Geodesic
A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS
Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 143-165

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
RAULF, NICOLE; STEIN, OLIVER. A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS. Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 143-165. doi: 10.1017/S0017089516000094
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[1] 1. Bauer-Price, P., The Selberg trace formula for PSL (2, ) for imaginary quadratic number fields K of arbitrary class number, Bonner Mathematische Schriften, vol. 221 (Universität Bonn, Mathematisches Institut, Bonn, 1991). Google Scholar

[2] 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] 3. Borcherds, R., Automorphic forms with singularities on Grassmannians, Inv. Math. 132 (1998), 491–562. Google Scholar | DOI

[4] 4. Borcherds, R., Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), 319–366. Google Scholar | DOI

[5] 5. Brown, K., Cohomology of groups, Graduate Texts in Mathematics, vol. 87 (Springer-Verlag, Berlin, 1982). Google Scholar

[6] 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). Google Scholar | DOI

[7] 7. Bruinier, J. H., On the rank of Picard groups of modular varieties attached to orthogonal groups, Compos. Math. 133 (2002), 49–63. Google Scholar | DOI

[8] 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] 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] 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] 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] 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] 13. Efrat, I. Y., The Selberg trace formula for PSL(), Mem. Amer. Math. Soc. 65 (359) (1987). Google Scholar

[14] 14. Eholzer, W. and Skoruppa, N.-P., Modular invariance and uniqueness of conformal characters, Commun. Math. Phys. 174 (1995), 117–136. Google Scholar | DOI

[15] 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] 16. Hagemeier, H., Automorphe Produkte singulären Gewichts, Dissertation (TU Darmstadt 2010). Google Scholar

[17] 17. Hejhal, D., The Selberg trace formula for PSL(2, ), volume 1, Lecture Notes in Mathematics, (Springer Verlag, Berlin 1976). Google Scholar

[18] 18. Hejhal, D., The Selberg trace formula for PSL(2, ), volume 2, Lecture Notes in Mathematics (Springer Verlag, Berlin, 1983). Google Scholar

[19] 19. Hijikata, H., Explicit formula of the traces of Hecke operators for Γ(N), J. Math. Soc. Japan 26 (1974), 56–82. Google Scholar

[20] 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). Google Scholar | DOI

[21] 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] 22. Kalinin, V. L., A trace formula for Hecke operators, Math. USSR Sbornik, 47 (2) (1984). Google Scholar

[23] 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] 24. Raulf, N., Traces of Hecke operators acting on three-dimensional hyperbolic space, J. Reine Angew. Math. 591 (2006), 111–148. Google Scholar

[25] 25. Sarnak, P., Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229–247. Google Scholar

[26] 26. Sarnak, P., The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (3–4), (1983), 253–295. Google Scholar

[27] 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] 28. Scheithauer, N. R., The Weil representation of SL() and some applications, Int. Math. Res. Not. 8 (2009), 1488–1545. Google Scholar

[29] 29. Scheithauer, N. R., Moonshine for Conway's group (Habilitationsschrift, Ruprecht-Karls-Universität Heidelberg, 2004). Google Scholar

[30] 30. Scheithauer, N. R., On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164, 641–678 (2006). Google Scholar | DOI

[31] 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] 32. Serre, J.-P., Répartition asymptotique des valeurs propres de l'opérateur de Hecke T , J. Amer. Math. Soc. 10 (1) (1997), 75–102. Google Scholar

[33] 33. Shimizu, H., On traces of Hecke operators, J. Fac. Sci. Univ. Tokyo 10 (1963), 1–19. Google Scholar

[34] 34. Shintani, T., On the construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126. Google Scholar

[35] 35. Skoruppa, N.-P. and Zagier, D., A trace formula for Jacobi forms, J. Reine Angew. Math. 393 (1989), 168–198. Google Scholar

[36] 36. Strömberg, F., Weil representations associated to finite quadratic modules, Math. Z. 275 (2013), 509–527. Google Scholar

[37] 37. Teruji, T., The character of the Weil representation, J. Lond. Math. Soc., II. Ser. 77 (1) (2008), 221–239. Google Scholar

[38] 38. Werner, F., Discriminant forms and Hecke operators, Master Thesis (TU Darmstadt, 2010). Google Scholar

[39] 39. Zagier, D., The Eichler-Selberg trace formula on SL(), 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] 40. Zagier, D., Correction to “the Eichler-Selberg trace fomula on SL()”, in Modular functions of one variable VI Lecture Notes in Mathematics, vol. 627 (Springer-Verlag, Berlin, 1977), 171–173. Google Scholar

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