We study a proposal of D'Hoker and Phong for the chiral superstring measure
for genus three. A minor modification of the constraints they impose on certain
Siegel modular forms leads to a unique solution. We reduce the problem of
finding these modular forms, which depend on an even spin structure, to finding
a modular form of weight 8 on a certain subgroup of the modular group. An
explicit formula for this form, as a polynomial in the even theta constants, is
given. We checked that our result is consistent with the vanishing of the
cosmological constant. We also verified a conjecture of D'Hoker and Phong on
modular forms in genus 3 and 4 using results of Igusa.Comment: 25 page
