Functiones et Approximatio Commentarii Mathematici

Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$

Nigel Watt

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Abstract

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a~Hecke congruence subgroup $\Gamma\leq SL(2,{\mathbb Z}[i])$, and correspond to exceptional eigenvalues of the Laplace operator on the space $L^2(\Gamma\backslash SL(2,{\mathbb C})/SU(2))$. These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a~proof of one such application, which is an upper bound for a~sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-functions with groessencharakters). Our proofs make extensive use of Lokvenec-Guleska's generalisation of the Bruggeman-Motohashi summation formulae for $PSL(2,{\mathbb Z}[i])\backslash PSL(2,{\mathbb C})$. We also employ a~bound of Kim and Shahidi for the first eigenvalues of the relevant Laplace operators, and an `unweighted' spectral large sieve inequality (our proof of which is to appear separately).

Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 2 (2013), 213-376.

Dates
First available in Project Euclid: 18 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1371586157

Digital Object Identifier
doi:10.7169/facm/2013.48.2.4

Mathematical Reviews number (MathSciNet)
MR3100141

Zentralblatt MATH identifier
1277.11041

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11L05: Gauss and Kloosterman sums; generalizations 11L07: Estimates on exponential sums 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11N13: Primes in progressions [See also 11B25] 11N35: Sieves 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R44: Distribution of prime ideals [See also 11N05] 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 44A15: Special transforms (Legendre, Hilbert, etc.)

Keywords
spectral theory large sieve mean value Hecke congruence group Gaussian number field Gaussian integers sum formula automorphic form cusp form non-holomorphic modular form Fourier coefficient Kloosterman sum inverse Bessel transform eigenvalue conjecture grössencharakter Hecke character

Citation

Watt, Nigel. Weighted spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,\mathbb{Z}[i])$. Funct. Approx. Comment. Math. 48 (2013), no. 2, 213--376. doi:10.7169/facm/2013.48.2.4. https://projecteuclid.org/euclid.facm/1371586157


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References

  • T.M. Apostol, Mathematical Analysis, 2nd edition, World Student Series, Addison-Wesley, Reading MA, 1974.
  • R.W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), 1–18.
  • R.W. Bruggeman and R.J. Miatello, Estimates of Kloosterman sums for groups of real rank one, Duke Math. J. 80 (1995), 105–137.
  • R.W. Bruggeman and Y. Motohashi, Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field, Functiones et Approximatio 31 (2003), 23–92.
  • J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982), 219–288.
  • J.-M. Deshouillers and H. Iwaniec, Power mean values of the Riemann zeta-function, Mathematika 29 (1982), 202–212.
  • J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Springer Monographs in Mathematics, 1997.
  • G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, (5th edition), Oxford University Press 1979 (reprinted with corrections in 1983 and 1984).
  • M.N. Huxley, The large sieve inequality for algebraic number fields, Mathematika 15 (1968), 178–187.
  • A. Ivić, The Riemann Zeta-Function: Theory and Applications, Dover Publications, Inc., 2003.
  • H. Iwaniec, Spectral Theory of Automorphic Functions, lecture notes (manuscript), Rutgers University, 1987.
  • H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics 17, AMS, Providence RI, 1997.
  • H.H. Kim, On local $L$-functions and normalized intertwining operators, Canad. J. Math. 57 (2005), no. 3, 535–597.
  • H.H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177–197.
  • N.V. Kuznetsov, Petersson hypothesis for forms of weight zero and Linnik hypothesis, preprint no. 02, Khabarovsk Complex Res. Inst., East Siberian Branch Acad. Sci. USSR, Khabarovsk, 1977 (in Russian).
  • N.V. Kuznetsov, Petersson hypothesis for parabolic forms of weight zero and Linnik hypothesis. Sums of Kloosterman sums, Mat. Sbornik. 111(153) (1980), no. 3, 334–383.
  • S. Lang, Real Analysis, (2nd edn.), Addison-Wesley, Reading MA, 1983.
  • R.P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544, Springer-Verlag, Berlin, 1976.
  • H. Lokvenec-Guleska, Sum Formula for SL${}_2$ over Imaginary Quadratic Number Fields, thesis (in English, with summaries in Dutch and Macedonian), University of Utrecht, 2004.
  • O. Ramaré and R. Rumely, Primes in arithmetic progressions, Math. Comp. 65, No. 213, Jan. 1996, 397–425.
  • N. Watt, Kloosterman sums and a mean value for Dirichlet polynomials, J. Number Theory 53 (1995), 179–210.
  • N. Watt, Spectral large sieve inequalities for Hecke congruence subgroups of $SL(2,{\Bbb Z}[i])$, preprint submitted for publication, arXiv:1302.3112v1 [math.NT], 2013.
  • N. Watt, Weighted fourth moments of Hecke zeta functions with groessencharakters, preprint in preparation.
  • E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Fourth edn. Cambridge University Press, 1927.