Listing superspecial curves of genus three using Richelot isogeny graphs

2024/01/19 by Ryo Ohashi, Hiroshi Onuki, Ohashi, Ryo +7
Computer Science · Mathematics · #14K02 #14K25 (Primary) 14H45 #14Q05 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2401.10500

openalex publication_date 2024/01/19 · openalex created_date 2024/01/23 · openalex updated_date 2026/07/28

Abstract

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic p is known, the number of hyperelliptic ones among them has not been determined even for small p. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in \mathbbF_p². By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes 11 ≤ p < 100.

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