Remark 63.30.1. Here are some observations concerning this notion.

If modeling projective curves then we can use cohomology and we don't need factor $q^ n$.

The only examples I know are $\Gamma = \pi _1(X, \overline\eta )$ where $X$ is smooth, geometrically irreducible and $K(\pi , 1)$ over finite field. In this case $q = (\# k)^{\dim X}$. Modulo the proposition, we proved this for curves in this course.

Given the integer $q$ then the sets $S_ d$ are uniquely determined. (You can multiple $q$ by an integer $m$ and then replace $S_ d$ by $m^ d$ copies of $S_ d$ without changing the formula.)

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