The Sato-Tate group of a motive over a number field is a compact (but often disconnected) Lie group which conjecturally governs the variation of Euler factors of the L-function associated to this motive. For example, for (the 1-motive of) a generic abelian variety of dimension $g$, this group is $\mathrm{USp}(2g)$.

For an elliptic curve $E$ over a number field $K$, the Sato-Tate group equals $\mathrm{SU}(2)$ if $E$ does not have complex multiplication. Otherwise, one gets $\mathrm{SO}(2)$ if $E$ has CM defined over $K$, and the normalizer of $\mathrm{SO}(2)$ in $\mathrm{SU}(2)$ if $E$ has CM not defined over $K$.

For an abelian surface $A$ over a number field $K$, there are 52 possible Sato-Tate groups as classified by Fité-Kedlaya-Rotger-Sutherland [arXiv:1110.6638, 10.1112/S0010437X12000279, MR:2982436]; of these, 34 can occur for $K = \mathbb{Q}$. The typical case $\mathrm{USp}(4)$ occurs when $\mathrm{End}(A_{\overline{K}}) = \mathbb{Z}$.

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- Last edited by Kiran S. Kedlaya on 2019-04-20 14:00:14

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