  It is well known that, fixed an even, unimodular, positive definite quadratic
form, one can construct a modular form in each genus; this form is called the
theta series associated to the quadratic form. Varying the quadratic form, one
obtains the ring of stable modular forms. We show that the differences of theta
series associated to specific pairs of quadratic forms vanish on the locus of
hyperelliptic Jacobians in each genus. In our examples, the quadratic forms
have rank 24, 32 and 48. The proof relies on a geometric result about the
boundary of the Satake compactification of the hyperelliptic locus. We also
study the monoid formed by the moduli space of all principally polarised
abelian varieties, the operation being the product of abelian varieties. We use
this construction to show that the ideal of stable modular forms vanishing on
the hyperelliptic locus in each genus is generated by differences of theta
series.
