The asymptotic distribution of Elkies primes for reductions of abelian varieties is Gaussian

TL;DR

This paper extends the concept of Elkies primes to abelian varieties with real multiplication, showing their distribution converges to a Gaussian, which impacts point counting algorithms for these varieties.

Contribution

It generalizes Elkies primes to abelian varieties with real multiplication and proves their distribution is Gaussian, refining previous elliptic curve results.

Findings

Elkies primes for abelian varieties follow a Gaussian distribution

Distribution convergence is proven under large Galois image assumptions

Implications for the complexity of SEA point counting algorithms

Abstract

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let $A$ be an abelian variety with RM over a number field whose attached Galois representation has large image. Then the number of Elkies primes (in a suitable range) for reductions of $A$ modulo primes converges weakly to a Gaussian distribution around its expected value. This refines and generalizes results obtained by Shparlinski and Sutherland in the case of non-CM elliptic curves, and has implications for the complexity of the SEA point counting algorithm for abelian surfaces over finite fields.