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 (Theorem
3.1). 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 (Theorem 3.2).