Enumerating hyperelliptic curves over finite fields in quasilinear time

2024/01/27 by Everett W. Howe, Howe, Everett W. · 1 voice · 2 citations
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2401.15255

Abstract

We present an algorithm that, for every fixed genus g, will enumerate all hyperelliptic curves of genus g over a finite field k of odd characteristic in quasilinear time; that is, the time required for the algorithm is \widetildeO(q^2g-1), where q=#k. Such an algorithm already exists in the case g=2, thanks to work of Mestre and Cardona and Quer, and in the case g=3, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.

Cited by

• Hyperelliptic Jacobians in Isogeny Classes of Abelian Threefolds Over Finite Fields
• Heuristics for (ir)reducibility of p-rank strata of the moduli space of hyperelliptic curves

Discussions

• Still hoping to find the old Math Twitter community somewhere. Just in case it’s here, let me link to my newest paper… For every fixed genus g, I give an elementary algorithm that enumerates all the [bsky, 12 points, 1 comments]

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