The trace of Frobenius of an elliptic curve and its the number of points over the finite field it is over are both integers, $a_p$ and $p$, satisfying the relation $a_p^2\leq 4p$.
It seems to me that a nonstandard angle should come from is a pair consequence of nonstandard numbers, $a_u$ and $u$, satisfying $a_u^2\leq 4u$. Then with the natural map from nonstandard integers to nonstantard realsgeneralized Sato-Tate conjecture, $a_u/2\sqrt{u}$ would be a nonstandard real number in the interval $[-1,1]$, thus that given a real number, and the $\cos$ of some real number.
In particular you're working over an ultrafilter non-CM elliptic curve over the set of primes, so you know what you want $u$ to be already. So the question is for ways to find nonstandard numbers $a_u$ satisfying this inequality.
The Tate module allows you to compute\mathbb Q$, from an any element of $\sigma \in Gal(\bar{\mathbb Q}/\mathbb Q)$, an element of and a real number $\hat{\mathbb Z}$, \in[−1,1]$, one can construct an ultrafilter on the trace, primes such that if $\sigma=Frob_p$, is equal Frobenius converges to $a_p$ everywhere except $\mathbb Z_p$. Since $\hat{\mathbb Z}$ is a compactification that element and the angle of an integers, there is a map Frobenius converges to it from any notion of nonstandard integersthat real number.
Thus any notion that sends automorphisms of $\bar{\mathbb Q}$, or $\mathbb C$, attempt to angles or traces of Frobenius, should probably form a commutative diagram with the trace answer this question must somehow make use of the Tate module and that maptranscendentals.
The reason this cannot serve as a definition on its own is I have no idea how one might do thatas far as .
(I know this map would not be injectivehad a different answer earlier, nor would the traces need to satisfy the relevant inequality.but this comment is a better one.)
