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A Sato-Tate group of weight $w$ and degree $d$ is a subgroup of either:

  • $\mathrm{USp}(d)$, if $w$ is odd (in which case $d$ must be even);
  • $\mathrm{O}(d)$, if $w$ is even (in which case $d$ may be even or odd).

Here $\mathrm{USp}(d)\subseteq \mathrm{GL}_d(\mathbb{C})$ denotes the group of $d\times d$ complex matrices that are both unitary ($\bar A A^{\rm t}=I$) and symplectic ($A^{\rm t}\Omega A = \Omega$ for some fixed symplectic form $\Omega$), while $\mathrm{O}(d)\subseteq\mathrm{GL}_d(\mathbb{C})$ is the group of $d\times d$ orthogonal matrices ($AA^{\rm t}=I$). Both are classical real Lie groups.

In either case, the group $\mathrm{USp}(d)$ or $\mathrm{O}(d)$ is called the ambient group of the Sato-Tate group.