https://hal.inria.fr/inria-00420605Smith, BenjaminBenjaminSmithTANC - Algorithmic number theory for cryptology - LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en AutomatiqueFamilies of Explicit Isogenies of Hyperelliptic JacobiansHAL CCSD2010Isogenyhyperelliptic curveJacobian[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]Smith, BenjaminDavid Kohel and Robert Rolland2009-09-29 14:07:462023-03-24 14:52:522009-09-29 20:37:13enConference papershttps://hal.inria.fr/inria-00420605/documentapplication/pdf1We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically absolutely simple, and describe the kernels of the isogenies. 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.