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

2024/11/27 by Benoist, Alexandre, Kieffer, Jean
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.18171

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.

Related

• The Counting function for Elkies primes
• Counting points on abelian surfaces over finite fields with Elkies's method
• On the Product of Small Elkies Primes
• Almost prime values of the order of abelian varieties over finite fields
• Positive density of primes of ordinary reduction for abelian varieties of simple signature