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Computing \(L\)-series of geometrically hyperelliptic curves of genus three. (English) Zbl 1404.11143

Summary: Let \(C/\mathbb Q\) be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of \(\mathbb Q\), but may not have a hyperelliptic model of the usual form over \(\mathbb Q\). We describe an algorithm that computes the local zeta functions of \(C\) at all odd primes of good reduction up to a prescribed bound \(N\). The algorithm relies on an adaptation of the ‘accumulating remainder tree’ to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

MSC:

11Y16 Number-theoretic algorithms; complexity
11G20 Curves over finite and local fields
11M38 Zeta and \(L\)-functions in characteristic \(p\)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

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