Skip to main content
SpringerLink
Log in

Discrete series multiplicities for classical groups over \(\mathbf {Z}\) and level 1 algebraic cusp forms

  • Published:
Publications mathématiques de l'IHÉS Aims and scope Submit manuscript

Abstract

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group \(G\) over \(\mathbf {Z}\), and provide numerical applications in absolute rank \(\leq 8\). Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of \(\mathrm{GL}_{n}\) over \(\mathbf {Q}\) (\(n\) arbitrary) whose motivic weight is \(\leq 24\).

In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on \(\mathrm{GL}_{n}\), and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of \(G\), for an appropriate test function.

Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for \(\mathrm{Sp}_{2g}(\mathbf {Z})\): we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Adams and J. F. Johnson, Endoscopic groups and packets of nontempered representations, Compos. Math., 64 (1987), 271–309.

    MathSciNet  MATH  Google Scholar 

  2. N. Arancibia, C. Moeglin and D. Renard, Paquets d’Arthur des groupes classiques et unitaires, Ann. Fac. Sci. Toulouse Math. (6), 27 (2018), 1023–1105.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Arthur, The invariant trace formula. II. Global theory, J. Am. Math. Soc., 1 (1988), 501–554.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Arthur, The \(L^{2}\)-Lefschetz numbers of Hecke operators, Invent. Math., 97 (1989), 257–290.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic groups, American Mathematical Society Colloquium Publications, vol. 61, Am. Math. Soc., Providence, 2013.

    MATH  Google Scholar 

  6. J. Bergström, C. Faber and G. van der Geer, Siegel modular forms of degree two and three, 2017, retrieved June 2019, http://smf.compositio.nl.

  7. R. E. Borcherds, E. Freitag and R. Weissauer, A Siegel cusp form of degree 12 and weight 12, J. Reine Angew. Math., 494 (1998), 141–153, dedicated to Martin Kneser on the occasion of his 70th birthday.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Böcherer, Siegel modular forms and theta series, in Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., vol. 49, pp. 3–17, Am. Math. Soc., Providence, 1989.

    Chapter  Google Scholar 

  9. G. Chenevier, An automorphic generalization of the Hermite-Minkowski theorem, Duke Math. J., to appear.

  10. G. Chenevier, The characteristic masses of Niemeier lattices, preprint, 2020.

  11. G. Chenevier, Subgroups of \(\mathrm{Spin}(7)\) or \(\mathrm{SO}(7)\) with each element conjugate to some element of \(\mathrm{G}_{2}\) and applications to automorphic forms, Doc. Math., 24 (2019), 95–161.

    MathSciNet  MATH  Google Scholar 

  12. G. Chenevier and J. Lannes, Automorphic Forms and Even Unimodular Lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 69, Springer, Berlin, 2019.

    Book  MATH  Google Scholar 

  13. B. Conrad, Reductive group schemes, in Autour des schémas en groupes. Vol. I, Panor. Synthèses, vol. 42/43, pp. 93–444, Soc. Math. France, Paris, 2014.

    Google Scholar 

  14. G. Chenevier and D. Renard, Level One Algebraic Cusp Forms of Classical Groups of Small Rank, Mem. Am. Math. Soc., 237 (2015), no. 1121, v+122.

    MATH  Google Scholar 

  15. G. Chenevier and O. Taïbi, Siegel modular forms of weight 13 and the Leech lattice, preprint, 2019.

  16. G. Chenevier and O. Taïbi, Tables and source of some computer programs used in this paper, 2019, https://gaetan.chenevier.perso.math.cnrs.fr/levelone/, https://otaibi.perso.math.cnrs.fr/levelone/, or the Electronic Supplementary Material published online by Springer along with this article.

  17. F. Cléry and G. van der Geer, On vector-valued Siegel modular forms of degree 2 and weight \((j,2)\), Doc. Math., 23 (2018), 1129–1156, with two appendices by Gaëtan Chenevier.

