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arXiv · 1402.6579

Deterministically generating Picard groups of hyperelliptic curves over finite fields

Michiel Kosters
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Abstract

Let $\epsilon>0$. 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 $y^2+h(x)y=f(x)$ such that the $x$-coordinate map is ramified at $\infty$. In time $O(g^{2+\epsilon} q^{1/2+\epsilon})$ the algorithm outputs a set of generators of the Picard group $\mathrm{Pic}^0_k(C)$. 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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Michiel Kosters. 2014-02-26. Deterministically generating Picard groups of hyperelliptic curves over finite fields. https://arxiv.org/abs/1402.6579

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