  We study integrality of instanton numbers (genus zero Gopakumar - Vafa
invariants) for quintic and other Calabi-Yau manifolds. We start with the
analysis of the case when the moduli space of complex structures is
one-dimensional; later we show that our methods can be used to prove
integrality in general case. We give an expression of instanton numbers in
terms of Frobenius map on $p$-adic cohomology ; the proof of integrality is
based on this expression.
