On the Ramanujan Vector Field modulo $p$

TL;DR

This paper computes the p-th power of the Ramanujan vector field modulo p, verifies a conjecture for it, and describes the supersingular locus in characteristic p using both classical and novel methods.

Contribution

It provides explicit equations for the p-th power of the Ramanujan vector field and verifies a conjecture, linking modular forms and supersingular elliptic curves in characteristic p.

Findings

Explicit equations for the p-th power of the Ramanujan vector field.

Verification of a conjecture by Shepherd-Barron and Ekedahl.

Description of the supersingular locus as a singular set of vector fields.

Abstract

For every prime $p \geq 5$, we compute the $p$-th power of the Ramanujan vector field that arises from the differential relations discovered by Ramanujan for the Eisenstein series $E_2,E_4$ and $E_6$. Our method results in explicit equations for the $p$-th power and uses classical results of Serre and Swinnerton-Dyer about modular forms modulo $p$. From this, we verify that a general conjecture by Sheperd-Barron and Ekedahl is valid for the Ramanujan vector field. Furthermore, we consider the affine realization of a certain moduli space of elliptic curves where the Ramanujan vector field is defined, and describe - in characteristic $p$ - the locus given by supersingular elliptic curves in two ways: a classical one - using equations for the supersingular polynomial - and a new one as the singular set of some vector fields. Additionally, we prove that the Ramanujan vector field is transversal to this locus.