arXiv subjects
Publications and source records attributed to Sam Schiavone.
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 $\mathbb{Q}$ with endomorphism algebra $\mathbb{Z}$ and additional Hodge tensors.
We show how Hilbert modular forms can be used in the constructive inverse Galois problem over the rationals. In particular, we prove that the transitive permutation group 17T7, isomorphic to a split extension of C_2 by PSL_2(FF_16), is a Galois group over the rationals and exhibit an explicit degree 17 polynomial with this Galois group. 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 a fourfold attached to a Hilbert modular form. Building upon work of Dembele, we describe a method for reconstructing a period matrix attached to a Hilbert modular form, and we use it to construct the 2-isogeny polynomial. We also rigorously identify the relevant fourfold as the Jacobian of a genus 4 Shimura curve and compute explicit equations for this curve.
A database of abstract groups has been added to the L-functions and Modular Forms Database (LMFDB), available at https://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.
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.
We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.
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 involves the use of hyperplane sections of the Kummer variety of $Y$ whose desingularization is isomorphic to $X$, whereas the second is based on interpolation methods involving numerical results over $\mathbb{C}$ that are proved to be correct over general fields a posteriori. As an application, we find a twist of a Jacobian over $\mathbb{Q}$ that admits a rational 70-torsion point.
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.
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.