Moments of Gaussian periods and modified Fermat curves

Abstract We use supercharacter theory to study moments of Gaussian periods. For $$p-1=dk$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> 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.