University of Cambridge > > Isaac Newton Institute Seminar Series > COMPUTATIONAL APPROACH TO THE SCHOTTKY PROBLEM


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CATW04 - Complex analysis: techniques, applications and computations - perspectives in 2023

We present a computational approach to the classical Schottky problem based on Fay’s trisecant identity for genus g ≥ 4. For a given Riemann matrix B ∈ Hg, 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 B ∈ Hg. 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. This is work with E. Brandon de Leon and J. Frauendiener.

This talk is part of the Isaac Newton Institute Seminar Series series.

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