  A main issue in superstring theory are the superstring measures. D'Hoker and
Phong showed that for genus two these reduce to measures on the moduli space of
curves which are determined by modular forms of weight eight and the bosonic
measure. They also suggested a generalisation to higher genus. We showed that
their approach works, with a minor modification, in genus three and we
announced a positive result also in genus four. Here we give the modular form
in genus four explicitly. Recently S. Grushevsky published this result as part
of a more general approach.