    MathSciNet  MATH  Google Scholar 

  18. W. Duke and Ö. Imamoḡlu, Siegel modular forms of small weight, Math. Ann., 310 (1998), 73–82.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Fermigier, Annulation de la cohomologie cuspidale de sous-groupes de congruence de \(\mathrm{GL}_{n}(\mathbf{Z})\), Math. Ann., 306 (1996), 247–256.

    Article  MathSciNet  MATH  Google Scholar 

  20. U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., 44 (1985), 463–471.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Freitag, Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe, Invent. Math., 30 (1975), 181–196.

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Freitag, Stabile Modulformen, Math. Ann., 230 (1977), 197–211.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Freitag, Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koeffizienten, Math. Ann., 258 (1981/82), 419–440.

    Article  MATH  Google Scholar 

  24. B. H. Gross and C. T. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, J. Algebra, 257 (2002), 265–290.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. H. Gross, On the motive of a reductive group, Invent. Math., 130 (1997), 287–313.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Gelbart and F. Shahidi, Boundedness of automorphic \(L\)-functions in vertical strips, J. Am. Math. Soc., 14 (2001), 79–107.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Howe, Automorphic forms of low rank, in Noncommutative Harmonic Analysis and Lie Groups, Lecture Notes in Math., vol. 880, Marseille, 1980, pp. 211–248, Springer, Berlin–New York, 1981.

    Chapter  Google Scholar 

  28. T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree \(2n\), Ann. Math. (2), 154 (2001), 641–681.

    Article  MathSciNet  MATH  Google Scholar 

  29. N. Jacobson, A note on Hermitian forms, Bull. Am. Math. Soc., 46 (1940), 264–268.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika, Rankin-Selberg convolutions, Am. J. Math., 105 (1983), 367–464.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Am. J. Math., 103 (1981), 777–815.

    Article  MathSciNet  MATH  Google Scholar 

  32. O. D. King, A mass formula for unimodular lattices with no roots, Math. Comput., 72 (2003), 839–863.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. W. Knapp, Local Langlands correspondence: the Archimedean case, in Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, pp. 393–410, Amer. Math. Soc., Providence, 1994.

    Chapter  Google Scholar 

  34. M.-A. Knus, Quadratic and Hermitian Forms Over Rings, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer, Berlin, 1991.

    Book  MATH  Google Scholar 

  35. W. Kohnen and R. Salvati Manni, Linear relations between theta series, Osaka J. Math., 41 (2004), 353–356.

    MathSciNet  MATH  Google Scholar 

  36. K. Koike and I. Terada, Young-diagrammatic methods for the representation theory of the classical groups of type \(B_{n}\), \(C _{n}\), \(D_{n}\), J. Algebra, 107 (1987), 466–511.

    Article  MathSciNet  MATH  Google Scholar 

  37. G. Lachaussée, Ph. D. dissertation, Paris-Saclay university, forthcoming.

  38. T. Mégarbané, Traces des opérateurs de Hecke sur les espaces de formes automorphes de SO7, SO8 ou SO9 en niveau 1 et poids arbitraire, J. Théor. Nr. Bordx., 30 (2018), 239–306.

    Article  MATH  Google Scholar 

  39. J.-F. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compos. Math., 58 (1986), 209–232.

    MATH  Google Scholar 

  40. S. D. Miller, The highest lowest zero and other applications of positivity, Duke Math. J., 112 (2002), 83–116.

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Mizumoto, Poles and residues of standard \(L\)-functions attached to Siegel modular forms, Math. Ann., 289 (1991), 589–612.

    Article  MathSciNet  MATH  Google Scholar 

  42. S. Mizumoto, Erratum to: Poles and residues of standard \(L\)-functions attached to Siegel modular forms, personal communication, 2019.

  43. C. Moeglin and D. Renard, Sur les paquets d’arthur de \(\mathrm{Sp}(2n,\mathbf {R})\) contenant des modules unitaires de plus haut poids, scalaires, Nagoya Math. J. (2019). https://doi.org/10.1017/nmj.2019.15.

