Maybe my question wasn't clear enough as stated. The comparison theorems between cohomology theories imply that there are (at least two) ways of of counting points on curves over finite fields with cohomology. Etale (equiv. using the Tate module) cohomology or Monsky-Washnitzer are two examples, and of course they compute the same number of points on the curve. What's interesting to me is that, by easy topological reasons, there are also nonstandard elliptic curve angles coming from ultrafilters on the set of primes. But nobody knows what these nonstandard angles really mean!

I'm pretty sure that what you described as "cos of some real number" is exactly what I would call $\cos(\theta_u)$ in my question. It's just a question of when you want to make the transition from nonstandard back to standard. I agree that the Tate module approach won't work... generic automorphisms of $\bar Q$ won't act nicely. I would be more hopeful about a purely $p$-adic approach, like Monsky-Washnitzer cohomology but with $p$-adic numbers replaced by Laurent series over ultra-finite fields.

Here's some heuristics for predicting how many values of $z$ are needed: the solutions to $x^2 + y^2 + z^2 = n$ are roughly equidistributed on the sphere, and the number of solutions is proportional to the class number $h(-4n)$, which grows like $\sqrt{n}$. So consider $C \cdot \sqrt{n}$ points equidistributed on the sphere of area $4 \pi n$. What is the statistical expectation of the smallest (positive) $z$-coordinate among these points?

You're right - it was misleading for me to describe the recursive definition, when it requires a more complicated formula to define $(x,c^x)$. I didn't follow the golden rule: look for every instance of "Of course" -- that's where the mistakes are.

You might want to see the closely related question I asked: mathoverflow.net/questions/25929/u3-sato-tate-measure My perspective now is that a closed form for the measure is messy, and it's better to compute moments instead. There are closed forms for the first few moments for the SU(3) Sato-Tate measure (I remember the number 12 since it agreed with my student's numerical evidence).

Better to take a sheaf of Frechet spaces, methinks. Or, take the base to be a measure space and consider Hilbert space bundles over that.

I think this is wide wide open.

This answer really belongs as a comment on Broussous's answer.

It's the splitting field over $Q$ of the polynomial $x^4-x^3-7*x^2+2*x+9$. See the PDF printout of the SAGE code on my webpage for more.

Nope - I think that's just where it ends. Basically, I numerically checked the functional equation of the degree 3 Artin L-function. The bibliography is on p.16, and afterwards is an appendix. I'll check the print copy in my office to make sure it's all there, but I think that's it.

monodromy.com is a real website. But holonomy.com is not. :)