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Constructing genus-3 hyperelliptic Jacobians with CM

Part of: Arithmetic algebraic geometry Computational aspects in algebraic geometry Curves

Published online by Cambridge University Press:  26 August 2016

Jennifer S. Balakrishnan ,
Sorina Ionica ,
Kristin Lauter  and
Christelle Vincent
Jennifer S. Balakrishnan
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom email balakrishnan@maths.ox.ac.uk
Sorina Ionica
Affiliation:
Laboratoire MIS, Université de Picardie Jules Verne, 33 Rue Saint Leu, 80000 Amiens, France email sorina.ionica@m4x.org
Kristin Lauter
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98062, USA email klauter@microsoft.com
Christelle Vincent
Affiliation:
Department of Mathematics and Statistics, The University of Vermont, 16 Colchester Avenue, Burlington, VT 05401, USA email christelle.vincent@uvm.edu

Abstract

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Given a sextic CM field $K$ , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$ , and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field  $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo  $p$ .

MSC classification

Primary: 11G15: Complex multiplication and moduli of abelian varieties
Secondary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 14H42: Theta functions; Schottky problem 14Q05: Curves

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