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The orthogonality of Hecke eigenvalues

Published online by Cambridge University Press:  04 December 2007

Henryk Iwaniec  and
Xiaoqing Li
Henryk Iwaniec
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA iwaniec@math.rutgers.edu
Xiaoqing Li
Affiliation:
Department of Mathematics, Columbia University, MC 4423, New York, NY 10027, USA xl29@math.buffalo.edu Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA
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Abstract

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In this paper, we study the orthogonalities of Hecke eigenvalues of holomorphic cusp forms. An asymptotic large sieve with an unusually large main term for cusp forms is obtained. A family of special vectors formed by products of Kloosterman sums and Bessel functions is constructed for which the main term is exceptionally large. This surprising phenomenon reveals an interesting fact: that Fourier coefficients of cusp forms favor the direction of products of Kloosterman sums and Bessel functions of compatible type.

Keywords

orthogonalityHecke eigenvaluesholomorphic cusp formslarge sieveKloosterman sumsBessel functions11F0311F1111F1211F30
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 3 , May 2007 , pp. 541 - 565
Copyright
Foundation Compositio Mathematica 2007