# General derivative Thomae formula for singular half-periods

## TL;DR

This paper generalizes Thomae's formula for hyperelliptic curves, expressing derivatives of theta functions with singular characteristics in terms of branch points and period matrices, and explores their structural properties.

## Contribution

It introduces a general Thomae formula for derivatives of theta functions with arbitrary multiplicity and analyzes the structure of derivative theta constants across genera.

## Key findings

- Second derivative theta constants form rank three matrices in any genus.
- The structure of third derivative theta constants is characterized, with conjectures for higher multiplicities.
- Generalized Bolza formulas are derived as a byproduct.

## Abstract

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order $3$ tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct a generalization of Bolza formulas are deduced.