Purely inseparable Richelot isogenies

TL;DR

This paper demonstrates that supersingular genus-2 curves over algebraically-closed fields of characteristic 2 have infinitely many Richelot isogenies, contrasting with the finite cases for non-supersingular or different characteristic curves.

Contribution

It establishes the exact number of Richelot isogenies between supersingular genus-2 curves in characteristic 2 and provides explicit constructions for these isogenies.

Findings

Infinitely many Richelot isogenies from a supersingular genus-2 curve in characteristic 2.

Exactly sixty Richelot isogenies between two arbitrary supersingular genus-2 curves, unless special isomorphism conditions apply.

Explicit methods to construct all Richelot isogenies between supersingular genus-2 curves.

Abstract

We show that if $C$ is a supersingular genus-$2$ curve over an algebraically-closed field of characteristic $2$, then there are infinitely many Richelot isogenies starting from $C$. This is in contrast to what happens with non-supersingular curves in characteristic $2$, or to arbitrary curves in characteristic not $2$: In these situations, there are at most fifteen Richelot isogenies starting from a given genus-$2$ curve. More specifically, we show that if $C_1$ and $C_2$ are two arbitrary supersingular genus-$2$ curves over an algebraically-closed field of characteristic $2$, then there are exactly sixty Richelot isogenies from $C_1$ to $C_2$, unless either $C_1$ or $C_2$ is isomorphic to the curve $y^2 + y = x^5$. In that case, there are either twelve or four Richelot isogenies from $C_1$ to $C_2$, depending on whether $C_1$ is isomorphic to $C_2$. (Here we count Richelot isogenies up to isomorphism.) We give explicit constructions that produce all of the Richelot isogenies between two supersingular curves.