## Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem (2001)

Citations: | 14 - 3 self |

### BibTeX

@TECHREPORT{Cojocaru01cyclicityof,

author = {Alina Carmen Cojocaru and M. Ram Murty},

title = {Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem},

institution = {},

year = {2001}

}

### OpenURL

### Abstract

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