Knightly, Andrew; Li, Charles

On the Distribution of Satake Parameters for Siegel Modular Forms

Doc. Math. 24, 677-747 (2019)
DOI: 10.25537/dm.2019v24.677-747
Communicated by Don Blasius

Summary

We prove a harmonically weighted equidistribution result for the \(p\)-th Satake parameters of the family of automorphic cuspidal representations of \(\operatorname{PGSp}(2n)\) of fixed weight \(\mathtt{k}\) and prime-to-\(p\) level \(N\to \infty\). The main tool is a new asymptotic Petersson formula for \(\operatorname{PGSp}(2n)\) in the level aspect.

Mathematics Subject Classification

11F46, 11F25, 11F30, 11F70, 11F72

Keywords/Phrases

Siegel modular forms, Petersson formula, Satake parameters

References

  • 1. M. Asgari and R. Schmidt, Siegel modular forms and representations, Manuscripta Math. 104 (2001), no. 2, 173-200. DOI 10.1007/PL00005869; zbl 0987.11037; MR1821182.
  • 2. V. Blomer, J. Buttcane, and N. Raulf, A Sato-Tate law for $GL(3)$, Comm. Math. Helv. 89 (2014), no. 4, 895-919. DOI 10.4171/cmh/337; zbl 1317.11053; MR3284298.
  • 3. B. J. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc., 43 (1968), 57-60. DOI 10.1112/jlms/s1-43.1.57; zbl 0183.25503; MR0230682.
  • 4. V. Blomer, Spectral summation formula for GSp(4) and moments of spinor L-functions, J. Eur. Math. Soc., to appear. https://arxiv.org/abs/1602.00780.
  • 5. R. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45 (1978), no. 1, 1-18. DOI 10.1007/BF01406220; zbl 0351.10019; MR0472701.
  • 6. R. Bruggeman and R. Miatello, Eigenvalues of Hecke operators on Hilbert modular groups, Asian J. Math. 17 (2013), no. 4, 729-757. DOI 10.4310/ajm.2013.v17.n4.a10; zbl 1286.11068; MR3152262; arxiv 0912.1692.
  • 7. A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition. Mathematical Surveys and Monographs, 67. Amer. Math. Soc., Providence, RI, 2000. zbl 0980.22015; MR1721403.
  • 8. D. Bump, Lie Groups, Springer-Verlag, New York, 2004. zbl 1053.22001; MR2062813.
  • 9. P. Cartier, Representations of $\mathfrak{p}$-adic groups: a survey, p. 111-155, Proc. Sympos. Pure Math. XXXIII Part 1, Amer. Math. Soc., Providence, 1979. zbl 0421.22010; MR0546593.
  • 10. M. Chida, H. Katsurada, and K. Matsumoto, On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups, Abh. Math. Semin. Univ. Hambg. 84 (2014), no. 1, 31-47. DOI 10.1007/s12188-013-0087-x; zbl 1371.11088; MR3197011; arxiv 1111.4648.
  • 11. M. Dickson, Local spectral equidistribution for degree 2 Siegel modular forms in level and weight aspects, Int. J. Number Theory 11 (2015), no. 2, 341-396. DOI 10.1142/S1793042115500190; zbl 1396.11075; MR3325425; arxiv 1312.5584.
  • 12. W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991. zbl 0744.22001; MR1153249.
  • 13. B. Gross, On the Satake isomorphism. In: Galois representations in arithmetic algebraic geometry (Durham, 1996), 223-237, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998. zbl 0996.11038; MR1696481.
  • 14. C. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61, (1955). 474-523. DOI 10.2307/1969810; zbl 0066.32002; MR0069960.
  • 15. Harish-Chandra, Representations of semisimple Lie groups VI. Integrable and square-integrable representations, Amer. J. Math. 78 (1956), 564-628. DOI 10.2307/2372674; zbl 0072.01702; MR0082056.
