Thomae-Weber Formula: Algebraic Computations of Theta Constants

Turku Ozlum Celik

doi:10.1093/imrn/rnz302

Abstract

Abstract We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with 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 level 2 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.

Highlights

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
Computations of theta constants are closely related to a classical problem that asks which complex principally polarized abelian varieties arise as Jacobian varieties of curves
Mumford showed that a principally polarized abelian variety can be written as an intersection of explicit quadrics in a projective space [15]. The coefficients of these quadrics are determined by theta constants denoted by θ[q](τ ), where τ is a Riemann matrix for a specific choice of bases of regular differentials and homology and

Summary

Introduction

Computations of theta constants are closely related to a classical problem that asks which complex principally polarized abelian varieties arise as Jacobian varieties of curves. Mumford showed that a principally polarized abelian variety can be written as an intersection of explicit quadrics in a projective space [15] The coefficients of these quadrics are determined by theta constants denoted by θ[q](τ ), where τ is a Riemann matrix for a specific choice of bases of regular differentials and homology and [q]. A complete 2-level structure of C is represented via the defining equation(s) of the image of C under the canonical embedding and certain divisors on the curve with a suitable labeling as follows. Such divisors are called theta characteristic divisors. We exhibit Example 4.1 to apply the way and use the example for an explicit computation with our algorithm

Theta Characteristics

Computation of Theta Constants

Applying the Algorithm in genus 4