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Genus-2 curves and Jacobians with a given number of points

Part of: Arithmetic algebraic geometry Abelian varieties and schemes Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  01 February 2015

Reinier Bröker ,
Everett W. Howe ,
Kristin E. Lauter  and
Peter Stevenhagen
Reinier Bröker
Affiliation:
Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA email reinier@math.brown.edu
Everett W. Howe
Affiliation:
Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, USA email however@alumni.caltech.edu
Kristin E. Lauter
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA email klauter@microsoft.com
Peter Stevenhagen
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands email psh@math.leidenuniv.nl

Abstract

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We study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.

In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.

On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.

In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.

Supplementary materials are available with this article.

MSC classification

Primary: 14K22: Complex multiplication
Secondary: 11G15: Complex multiplication and moduli of abelian varieties 11G20: Curves over finite and local fields 14G15: Finite ground fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 170 - 197
Copyright
© The Author(s) 2015 

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