| Our main focus in this thesis is on the cyclicity of the group of points of the reduction modulo primes p of an elliptic curve defined over the rationals. More precisely, given an elliptic curve E defined over , we investigate the asymptotic behaviour of the number f( x, ) of primes p ≤ x such that the group of points of the reduction of E modulo p is cyclic.; An asymptotic formula of the form f(x, ) ∼ ∫Eπ(x), where ∫E is some constant depending on E and π(x) is the number of primes ≤ x, was first obtained by Jean-Pierre Serre in 1976, under the assumption of a generalized Riemann hypothesis (denoted GRH). Important work on the asymptotic behaviour of f(x, ) was further done by Ram Murty in 1979 and 1987, and by Rajiv Gupta and Ram Murty in 1990.; In this thesis we consider the problems of obtaining an unconditional asymptotic formula for f(x, ), if possible, and of providing explicit error terms for such a formula. The different properties of elliptic curves with or without complex multiplication (denoted CM) lead us to different analyses and results in the two situations. In the case of a non-CM elliptic curve we obtain an effective asymptotic formula for f(x, ) under the assumption of a quasi-GRH. In the case of a CM elliptic curve we obtain an unconditional effective asymptotic formula for f(x, ). Using the ideas involved in the proofs of these results we make significant improvements in the size of the error terms in the asymptotic formulae for f(x, ), under GRH. Consequently we obtain interesting upper estimates for the smallest prime p for which the group of points of E modulo p is cyclic.; Observing that the cyclicity of the group of points of E modulo p is ensured if the group has square-free order, we are naturally led to considering the problem of finding (effective) asymptotic formulae for the number h(x, ) of primes p ≤ x for which the order of the group E modulo p is square-free (apart possibly from the part composed of primes of bad reduction for E). (Abstract shortened by UMI.)... |
