modular forms - Fourier coefficients for elliptic curves on average

As others have mentioned, if $p$ is fixed then you're really looking at elliptic curves over a fixed finite field.

From some points of view an interesting variant would be to look at elliptic curves say $E_{a,b}:y^2 = x^3 + ax + b$ where $a$ and $b$ vary over integers in a box, say $|a| leq A$ and $|b| leq B$ and relatively small compared to $p$. The one might try to find asymptotic results that hold as $p$, $A$, $B$ get large together. If $A$ and $B$ aren't too big then this is giving more information about individual curves. For example, in bounding the average analytic rank of elliptic curves it is important to get a good bound on $$frac{1}{AB} sum_{p < P} sum_{|a| leq A} sum_{|b| leq B} a_P(E_{a,b})$$ with $A$ and $B$ as small as possible. For example, see A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109(3), 445–472 (1992).

In a different but related direction, there is a paper of David and Pappalardi, Average Frobenius distributions of elliptic curves (it's the fourth from the bottom) on this subject. They get a kind of Lang-Trotter on average, so they are varying both $p$ and the coefficients defining the elliptic curves. Stephan Baier later made some improvements on this problem here.