Deterministically generating Picard groups of hyperelliptic curves over finite fields

Let ϵ>0ϵ>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)y2+h(x)y=f(x) such that the x  -coordinate map is ramified at ∞. In time O(g2+ϵq1/2+ϵ)O(g2+ϵq1/2+ϵ) the algorithm outputs a set of generators of the Picard group Pick0(C). This extends results which others have obtained when g=1g=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.