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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.

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