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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).

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