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

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 with respect to the chosen RM. 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.