Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. In 1988, Neal Koblitz formulated a conjecture regarding the number of primes p ¡ x for which the group of points of Ep has prime order. The question, still open, may be viewed as an analogue of the twin prime conjecture. I will present a result that says that Koblitz’s conjecture is true on average over a two-dimensional family of elliptic curves. This is joint work with Antal Balog (Budapest) and Chantal David (Montreal).