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

Research paper by Sean Ballentine, Aurore Guillevic, Elisa Lorenzo García, Chloe Martindale, Maike Massierer, Benjamin Smith, Jaap Top

Indexed on: 08 Jan '17Published on: 08 Jan '17Published in: arXiv - Mathematics - Number Theory


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 by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.