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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn059, 29 pages, doi:10.1093/imrn/rnn059 published on June 13, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Sums of Hecke Eigenvalues over Values of Quadratic Polynomials

Valentin Blomer

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada

Correspondence: Correspondence to be sent to: vblomer{at}math.toronto.edu

Let Formula be a cusp form for {Gamma}0(N), weight k ≥ 4, and character {chi}. Let Formula be a quadratic polynomial. It is shown that


Formula 1

for some constant c = c(f, q). The constant vanishes in many (but not all) cases, for example, if k is even or if {Delta} = s2 – 4t > 0. On the way, a Kuznetsov formula for half-integral weight and entries having different sign is derived.



References

  1. Biró A. Cycle integrals of Maass forms of weight 0 and Fourier coefficients of Maass forms of weight 1/2. Acta Arithmetica (2000) 94:103–52.[ISI]
  2. Blomer V. On the central value of symmetric square L-functions. Mathematische Zeitschrift. forthcoming.
  3. Bykovskii V. A. Spectral decompositions of certain automorphic functions and their number-theoretic applications. Journal of Soviet Mathematics (1987) 36:8–21.[CrossRef]
  4. Deshouillers J.-M., Iwaniec H. Kloosterman sums and Fourier coefficients of cusp forms. Inventiones Mathematicae (1982) 70:219–88.[CrossRef][ISI]
  5. Duke W., Friedlander J., Iwaniec H. The subconvexity problem for Artin L-functions. Inventiones Mathematicae (2002) 149:489–507.[CrossRef][ISI]
  6. Friedlander J., Iwaniec H. A polynomial divisor problem. Journal fur die Reine und Angewandte Mathematik (2006) 601:109–37.[ISI]
  7. Gradshteyn I. S., Ryzhik I. M. Tables of Integrals, Series, and Products (1994) 5th ed. New York: Academic Press.
  8. Hooley C. On the number of divisors of quadratic polynomials. Acta Arithmetica (1963) 110:97–114.
  9. Iwaniec H. Fourier coefficients of modular form of half-integral weight. Inventiones Mathematicae (1987) 87:385–401.[CrossRef][ISI]
  10. Iwaniec H., Kowalski E. Analytic Number Theory (2004) Providence, RI: American Mathematical Society. American Mathematical Society Colloquium Publications 53.
  11. Kim H. "Functoriality for the exterior square of GL4 and the symmetric fourth of GL2," with Appendix 1 by D. Ramakrishnan and Appendix 2 by H. Kim and P. Sarnak. In: Journal of the American Mathematical Society (2003) 16:139–83.[CrossRef][ISI]
  12. Kim H. Functoriality and number of solutions of congruences. Acta Arithmetica (2007) 128:235–43.[ISI]
  13. Motohashi Y. Spectral Theory of the Riemann Zeta Function (1997) Cambridge, United Kingdom: Cambridge University Press.
  14. Proskurin N. V. On the general Kloosterman sums. Journal of Mathematical Sciences (2005) 129:3874–89.[CrossRef]
  15. Serre J.-P., Stark H. M. Modular forms of weight 1/2. In: Modular Functions of One Variable VI (1977) Springer: New York. 27–67. Lecture Notes in Mathematics 627.
  16. Shimura G. On modular forms of half-integral weight. Annals of Mathematics (1973) 97:440–81.[CrossRef][ISI]
  17. Shimura G. On the holomorphy of certain Dirichlet series. Proceedings of the London Mathematical Society (1975) 31(3):79–98.[CrossRef][ISI]
  18. Sneddon I. N. The Use of Integral Transforms (1972) New York: McGraw Hill.
  19. Templier N. Points spéciaux et valeurs spéciales de fonctions L. PhD thesis, in preparation.
  20. Tipu V. Polynomial divisor problems. (2008) Ph.D. thesis, University of Toronto.

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This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
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