2010/09/08 by Benjamin Smith
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Dimension (graph theory) #Isogeny #Isomorphism (crystallography) #Jacobian matrix and determinant #Kernel (algebra) #Moduli #Multiplication (music) #math.NT
paper · pdf · doi:10.1112/s1461157010000410
published as LMS J. Comput. Math. 14 (2011) 179-199
arxiv created 2010/09/08 · openalex publication_date 2011/08/01 · openalex created_date 2016/06/24 · arxiv updated 2019/02/20 · openalex updated_date 2026/08/05
Abstract 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 ( x 1 )− g ( x 2 ) is reducible. Supplementary materials are available with this article.