Question

Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a_p occur? Are there any conjectures?

Answer 1

There is a very clear discussion of some results in this direction in section 1 of Lenstra's paper "Factoring Integers with Elliptic Curves". I'll attempt to summarize.

There are finitely many (about $2p$) isomorphism classes elliptic curves over $\mathbb{F}_p$. Most sampling methods choose the isomorphism class of $E$ with probability $1/|\mathrm{Aut}(E)|$. For example (in characteristic not $2$ or $3$), suppose you pick $a$ and $b$ in $\mathbb{F}_p$ at random and generate the curve $y^2=x^3+ax+b$, discarding it if it is singular. Then the number of $(a,b)$ for which you will generate a curve isomorphic to $E$ is $(p-1)/|\mathrm{Aut}(E)|$. I imagine sampling by height will have the same effect.

Weighting by automorphism groups, the number of $E$ for which $a_p=t$ is $H(t^2-4p)$, where $H(D)$ is the Kronecker class number. For $t = 2 \alpha \sqrt{p}$ with $\alpha$ fixed and $<1$, we are looking at $H(-\Delta)$ for $\Delta = 4(1-\alpha^2) p$. $H(-\Delta)$ is $\Delta^{1/2+o(1)}$, but the oscillations are large enough to swamp the effect of $\alpha$.

In some moral sense, one wants to say that we are converging on the distribution propositional to $\sqrt{1-\alpha^2}$ as $p \to \infty$. In particular, it is true that the moments are approaching the moments of this semicircular distribution; see Birch.

Answer 2

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.

Answer 3

Wouter Castryck spoke about this at a GTEM workshop in Warwick. He considers isomorphism classes of elliptic curves over $\Bbb F_p$. His results are written up here:

http://wis.kuleuven.be/algebra/hubrechts/DistributionTraces.pdf

Of course there are a finite number of isomorphism classes of elliptic curves over a given finite field. So to answer your question for say elliptic curves over $\Bbb Q$ ordered by height, you would have to understand the distribution of the image of mod p reduction.

I hope this at least gives a start.