Title: Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication,
Abstract: 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.