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Purely inseparable Richelot isogenies

Bradley W. Brock, Everett W. Howe

TL;DR

The paper analyzes Richelot isogenies between supersingular genus-$2$ curves in characteristic $2$, showing there are exactly $60$ isogenies between any two such curves with nonzero supersingular invariants, $12$ when exactly one invariant is zero, and $4$ when both are zero, up to isomorphism. It recasts the problem through polarizations on $E\times E$ (the square of the unique supersingular elliptic curve) and uses double coset counts in the automorphism group of $(E\times E, P\circ M)$ to derive the enumerative results, with the extra complication of the special curve $y^2+y=x^5$. The paper provides explicit constructions—dihedral, degenerate, Frobenius, and Verschiebung—to realize all Richelot isogenies, and develops a parametrization of dihedral diagrams that connects invariants to concrete models. The results illuminate the rich inseparable isogeny structure in char $2$ and yield practical tools for constructing all Richelot isogenies between supersingular Jacobians. Overall, the work extends understanding of isogeny graphs in characteristic $2$ and provides explicit, computable descriptions of all purely inseparable Richelot isogenies for 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.

Purely inseparable Richelot isogenies

TL;DR

The paper analyzes Richelot isogenies between supersingular genus- curves in characteristic , showing there are exactly isogenies between any two such curves with nonzero supersingular invariants, when exactly one invariant is zero, and when both are zero, up to isomorphism. It recasts the problem through polarizations on (the square of the unique supersingular elliptic curve) and uses double coset counts in the automorphism group of to derive the enumerative results, with the extra complication of the special curve . The paper provides explicit constructions—dihedral, degenerate, Frobenius, and Verschiebung—to realize all Richelot isogenies, and develops a parametrization of dihedral diagrams that connects invariants to concrete models. The results illuminate the rich inseparable isogeny structure in char and yield practical tools for constructing all Richelot isogenies between supersingular Jacobians. Overall, the work extends understanding of isogeny graphs in characteristic and provides explicit, computable descriptions of all purely inseparable Richelot isogenies for supersingular genus- curves.

Abstract

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

Paper Structure

This paper contains 10 sections, 19 theorems, 73 equations, 1 table.

Table of Contents

  1. Introduction
  2. Supersingular genus-2 curves in characteristic 2
  3. The theorem and three lemmas
  4. Proof of the theorem
  5. Explicit constructions for purely inseparable Richelot isogenies
  6. The dihedral construction
  7. The degenerate construction
  8. Frobenius and Verschiebung
  9. Parametrizing dihedral diagrams
  10. Counting isogenies

Key Result

Theorem 1.1

Let $C_1$ and $C_2$ be two supersingular genus-$2$ curves over $k$ with polarized Jacobians $(J_1,\lambda_1)$ and $(J_2,\lambda_2)$. If neither $C_1$ nor $C_2$ is isomorphic to the curve $y^2 + y = x^5$, then there are exactly sixty Richelot isogenies from $(J_1,\lambda_1)$ to $(J_2,\lambda_2)$, up

Theorems & Definitions (40)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 30 more