The Lang-Trotter Conjecture on Average

For an elliptic curve E over Q and an integer r let πr E(x) be the number of primes p ≤ x of good reduction such that the trace of the Frobenius morphism of E/Fp equals r. We consider the quantity π r E(x) on average over certain sets of elliptic curves. More in particular, we establish the following: If A,B > x1/2+ε and AB > x3/2+ε, then the arithmetic mean of πr E(x) over all elliptic curves E : y 2 = x3 + ax+ b with a, b ∈ Z, |a| ≤ A and |b| ≤ B is ∼ Cr √ x/ log x, where Cr is some constant depending on r. This improves a result of C. David and F. Pappalardi. Moreover, we establish an “almost-all” result on πr E(x).