Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

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 , a new class of vector-valued modular forms, defined on the Teichmüller 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 is proportional to the square root of the products of Thetanullwerte <!-- MATH $\chi _{68}$ --> , 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, <!-- MATH $\chi _{68}$ --> and the theta series corresponding to the even unimodular lattices <!-- MATH $E_8\oplus E_8$ --> and . We also find, for , a functional relation between the singular component of the theta divisor and the Riemann period matrix.