Ramanujan vector field

TL;DR

This paper proves the existence of invariant algebraic curves for the Ramanujan vector field for primes not equal to 2 or 3, and interprets these curves via the Cartier operator on elliptic curves, with implications for Calabi-Yau modular forms.

Contribution

It establishes the existence of invariant algebraic curves for the Ramanujan vector field and provides a moduli space interpretation using the Cartier operator, extending understanding in modular forms and algebraic geometry.

Findings

Existence of invariant algebraic curves for primes p ≠ 2,3.

Moduli space interpretation via Cartier operator.

Potential approach to proving integrality of q-expansions in Calabi-Yau theories.

Abstract

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.