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<p>We use supercharacter theory to study moments of Gaussian periods. For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1090_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(p-1=dk\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo>=</mo> <mi>d</mi> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> and fixed <i>k</i>, we compute the fourth absolute moments for all but finitely many primes <i>p</i>. For <i>d</i> fixed, we relate the fourth absolute moments to the number of rational points on modified Fermat curves. For small <i>d</i>, this relation is in terms of a single curve. For larger <i>d</i>, we provide both exact formulas using families of modified Fermat curves and bounds via Hasse–Weil.</p>

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Moments of Gaussian periods and modified Fermat curves

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月18日

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  • Stephan Ramon Garcia,
  • Brian Lorenz,
  • George Todd

摘要

We use supercharacter theory to study moments of Gaussian periods. For \(p-1=dk\) p - 1 = d k 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.