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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
Abstract

Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Frobenius fields for elliptic curves
 Amer. J. Math
"... Let E/Q be an elliptic curve over the field of rational numbers, with End ¯ Q(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E ov ..."
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Cited by 5 (3 self)
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Let E/Q be an elliptic curve over the field of rational numbers, with End ¯ Q(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K; x) be the number of primes p ≤ x such that the field extension Q(πp(E)) is the fixed imaginary quadratic field K. We present upper bounds for ΠE(K; x) obtained using two different approaches. The first one, inspired from work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the elliptic curve E. The second one, inspired from work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained using the first approach are better, ΠE(K; x) ≪ x 4/5 /(log x) 1/5, and are the best known so far. The bounds obtained using the second approach are weaker, but are ∗ also affiliated with the Institute of Mathematics of the Romanian Academy 1 independent of the number field K, a property which is essential for other applications. All