On the refined Koblitz conjecture

Abstract

Let p be a prime, E be a non-CM elliptic curve over $Q$, and $N_p$ be the number of points of E over $F_p$. Given $t\in N$, we are concerned with the asymptotic formula for the set of primes for which $N_p/t$ 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 $N_p$, 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.