Dissertation Talk: Numerical Algorithms for the Classical and Tropical Schottky Problem
Seminar: Dissertation Talk: CS | March 6 | 2-3 p.m. | 310 Soda Hall
The Schottky problem asks to characterize Jacobians of curves amongst abelian varieties, and has a complete solution only in genus up to four. We describe numerical methods for deciding whether a symmetric matrix defines a genus four Jacobian, and if so, to compute the curve and its canonical embedding. We then discuss weak solutions to the Schottky problem in genus five, where a complete solution is still unknown. For computations related to the Schottky problem, we develop a Julia package for numerical evaluations of the Riemann theta function. We also present a solution to a variant of the Schottky problem in genus five, for Jacobians with a vanishing theta null. We next discuss the tropical Schottky problem, which is the analogue of the Schottky problem in the combinatorial setting of tropical geometry. We describe the process of going from a curve to its tropical Jacobian, and we present solutions to the tropical Schottky problem in genus four. We also relate the classical and tropical solutions to the genus four Schottky problem, by tropicalizing the Schottky-Igusa modular form.