On the Ramanujan Vector Field modulo $p$
Frederico Bianchini
TL;DR
This work analyzes the Ramanujan vector field $R$ modulo primes $p\ge5$ on an affine moduli space of elliptic curves, giving an explicit formula for its $p$-th power $R^p$ and proving that the associated $p$-curvature is nonzero. The computation expresses $R^p$ in the $R,F,H$ basis with coefficients determined by reductions of the modular-polynomial pair $(A,B)$ to $\tilde A,\tilde B$, tying them to $E_{p-1}$ and $E_{p+1}$. The paper further identifies the supersingular locus $SS_p$ inside the moduli space as the zero set of $\tilde A(e_4,e_6)$ modulo $p$, proves that $SS_p$ has smooth irreducible components, and shows it coincides with the singular set of the vector-field modification $R^p + \left(\frac{\tilde B}{12}\right)^2 F$, with the Ramanujan vector field transversal to $SS_p$. By combining elementary congruence results of Serre and Swinnerton-Dyer with explicit factorizations of $\tilde A$ and $\tilde B$, the authors provide a concrete, uniform description of these phenomena for all $p\ge5$.
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.