Sato-Tate group (reviewed)

The Sato-Tate group of a motive $X$ is a compact Lie group $G$ containing (as a dense subset) the image of a representation that maps Frobenius elements to conjugacy classes. When $X$ is an Artin motive, $G$ corresponds to the image of the Artin representation; when $X$ is an abelian variety over a number field, one can define $G$ in terms of an $\ell$-adic Galois representation attached to $X$.

For motives of even weight $w$ and degree $d$, the Sato-Tate group is a compact subgroup of the orthogonal group $\mathrm{O}(d)$. For motives of odd weight $w$ and even degree $d$, the Sato-Tate group is a compact subgroup of the unitary symplectic group $\mathrm{USp}(d)$. For motives $X$ arising as abelian varieties, the weight is always $w=1$ and the the degree is $d=2g$, where $g$ is the dimension of the variety.

The simplest case is when $X$ is an elliptic curve $E/\Q$, in which case $G$ is either $\mathrm{SU}(2)=\mathrm{USp}(2)$ (the generic case), or $G$ is $N(\mathrm{U}(1))$, the normalizer of the subgroup $\mathrm{U}(1)$ of diagonal matrices in $\mathrm{SU}(2)$, which contains $\mathrm{U}(1)$ with index 2.

The generalized Sato-Tate conjecture states that when ordered by norm, the sequence of images of Frobenius elements under this representation is equidistributed with respect to the pushforward of the Haar measure of $G$ onto its set of conjugacy classes. This is known for all elliptic curves over totally real number fields (including $\mathbb{Q}$) or CM fields.