  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.
