The generalized Sato–Tate conjecture for generic curves

Let $C$ be a smooth projective curve of genus $g$. Assume that $C$ is in the generic family. For each $k=1,2,\dots,g$, let $a_k(m;g)$ denote the $m$-th moment of the corresponding Frobenius coefficient distribution, and let $c_k(m;g)$ denote the $m$-th moment of the coefficient $X_k$ of the characteristic polynomial of a Haar-random matrix in $\operatorname{USp}(2g)$. Generalized Sato–Tate conjecture. For each $k=1,2,\dots,g$ and $m\geq 0$, we have

$$a_k(m;g)=c_k(m;g).$$

This asserts that the coefficient distributions arising from curves in the generic family agree with the corresponding distributions for random matrices in $\operatorname{USp}(2g)$; the source presents this as the generalized Sato–Tate conjecture.