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

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