  In this article we prove that for all primes $p\not=2,3$, the Ramanujan
vector field has an invariant algebraic curve and then we give a moduli space
interpretation of this curve in terms of Cartier operator acting on the de Rham
cohomology of elliptic curves. The main ingredients of our study are due to
Serre, Swinnerton-Dyer and Katz in 1973. We aim to generalize this for the
theory of Calabi-Yau modular forms, which includes the generating function of
genus $g$ Gromov-Witten invariants. The integrality of $q$-expansions of such
modular forms is still a main conjecture which has been only established for
special Calabi-Yau varieties, for instance those whose periods are
hypergeometric functions. For this the main tools are Dwork's theorem. We
present an alternative project which aims to prove such integralities using
modular vector fields and Gauss-Manin connection in positive characteristic.
