publications

2025

1. 

   Mumford-type Shimura curves contained in the Torelli locus

   Thomas Bouchet, Jeroen Hanselman, Andreas Pieper, and 1 more author

   Preprint, arXiv:2510.00093 [math.AG], 2025

   Abs arXiv

   In this article we give explicit equations for two Shimura families of genus 4 curves whose Jacobians are abelian fourfolds of Mumford type. These are the first explicit examples of abelian varieties over QQ with endomorphism algebra ZZ and additional Hodge tensors.

2. 

   17T7 is a Galois group over the rationals

   Raymond Bommel, Edgar Costa, Noam D. Elkies, and 3 more authors

   Preprint, arXiv:2411.07857 [math.NT], 2025

   Abs arXiv

   We prove that the transitive permutation group \textsf17T7, isomorphic to a split extension of C_2 by \PSL_2(\F_16), is a Galois group over the rationals. The group arises from the field of definition of the 2-torsion on an abelian fourfold with real multiplication defined over a real quadratic field. We find such fourfolds using Hilbert modular forms. Finally, building upon work of Dembélé, we show how to conjecturally reconstruct a period matrix for an abelian variety attached to a Hilbert modular form; we then use this to exhibit an explicit degree 17 polynomial with Galois group \textsf17T7.

2024

1. 

   Creating a dynamic database of finite groups

   Lewis Combes, John W. Jones, Jennifer Paulhus, and 3 more authors

   Preprint, arXiv:2409.09189 [math.GR], 2024

   Abs arXiv

   A database of abstract groups has been added to the \emphL-functions and Modular Forms Database (LMFDB), available at \urlhttps://www.lmfdb.org/Groups/Abstract/. We discuss the functionality of the database and what makes it distinct from other available databases of abstract groups. We describe solutions to mathematical problems we encountered while creating the database, as well as connections between the abstract groups database and other collections of objects in the LMFDB.

2. 

   Equations of genus 4 curves from their theta constants

   Jeroen Hanselman, Andreas Pieper, and Sam Schiavone

   Preprint, arXiv:2402.03160 [math.AG], 2024

   Abs arXiv

   In this article we give explicit formulas for the equations of a generic genus 4 curve in terms of its theta constants. The method uses the Prym construction and the beautiful classical geometry around it.

3. 

   A database of basic numerical invariants of Hilbert modular surfaces

   Eran Assaf, Angelica Babei, Ben Breen, and 8 more authors

   In LuCaNT: LMFDB, computation, and number theory. Conference, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island, USA, July 10–14, 2023, 2024

   Abs DOI arXiv

   We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.

2021

1. 

   Gluing curves of genus 1 and 2 along their 2-torsion

   Jeroen Hanselman, Sam Schiavone, and Jeroen Sijsling

   Math. Comput., 2021

   Abs DOI arXiv

   Let X (resp. Y) be a curve of genus 1 (resp. 2) over a base field k whose characteristic does not equal 2. We give criteria for the existence of a curve Z over k whose Jacobian is up to twist (2,2,2)-isogenous to the products of the Jacobians of X and Y. Moreover, we give algorithms to construct the curve Z once equations for X and Y are given. The first of these is based on interpolation methods involving numerical results over \C that are proved to be correct over general fields a posteriori, whereas the second involves the use of hyperplane sections of the Kummer variety of Y whose desingularization is isomorphic to X. As an application, we find a twist of a Jacobian over \Q that admits a rational 70-torsion point.

2019

1. 

   A database of Belyi maps

   Michael Musty, Sam Schiavone, Jeroen Sijsling, and 1 more author

   In ANTS XIII. Proceedings of the thirteenth algorithmic number theory symposium, University of Wisconsin-Madison, WI, USA, July 16–20, 2018, 2019

   Abs DOI arXiv

   We use a numerical method to compute a database of three-point branched covers of the complex projective line of small degree. We report on some interesting features of this data set, including issues of descent.

2. 

   On Algebras of Low Rank and on Belyi Maps

   Samuel Schiavone

   2019

   Abs URL PDF

   This thesis comes in two parts. The first concerns the classification of algebras of low rank. The main goal of this part is to study the moduli space of rank 3 algebras. Our investigations lead to a generalization of a theorem of Levin that shows that rank 3 algebras over an integral domain occur in essentially two types. We extend this result to rank 3 algebras over a commutative ring, and then to sheaves of rank 3 algebras over a scheme. In the second part we describe a method for computing Belyi maps. In 1984, Grothendieck described an action of the absolute Galois group of the rational numbers on the set of isomorphism classes of Belyi maps. Thus Belyi maps provide a tantalizing possibility of better understanding this important group. We explain in detail the steps used to compute Belyi maps defined on elliptic and hyperelliptic curves. We conclude with our progress in computing an exhaustive catalogue of Belyi maps, and make some basic observations about the Galois action on these maps.

2014

1. 

   Numerical calculation of three-point branched covers of the projective line

   Michael Klug, Michael Musty, Sam Schiavone, and 1 more author

   LMS J. Comput. Math., 2014

   Abs DOI arXiv

   We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.