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The weight $w$ of the Sato-Tate group $G$ of a motive $X$ is determined by the cohomology group $H^w(X,\mathbb{Q}_\ell)$ used to define $G$. The roots of the characteristic polynomials of Frobenius are then $q$-weil numbers $\alpha$ of weight $w$, meaning that $|\iota(\alpha)|=q^w/2$ for every embedding $\iota\colon \mathbb{Q}(\alpha)\to \mathbb{C}$.

Knowl status:
  • Review status: beta
  • Last edited by Andrew V. Sutherland on 2016-05-03 23:19:02.328000