Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula

TL;DR

This paper introduces a new class of vector-valued Siegel modular forms derived from Mumford forms, revealing connections to the Schottky problem, Klein's formula, and theta relations for genus 4 curves.

Contribution

It demonstrates the emergence of vector-valued modular forms from Mumford forms for genus ≥ 4 and links them to classical formulas and theta relations, advancing understanding of the Schottky problem.

Findings

Discriminant of the quadric is proportional to the square root of Thetanullwerte products.

Coefficients of the quadric are derivatives of the Schottky-Igusa form at the Jacobian locus.

Established a relation between the theta divisor's singular component and the Riemann period matrix.

Abstract

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.