The first p-jet space of an elliptic curve: global functions and lifts of Frobenius

TL;DR

This paper investigates the structure of the first p-jet space of an elliptic curve, establishing that non-constant global functions and Frobenius lifts exist only if the elliptic curve admits a Frobenius lift, revealing deep connections between jet spaces and Frobenius structures.

Contribution

It proves a fundamental restriction on the existence of global functions and Frobenius lifts on the first p-jet space of elliptic curves, linking these properties to the curve's own Frobenius liftability.

Findings

No non-constant global functions on the first p-jet space unless the curve has a Frobenius lift.

No lifts of Frobenius on the jet space unless the elliptic curve admits a Frobenius lift.

Establishes a criterion connecting jet space properties with the elliptic curve's Frobenius liftability.

Abstract

We prove that there are no non-constant global functions and no lifts of Frobenius on the first $p$-jet space on an elliptic curve unless the elliptic curve itself has a lift of Frobenius.