On the refined Koblitz conjecture

Let p be a prime, E be a non-CM elliptic curve over Q, and N<inf>p</inf> be the number of points of E over F<inf>p</inf>. Given t∈N, we are concerned with the asymptotic formula for the set of primes for which N<inf>p</inf>/t is a prime. The asymptotic constant was first conjectured by Koblitz for t=1 and the conjecture was later refined by Zywina. Assuming an elliptic analogue of the Elliott-Halberstam conjecture and a conjecture on the average order of growth of N<inf>p</inf>, this paper arrives at the conjectured constant, using techniques from classical analytic number theory. This is the first result where the conjectured constant is conditionally determined.