Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

Sean Ballentine, Aurore Guillevic, Elisa Lorenzo-Garcia, Chloe Martindale, Maike Massierer, Benjamin Smith, Jaap Top

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

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 chapter, 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.
Original languageEnglish
Title of host publicationAlgebraic Geometry for Coding Theory and Cryptography
Chapter3
Pages63-94
Number of pages31
Publication statusPublished - 16 Nov 2017

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