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For an elliptic curve E/Q without complex multiplication we study thedistribution of Atkin and Elkies primes l, on average, over all good reductionsof E modulo primes p. We show that, under the Generalised Riemann Hypothesis,for almost all primes p there are enough small Elkies primes l to ensure thatthe Schoof-Elkies-Atkin point-counting algorithm runs in (log p)^(4+o(1))expected time.

On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average