The **Sato-Tate group** of a motive $X$ is a compact Lie group $G$ containing (as a dense subset) the image of a representation that maps Frobenius elements to conjugacy classes. When $X$ is an Artin motive, $G$ corresponds to the image of the Artin representation; when $X$ is an abelian variety over a number field, one can define $G$ in terms of an $\ell$-adic Galois representation attached to $X$.

For motives of even weight $w$ and degree $d$, the Sato-Tate group is a compact subgroup of the orthogonal group $\mathrm{O}(d)$. For motives of odd weight $w$ and even degree $d$, the Sato-Tate group is a compact subgroup of the unitary symplectic group $\mathrm{USp}(d)$. For motives $X$ arising as abelian varieties, the weight is always $w=1$ and the the degree is $d=2g$, where $g$ is the dimension of the variety.

The simplest case is when $X$ is an elliptic curve $E/\Q$, in which case $G$ is either $\mathrm{SU}(2)=\mathrm{USp}(2)$ (the generic case), or $G$ is $N(\mathrm{U}(1))$, the normalizer of the subgroup $\mathrm{U}(1)$ of diagonal matrices in $\mathrm{SU}(2)$, which contains $\mathrm{U}(1)$ with index 2.

The generalized Sato-Tate conjecture states that when ordered by norm, the sequence of images of Frobenius elements under this representation is equidistributed with respect to the pushforward of the Haar measure of $G$ onto its set of conjugacy classes. This is known for all elliptic curves over totally real number fields (including $\mathbb{Q}$) or CM fields.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2019-05-02 23:26:44

**Referred to by:**

- ec.invariants
- ec.q.invariants
- g2c.st_group
- rcs.cande.st_group
- st_group.1.2.1.2.1a.bottom
- st_group.1.2.3.1.1a.top
- st_group.1.4.1.48.48a.bottom
- st_group.1.4.10.1.1a.top
- st_group.embedding
- st_group.hodge_circle
- st_group.label
- st_group.moment_matrix
- st_group.moment_simplex
- st_group.moments
- st_group.rational
- st_group.search_input
- st_group.subgroups
- st_group.summary
- st_group.supgroups
- lmfdb/ecnf/templates/ecnf-curve.html (line 161)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 184)
- lmfdb/sato_tate_groups/main.py (line 959)
- lmfdb/sato_tate_groups/main.py (line 966)

**History:**(expand/hide all)

- 2021-01-16 14:29:36 by Andrew Sutherland
- 2021-01-01 15:27:05 by Andrew Sutherland
- 2021-01-01 15:26:53 by Andrew Sutherland
- 2019-05-02 23:26:44 by Kiran S. Kedlaya (Reviewed)
- 2019-05-02 23:24:49 by Kiran S. Kedlaya
- 2019-03-09 14:52:45 by Andrew Sutherland

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