Computational approach to the Schottky problem

TL;DR

This paper introduces a computational method using Fay's trisecant identity to identify Jacobian varieties among Riemann matrices, providing a numerical approach to the classical Schottky problem for higher genus cases.

Contribution

The paper develops a novel numerical algorithm based on Fay's identity to solve the Schottky problem, extending previous methods and comparing results with the Schottky-Igusa form.

Findings

Algorithm successfully identifies Jacobian varieties in genus 4.

Residuals from Fay's identity match those from the Schottky-Igusa form in genus 4.

Method's applicability in higher genera (5-7) is discussed with known examples.

Abstract

We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus $g\geq 4$. For a given Riemann matrix $\mathbb{B}\in\mathbb{H}^{g}$, the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The theta functions in terms of which these secants are expressed depend on the Abel maps of four arbitrary points on a Riemann surface. However, there is no concept of an Abel map for general $\mathbb{B} \in \mathbb{H}^{g}$. To establish linear dependence of the secants, four components of the vectors entering the theta functions can be chosen freely. The remaining components are determined by a Newton iteration to minimize the residual of the Fay identity. Krichever's theorem assures that if this residual vanishes within the finite numerical precision for a generic choice of input data, then the Riemann matrix is with this numerical precision the period matrix of a Riemann surface. The algorithm is compared in genus 4 for some examples to the Schottky-Igusa modular form, known to give the Jacobi locus in this case. It is shown that the same residuals are achieved by the Schottky-Igusa form and the approach based on the Fay identity in this case. In genera 5, 6 and 7, we discuss known examples of Riemann matrices and perturbations thereof for which the Fay identity is not satisfied.