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On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average
Published online by Cambridge University Press: 01 April 2015
Abstract
For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$, on average, over all good reductions of $E$ modulo primes $p$. We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.
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- © The Author(s) 2015
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