  In this paper we study the expressions of the superstring chiral measures for
$g\leq 5$. We obtain certain new expressions which are functions of higher
powers of theta constants. For $g=3$ we show that the measures can be written
in terms of fourth power of theta constants and for $g=4$ in terms of squares
of theta constants. In both cases the forms $\Xi_8^{(g)}[0^{(g)}]$ appearing in
the expression of the measures are defined on the whole Siegel upper half
space. Instead, for $g=5$ we find a form $\Xi_8^{(5)}[0^{(5)}]$ which is a
polynomial in the classical theta constants, well defined on the Siegel upper
half space and satisfying some suitable constraints on the moduli space of
curves (and not on the whole Siegel upper half space) that could be a candidate
for the genus five superstring measure. Moreover, we discuss the problem of the
uniqueness of this form in genus five. We also determine the dimension of
certain spaces of modular forms and reinterpret the vanishing of the
cosmological constant in terms of group representations.
