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The degree $d$ of a Sato-Tate group is the degree of the characteristic polynomials of its elements, equivalently, the dimension of the $d\times d$ matrices it contains.

For an abelian variety $A$ over a number field, the degree $d$ of its Sato-Tate group is twice its dimension $g$ as an abelian variety (if $A=\mathrm{Jac}(C)$ is the Jacobian of a curve $C$, then $g$ is also the genus of $C$). The degree $d=2g$ is then also the degree of the characteristic polynomials of the Frobenius endomorphism of the reductions of $A$ modulo good primes.

For Artin motives, the degree of the Sato-Tate group is the same as the degree of the Artin representation (as a permutation group).

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  • Last edited by Kiran S. Kedlaya on 2018-06-20 04:06:54
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