Invariant Frobenius lifts and deformation of the Hasse invariant

TL;DR

This paper investigates Frobenius lifts on affine elliptic curves with ordinary reduction, demonstrating their preservation of invariant 1-forms mod p and exploring infinitesimal deformations mod p^2 related to the Hasse invariant.

Contribution

It introduces a method to lift Frobenius actions on elliptic curves and extends the mod p results to mod p^2, revealing new p-adic deformations of the Hasse invariant.

Findings

Frobenius lifts preserve invariant 1-forms mod p.

Existence of mod p^2 analogues after removing 2-torsion sections.

Identification of a p-adic deformation of the Hasse polynomial.

Abstract

We show that the $p$-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts whose "normalized" action on $1$-forms preserves mod $p$ the space of invariant $1$-forms. We next show that, after removing the $2$-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod $p$ result has a mod $p^2$ analogue. While the "eigenvalues" mod $p$ are given by the reciprocal of the Hasse polynomial, the "eigenvalues" mod $p^2$ are given by an appropriate $\d$-modular function whose reciprocal is a $p$-adic deformation of the Hasse polynomial.