New generalisation of Jacobi's derivative formula

TL;DR

This paper introduces a new generalization of Jacobi's derivative formula by deriving theta relations from the Thomae formula, linking derivatives of theta functions to hyperelliptic Riemann surface branch points.

Contribution

It presents a novel general Thomae formula and a comprehensive set of theta relations, including linear, bilinear, and trilinear forms, extending the understanding of theta derivatives.

Findings

Derived new theta relations from the Thomae formula.

Connected theta derivatives to hyperelliptic Riemann surface branch points.

Showed how the Schottky identity follows from these relations.

Abstract

A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with singular half-period characteristics) in terms of branch points and the period matrix of a hyperelliptic Riemann surface. The new theta relations contain (i) linear relations on the vector space of first order theta derivatives which are arranged in gradients, (ii) relations between second order theta derivatives and symmetric bilinear forms on the vector space of the gradients, (iii) relations between third order theta derivatives and symmetric trilinear forms on the vector space of the gradients, and (iv) a conjecture regarding higher order theta derivatives. It is shown how the Schottky identity (in the hyperelliptic case) is derived from the obtained relations.