On the Petersson scalar product of arbitrary modular forms
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- by Vicentiu Pasol and Alexandru A. Popa PDF
- Proc. Amer. Math. Soc. 142 (2014), 753-760 Request permission
Abstract:
We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\geqslant 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $\Gamma _1(N)$ and $k>2$. For $k=2$ we give examples when it is degenerate and when it is nondegenerate.References
- Siegfried Böcherer and Francesco Ludovico Chiera, On Dirichlet series and Petersson products for Siegel modular forms, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 801–824 (English, with English and French summaries). MR 2427511, DOI 10.5802/aif.2370
- F. L. Chiera, On Petersson products of not necessarily cuspidal modular forms, J. Number Theory 122 (2007), no. 1, 13–24. MR 2287108, DOI 10.1016/j.jnt.2006.03.003
- Anton Deitmar and Nikolaos Diamantis, Automorphic forms of higher order, J. Lond. Math. Soc. (2) 80 (2009), no. 1, 18–34. MR 2520375, DOI 10.1112/jlms/jdp015
- Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
- Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR 1942691, DOI 10.1090/gsm/053
- Philippe Michel and Akshay Venkatesh, The subconvexity problem for $\textrm {GL}_2$, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271. MR 2653249, DOI 10.1007/s10240-010-0025-8
- V. Pasol and A. A. Popa, Modular forms and period polynomials, Proc. Lond. Math. Soc. (2013), DOI 10.1112/plms/pdt003.
- 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
- Goro Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804. MR 434962, DOI 10.1002/cpa.3160290618
- Don Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 415–437 (1982). MR 656029
Additional Information
- Vicentiu Pasol
- Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
- Email: vicentiu.pasol@imar.ro
- Alexandru A. Popa
- Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
- MR Author ID: 792375
- Email: alexandru.popa@imar.ro
- Received by editor(s): April 3, 2012
- Published electronically: November 19, 2013
- Additional Notes: The first author was partially supported by the CNCSIS grant PD-171/28.07.2010
The second author was partially supported by the European Community grant PIRG05-GA-2009-248569 - Communicated by: Ken Ono
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 753-760
- MSC (2010): Primary 11F11
- DOI: https://doi.org/10.1090/S0002-9939-2013-11815-0
- MathSciNet review: 3148510