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

TL;DR

This paper extends Atkin-style point-counting methods to genus-2 hyperelliptic curves with maximal real multiplication, leveraging newly computed modular polynomials to enhance algorithm efficiency.

Contribution

It proves Atkin-style results for genus-2 Jacobians with maximal real multiplication, enabling improved point-counting algorithms using recent modular polynomial computations.

Findings

Proved Atkin-style results for genus-2 Jacobians with real multiplication.

Utilized new modular polynomials to improve point-counting algorithms.

Aimed to enhance practicality of point counting on genus-2 curves.

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.