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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
Cyclicity of CM elliptic curves modulo p
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. ..."
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Cited by 3 (1 self)
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Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.
On the rrank Artin Conjecture II
"... For any finitely generated subgroup \Gamma of Q we compute a formula for the density of the primes for which the reduction modulo p of \Gamma contains a primitive root modulo p. We use this to conjecture a characterization of "optimal" subgroups (i.e. subgroups that have maximal density). We also ..."
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For any finitely generated subgroup \Gamma of Q we compute a formula for the density of the primes for which the reduction modulo p of \Gamma contains a primitive root modulo p. We use this to conjecture a characterization of "optimal" subgroups (i.e. subgroups that have maximal density). We also improve the error term in the asymptotic formula of [9, Theorem 1.1]. 1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS Let \Gamma be a finitely generated multiplicative subgroup of Q . We denote by supp(\Gamma) the (finite) set of those primes p such that p (a) 6= 0 for some a 2 \Gamma. For all primes p not in supp(\Gamma), we consider the reduction \Gamma p of \Gamma. Precisely, \Gamma p is the subgroup of F p obtained by reducing modulo p all the elements of \Gamma. We let N \Gamma (x) = # \Phi p x; p = 2 supp(\Gamma) j \Gamma p = F p \Psi : (1) The statement that N hai (x) !1 as x !1 (when a is an integer 6= 0, \Sigma1 and not a perfect square), is known as the Artin C...