A Thomae-like Formula: Algebraic Computations of Theta Constants

TL;DR

This paper introduces an algebraic approach to compute theta constants for non-hyperelliptic curves of genus 4, linking geometric properties of the curves with theta function computations.

Contribution

It develops a new algebraic method for calculating theta constants using the geometry of non-hyperelliptic curves, especially those on singular quadrics from del Pezzo surfaces.

Findings

Algebraic formulas for theta constants of genus 4 curves

Application to curves on singular quadrics from del Pezzo surfaces

Complete 2-level structure analysis of such curves

Abstract

We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated to a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete 2-level structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.