Deterministically generating Picard groups of hyperelliptic curves over finite fields

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• 作者：Michiel Kosters ^{kosters@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11G20 ; 14H25 ; 14C22 ; 11R58
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：739-758
• 全文大小：419 K

文摘

Let ϵ>0$ϵ>0$. 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)$y^2+h(x)y=f(x)$ such that the x  -coordinate map is ramified at ∞. In time O(g^2+ϵq^1/2+ϵ)$O(g^{2+ϵ}{q}^{1/2+ϵ})$ the algorithm outputs a set of generators of the Picard group c3078dd71f995f255a">${\mathrm{Pic}}_k^0(C)$. This extends results which others have obtained when g=1$g=1$.

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.