A new approach in constructing isogenies of elliptic curves in characteristic three

Abstract

Given an elliptic curve ${\mathcal E}$ over a field $K$ it is a challenging problem to write down explicit elements of its endomorphism ring ${\rm End}({\mathcal E});$ the problem amounts to find all possible solutions to a functional equation in the field of rational functions $K(X).$ Instead of attempting to describe them directly, we look first for solutions in the larger field of Laurent power series $K((X))$, which we call them {\em formal endomorphisms}. We show that the set of separable formal endomorphisms naturally identifies with a subset of $\frac{1}{X}K[[X]]-$rational points of a plane cubic defined over $K((X)).$ As a by-product, we present a method for finding all formal separable endomorphisms in characteristic $3$. %and an efficient test for determining if a given formal solution is actually rational, yielding to an endomorphism of the given curve.