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

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.