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On integral kernels for Dirichlet series associated to Jacobi forms

2014; Wiley; Volume: 90; Issue: 1 Linguagem: Inglês

10.1112/jlms/jdu016

1469-7750

Yves Martin,

Advanced Mathematical Identities

Journal of the London Mathematical SocietyVolume 90, Issue 1 p. 67-88 Articles On integral kernels for Dirichlet series associated to Jacobi forms Yves Martin, Corresponding Author Yves Martin [email protected] Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile[email protected]Search for more papers by this author Yves Martin, Corresponding Author Yves Martin [email protected] Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile[email protected]Search for more papers by this author First published: 20 May 2014 https://doi.org/10.1112/jlms/jdu016Citations: 2 2010 Mathematics Subject Classification 11F50 (primary), 11F66, 11F27 (secondary). This research was supported in part by the FONDECYT grant no. 1121064. Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract Every Jacobi cusp form of weight k and index m over SL 2 ( Z ) ⋉ Z 2 is in correspondence with 2 m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2 m Dirichlet series via the Petersson inner product. We show that those kernels are Jacobi cusp forms and express them in terms of Jacobi Poincaré series. As an application, we give a new proof of the analytic continuation and functional equations satisfied by the Dirichlet series mentioned above. References 1.R. Berndt, ‘ L-functions for Jacobi forms à la Hecke’, Manuscripta Math., 84 (1994) 101– 112. 2.H. Cohen, ‘ Sur certaines sommes de séries liées aux périodes de formes modulaires’, Séminaire de théorie de nombres (Grenoble, 1981). 3.N. Diamantis, C. O'Sullivan, ‘Kernels of L-functions of cusp forms’, Math. Ann., 346 (2010) 897– 929. 4.M. Eichler, D. Zagier, The theory of Jacobi forms (Birkhauser, Boston, 1985). 5.S. Fukuhara, ‘Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials’, J. reine angew. Math., 607 (2007) 163– 216. 6.S. Fukuhara, Y. Yang, ‘Period polynomials and explicit formulas for Hecke operators on Γ 0 ( 2 ) ’, Math. Proc. Cambridge Philos. Soc., 146 (2009) 321– 350. 7.A. Good, ‘Dirichlet and Poincaré series’, Glasgow Math. J., 27 (1985) 39– 56. 8.I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products (Academic Press, San Diego, CA, 1994). 9.B. Gross, W. Kohnen, D. Zagier, ‘Heegner points and derivatives of L-series. II’, Math. Ann., 278 (1987) 497– 562. 10.E. Hecke, ‘Uber die Bestimmung Dirichletscher Reichen durch ihre Funktionalgleichung’, Math. Ann., 112 (1936) 664– 699 (Werke 33). 11.H. Iwaniec, Spectral methods of automorphic forms, Graduate Studies in Math. 53 (American Mathematical Society, Providence, RI, 1997). 12.J. Jorgenson, C. O'Sullivan, ‘Unipotent vector bundles and higher-order non-holomorphic Eisenstein series’, J. Théor. Nombres Bordeaux, 20 (2008) 131– 163. 13.M. Knopp, S. Robins, ‘Easy proofs of Riemann's functional equation for ζ ( s ) and of Lipschitz summation’, Proc. Amer. Math. Soc., 129 (2001) 1915– 1922. 14.W. Kohnen, ‘Nonvanishing of Hecke L-functions associated to cusp forms inside the critical strip’, J. Number Theory, 67 (1997) 182– 189. 15.W. Kohnen, J. Sengupta, ‘On Koecher–Maass series of Siegel modular forms’, Math. Z., 242 (2002) 149– 157. 16.W. Kohnen, D. Zagier, ‘Modular forms with rational periods’, Modular forms (Durham 1983) (ed. R. A. Rankin; Wiley, New York, 1984). 17.S. Lang, Complex analysis (Springer, New York, NY, 1997). 18.Y. Martin, ‘A converse theorem for Jacobi forms’, J. Number Theory, 61 (1996) 181– 193. 19.H. Petersson, ‘Einheitliche Begründung der Vollständigkeitssatze für die Poincaréschen Reihen von reeller Dimension bei beliebigen Grenzkreisgruppen von erster Art’, Abh. Math. Sem. Hansischen Univ., 14 (1941) 22– 60. 20.N.-P. Skoruppa, D. Zagier, ‘A trace formula for Jacobi forms’, J. reine angew. Math., 393 (1989) 168– 198. Citing Literature Volume90, Issue1August 2014Pages 67-88 ReferencesRelatedInformation