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arXiv · 1605.04708

Computing L-series of geometrically hyperelliptic curves of genus three

David Harvey·Maike Massierer·Andrew V. Sutherland
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Abstract

Let C/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 Q, but may not have a hyperelliptic model of the usual form over 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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David Harvey, Maike Massierer, Andrew V. Sutherland. 2016-05-16. Computing L-series of geometrically hyperelliptic curves of genus three. https://doi.org/10.1112/s1461157016000383

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