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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**Knowl status:**

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

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**History:**(expand/hide all)

- 2021-05-11 23:41:54 by Kiran S. Kedlaya (Reviewed)
- 2021-05-01 10:16:14 by Andrew Sutherland
- 2021-01-25 13:52:13 by Andrew Sutherland
- 2021-01-24 15:39:36 by Andrew Sutherland
- 2021-01-01 15:20:49 by Andrew Sutherland
- 2019-04-20 14:11:50 by Kiran S. Kedlaya (Reviewed)
- 2019-04-20 14:11:32 by Kiran S. Kedlaya
- 2016-05-04 03:29:51 by Andrew Sutherland

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