Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof’s algorithm. While we are currently missing the tools we need to generalize Elkies’ methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication Sean Ballentine Department of Mathematics, University of Maryland, 4176 Campus Dr., College Park, MD 207424015, USA e-mail: [email protected] Aurore Guillevic Inria Nancy Grand Est, Equipe Caramba, 615 rue du jardin botanique, CS 20101, 54603 Villers-lèsNancy Cedex, France e-mail: [email protected] Elisa Lorenzo Garcı́a IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: [email protected] Chloe Martindale Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands e-mail: [email protected] Maike Massierer School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia e-mail: [email protected] Benjamin Smith INRIA and Laboratoire d’Informatique de l’École polytechnique (LIX), 91120 Palaiseau, France e-mail: [email protected] Jaap Top University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science, P.O. Box 407, 9700AK Groningen, The Netherlands e-mail: [email protected] 1 ar X iv :1 70 1. 01 92 7v 2 [ m at h. N T ] 1 9 A pr 2 01 7 by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.