Shifted convolution of cusp-forms with θ-series

Article

Abstract

We generalize the classical Voronoi formula for
$$r_{l}(n) = \#\{ (n_{1}, \ldots , n_{l}) \in \mathbf{Z}^{l}, n_{1}^{2} + \cdots + n_{l}^{2} = n \},$$
and as an application, we derive a sharp bound for the shifted convolution sum convolving the Fourier coefficients of holomorphic cusp forms with those of theta series.

Keywords

Shifted convolution Theta series and Poincare series Cusp form Voronoi formula 

Mathematics Subject Classification (2000)

11F11 11F27 11F30 11F37 

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References

  1. 1.
    Duke, W., Iwaniec, H.: Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes. Math. Ann. 286(4), 783–802 (1990) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Goldfeld, D.: Analytic and arithmetic theory of Poincaré series. Astérisque 61, 95–107 (1979) MathSciNetMATHGoogle Scholar
  3. 3.
    Good, A.: Cusp forms and eigenfunctions of the Laplacian. Math. Ann. 255(4), 523–548 (1981) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gradshtein, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1965) Google Scholar
  5. 5.
    Hafner, J.L.: Explicit estimates in the arithmetic theory of cusp forms and Poincaré series. Math. Ann. 264, 9–20 (1983) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Iwaniec, H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87(2), 385–401 (1987) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. Am. Math. Soc., Providence (1997) MATHGoogle Scholar
  8. 8.
    Luo, W., Sarnak, P.: Quantum variance for Hecke eigenforms. Ann. Sci. Ecole Norm. Super. (4) 37(5), 769–799 (2004) MathSciNetMATHGoogle Scholar
  9. 9.
    Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Proc. Symp. Pure Math. Theory of Numbers, Vol. VIII, pp. 1–15. AMS, Providence (1965) Google Scholar
  10. 10.
    Voronoi, G.: Sur une fonction transcendante et ses applications á la sommation de quelques séries. Ann. Sci. Ecole Norm. Super. 21(3), 207–267 (1904), 459–533 MathSciNetGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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