Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-12T07:12:15.243Z Has data issue: false hasContentIssue false
  • English
  • Français

SPECIAL VALUES OF SHIFTED CONVOLUTION DIRICHLET SERIES

Part of:Arithmetic algebraic geometryDiscontinuous groups and automorphic forms

Published online by Cambridge University Press:  16 June 2015

Michael H. Mertens and
Ken Ono
Michael H. Mertens
Affiliation:
Mathematisches Institut der Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany email mmertens@math.uni-koeln.de
Ken Ono
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30022, U.S.A. email ono@mathcs.emory.edu

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.

In a recent important paper, Hoffstein and Hulse [Multiple Dirichlet series and shifted convolutions, arXiv:1110.4868v2] generalized the notion of Rankin–Selberg convolution $L$-functions by defining shifted convolution$L$-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and “mixed mock modular” forms.

MSC classification

Primary:11F37: Forms of half-integer weight; nonholomorphic modular forms
Secondary:11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture11G05: Elliptic curves over global fields11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016, pp. 47 - 66
Copyright
Copyright © University College London 2015 

References

Blomer, V., Shifted convolution sums and subconvexity bounds for automorphic L-functions. Int. Math. Res. Not. IMRN 73 2004, 3905–3926.CrossRefGoogle Scholar
Borcherds, R. E., Automorphic forms with singularities on Grassmannians. Invent. Math. 132 1998, 491–562.CrossRefGoogle Scholar
Bringmann, K. and Ono, K., Lifting cusp forms to Maass forms with an application to partitions. Proc. Natl. Acad. Sci. USA 104(10) 2007, 3725–3731.CrossRefGoogle ScholarPubMed
Bruinier, J. H., Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors (Springer Lecture Notes in Mathematics 1780), Springer (2002).CrossRefGoogle Scholar
Bruinier, J. H. and Funke, J., On two geometric theta lifts. Duke Math. J. 1(125) 2004, 45–90.Google Scholar
Bruinier, J. H., Jenkins, P. and Ono, K., Hilbert class polynomials and traces of singular moduli. Math. Ann. 334 2006, 373–393.CrossRefGoogle Scholar
Bruinier, J. H., Rhoades, R. and Ono, K., Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann. 342 2008, 673–693.CrossRefGoogle Scholar
Cohen, H., Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217 1975, 271–285.CrossRefGoogle Scholar
Fay, J. D., Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 293–294 1977, 143–203.Google Scholar
Good, A., Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind. J. Number Theory 13 1981, 18–65.CrossRefGoogle Scholar
Good, A., Cusp forms and eigenfunctions of the Laplacian. Math. Ann. 255 1981, 523–548.CrossRefGoogle Scholar
Gross, B. H. and Zagier, D. B., Heegner points and derivatives of L-series. Invent. Math. 84 1986, 225–320.CrossRefGoogle Scholar
Hejhal, D. A., The Selberg Trace Formula for PSL(2, ℝ) (Springer Lecture Notes in Mathematics 1001), Springer (Berlin, 1983).CrossRefGoogle Scholar
Hoffstein, J. and Hulse, T. A., Multiple Dirichlet series and shifted convolutions. Preprint, 2011,arXiv:1110.4868v2.Google Scholar
Imamoğlu, Ö., Raum, M. and Richter, O., Holomorphic projections and Ramanujan’s mock theta functions. Proc. Natl. Acad. Sci. USA 111(11) 2014, 3961–3967.CrossRefGoogle ScholarPubMed
Iwaniec, H., Topics in Classical Automorphic Forms (Graduate Studies in Mathematics 17), American Mathematical Society (Providence, RI, 1997).CrossRefGoogle Scholar
Lau, Y.-K., Liu, J. and Ye, Y., Shifted convolution sums of Fourier coefficients of cusp forms. In Number Theory (Series on Number Theory and its Applications 2), World Scientific (Hackensack, NJ, 2007), 108–135.CrossRefGoogle Scholar
Lewis, J. and Zagier, D., Period functions for Maass wave forms. Ann. of Math. (2) 153 2001, 191–258.CrossRefGoogle Scholar
Mertens, M. H., Eichler–Selberg type identities for mixed mock modular forms. Preprint, 2014,arXiv:1404.5491.Google Scholar
Niebur, D., A class of nonanalytic automorphic functions. Nagoya Math. J. 52 1973, 133–145.CrossRefGoogle Scholar
Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory. Curr. Dev. Math. 2008 2009, 347–454.CrossRefGoogle Scholar
Rankin, R. A., Contributions to the theory of Ramanujan’s function 𝜏(n). Proc. Cambridge Philos. Soc. 35 1939, 357–372.CrossRefGoogle Scholar
Rankin, R. A., The construction of automorphic forms from the derivatives of a given form. J. Indian Math. Soc. (N.S.) 20 1956, 103–116.Google Scholar
Selberg, A., Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch. Math. Naturvid. 43 1940, 47–50.Google Scholar
Selberg, A., On the Estimation of Fourier Coefficients of Modular Forms (Proceedings of Symposia in Applied Mathematics VIII), American Mathematical Society (Providence, RI, 1965), 1–15.Google Scholar
Sturm, J., Projections of C ∞ automorphic forms. Bull. Amer. Math. Soc. (N.S.) 2(3) 1980, 435–439.CrossRefGoogle Scholar
Zagier, D., Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci. 104(1) 1994, 57–75.CrossRefGoogle Scholar
Zagier, D., Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann–Ono]. In Séminaire Bourbaki, 60ème année, 2007/2008 (Astérisque 326), Société Mathématique de France (2009), Exposé 986, 143–164.Google Scholar
Zwegers, S., Mock theta functions. PhD Thesis, Universiteit Utrecht, 2002.Google Scholar