    Article  Google Scholar 

  44. C. Moeglin and J.-L. Waldspurger, Le spectre résiduel de \(\mathrm{GL}(n)\), Ann. Sci. Ec. Norm. Super., 22 (1989), 605–674.

    Article  MATH  Google Scholar 

  45. C. Moeglin and J.-L. Waldspurger, Décomposition spectrale et séries d’Eisenstein, Progress in Mathematics, vol. 113, Birkhäuser Verlag, Basel, 1994, Une paraphrase de l’Écriture. [A paraphrase of Scripture].

    MATH  Google Scholar 

  46. G. Poitou, Minorations de discriminants (d’après A. M. Odlyzko), Séminaire Bourbaki, Vol. 1975/76 28ème année, Exp. No. 479, pp. 136–153, Springer, Berlin, 1977, Lecture Notes in Math., 567.

    MATH  Google Scholar 

  47. G. Poitou, Sur les petits discriminants, in Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1 (French), vol. 6, p. 18, Secrétariat Math., Paris, 1977

    Google Scholar 

  48. S. Rallis, Langlands’ functoriality and the Weil representation, Am. J. Math., 104 (1982), 469–515.

    Article  MathSciNet  MATH  Google Scholar 

  49. S. Rallis, On the Howe duality conjecture, Compos. Math., 51 (1984), 333–399.

    MathSciNet  MATH  Google Scholar 

  50. H. L. Resnikoff, Automorphic forms of singular weight are singular forms, Math. Ann., 215 (1975), 173–193.

    Article  MathSciNet  MATH  Google Scholar 

  51. W. A. Stein, et al., Sage Mathematics Software (Version 6.1.1), The Sage Development Team, 2014, http://www.sagemath.org.

  52. J.-P. Serre, Cohomologie des groupes discrets, in Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, vol. 70, pp. 77–169, Princeton University Press, Princeton, 1971.

    Google Scholar 

  53. C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1969 (1969), 87–102.

    MathSciNet  MATH  Google Scholar 

  54. O. Taïbi, Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 269–344.

    Article  MathSciNet  MATH  Google Scholar 

  55. O. Taïbi, Arthur’s multiplicity formula for certain inner forms of special orthogonal and symplectic groups, J. Eur. Math. Soc., 21 (2019), 839–871.

    Article  MathSciNet  MATH  Google Scholar 

  56. J. Tate, Number theoretic background, in Automorphic Forms, Representations and \(L\)-Functions, Part 2 (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. Pure Math. vol. XXXIII, pp. 3–26, Am. Math. Soc., Providence, 1979.

    Chapter  Google Scholar 

  57. G. van der Geer, Siegel modular forms and their applications, in The 1-2-3 of Modular Forms, Universitext, pp. 181–245, Springer, Berlin, 2008.

    Chapter  MATH  Google Scholar 

  58. G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc., 3 (1963), 1–62.

    Article  MathSciNet  MATH  Google Scholar 

  59. R. Weissauer, Vektorwertige Siegelsche Modulformen kleinen Gewichtes, J. Reine Angew. Math., 343 (1983), 184–202.

    MathSciNet  MATH  Google Scholar 

  60. H. Zassenhaus, On the spinor norm, Arch. Math., 13 (1962), 434–451.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, Université Paris-Saclay, Orsay, France

    Gaëtan Chenevier

  2. CNRS, École Normale Supérieure de Lyon, Lyon, France

    Olivier Taïbi

Authors
  1. Gaëtan Chenevier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olivier Taïbi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Gaëtan Chenevier and Olivier Taïbi are supported by the C.N.R.S. and by the project ANR-14-CE25.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(ZIP 3.5 MB)

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chenevier, G., Taïbi, O. Discrete series multiplicities for classical groups over \(\mathbf {Z}\) and level 1 algebraic cusp forms. Publ.math.IHES 131, 261–323 (2020). https://doi.org/10.1007/s10240-020-00115-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10240-020-00115-z

Search

Navigation