Abstract. Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication.

Sieve theory has a rich and romantic history. The ancient question of whether there exist infinitely many twin primes (primes p such that p+2 is also prime), and Goldbach's conjecture that every even number can be written as the sum of two prime numbers, have been two of the problems that have inspired the development of the theory. This book provides a motivated introduction to sieve theory. Rather than focus on technical details which can obscure the beauty of the theory, the authors focus on examples and applications, developing the theory in parallel. The text can be used for a senior level undergraduate course or an introductory graduate course in analytic number theory, and non-experts can gain a quick introduction to the techniques of the subject.

Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over F p is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves.Koblitz's conjecture is still widely open. In this paper we prove that Koblitz's conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam, where the average is taken over twin primes and their differences.

Abstract1 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(F p ) 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 = p E for which the group E(F p ) is cyclic and we show that, under the generalized Riemann hypothesis,4+ε if E is without complex multiplication, and p E = O (log N ) 2+ε if E is with complex multiplication, for any 0 < ε < 1.

Abstract. 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(π p ) = K, where π p denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is O E (x 17/18 log x), and unconditionally we show that this number is O E,K x(log log x) 13/12 (log x) 25/24 . We also prove that the number of imaginary quadratic fields K, with − disc K ≤ x and of the form K = Q(π p ), is ≫ E log log log x for x ≥ x 0 (E). These results represent progress towards a 1976 Lang-Trotter conjecture.