# Computational approach to the Schottky problem

- Authors
- Publication Date
- Jul 05, 2023
- Source
- HAL-Descartes
- Keywords
- Language
- English
- License
- Unknown
- External links

## Abstract

We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus ggeq4. For a given Riemann matrix \Omega\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 given depend on the Abel maps of four arbitrary points on a Riemann surface. To establish linear dependence of the secants, four components of these Abel maps are conveniently chosen. 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 in the Jacobi locus. The algorithm is compared in genus 4 for some examples to the Schottky-Igusa modular form known togive the Jacobi locus 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.To illustrate the importance of Jacobian varieties in areas outside of mathematics, we present algebro-geometric solutions of the Ernst equation that solve the Einstein field equations for stationary axisymmetric spacetimes. These solutions have not been extensively studied in simulations, due to the computational cost of calculating theta functions. Thus, we present a series of numerical techniques in order to approach the numerical simulations efficiently, which would allow further studies of these solutions.