Linear independence of Poincaré series of exponential type via non-analytic methods
HTML articles powered by AMS MathViewer
- by Siegfried Böcherer and Soumya Das PDF
- Trans. Amer. Math. Soc. 367 (2015), 1329-1345 Request permission
Abstract:
Given a finite set $\{T\}$ of symmetric, positive definite, half-integral $n$ by $n$ matrices over $\mathbf {Z}$ which are inequivalent under the action of $\mathrm {GL}(n, \mathbf {Z})$, we show that the corresponding set of Poincaré series $\{ P_k^n(T) \}$ attached to them are linearly independent for weights $k$ in infinitely many arithmetic progressions. We also give a quite explicit description of those arithmetic progressions for all even degrees, when the matrices $T$ have no improper automorphisms and their level is an odd prime. Our main tools are theta series with simple harmonic polynomials as coefficients and techniques familiar from the theory of modular forms mod $p$.References
- Scott Ahlgren, On the irreducibility of Hecke polynomials, Math. Comp. 77 (2008), no. 263, 1725–1731. MR 2398790, DOI 10.1090/S0025-5718-08-02078-4
- Anatolij N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286, Springer-Verlag, Berlin, 1987. MR 884891, DOI 10.1007/978-3-642-70341-6
- A. N. Andrianov and V. G. Zhuravlëv, Modular forms and Hecke operators, Translations of Mathematical Monographs, vol. 145, American Mathematical Society, Providence, RI, 1995. Translated from the 1990 Russian original by Neal Koblitz. MR 1349824, DOI 10.1090/mmono/145
- A. N. Andrianov, The action of Hecke operators on Maass theta series and zeta functions, Algebra i Analiz 19 (2007), no. 5, 3–36 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 5, 675–698. MR 2381939, DOI 10.1090/S1061-0022-08-01015-7
- R. Bauer, Über die methodische Bedeutung von Eisensteinschen und Poincaréschen Reihen in der Theorie der Siegelschen Modulformen, Diplomarbeit Mathematisches Institut Universität Freiburg, 1988 (unpublished).
- Siegfried Böcherer, On the Hecke operator $U(p)$, J. Math. Kyoto Univ. 45 (2005), no. 4, 807–829. With an appendix by Ralf Schmidt. MR 2226631, DOI 10.1215/kjm/1250281658
- Siegfried Böcherer and Shoyu Nagaoka, On mod $p$ properties of Siegel modular forms, Math. Ann. 338 (2007), no. 2, 421–433. MR 2302069, DOI 10.1007/s00208-007-0081-7
- Siegfried Böcherer and Shoyu Nagaoka, On Siegel modular forms of level $p$ and their properties mod $p$, Manuscripta Math. 132 (2010), no. 3-4, 501–515. MR 2652444, DOI 10.1007/s00229-010-0357-1
- S. Böcherer and S. Nagaoka, On p-adic properties of Siegel modular forms. arXiv:1305.2813[math.NT]
- Siegfried Böcherer, Jens Funke, and Rainer Schulze-Pillot, Trace operator and theta series, J. Number Theory 78 (1999), no. 1, 119–139. MR 1706909, DOI 10.1006/jnth.1999.2404
- Youngju Choie, Winfried Kohnen, and Ken Ono, Linear relations between modular form coefficients and non-ordinary primes, Bull. London Math. Soc. 37 (2005), no. 3, 335–341. MR 2131386, DOI 10.1112/S0024609305004285
- Ulrich Christian, Über Hilbert-Siegelsche Modulformen und Poincarésche Reihen, Math. Ann. 148 (1962), 257–307 (German). MR 145105, DOI 10.1007/BF01451138
- Soumya Das and Jyoti Sengupta, Nonvanishing of Siegel Poincaré series, Math. Z. 272 (2012), no. 3-4, 869–883. MR 2995143, DOI 10.1007/s00209-011-0961-0
- Soumya Das and Satadal Ganguly, Linear relations among Poincaré series, Bull. Lond. Math. Soc. 44 (2012), no. 5, 988–1000. MR 2975157, DOI 10.1112/blms/bds028
- Soumya Das, Winfried Kohnen, and Jyoti Sengupta, Nonvanishing of Siegel-Poincaré series II, Acta Arith. 156 (2012), no. 1, 75–81. MR 2997572, DOI 10.4064/aa156-1-6
- A. Evdokimov, Action of the irregular Hecke operator of index $p$ on the theta series of a quadratic form. J. Soviet Math. 38 (1977), 141–155.
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- Yoshiyuki Kitaoka, Representations of quadratic forms and their application to Selberg’s zeta functions, Nagoya Math. J. 63 (1976), 153–162. MR 424688
- Helmut Klingen, Zur Transformationstheorie von Thetareihen indefiniter quadratischer Formen, Math. Ann. 140 (1960), 76–86 (German). MR 122802, DOI 10.1007/BF01359987
- Helmut Klingen, Über Poincarésche Reihen vom Exponentialtyp, Math. Ann. 234 (1978), no. 2, 145–157 (German). MR 480357, DOI 10.1007/BF01420965
- Helmut Klingen, Introductory lectures on Siegel modular forms, Cambridge Studies in Advanced Mathematics, vol. 20, Cambridge University Press, Cambridge, 1990. MR 1046630, DOI 10.1017/CBO9780511619878
- Wen Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315. MR 369263, DOI 10.1007/BF01344466
- Hans Maass, Über die Darstellung der Modulformen $n$-ten Grades durch Poincarésche Reihen, Math. Ann. 123 (1951), 125–151 (German). MR 43128, DOI 10.1007/BF02054945
- Hans Maass, Konstruktion von Spitzenformen beliebigen Grades mit Hilfe von Thetareihen, Math. Ann. 226 (1977), no. 3, 275–284. MR 444575, DOI 10.1007/BF01362431
- Hans Petersson, Über Heckesche Operatoren, Poincarésche Reihen und eine Siegelsche Konstruktion, Acta Arith. 24 (1973), 411–434 (German). MR 337792, DOI 10.4064/aa-24-4-411-434
- R. A. Rankin, The vanishing of Poincaré series, Proc. Edinburgh Math. Soc. (2) 23 (1980), no. 2, 151–161. MR 597120, DOI 10.1017/S0013091500003035
- Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
- Goro Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 17 (1975), 261–268. MR 485706
- H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
- J. R. Wilton, Congruence Properties of Ramanujan’s Function tau(n), Proc. London Math. Soc. (2) 31 (1930), no. 1, 1–10. MR 1577449, DOI 10.1112/plms/s2-31.1.1
Additional Information
- Siegfried Böcherer
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan
- Email: boecherer@t-online.de
- Soumya Das
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India
- Email: somu@math.tifr.res.in, soumya.u2k@gmail.com
- Received by editor(s): September 30, 2012
- Received by editor(s) in revised form: May 25, 2013
- Published electronically: July 25, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 1329-1345
- MSC (2010): Primary 11F30; Secondary 11F46
- DOI: https://doi.org/10.1090/S0002-9947-2014-06204-1
- MathSciNet review: 3280046