Component group of a Sato-Tate group (reviewed)

The component group of a Sato-Tate group $G$ is the quotient $G/G^0$ of $G$ by its identity component $G^0$, which is a normal subgroup of finite index.

If $G$ is the Sato-Tate group of a motive defined over a number field $K$, then the component group $G/G^0$ is canonically isomorphic to the Galois group of a finite extension $L/K$. For the motive attached to an abelian variety $A$ of dimension at most 3, $L$ is the smallest field over which all endomorphisms of $A_{\overline{K}}$ (the base change of $A$ to an algebraic closure of $K$) are defined.

Component groups are named according to their isomorphism type. This does not determine them uniquely; one can obtain a more explicit description by examining the list of generators. Common notations used in component group names include:

• $C_n$, the cyclic group of order $n$;
• $D_n$, the dihedral group of order $2n$;
• $A_n$, the alternating group on $n$ letters;
• $S_n$, the symmetric group on $n$ letters.