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?



Fourier coefficients for elliptic curves on average
There is a very clear discussion of some results in this direction in section 1 of Lenstra's paper ["Factoring Integers with Elliptic Curves"](http://www.jstor.org/stable/1971363). 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](http://jlms.oxfordjournals.org/cgi/reprint/s1-43/1/57).


