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arXiv · 2411.18171

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

Alexandre Benoist·Jean Kieffer
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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.

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Alexandre Benoist, Jean Kieffer. 2025-11-25. The asymptotic distribution of Elkies primes for reductions of abelian varieties is Gaussian. https://arxiv.org/abs/2411.18171

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