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This is a question which has bounced around my head over the past few years. At the same time, I am answering Riemann hypothesis for zeta function of algebraic curves over fields of infinite characteristic. 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?

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It is a consequence of the generalized Sato–Tate conjecture, that given a non-CM elliptic curve over $\mathbb Q$, any element of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$, 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.

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  • $\begingroup$ 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. $\endgroup$
    – Marty
    Commented Aug 10, 2012 at 22:41
  • $\begingroup$ My understanding from reading the wikipedia page is that for smooth proper varieties like elliptic curves, Monsky-Washnitzer cohomology is the same as Grothendieck's crystalline cohomology. Since crystalline cohomology is De Rham cohomology in characteristic zero, I am worried that any version of the construction would be the same as De Rham cohomology and thus be uninteresting. $\endgroup$
    – Will Sawin
    Commented Aug 10, 2012 at 23:40
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    $\begingroup$ Just to sound the death knell: It seems clear, although Sato-Tate-type theorems might not actually be strong enough currently, that given an elliptic curve, any element of $Gal(\bar{\mathbb Q}/Q)$, and a real number $[-1,1]$, 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. $\endgroup$
    – Will Sawin
    Commented Aug 11, 2012 at 21:33
  • $\begingroup$ 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! $\endgroup$
    – Marty
    Commented Aug 18, 2012 at 23:38
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    $\begingroup$ My claim implies that you cannot determine $\cos(\theta_u)$ number using just the elliptic curve and an automorphism of $\bar{\mathbb Q}$. Thus you must somehow take advantage of the action on the larger field, which I have no idea how to do. $\endgroup$
    – Will Sawin
    Commented Aug 19, 2012 at 4:03

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