Purely inseparable Richelot isogenies

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.

31 pages. We simplified some proofs and calculations by changing the model we use for generic supersingular genus-2 curves. We also include as an ancillary file a collection of Magma routines that the reader can use to verify some calculations made in the paper