Sato-Tate group identity component (reviewed)

The Sato-Tate group of an abelian variety $A$ is a compact Lie group that is not necessarily connected; it may have a finite number of distinct components. The component containing the identity element of the Sato-Tate group $\mathrm{ST}(A)$ is the identity component, denoted $\mathrm{ST}^0(A)$; it acts as the identity of the component group $\mathrm{ST}(A)/\mathrm{ST}^0(A)$.

For an abelian variety $A$ of dimension 2, including the Jacobian of a genus 2 curve, there are six possibilities for the identity component $\mathrm{ST}^0(A)$ (up to conjugacy), corresponding to the six possibilities for its real geometric endomorphism algebra $\mathrm{End}(A_{\overline{\Q}})\otimes\R$ (up to isomorphism). These are listed below:

• $\mathrm{U}(1)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \mathrm{M}_2(\C)$, with real dimension 1; this occurs when $A_{\overline{\Q}}$ is isogenous to the square of a CM elliptic curve.
• $\mathrm{SU}(2)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \mathrm{M}_2(\R)$, with real dimension 3; this occurs when $A_{\overline{\Q}}$ is isogenous to the square of a non-CM elliptic curve.
• $\mathrm{U}(1)\times\mathrm{U}(1)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \C\times \C$, with real dimension 2; this occurs both when $A_{\overline{\Q}}$ is isogenous to a product of non-isogenous CM elliptic curves and when $A_{\overline{\Q}}$ is a simple abelian surface with CM.
• $\mathrm{U}(1)\times\mathrm{SU}(2)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \R\times\C$, with real dimension 4; this occurs when $A_{\overline{\Q}}$ is isogenous to the product of a CM elliptic curve with a non-CM ellliptic curve.
• $\mathrm{SU}(2)\times\mathrm{SU}(2)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \R\times\R$, with real dimension 6; this occurs both when $A_{\overline{\Q}}$ is isogenous to a product of non-isogenous non-CM elliptic curves and when $A_{\overline{\Q}}$ is a simple abelian surface with RM.
• $\mathrm{USp}(4)$, corresponding to $\mathrm{End}(A_{\overline{\Q}})\otimes\R\simeq \R$, with real dimension 10; this is the generic case.

Here CM denotes complex multiplication, and RM denotes real multiplication.

For arbitrary $A$, there is a canonical surjective homomorphism from the component group $\mathrm{ST}(A)/\mathrm{ST}^0(A)$ to the group $\mathrm{Gal}(K/\Q)$, where $K$ is the minimal field over which all endomorphisms of $A_{\overline{\Q}}$ can be defined. For $A$ of dimension $\leq 3$, this homomorphism is an isomorphism.