Chapter 26 The Sato–Tate distribution
\(\ell\)\(X\)\(C(X)\)\(X \to \CC\text{;}\)\(\mu\)\(X\text{,}\)\(C(X) \to \CC\)\(x_1, x_2, \dots\)\(X\)\(\mu\)\(f\text{,}\)\(X\)\(K\)\(X\)\(G\)\(K\)\(X\)\(K\text{.}\)\(x_1, x_2, \dots\)\(\mu\)\(\chi: G \to \CC\)\(G\text{,}\)\(\chi\)\(L\)\(K\)\(X\)\(x_1, x_2, \dots\)\(X\text{,}\)\(x_i \to N(x_i)\)\(\Real(s) > 1\text{,}\)\(\Real(s) \geq 1\)\(\Real(s) \geq 1\)\(s=1\text{.}\)\(\rho\)\(K\)\(\chi\text{.}\)\(\rho(x_i)\)\(L(s,\rho)\)\(\Real(s) > 1\text{,}\)\(\Real(s) \geq 1\)\(\Real(s) \geq 1\)\(s=1\text{.}\)\(x_i\)\(N(x_i) \leq n\)\(n/\log n\)\(n
\to \infty\text{.}\)\(\chi\)\(G\text{,}\)\(-c(\chi)\)\(L(s,\rho)\)\(s=1\text{.}\)\(c\)\(n \in \ZZ\text{,}\)\(c\)\(i\)\(N(x_i) \leq c\text{.}\)\(x_i\)\(c(\chi) = 0\)\(\chi\text{.}\)\(E\)\(\alpha_p\)\(x^2 - a_p x + p\)\(\arg(\alpha_p/\sqrt{p})\)\([0, \pi]\)\(\frac{2}{\pi} \sin^2 \theta d\theta\text{.}\)\(E\)\(E\)\(E\)\(E\)\(\CC\text{,}\)\(E\)\(\CC\)\(E\)\(j(E) \notin \ZZ\text{.}\)\(E\)\(j\)\(E\)\(K = SU(2)\text{,}\)\(2 \times 2\)\(X\)\(\alpha_p\)\(x_p\)\(X\)\(\theta = \arg(\alpha_p/\sqrt{p})\text{.}\)\(X\)\(x_p\)\(K\)\(1/2\text{.}\)\(P_n(T)\)\(1\)\(\alpha_p^n, \alpha_p^{n-1}
\overline{\alpha_p}, \dots, \overline{\alpha_p}^n\text{.}\)\(j(E) \notin \ZZ\text{,}\)\(\CC\text{.}\)\(\Real(s) > 3/2\text{,}\)\(\Real(s) \geq 3/2\text{.}\)\(\alpha_1, \dots, \alpha_m\)\(1, \alpha_1, \dots, \alpha_m\)\(\QQ\text{.}\)\(x_n = (n \alpha_1, \dots, n \alpha_m) \in (\RR/\ZZ)^m\)\(\log n\)\(\RR/\ZZ\text{.}\)
