The **fourth trace moment** of a Sato-Tate group $G$ is the expected value of the fourth power of the trace of a random element of $G$ (under the Haar measure).

By a result known as "Larsen's alternative" [MR:2058618], the fourth trace moment is less than or equal to 5 if and only if $G$ is irreducible. and for $G$ of odd weight, the fourth trace moment is 3 when $G=\mathrm{USp}(2g)$ and greater than 3 otherwise.

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- Last edited by Andrew Sutherland on 2021-03-19 21:30:05

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