families of K3 surfaces over $\mathbb Q$ of Picard number $19$ by relating them to the symmetric square of an elliptic curve. The goal of this project is to establish the modularity of additional families of K3 surfaces with high Picard number. For example, we will be interested in families of Picard number $19$ associated to abelian surfaces whose endomorphism ring is an order in a nontrivial quaternion algebra or that have endomorphisms not defined over the base field. We would also like to find examples of elements of these families that have small conductor, so as to check by computing the eigenvalues of the Hecke eigenforms explicitly in terms of the cohomology of arithmetic groups. Time permitting, we also intend to consider
families of CalabiYau threefolds with $h^{2,1} = 1$, for example those coming from hypergeometric motives. Some special members of these families have reducible Galois representations on $H^3$, possibly after restricting to the Galois group of a quadratic field; it would be very interesting to prove a modularity theorem for the general member of such a family and see how the modular forms and representations specialize in the reducible cases.
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 elltorsion 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 Galoismodule 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 elltorsion 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 nonspecialists, 2) implement the results into a computer algebra, 3) theoretically extend the cluster approach e.g. to nonsemistable 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.
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. Michael Stoll
Project: Torsion packets on genus 2 curves
Description: A torsion packet on a curve C is a set of (geometric) points on C such that the difference of any two points in the set gives a point of finite order on the Jacobian of C. For a given curve of genus g ≥ 2 in characteristic zero, the size of a torsion packet is finite and bounded, and conjecturally, there should be a uniform bound on the size of a torsion packet that depend only on the genus g. It is then an interesting question what might be the largest size of a torsion packet on a curve of genus 2. Poonen has shown in [Poo00] that there are infinitely many pairwise nonisomorphic genus 2 curves over Q that have a torsion packet of size (at least) 22. They belong to the oneparameter family y^{2} = x^{6} + x^{3} + t of curves with automorphism group D_{6}, whose hyperelliptic torsion packet (the torsion packet that contains the Weierstrass points) generically has size 10. I have found one example in this family with a torsion packet of size 34 (it has an additional orbit of size 12 of torsion points on the curve).
One part of this project could consist in a more systematic study of the family mentioned above. It is very likely out of reach to show that 34 is indeed the maximum size of a (hyperelliptic) torsion packet on a genus 2 curve, but it should be feasible to show computationally that no other examples exist within a fairly large range of parameter values. One might perhaps even find some other interesting examples.
Another part could be to consider other families of genus 2 curves with large automorphism groups, to see how large their torsion packets can get.
It would be useful for this project to produce a Magma implementation of Poonen’s algorithm for determining torsion packets (see [Poo01]). A further part of this project could be to extend the implementation to hyperelliptic genus 3 curves and/or plane quartics, and to investigate torsion packets in genus 3.
[Poo00] Bjorn Poonen, Genustwo curves with 22 torsion points, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 7, 573–576, DOI 10.1016/S07644442(00)002226
[Poo01] Bjorn Poonen, Computing torsion points on curves, Experiment. Math. 10 (2001), no. 3, 449–465.
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^1isolated 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.
