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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

Igor E. Shparlinski
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney NSW 2052, Australia email [email protected]
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email [email protected]

Abstract

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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.

Type
Research Article
Copyright
© The Author(s) 2015 

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