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RANKIN–SELBERG CONVOLUTIONS OF NONCUSPIDAL HALF-INTEGRAL WEIGHT MAASS FORMS IN THE PLUS SPACE

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 May 2018

YOSHINORI MIZUNO
YOSHINORI MIZUNO*
Affiliation:
Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1, Minami-josanjima-cho, Tokushima, 770-8506, Japan email mizuno.yoshinori@tokushima-u.ac.jp
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Abstract

The author gives the analytic properties of the Rankin–Selberg convolutions of two half-integral weight Maass forms in the plus space. Applications to the Koecher–Maass series associated with nonholomorphic Siegel–Eisenstein series are given.

MSC classification

Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Article
Information
Nagoya Mathematical Journal , Volume 237 , March 2020 , pp. 127 - 165
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

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References

Arakawa, T., Dirichlet series related to the Eisenstein series on the Siegel upper half-plane , Comment. Math. Univ. St. Pauli 27(1) (1978), 29–42.Google Scholar
Arakawa, T., Ibukiyama, T. and Kaneko, M., Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, Tokyo, 2014, xii+274 pp. With an appendix by Don Zagier.Google Scholar
Böcherer, S., Bemerkungen über die Dirichletreihen von Koecher und Maass , Mathematica Göttingensis, Schriftenreihe des SFB Geometrie und Analysis, Heft 68 (1986), 36.Google Scholar
Biro, A., A relation between triple products of weight 0 and weight 1/2 cusp forms , Israel J. Math. 182 (2011), 61–101.Google Scholar
Chinta, G., Friedberg, S. and Hoffstein, J., “ Multiple Dirichlet series and automorphic forms ”, in Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Proc. Sympos. Pure Math. 75 , American Mathematical Society, Providence, RI, 2006, 3–41.Google Scholar
Chinta, G. and Gunnells, P., Weyl group multiple Dirichlet series constructed from quadratic characters , Invent. Math. 167(2) (2007), 327–353.Google Scholar
Cohen, H., Sums involving the values at negative integers of L-functions of quadratic characters , Math. Ann. 217 (1975), 271–285.Google Scholar
Diamantis, N. and Goldfeld, D., A converse theorem for double Dirichlet series and Shintani zeta functions , J. Math. Soc. Japan 66(2) (2014), 449–477.Google Scholar
Duke, W. and Imamoḡlu, Ö., A converse theorem and the Saito–Kurokawa lift , Int. Math. Res. Not. IMRN (7) (1996), 347–355.Google Scholar
Goldfeld, D. and Hoffstein, J., Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series , Invent. Math. 80(2) (1985), 185–208.Google Scholar
Hashim, A. and Ram Murty, M., On Zagier’s cusp form and the Ramanujan 𝜏 function , Proc. Indian Acad. Sci. Math. Sci. 104(1) (1994), 93–98.Google Scholar
Ibukiyama, T. and Katsurada, H., “ Koecher–Maass series for real analytic Siegel Eisenstein series ”, in Automorphic Forms and Zeta Functions, Proceedings of the conference in memory of Tsuneo Arakawa, World Scientific, Hackensack, NJ, USA, 2006, 170–197.Google Scholar
Ibukiyama, T. and Saito, H., On zeta functions associated to symmetric matrices, II: functional equations and special values , Nagoya Math. J. 208 (2012), 265–316.Google Scholar
Katok, S. and Sarnak, P., Heegner points, cycles and Maass forms , Israel J. Math. 84(1–2) (1993), 193–227.Google Scholar
Kaufhold, G., Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades , Math. Ann. 137 (1959), 454–476.Google Scholar
Lebedev, N., Special Functions and their Applications, Revised edition (ed. Silverman, R. A.) Dover Publications, Inc., New York, 1972, xii+308 pp. translated from the Russian. Unabridged and corrected republication.Google Scholar
Kohama, H. and Mizuno, Y., Kernel functions of the twisted symmetric square of elliptic modular forms , Mathematika 64 (2018), 184–210.Google Scholar
