The Sato–Tate conjecture for Frobenius conjugacy classes of K3 surfaces

Let $X$ be as above, let $i$ be as above, and let $S$ be the finite set of bad primes. For each good prime $p\in\mathbb P\setminus S$, let $x_p\in\mathop{\operatorname{Cl}}(\mathop{\operatorname{ST}}^i(X))$ be the conjugacy class of the semisimple part of the Frobenius element. Let $\pi\colon \mathop{\operatorname{ST}}^i(X)\to\mathop{\operatorname{Cl}}(\mathop{\operatorname{ST}}^i(X))$ be the canonical map, and let $\pi_*\mu_{\operatorname{Haar}}$ be the pushforward of normalized Haar measure. The Sato–Tate conjecture. The sequence $(x_p)_{p\in\mathbb P\setminus S}$, with the good primes ordered normally, is equidistributed with respect to $\pi_*\mu_{\operatorname{Haar}}$; equivalently, the empirical measures converge weakly to this measure. This predicts the distribution of Frobenius conjugacy classes through the compact Sato–Tate group; the statement is presented as a conjecture and is not resolved in the supplied text.