If $a_i$ denotes the $i$th coefficient of the characteristic polynomial $\sum a_ix^i$ of a random matrix $g$ in a Sato-Tate group $G$, then the **probability**
\[
\mathrm{P}[a_i=n]
\]
denotes the measure of the point-mass $n$ in the image $X$ of the continuous map $g\mapsto a_i$ under the pushforward of the normalized Haar measure of $G$ to the compact real interval $X$.

This probability is zero unless $n$ is one of a finite set of integers, and it is always a rational number whose denominator divides the order of the component group (if we restrict to a single component and renormalize the measure, the probabiliy is always 0 or 1).

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2019-04-20 14:11:50

**Referred to by:**

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

- 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

**Differences**(show/hide)