The Lang-Trotter Conjecture on Average

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+\epsilon}$ and $AB>x^{3/2+\epsilon}$, then the arithmetic mean of $\pi_E^r(x)$ over all elliptic curves $E$ : $y^2=x^3+ax+b$ with $a,b\in \intz$, $|a|\le A$ and $|b|\le B$ is $\sim C_r\sqrt{x}/\log x$, 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 $\pi_E^r(x)$.