  We use supercharacter theory to study moments of Gaussian periods. For
$p-1=dk$ and fixed $k$, we compute the fourth absolute moments for all but
finitely many primes $p$. For $d$ fixed, we relate the fourth absolute moments
to the number of rational points on modified Fermat curves. For small $d$, this
relation is in terms of a single curve. For larger $d$, we provide both exact
formulas using families of modified Fermat curves and bounds via Hasse--Weil.
