Inseparable endomorphisms and rank-2 sublattices of the Gross lattice

TL;DR

This paper explores the relationship between inseparable endomorphisms of supersingular elliptic curves and rank-2 sublattices of the Gross lattice, clarifying conditions under which a known correspondence holds.

Contribution

It establishes a precise criterion involving the trace of inseparable endomorphisms for the correspondence to hold, answering a question posed by Love.

Findings

The correspondence holds when the trace of the inseparable endomorphism is zero.

The correspondence may not hold if the trace is non-zero.

The paper recasts the problem in terms of inseparable endomorphisms, providing new insights.

Abstract

We answer a question posed by Love asking about a correspondence between isogenies from a supersingular elliptic curve to its Frobenius base-change and rank-2 sublattices of its Gross lattice. We recast the question as one about the inseparable endomorphisms of the curve, and show that the correspondence holds when the trace of the endomorphism is zero, and may not hold otherwise.