Invariant Frobenius lifts and deformation of the Hasse invariant

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.