Hostname: page-component-7f64f4797f-g5lq8 Total loading time: 0.001 Render date: 2025-11-04T15:41:47.482Z Has data issue: false hasContentIssue false
  • English
  • Français

A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 June 2016

NICOLE RAULF  and
OLIVER STEIN
NICOLE RAULF
Affiliation:
Université de Lille 1, U.M.R. CNRS 8524, U.F.R. de Mathématiques, Cité Scientifique, F-59655 Villeneuve d'Ascq Cédex e-mail: nicole.raulf@math.univ-lille1.fr
OLIVER STEIN
Affiliation:
Ostbayerische Technische Hochschule Regensburg, Fakultät für Informatik und Mathematik, Universitätsstraße 31, 93053 Regensburg, Germany e-mail: oliver.stein@hs-regensburg.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a ready to compute trace formula for Hecke operators on vector-valuedmodular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

MSC classification

Primary: 11F27: Theta series; Weil representation; theta correspondences
Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11F72: Spectral theory; Selberg trace formula

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 143 - 165
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Bauer-Price, P., The Selberg trace formula for PSL (2, $\mathcal{O}_K$ ) for imaginary quadratic number fields K of arbitrary class number, Bonner Mathematische Schriften, vol. 221 (Universität Bonn, Mathematisches Institut, Bonn, 1991).Google Scholar
2. Berndt, B., Evans, R. and Williams, K., Gauss and Jacobi sums, Canadian Mathematical Society series of monographs and advanced texts, vol. 21 (Wiley, New York, NY, 1998).Google Scholar
3. Borcherds, R., Automorphic forms with singularities on Grassmannians, Inv. Math. 132 (1998), 491–562.CrossRefGoogle Scholar
4. Borcherds, R., Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), 319–366.CrossRefGoogle Scholar
5. Brown, K., Cohomology of groups, Graduate Texts in Mathematics, vol. 87 (Springer-Verlag, Berlin, 1982).Google Scholar
6. Bruinier, J. H., Borcherds products on O(2, l) and Chern classes of Heegner divisors, Springer Lecture Notes in Mathematics vol. 1780 (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
7. Bruinier, J. H., On the rank of Picard groups of modular varieties attached to orthogonal groups, Compos. Math. 133 (2002), 49–63.CrossRefGoogle Scholar
8. Bruinier, J. H. and Bundschuh, M., On Borcherds products associated with lattices of prime discriminant, Ramanujan J. 7 (2003), 49–61.Google Scholar
9. Bruinier, J. H. and Ono, K., Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, preprint (2011).Google Scholar
10. Bruinier, J. H. and Stein, O., The Weil representation and Hecke operators on vector valued modular forms, Math. Z. 264, 249–270.Google Scholar
11. Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics vol. 228 (Springer-Verlag, New York, 2005)Google Scholar
12. Ebeling, W., Lattices and codes, A course partially based on lectures by Hirzebruch, F., Second revised edition, Advanced Lectures in Mathematics (Vieweg, Braunschweig, 2002).Google Scholar
13. Efrat, I. Y., The Selberg trace formula for PSL2( $\mathbb{R}^n$ ), Mem. Amer. Math. Soc. 65 (359) (1987).Google Scholar
14. Eholzer, W. and Skoruppa, N.-P., Modular invariance and uniqueness of conformal characters, Commun. Math. Phys. 174 (1995), 117–136.CrossRefGoogle Scholar
15. Elstrodt, J., Grunewald, F. and Mennicke, J., Groups acting on hyperbolic space, Harmonic analysis and number theory, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 1998).Google Scholar
16. Hagemeier, H., Automorphe Produkte singulären Gewichts, Dissertation (TU Darmstadt 2010).Google Scholar
17. Hejhal, D., The Selberg trace formula for PSL(2, $\mathbb{R}$ ), volume 1, Lecture Notes in Mathematics, 548 (Springer Verlag, Berlin 1976).Google Scholar
18. Hejhal, D., The Selberg trace formula for PSL(2, $\mathbb{R}$ ), volume 2, Lecture Notes in Mathematics 1001 (Springer Verlag, Berlin, 1983).Google Scholar
19. Hijikata, H., Explicit formula of the traces of Hecke operators for Γ0(N), J. Math. Soc. Japan 26 (1974), 56–82.Google Scholar
20. Li, X.-J., On the trace of Hecke operators for Maass forms, Number theory (Ottawa, ON, 1996), 215–229, CRM Proc. Lecture Notes, vol. 19 (Amer. Math. Soc., Providence, RI, 1999).CrossRefGoogle Scholar
21. Milnor, J. and Husemöller, D., Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73 (Springer-Verlag, New York-Heidelberg, 1973).Google Scholar
22. Kalinin, V. L., A trace formula for Hecke operators, Math. USSR Sbornik, 47 (2) (1984).Google Scholar
23. Oesterlé, J., Sur la trace des operateurs de Hecke, Thèse de 3e cycle (Université de Paris-Sud, University in Orsay, France, 1977).Google Scholar
24. Raulf, N., Traces of Hecke operators acting on three-dimensional hyperbolic space, J. Reine Angew. Math. 591 (2006), 111–148.Google Scholar
25. Sarnak, P., Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229–247.Google Scholar
26. Sarnak, P., The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (3–4), (1983), 253–295.Google Scholar
27. Sarnak, P., Statistical properties of eigenvalues of the Hecke operators, Analytic number theory and Diophantine problems (Stillwater, OK, 1984), 321–331, Progr. Math., vol. 70 (Birkhäuser Boston, Boston, MA, 1987).Google Scholar
28. Scheithauer, N. R., The Weil representation of SL2( $\mathbb{Z}$ ) and some applications, Int. Math. Res. Not. 8 (2009), 1488–1545.Google Scholar
29. Scheithauer, N. R., Moonshine for Conway's group (Habilitationsschrift, Ruprecht-Karls-Universität Heidelberg, 2004).Google Scholar
30. Scheithauer, N. R., On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164, 641–678 (2006).CrossRefGoogle Scholar
31. Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. B 20 (1956), 47–87.Google Scholar
32. Serre, J.-P., Répartition asymptotique des valeurs propres de l'opérateur de Hecke Tp , J. Amer. Math. Soc. 10 (1) (1997), 75–102.Google Scholar
33. Shimizu, H., On traces of Hecke operators, J. Fac. Sci. Univ. Tokyo 10 (1963), 1–19.Google Scholar
34. Shintani, T., On the construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126.Google Scholar
35. Skoruppa, N.-P. and Zagier, D., A trace formula for Jacobi forms, J. Reine Angew. Math. 393 (1989), 168–198.Google Scholar
36. Strömberg, F., Weil representations associated to finite quadratic modules, Math. Z. 275 (2013), 509–527.Google Scholar
37. Teruji, T., The character of the Weil representation, J. Lond. Math. Soc., II. Ser. 77 (1) (2008), 221–239.Google Scholar
38. Werner, F., Discriminant forms and Hecke operators, Master Thesis (TU Darmstadt, 2010).Google Scholar
39. Zagier, D., The Eichler-Selberg trace formula on SL2( $\mathbb{Z}$ ), Appendix to S. Lang, Introduction to Modular Forms, Grundlehren d. math. Wiss., vol. 222 (Springer-Verlag, Berlin-Heidelberg-New York, 1976), 44–54.Google Scholar
40. Zagier, D., Correction to “the Eichler-Selberg trace fomula on SL2( $\mathbb{Z}$ )”, in Modular functions of one variable VI Lecture Notes in Mathematics, vol. 627 (Springer-Verlag, Berlin, 1977), 171–173.Google Scholar