Counting points on abelian surfaces over finite fields with Elkies's method

TL;DR

This paper extends Elkies's method to count points on p.p. abelian surfaces over finite fields, achieving improved complexity over Schoof's method under certain conditions, with practical numerical validation.

Contribution

It generalizes Elkies's method from elliptic curves to abelian surfaces, providing new algorithms with better asymptotic complexity for point counting.

Findings

Achieves Otilde(log^4 q) complexity for abelian surfaces with real multiplication.

Achieves Otilde(log^6 q) complexity for counting points over multiple primes.

Numerical experiments confirm the practical effectiveness of the proposed methods.

Abstract

We generalize Elkies's method, an essential ingredient in the SEA algorithm to count points on elliptic curves over finite fields of large characteristic, to the setting of p.p. abelian surfaces. Under reasonable assumptions related to the distribution of Elkies primes, we obtain improvements over Schoof's method in two cases. If the abelian surface A over Fq has RM by a fixed quadratic field F, we reach the same asymptotic complexity Otilde(log4 q) as the SEA algorithm up to constant factors depending on F. If A is defined over a number field, we count points on A modulo sufficiently many primes in Otilde(log6 q) binary operations on average. Numerical experiments demonstrate the practical usability of our methods.