TITLE: Where do nonstandard elliptic curve angles come from?
QUESTION [6 upvotes]: This is a question which has bounced around my head over the past few years.  At the same time, I am answering https://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?

REPLY [2 votes]: It is a consequence of the generalized Sato-Tate conjecture, that given a non-CM elliptic curve over $\mathbb Q$, any element of $Gal(\bar{\mathbb Q}/\mathbb Q)$, and a real number $\in[−1,1]$, one can construct an ultrafilter on the primes such that Frobenius converges to that element and the angle of Frobenius converges to that real number.
Thus any attempt to answer this question must somehow make use of the transcendentals. I have no idea how one might do that.
EDIT: By ACL's answer to my question, the nonstandard angle of Frobenius is totally independent from all first-order statements about the nonstandard elliptic curve.