Vector-valued modular forms from the Mumford forms, 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 vectorvalued modular forms, defined on the Teichm̈uller 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 χ68, 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, χ68 and the theta series corresponding to the even unimodular lattices E8 ⊕ E8 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. © 2012 American Mathematical Society.