Algorithms to enumerate superspecial Howe curves of genus four

Abstract: A Howe curve (so named by Senda and the first two authors) is a curve of genus $4$ obtained as the fiber product of two genus-$1$ double covers of ${\mathbf P}^1$. In this paper, we present a simple algorithm for testing isomorphism of Howe curves, and we propose two main algorithms for finding and enumerating these curves: One involves solving multivariate systems coming from Cartier--Manin matrices, while the other uses Richelot isogenies of curves of genus $2$. Comparing the two algorithms by implementation and by complexity analyses, we conclude that the latter enumerates curves more efficiently. However, in order to say that the latter strategy outputs all superspecial Howe curves, we require a conjecture that all superspecial curves of genus $2$ in characteristic $p>2$ are connected by a path of Richelot isogenies. Given a prime $p$, the algorithm verifies this conjecture before producing output.

algebraic geometrynumber theory

Audience: researchers in the topic

( chat | paper | slides | video )

Comments: The slides used in the pre-recorded video can be found here.

Chairs: Brendan Creutz and Felipe Voloch

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