Let $E$ be an elliptic curve (without CM) over a number field $K.$ Is it known that $a(\mathfrak{p})=N(\mathfrak{p})+1-|E(\mathbb{F}_{\mathfrak{p}})|$ is neither zero, not $2\sqrt{N(\mathfrak{p})}$ for almost all primes (in the sense of density) ? In literature I know, Sato-Tate holds for totally real number fields with some additional conditions on the elliptic curve. I am actually asking for this particular case of length zero, but for arbitrary number fields.
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$\begingroup$ Of course $a(p)$ is never irrational, so your second sentence is equivalent to $a(p)$ is non-zero for almost all primes -- or I misunderstand something. For curves with complex multiplication that is very wrong and for the others it depends on what you mean by "almost all primes". Maybe you could improve the question. $\endgroup$– Chris WuthrichCommented Mar 4, 2020 at 8:15
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$\begingroup$ Actually, yes assume E is without CM, and when we are looking over arbitary number fields, $a(\mathfrak{p})=N(\mathfrak{p})+1-|E(\mathbb{F}_{\mathfrak{p}})|$. I m asking for density of prime ideals $\mathfrak{p}$ in $\mathcal{O}_K$ such that $a(\mathfrak{p})$ is neither zero, nor $N(\mathfrak{p})^{1/2}$, and $\mathbb{F}_{\mathfrak{p}}$ is the residue field at $\mathfrak{p}$. $\endgroup$– AnonymousCommented Mar 4, 2020 at 8:46
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$\begingroup$ I think it would be good to modify the question accordingly. $\endgroup$– Chris WuthrichCommented Mar 4, 2020 at 8:51
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$\begingroup$ I just did, does it look good now ? $\endgroup$– AnonymousCommented Mar 4, 2020 at 8:53
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$\begingroup$ Yes, by the way mathoverflow.net/questions/115555/… is related $\endgroup$– Chris WuthrichCommented Mar 4, 2020 at 8:56
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