The Lang-Trotter Conjecture on Average

For an elliptic curve E over Q and an integer, r let π^r_E (x) be the number of primes p ≤ x of good reduction such that the trace of the Frobenius morphism of E/F_p 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 > x^1/2+ε and AB > x^3/2+ε, then the arithmetic mean of π^r_E (x) over all elliptic curves E : y² = x³ + ax + b with a, b ∈ Z, |a| ≤ A and |b| ≤ B is ∼ Cr √ x/ log x, where C_r is some constant depending on r. This improves a result of C. David and F. Pappalardi. Moreover, we establish an “almost-all” result on π^r_E (x).