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Computing $L$ -series of geometrically hyperelliptic curves of genus three

  • David Harvey (a1), Maike Massierer (a2) and Andrew V. Sutherland (a3)

Abstract

Let $C/\mathbf{Q}$ 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 $\mathbf{Q}$ , but may not have a hyperelliptic model of the usual form over $\mathbf{Q}$ . 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.

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References

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