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A Sato-Tate group $G$ of degree $d$ is said to be rational if for every irreducible character $\chi$ of $\GL_d(\C)$ and every component $H$ of $G$ we have $$ \int_H \chi(g)\mu_H \in \Z, $$ where $\mu_H$ denotes the restriction to $H$ of the Haar measure $\mu$ on $G$, normalized so that $\mu_H(H)=1$. If $G$ is the Sato-Tate group of an arithmetic L-function $$ L(s)=\prod_{\mathfrak p}L_\mathfrak{p}(N(\mathfrak{p})^{-s})^{-1}, $$ this means that the polynomials $L_{\mathfrak p}(T)$ have integer coefficients, equivalently, the $L$-function $L(s)$ is rational.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2021-01-01 15:02:58
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