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Elliptic curves with a given number of points over finite fields
Given an elliptic curve and a positive integer , we consider the
problem of counting the number of primes for which the reduction of
modulo possesses exactly points over . On average (over a
family of elliptic curves), we show bounds that are significantly better than
what is trivially obtained by the Hasse bound. Under some additional
hypotheses, including a conjecture concerning the short interval distribution
of primes in arithmetic progressions, we obtain an asymptotic formula for the
average.Comment: A mistake was discovered in the derivation of the product formula for
K(N). The included corrigendum corrects this mistake. All page numbers in the
corrigendum refer to the journal version of the manuscrip
On the Distribution of Atkin and Elkies Primes
Given an elliptic curve E over a finite field F_q of q elements, we say that
an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a
square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of
F_q-rational points on E; otherwise ell is called an Atkin prime. We show that
there are asymptotically the same number of Atkin and Elkies primes ell < L on
average over all curves E over F_q, provided that L >= (log q)^e for any fixed
e > 0 and a sufficiently large q. We use this result to design and analyse a
fast algorithm to generate random elliptic curves with #E(F_p) prime, where p
varies uniformly over primes in a given interval [x,2x].Comment: 17 pages, minor edit
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