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

Part of: Computational number theory Zeta and $L$-functions: analytic theory Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 August 2016

David Harvey ,
Maike Massierer  and
Andrew V. Sutherland
David Harvey
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email d.harvey@unsw.edu.au
Maike Massierer
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia email maike@unsw.edu.au
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA email drew@math.mit.edu

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.

MSC classification

Primary: 11G20: Curves over finite and local fields
Secondary: 11Y16: Algorithms; complexity 11M38: Zeta and $L$-functions in characteristic $p$ 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)

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