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

Published online by Cambridge University Press:  26 August 2016

David Harvey
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email [email protected]
Maike Massierer
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email [email protected]
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA email [email protected]

Abstract

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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.

Type
Research Article
Copyright
© The Author(s) 2016 

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