Luo, W., Rudnick, Z. and Sarnak, P., The variance of arithmetic measures associated to closed geodesics on the modular surface , J. Mod. Dyn. 3(2) (2009), 271–309.Google Scholar
Maass, H., Konstruktion ganzer Modulformen halbzahliger Dimension mit V-Multiplikatoren in einer und zwei Variablen , Abhandlungen aus dem Mathematischen Seminar der Hansischen Universitat 12 (1937), 133–162.Google Scholar
Maass, H., Siegel’s Modular Forms and Dirichlet Series, Lecture Notes in Mathematics. 216 , Springer, Berlin–New York, 1971, v+328 pp.Google Scholar
Matthes, R., Rankin–Selberg method for real analytic cusp forms of arbitrary real weight , Math. Z. 211(1) (1992), 155–172.Google Scholar
Miyake, T., Modular Forms, x+335 pp. Springer, Berlin, 1989.Google Scholar
Mizuno, Y., The Rankin–Selberg convolution for real analytic Cohen’s Eisenstein series of half integral weight , J. Lond. Math. Soc. (2) 78 (2008), 183–197.Google Scholar
Mizuno, Y., Koecher–Maass series for positive definite Fourier coefficients of real analytic Siegel–Eisenstein series of degree 2 , Bull. Lond. Math. Soc. 41 (2009), 1017–1028.Google Scholar
Mizuno, Y., Dirichlet series associated with square of class numbers of binary quadratic forms , Math. Z. 272(3–4) (2012), 1115–1135.Google Scholar
Mizuno, Y., On characterization of Siegel cusp forms of degree 2 by the Hecke bound , Mathematika 61(1) (2015), 89–100.Google Scholar
Müller, W., The Rankin–Selberg method for non-holomorphic automorphic forms , J. Number Theory 51(1) (1995), 48–86.Google Scholar
Narkiewicz, W., Number Theory, xii, 371 p World Scientific, Singapore, 1983, Transl. from the Polish by S. Kanemitsu.Google Scholar
Pitale, A., Jacobi Maass forms , Abh. Math. Semin. Univ. Hambg. 79(1) (2009), 87–111.Google Scholar
Rademacher, H., On the Phragmén–Lindelöf theorem and some applications , Math. Z. 72 (1959/1960), 192–204.Google Scholar
Sato, F., Zeta functions of (SL2 × SL2 × GL2, M 2⊕M 2) associated with a pair of Maass cusp forms , Comment. Math. Univ. St. Pauli 55(1) (2006), 77–95.Google Scholar
Shimura, G., On modular forms of half integral weight , Ann. of Math. (2) 97 (1973), 440–481.Google Scholar
Shimura, G., On the holomorphy of certain Dirichlet series , Proc. Lond. Math. Soc. (3) 31(1) (1975), 79–98.Google Scholar
Shimura, G., Elementary Dirichlet Series and Modular Forms, Springer Monographs in Mathematics, Springer, New York, 2007, viii+147 pp. ISBN: 978-0-387-72473-7.Google Scholar
Siegel, C., Die Funktionalgleichungen einiger Dirichletscher Reihen , Math. Z. 63 (1956), 363–373.Google Scholar
Siegel, C., Advanced Analytic Number Theory, Second edition, Tata Institute of Fundamental Research Studies in Mathematics. 9 , v+268 pp. Tata Institute of Fundamental Research, Bombay, 1980.Google Scholar
Suzuki, T., Distributions with automorphy and Dirichlet series , Nagoya Math. J. 73 (1979), 157–169.Google Scholar
Sturm, J., Special values of zeta functions, and Eisenstein series of half integral weight , Amer. J. Math. 102(2) (1980), 219–240.Google Scholar
Wen, J., Shintani zeta functions and Weyl group multiple Dirichlet series. Thesis (Ph.D.) State University of New York at Stony Brook. 2014. 64 pp. ISBN: 978-1321-11773-8 (See also arXiv:1311.2132, Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series).Google Scholar
Zagier, D., Nombres de classes et formes modulaires de poids 3/2 , C. R. Acad. Sci. Paris Ser. A-B 281(21, Ai) (1975), A883–A886.Google Scholar
Zagier, D., “ Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields ”, in Modular Functions of One Variable VI, Lecture Notes in Mathematics. 627 , Springer, Berlin, 1977, 105–169.Google Scholar
Zagier, D., The Rankin–Selberg method for automorphic functions which are not of rapid decay , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 415–437.Google Scholar
Zagier, D., “ Eisenstein series and the Riemann zeta function ”, in Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math. 10 , Tata Inst. Fundamental Res., Bombay, 1981, 275–301.Google Scholar