  • 16. T. Haines, R. Kottwitz, and A. Prasad, Iwahori-Hecke algebras, J. Ramanujan Math. Soc., 25 (2010), no. 2, 113-145. zbl 1202.22013; MR2642451; arxiv math/0309168.
  • 17. H. Hecht and W. Schmid, On integrable representations of a semisimple Lie group, Math. Ann. 220 (1976), no. 2, 147-149. DOI 10.1007/BF01351699; zbl 0363.22008; MR0399358.
  • 18. Hua L. K., Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. zbl 0483.10001; MR0665428.
  • 19. A. E. Ingham, An integral which occurs in statistics, Proc. Cambridge Philos. Soc. 29, (1933), 1-26. zbl 0007.00701.
  • 20. H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations I \& II, Amer. J. Math. 103 (1981), 499-588 \& 777-815. DOI 10.2307/2374103; zbl 0473.12008; MR0623137.
  • 21. Y. Kitaoka, Fourier coefficients of Siegel cusp forms of degree two, Nagoya Math. J. 93 (1984), 149-171. DOI 10.1017/S0027763000020778; zbl 0531.10031; MR0738922.
  • 22. A. Knapp, Representation theory of semisimple groups, an overview based on examples, Princeton University Press, Princeton, NJ, 1986. zbl 0604.22001; MR0855239.
  • 23. Klingen, H Introductory lectures on Siegel modular forms. Cambridge Studies in Advanced Mathematics, 20. Cambridge University Press, Cambridge, 1990. zbl 0693.10023; MR1046630.
  • 24. A. Knightly and C. Li, Traces of Hecke operators, Mathematical Surveys and Monographs, 133. Amer. Math. Soc., 2006. zbl 1120.11024; MR2273356.
  • 25. A. Knightly and C. Li, Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms, in Modular forms on Schiermonnikoog, Cambridge University Press, 2008. DOI 10.1017/cbo9780511543371.011; zbl 1228.11062; MR2512361.
  • 26. A. Knightly and C. Li, Kuznetsov's trace formula and the Hecke eigenvalues of Maass forms, Mem. Amer. Math. Soc. 224 (2013), no. 1055. DOI 10.1090/s0065-9266-2012-00673-3; zbl 1314.11038; MR3099744.
  • 27. A. Knightly and C. Reno, Weighted distribution of low-lying zeros of $GL(2)$ $L$-functions, Can. J. Math. 71 (2019), no. 1, 153-182. DOI 10.4153/CJM-2018-013-8; zbl 07027347; MR3928260; arxiv 1806.05869.
  • 28. E. Kowalski, A. Saha, and J. Tsimerman, A note on Fourier coefficients of Poincaré series, Mathematika 57 (2011), no. 1, 31-40. DOI 10.1112/S0025579310001804; zbl 1220.11063; MR2764154; arxiv 1008.2288.
  • 29. E. Kowalski, A. Saha, and J. Tsimerman, Local spectral equidistribution for Siegel modular forms and applications, Compos. Math. 148 (2012), no. 2, 335-384. DOI 10.1112/S0010437X11007391; zbl 1311.11037; MR2904191; arxiv 1010.3648.
  • 30. H. Kim, S. Wakatsuki, and T. Yamauchi, An equidistribution theorem for holomorphic Siegel modular forms for $GSp_4$ and its applications, preprint, 2016. https://arxiv.org/abs/1604.02036.
  • 31. C. Li, On the distribution of Satake parameters of $GL_2$ holomorphic cuspidal representations, Israel J. Math. 169 (2009), 341-373. DOI 10.1007/s11856-009-0014-0; zbl 1231.11054; MR2460909.
  • 32. H. Maass, Über die Darstellung der Modulformen $n$-ten Grades durch Poincarésche Reihen, Math. Ann. 123, (1951), 125-151. DOI 10.1007/bf02054945; zbl 0042.32001; MR0043128.
  • 33. D. Miličić, Asymptotic behaviour of matrix coefficients of the discrete series, Duke Math. J. 44 (1977), no. 1, 59-88. DOI 10.1215/s0012-7094-77-04403-9; zbl 0398.22022; MR0430164.
