This is a question which has bounced around my head over the past few years. At the same time, I am answering http://mathoverflow.net/questions/104421/riemann-hypothesis-for-zeta-function-of-algebraic-curves-over-fields-of-infinite with another question.

Let $E$ be an elliptic curve over $Q$. Let $u$ be an nonprincipal ultrafilter on the set of prime numbers.

For each prime $p$ (at which $E$ has good reduction, let's say), let $\pm \theta_p$ be the elliptic curve angle at $p$. In other words, $a_p = 2 \sqrt{p} \cdot \cos(\theta_p)$. Then, by the compactness of the interval $[-1,1]$, there is a nonstandard elliptic curve angle $\theta_u$ naturally associated to the set $(\theta_p)$ and $u$.

I've been wondering if there's any other way to produce these nonstandard angles. For example, let $\sigma$ be a "generic" field automorphism of the complex numbers $C$, in the sense that $(C,\sigma)$ is a model of $ACFA$. Can one produce an elliptic curve angle $\theta$ directly from the data $(E, C, \sigma)$?

What's so difficult here is that, in the transfer from characteristic $p$ to characteristic $0$, it is so difficult to figure out how to handle things like $\sqrt{p}$. The only hope, that I can see, would be to think of $a_p$ as a $p$-adic number (use $p$-adic cohomology), and then transfer the result to a Laurent series field (so $a_u$ might belong to $C((\varpi))$ and $a_u / \sqrt{\varpi}$ would be well-behaved). But this is all "pie in the sky" for now.

Any ideas? Anyone thought about RH in models of ACFA?