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Families of explicitly isogenous Jacobians of variable-separated curves

Part of: Abelian varieties and schemes Computational number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 August 2011

Benjamin Smith
Benjamin Smith*
Affiliation:
INRIA Saclay–Île-de-France, Laboratoire d’Informatique (LIX), École Polytechnique, 91128 Palaiseau Cedex, France (email: smith@lix.polytechnique.fr)

Abstract

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We construct six infinite series of families of pairs of curves (X,Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs (f,g) of polynomials such that f(x1)−g(x2) is reducible.

Supplementary materials are available with this article.

MSC classification

Secondary: 14K02: Isogeny 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11Y99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 179 - 199
Copyright
Copyright © London Mathematical Society 2011

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