  • 34. J. Matz and N. Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on $SL(n,\mathbf{R})/SO(n)$, preprint, 2016. https://arxiv.org/abs/1505.07285.
  • 35. H. Montgomery and R. Vaughan, Multiplicative number theory I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007. zbl 1142.11001; MR2378655.
  • 36. M. Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45. Academic Press, New York-London, 1972. zbl 0254.15009; MR0340283.
  • 37. A. Pitale, A. Saha, and R. Schmidt, On the standard $L$-function for $GSp_{2n}\times GL_1$ and algebraicity of symmetric fourth $L$-values for $GL(2)$, preprint, 2018. https://arxiv.org/abs/1803.06227.
  • 38. J. Rogawski, Representations of $GL(n)$ and division algebras over a $p$-adic field, Duke Math. J. 50 (1983), no. 1, 161-196. DOI 10.1215/S0012-7094-83-05006-8; zbl 0523.22015; MR0700135.
  • 39. W. Rudin, Functional Analysis, Second Edition, Int. Ser. Pure and App. Math., McGraw-Hill, Inc., New York, 1991. zbl 0867.46001; MR1157815.
  • 40. A. Saha, On ratios of Petersson norms for Yoshida lifts, Forum Math. 27 (2015), no. 4, 2361-2412. DOI 10.1515/forum-2013-0093; zbl 1383.11060; MR3365801; arxiv 1303.5246.
  • 41. F. Shahidi, Eisenstein series and automorphic $L$-functions, Colloquium Publications, 58. American Mathematical Society, Providence, RI, 2010. zbl 1215.11054; MR2683009.
  • 42. S. W. Shin, Automorphic Plancherel Density Theorem, Israel J. Math. 192 (2012), no. 1, 83-120. DOI 10.1007/s11856-012-0018-z; zbl 1300.22006; MR3004076.
  • 43. C. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2) 36 (1935), no. 3, 527-606. DOI 10.2307/1968644; zbl 0012.19703; MR1503238.
  • 44. C. Siegel, Symplectic geometry, Amer. J. Math. 65, (1943). 1-86. DOI 10.2307/2371774; zbl 0063.07003; MR0008094.
  • 45. P. Sarnak, S. W. Shin, and N. Templier, Families of $L$-functions and their symmetry, in: Families of automorphic forms and the trace formula, 531-578, Simons Symp., Springer, [Cham], 2016. DOI 10.1007/978-3-319-41424-9_13; zbl 06675153; MR3675175.
  • 46. S. W. Shin and N. Templier, Sato-Tate theorem for families and low-lying zeros of automorphic L-functions, Invent. Math. 203 (2016), no. 1, 1-177. DOI 10.1007/s00222-015-0583-y; zbl 06541949; MR3437869; arxiv 1208.1945.
  • 47. P. Trombi and V. S. Varadarajan, Asymptotic behaviour of eigen functions on a semisimple Lie group: the discrete spectrum, Acta. Math. 129 (1972), no. 3-4, 237-280. DOI 10.1007/BF02392217; zbl 0244.43006; MR0393349.
  • 48. E. P. van den Ban, Induced representations and the Langlands classification, in Representation Theory and Automorphic Forms, pp. 123-155, T. N. Bailey, A. W. Knapp, eds, Proceedings of Symposia in Pure Mathematics, Vol. 61, Amer. Math. Soc. 1997. zbl 0888.22009; MR1476496.
  • 49. F. Waibel, Moments of spinor $L$-functions and symplectic Kloosterman sums, preprint, 2018. https://arxiv.org/abs/1805.11502.
  • 50. F. Zhou, Weighted Sato-Tate Vertical Distribution of the Satake Parameter of Maass Forms on $PGL(N)$, Ramanujan J. 35 (2014), no. 3, 405-425. DOI 10.1007/s11139-013-9535-6; zbl 1326.11021; MR3274875; arxiv 1303.0889.

Affiliation

Knightly, Andrew
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA
Li, Charles
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

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