  Here we study bounds on the Frobenius-Schur exponent of spherical fusion
categories based on their global dimension generalizing bounds from the
representation theory of finite-dimensional quasi-Hopf algebras. Our main
result is that if the Frobenius-Schur exponent of a modular fusion category is
a prime power for some prime integer $p$, then it is bounded by the norm of its
global dimension when $p$ is odd, and four times the norm of its global
dimension when $p=2$; these bounds are optimal. If one assumes in addition
pseudounitarity, the categories achieving the optimal bound are completely
described and we attain similar bounds and classifications for arbitrary
spherical fusion categories. This proof includes an explicit classification of
modular fusion categories of Frobenius-Perron dimension $p^5$; all examples
which are not pointed are constructed explicitly from the Ising categories and
the representation theory of extraspecial $p$-groups.
