Deterministically generating Picard groups of hyperelliptic curves over finite fields

2014/02/26 by Michiel Kosters, Kosters, Michiel
Computer Science · Mathematics · Medicine · #11G20 #11R58 #14C22 #14H25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cancer Mechanisms and Therapy #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G20 #msc:11R58 #msc:14C22 #msc:14H25

paper · pdf · doi:10.48550/arxiv.1402.6579

15 pages

arxiv created 2014/02/26 · openalex publication_date 2014/02/26 · arxiv updated 2014/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ε>0. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve C of genus g over a finite field k of cardinality q given by y²+h(x)y=f(x) such that the x-coordinate map is ramified at ∞. In time O(g^2+ε q^1/2+ε) the algorithm outputs a set of generators of the Picard group Pic⁰_k(C). This extends results which others have obtained when g=1. In this article we introduce a combinatorial tool, the `shape parameter', which we use together with character sum estimates from class field theory to deduce the statement.

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