Solution of Mumford's second problem

TL;DR

This paper provides a complete explicit solution to Mumford's second problem, expressing theta derivatives with rational characteristics in terms of theta constants, including formulas and illustrative examples.

Contribution

It introduces a new explicit formula for theta derivatives with rational characteristics, advancing the understanding of their algebraic structure.

Findings

Derived an explicit formula for theta derivatives with rational characteristics.

Showed that theta derivatives are homogeneous of degree 3 in theta constants.

Provided examples illustrating the computation of these derivatives.

Abstract

A complete solution of Mumford's second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. Expressions for theta derivatives appear to be homogeneous of degree $3$ with respect to theta constants.