Where do nonstandard elliptic curve angles come from? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:17:02Z http://mathoverflow.net/feeds/question/104435 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104435/where-do-nonstandard-elliptic-curve-angles-come-from Where do nonstandard elliptic curve angles come from? Marty 2012-08-10T20:11:32Z 2013-02-14T23:45:54Z <p>This is a question which has bounced around my head over the past few years. At the same time, I am answering <a href="http://mathoverflow.net/questions/104421/riemann-hypothesis-for-zeta-function-of-algebraic-curves-over-fields-of-infinite" rel="nofollow">http://mathoverflow.net/questions/104421/riemann-hypothesis-for-zeta-function-of-algebraic-curves-over-fields-of-infinite</a> with another question.</p> <p>Let $E$ be an elliptic curve over $Q$. Let $u$ be an nonprincipal ultrafilter on the set of prime numbers. </p> <p>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 $u$.</p> <p>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 $(E, C, \sigma)$?</p> <p>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.</p> <p>Any ideas? Anyone thought about RH in models of ACFA?</p> http://mathoverflow.net/questions/104435/where-do-nonstandard-elliptic-curve-angles-come-from/104438#104438 Answer by Will Sawin for Where do nonstandard elliptic curve angles come from? Will Sawin 2012-08-10T21:04:25Z 2013-02-14T23:45:54Z <p>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.</p> <p>Thus any attempt to answer this question must somehow make use of the transcendentals. I have no idea how one might do that.</p> <p>EDIT: By <a href="http://mathoverflow.net/questions/121824/what-sets-of-primes-can-we-pick-out-with-first-order-statements/121847#121847" rel="nofollow">ACL's answer to my question</a>, the nonstandard angle of Frobenius is totally independent from all first-order statements about the nonstandard elliptic curve.</p>