Generalized Sato–Tate conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $g$ defined over $\mathbb{Q}$. For each prime $p$ of good reduction, write the normalized Euler factor as

$$\overline{L}_{p}(A,T)=\sum_{i=0}^{2g}a_i(A)(p)T^i.$$

Let $\operatorname{ST}(A)\subset\operatorname{USp}(2g)$ be the Sato–Tate group of $A$, and for $i=0,1,\dots,2g$ let

$$I_i=\left[-{2g\choose i},{2g\choose i}\right]$$

and let $\Phi_i\colon\operatorname{ST}(A)\to I_i$ send an element to the $i^{\text{th}}$ coefficient of its characteristic polynomial. Denote the Haar measure on $\operatorname{ST}(A)$ by $\mu(\operatorname{ST}(A))$ and its pushforward under $\Phi_i$ by $\Phi_{i*}(\mu(\operatorname{ST}(A)))$. Generalized Sato–Tate conjecture. For each $i=0,1,\dots,2g$, the values $a_i(A)(p)$ are equidistributed, with respect to increasing size, on $I_i$ with respect to $\Phi_{i*}(\mu(\operatorname{ST}(A)))$. This conjecture predicts that the Sato–Tate group governs the distribution of normalized Euler factors at the primes of good reduction. The form stated here was recently proved; see Serre (2011), Lectures on $N_X(p)$ for references.