  In his paper "Hodge integrals and degenerate contributions", Pandharipande
studied the relationship between the enumerative geometry of certain 3-folds
and the Gromov-Witten invariants. In some good cases, enumerative invariants
(which are manifestly integers) can be expressed as a rational combination of
Gromov-Witten invariants. Pandharipande speculated that the same combination of
invariants should yield integers even when they do not have any enumerative
significance on the 3-fold. In the case when the 3-fold is the product of a
complex surface and an elliptic curve, Pandharipande has computed this
combination of invariants on the 3-fold in terms of the Gromov-Witten
invariants of the surface. This computation yields surprising conjectural
predictions about the genus 0 and genus 1 Gromov-Witten invariants of complex
surfaces. The conjecture states that certain rational combinations of the genus
0 and genus 1 Gromov-Witten invariants are always integers. Since the
Gromov-Witten invariants for surfaces are often enumerative (as oppose to
3-folds), this conjecture can often also be interpreted as giving certain
congruence relations among the various enumerative invariants of a surface.
  In this note, we state Pandharipande's conjecture and we prove it for an
infinite series of classes in the case of the projective plane blown-up at 9
points. In this case, we find generating functions for the numbers appearing in
the conjecture in terms of quasi-modular forms. We then prove the integrality
of the numbers by proving a certain a congruence property of modular forms that
is reminiscent of Ramanujan's mod 5 congruences of the partition function.
