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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 ..."
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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
The square sieve and the Lang–Trotter conjecture
 Canadian Journal of Mathematics
, 2001
"... 1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a ce ..."
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Cited by 9 (3 self)
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1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a certain generalized Riemann hypothesis we show that this number is OE x 17 18 log x, and unconditionally. We also prove that the number we show that this number is OE,K � 13 x(log log x) 12 (log x) 25 24 of imaginary quadratic fields K, with − disc K ≤ x and of the form K = Q(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 LangTrotter conjecture. 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
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
A PROBLEM OF FOMENKO’S RELATED TO ARTIN’S CONJECTURE
, 2012
"... Let a be a natural number greater than 1. For each prime p, letia(p) denote the index of the group generated by a in F ∗ p. Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of i ..."
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Let a be a natural number greater than 1. For each prime p, letia(p) denote the index of the group generated by a in F ∗ p. Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of ia(p) is constant without using Conjecture A of Hooley. More precisely, we show upon GRH that for any α with 0 ≤ α<1, there is a positive constant cα> 0 such that X (log ia(p)) α ∼ cαπ(x), p≤x where π(x) is the number of primes p ≤ x. We also study related questions.