Density results for automorphic forms on Hilbert modular groups II
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- by Roelof W. Bruggeman and Roberto J. Miatello PDF
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Abstract:
We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\mathrm {SL}_2$ over a totally real number field $F$, with a discrete subgroup of Hecke type $\Gamma _0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see §1.2.4–1.2.13) and products of prescribed small intervals for all but one of the infinite places of $F$. The main tool in the derivation is a sum formula of Kuznetsov type (Sum formula for SL$_2$ over a totally real number field, Theorem 2.1).References
- R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), no. 1, 1–18. MR 472701, DOI 10.1007/BF01406220
- R. W. Bruggeman, R. J. Miatello, and I. Pacharoni, Density results for automorphic forms on Hilbert modular groups, Geom. Funct. Anal. 13 (2003), no. 4, 681–719. MR 2006554, DOI 10.1007/s00039-003-0427-6
- Roelof W. Bruggeman and Roberto J. Miatello, Sum formula for $\rm SL_2$ over a totally real number field, Mem. Amer. Math. Soc. 197 (2009), no. 919, vi+81. MR 2489364, DOI 10.1090/memo/0919
- Roelof W. Bruggeman and Roberto J. Miatello, Distribution of square integrable automorphic forms on Hilbert modular groups, Geometry, analysis and topology of discrete groups, Adv. Lect. Math. (ALM), vol. 6, Int. Press, Somerville, MA, 2008, pp. 19–39. MR 2464392
- Harold Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Differential Geometry 17 (1982), no. 2, 239–253. MR 664496
- J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature, Invent. Math. 52 (1979), no. 1, 27–93. MR 532745, DOI 10.1007/BF01389856
- Isaac Y. Efrat, The Selberg trace formula for $\textrm {PSL}_2(\textbf {R})^n$, Mem. Amer. Math. Soc. 65 (1987), no. 359, iv+111. MR 874084, DOI 10.1090/memo/0359
- Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,\,\textbf {R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197, DOI 10.1007/BFb0061302
- Jonathan Huntley, Spectral multiplicity on products of hyperbolic spaces, Proc. Amer. Math. Soc. 111 (1991), no. 1, 1–12. MR 1031667, DOI 10.1090/S0002-9939-1991-1031667-X
- Jonathan Huntley and David Tepper, A local Weyl’s law, the angular distribution and multiplicity of cusp forms on product spaces, Trans. Amer. Math. Soc. 330 (1992), no. 1, 97–110. MR 1053114, DOI 10.1090/S0002-9947-1992-1053114-X
- Aleksandar Ivić and Matti Jutila, On the moments of Hecke series at central points. II, Funct. Approx. Comment. Math. 31 (2003), 93–108. MR 2059539, DOI 10.7169/facm/1538186641
- Matti Jutila and Yoichi Motohashi, Uniform bound for Hecke $L$-functions, Acta Math. 195 (2005), 61–115. MR 2233686, DOI 10.1007/BF02588051
- Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
- Jean-Pierre Labesse and Werner Müller, Weak Weyl’s law for congruence subgroups, Asian J. Math. 8 (2004), no. 4, 733–745. MR 2127945
- Serge Lang, $\textrm {SL}_{2}(\textbf {R})$, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. MR 0430163
- E. Lapid, W. Müller, Spectral asymptotic for arithmetic quotients of $\mathrm {SL}(n,\mathbb {R})/\mathrm {SO}(n)$, arXiv:0711.2925v1 [math.RT]
- Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms, Geom. Funct. Anal. 17 (2007), no. 1, 220–251. MR 2306657, DOI 10.1007/s00039-006-0589-0
- Lizhen Ji, The Weyl upper bound on the discrete spectrum of locally symmetric spaces, J. Differential Geom. 51 (1999), no. 1, 97–147. MR 1703605
- Stephen D. Miller, On the existence and temperedness of cusp forms for $\textrm {SL}_3({\Bbb Z})$, J. Reine Angew. Math. 533 (2001), 127–169. MR 1823867, DOI 10.1515/crll.2001.029
- Werner Müller, Weyl’s law for the cuspidal spectrum of $\textrm {SL}_n$, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 347–352 (English, with English and French summaries). MR 2057162, DOI 10.1016/j.crma.2004.01.003
- A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Additional Information
- Roelof W. Bruggeman
- Affiliation: Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, NL-3508 TA Utrecht, Nederland
- MR Author ID: 42390
- Email: bruggeman@math.uu.nl
- Roberto J. Miatello
- Affiliation: FaMAF-CIEM, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
- MR Author ID: 124160
- Email: miatello@mate.uncor.edu
- Received by editor(s): October 17, 2008
- Published electronically: February 24, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 3841-3881
- MSC (2000): Primary 11F30, 11F41, 11F72, 22E30
- DOI: https://doi.org/10.1090/S0002-9947-10-04974-3
- MathSciNet review: 2601612