On the trace formula for Hecke operators on congruence subgroups

Popa, Alexandru

doi:10.1090/proc/13896

On the trace formula for Hecke operators on congruence subgroups
HTML articles powered by AMS MathViewer

by Alexandru A. Popa PDF

Proc. Amer. Math. Soc. 146 (2018), 2749-2764 Request permission

Abstract:

We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when one varies the weight or the level in the trace formula.

References

Avner Ash and Glenn Stevens, Modular forms in characteristic $l$ and special values of their $L$-functions, Duke Math. J. 53 (1986), no. 3, 849–868. MR 860675, DOI 10.1215/S0012-7094-86-05346-9
Yj. Choie and D. Zagier, Rational period functions for $\textrm {PSL}(2,\mathbf Z)$, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 89–108. MR 1210514, DOI 10.1090/conm/143/00992
Sharon Frechette, Ken Ono, and Matthew Papanikolas, Combinatorics of traces of Hecke operators, Proc. Natl. Acad. Sci. USA 101 (2004), no. 49, 17016–17020. MR 2114776, DOI 10.1073/pnas.0407223101
Klaus Haberland, Perioden von Modulformen einer Variabler and Gruppencohomologie. I, II, III, Math. Nachr. 112 (1983), 245–282, 283–295, 297–315 (German). MR 726861, DOI 10.1002/mana.19831120113
Kazuyuki Hatada, On the rationality of periods of primitive forms, Acta Arith. 40 (1981/82), no. 1, 9–40. MR 654020, DOI 10.4064/aa-40-1-9-40
Adriaan Herremans, A combinatorial interpretation of Serre’s conjecture on modular Galois representations, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 5, 1287–1321 (English, with English and French summaries). MR 2032935, DOI 10.5802/aif.1980
Loïc Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd $2$-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, pp. 59–94. MR 1322319, DOI 10.1007/BFb0074110
Vicenţiu Paşol and Alexandru A. Popa, Modular forms and period polynomials, Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 713–743. MR 3108829, DOI 10.1112/plms/pdt003
Alexandru A. Popa, On the trace formula for Hecke operators on congruence subgroups, II. Preprint, arXiv:1706.02691
Alexandru A. Popa and Don Zagier, An elementary proof of the Eichler-Selberg trace formula. Preprint, arXiv:1711.00327
Florin Rădulescu, Endomorphisms of spaces of virtual vectors fixed by a discrete group, Russian Math. Surveys 71 (2016), 291–343.
Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75–102 (French). MR 1396897, DOI 10.1090/S0894-0347-97-00220-8
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
Don Zagier, Hecke operators and periods of modular forms, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 321–336. MR 1159123
Don Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Sūrikaisekikenkyūsho K\B{o}kyūroku 843 (1993), 162–170. Research into automorphic forms and $L$ functions (Japanese) (Kyoto, 1992). MR 1296720