Explicit Frobenius lifts on elliptic curves

TL;DR

This paper provides explicit formulas for Frobenius lifts on elliptic curves, enabling efficient point counting algorithms and generalizing Mestre's AGM sequence to odd characteristic.

Contribution

It introduces explicit Frobenius lift formulas for elliptic curves in odd characteristic and extends Mestre's AGM sequence, with applications to point counting.

Findings

Explicit Frobenius lift formulas for ordinary elliptic curves

Generalization of Mestre's AGM sequence to odd characteristic

Development of an efficient point counting algorithm

Abstract

In this article we give explicit formulae for a lift of the relative Frobenius morphism between elliptic curves and show how one can compute this lift in the case of ordinary reduction in odd characteristic. Our theory can also be used in the case of supersingular reduction. By means of the explicit formulae that describe a Frobenius lift, we are able to generalize Mestre's 2-adic arithmetic geometric mean (AGM) sequence of elliptic curves to odd characteristic, and prove its convergence. As an application, we give an efficient point counting algorithm for ordinary elliptic curves which is based on the generalized AGM sequence.