Algebraic twists of modular forms and Hecke orbits

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• 作者：étienne Fouvry ; Emmanuel Kowalski ; Philippe Michel
• 关键词：Modular forms ; Fourier coefficients ; Hecke eigenvalues ; Hecke orbits ; horocycles ; ?adic Fourier transform ; Riemann Hypothesis over finite fields ; 11F11 ; 11F32 ; 11F37 ; 11T23 ; 11L05
• 刊名：Geometric And Functional Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：25
• 期：2
• 页码：580-657
• 全文大小：744 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8970

文摘

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ?/em>-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.