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.
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- Last edited by Andrew Sutherland on 2021-01-01 15:06:31
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- 2021-01-01 15:06:31 by Andrew Sutherland (Reviewed)
- 2021-01-01 15:06:13 by Andrew Sutherland
- 2018-06-20 04:20:27 by Kiran S. Kedlaya (Reviewed)