The **Sato-Tate group** of a curve refers to the Sato-Tate group of its Jacobian, which is a special case of the Sato-Tate group of a motive.

For genus 2 curves over $\mathbb{Q}$ there are 6 possibilities for the identity component of the Sato-Tate group and 34 possibilities for the full Sato-Tate group; these are labelled according to the classification in Fité-Kedlaya-Rotger-Sutherland, *Sato-Tate distributions and Galois endomorphism modules in genus 2* [arXiv:1110.6638, MR:2982436, 10.1112/S0010437X12000279]. The generic case is $\mathrm{USp}(4)$, which occurs if and only if the Jacobian of the curve has trivial endomorphism ring.

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2019-05-02 23:16:36

**Referred to by:**

- g2c.2916.b.11664.1.bottom
- g2c.galois_rep.non_maximal_primes
- g2c.st_group_identity_component
- rcs.rigor.g2c
- rcs.source.g2c
- st_group.1.2.B.1.1a.bottom
- lmfdb/genus2_curves/main.py (line 597)
- lmfdb/genus2_curves/main.py (line 767)
- lmfdb/genus2_curves/main.py (line 1058)
- lmfdb/genus2_curves/templates/g2c_curve.html (line 224)
- lmfdb/genus2_curves/templates/g2c_isogeny_class.html (line 71)
- lmfdb/lfunctions/templates/Lfunction.html (line 133)

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

- 2019-05-02 23:16:36 by Kiran S. Kedlaya (Reviewed)
- 2019-05-02 23:13:51 by Kiran S. Kedlaya
- 2018-05-24 16:53:52 by John Cremona (Reviewed)

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