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If $a_i:=a_i(g)$ denotes the $i$th coefficient of the characteristic polynomial $\sum a_ix^i$ of a random matrix $g$ in a Sato-Tate group $G$ with Haar measure $\mu$, then \[ \mathrm{Pr}[a_i=n] := \mu(\{g\in G|a_i(g)=n\}) \] can be viewed as the probability that a random $g\in G$ satisfies $a_i(g)=n$. This is a rational number equal to the proportion of connected components of G on which $a_i$ is identically equal to $n$, by Proposition 8.4.3.3 of Serre's Lecture's on $N_X(p)$ [10.1201/b11315 ]. This probability is necessarily zero unless $n$ is one of a finite set of integers.

Note that while $\mathrm{Pr}[a_i=n]=0$ implies that $\{g\in G:a_i(g) = n\}$ has measure zero, this set need not be empty (unless $G$ is finite).

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  • Last edited by Kiran S. Kedlaya on 2021-05-11 23:41:54
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