Petersson norms of Eisenstein series and Kohnen–Zagier’s formula

Yoshinori Mizuno

doi:10.5802/jtnb.1138

¹Graduate School of Technology, Industrial and Social Sciences Tokushima University 2-1, Minami-josanjima-cho Tokushima, 770-8506, Japan

Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 665-684

Abstract (Original language)
Résumé

The regularized Petersson norms of Eisenstein series of integral and half-integral weight are computed. We use these results to establish Kohnen–Zagier’s formula for Eisenstein series.

Les normes de Petersson régularisées des séries d’Eisenstein de poids entier et demi-entier sont calculées. Nous utilisons ces résultats pour établir la formule de Kohnen–Zagier pour les séries d’Eisenstein.

Article information
Cite this article

Received: 2018-10-25
Revised: 2020-07-03
Accepted: 2020-09-18
Published online: 2021-01-08

DOI: 10.5802/jtnb.1138

Classification: 11F37, 11F11
Keywords: Petersson norms, Eisenstein series, Kohnen–Zagier’s formula

Author's affiliations:

Yoshinori Mizuno ¹

¹ Graduate School of Technology, Industrial and Social Sciences Tokushima University 2-1, Minami-josanjima-cho Tokushima, 770-8506, Japan

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CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Yoshinori Mizuno. Petersson norms of Eisenstein series and Kohnen–Zagier’s formula. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 665-684. doi: 10.5802/jtnb.1138

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References
Cited by

[1] Scott Ahlgren; Nickolas Andersen; Detchat Samart A polyharmonic Maass form of depth 3/2 for ${SL}_{2} (ℤ)$ , J. Math. Anal. Appl., Volume 468 (2018) no. 2, pp. 1018-1042 | DOI | MR | Zbl

[2] Nickolas Andersen; Jeffrey C. Lagarias; Robert C. Rhoades Shifted polyharmonic Maass forms for $PSL (2, ℤ)$ , Acta Arith., Volume 185 (2018) no. 1, pp. 39-79 | DOI | MR | Zbl

[3] Kathrin Bringmann; Nikolaos Diamantis; Stephan Ehlen Regularized inner products and errors of modularity, Int. Math. Res. Not., Volume 2017 (2017) no. 24, pp. 7420-7458 | Zbl | MR

[4] Kathrin Bringmann; Amanda Folsom; Ken Ono; Larry Rolen Harmonic Maass forms and mock modular forms: theory and applications, Colloquium Publications, 64, American Mathematical Society, 2017 | MR | Zbl

[5] Kathrin Bringmann; Pavel Guerzhoy; Zachary Kent; Ken Ono Eichler-Shimura theory for mock modular forms, Math. Ann., Volume 355 (2013) no. 3, pp. 1085-1121 | DOI | MR | Zbl

[6] Jan H. Bruinier; Jens Funke On two geometric theta lifts, Duke Math. J., Volume 125 (2004) no. 1, pp. 45-90 | DOI | MR | Zbl

[7] Jan H. Bruinier; Ken Ono; Robert C. Rhoades Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues, Math. Ann., Volume 342 (2008) no. 3, pp. 673-693 corrigendum in ibid. 345 (2009), no. 1, p. 31 | DOI | MR | Zbl

[8] Henri Cohen Sums involving the values at negative integers of $L$ -functions of quadratic characters, Math. Ann., Volume 217 (1975), pp. 271-285 | DOI | MR

[9] William Duke; Özlem Imamoḡlu A converse theorem and the Saito–Kurokawa lift, Int. Math. Res. Not., Volume 1996 (1996) no. 7, pp. 347-355 | DOI | MR | Zbl

[10] William Duke; Özlem Imamoḡlu; ’A’rpád Tóth Real quadratic analogs of traces of singular moduli, Int. Math. Res. Not., Volume 2011 (2011) no. 13, pp. 3082-3094 | MR | Zbl

[11] William Duke; Özlem Imamoḡlu; ’A’rpád Tóth Regularized inner products of modular functions, Ramanujan J., Volume 41 (2016) no. 1-3, pp. 13-29 | DOI | MR | Zbl

[12] Dorian Goldfeld; Jeffrey Hoffstein Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet $L$ -series, Invent. Math., Volume 80 (1985), pp. 185-208 | DOI | MR | Zbl

[13] Benedict Gross; Winfried Kohnen; Don Zagier Heegner points and derivatives of $L$ -series. II, Math. Ann., Volume 278 (1987), pp. 497-562 | DOI | MR | Zbl

[14] S. Hayashida; Yoshinori Mizuno Petersson norms of Jacobi–Eisenstein series and Gross–Kohnen–Zagier’s formula (preprint)

[15] Sebastián Herrero; Anna-Maria von Pippich Mock modular forms whose shadows are Eisenstein series of integral weight, Math. Res. Lett., Volume 27 (2020) no. 2, pp. 435-463 | DOI | MR | Zbl

[16] Tomoyoshi Ibukiyama; Hiroshi Saito On zeta functions associated to symmetric matrices, II: Functional equations and special values, Nagoya Math. J., Volume 208 (2012), pp. 265-316 | DOI | MR | Zbl

[17] Daeyeol Jeon; Soon-Yi Kang; Chang Heon Kim Weak Maass–Poincare series and weight 3/2 mock modular forms, J. Number Theory, Volume 133 (2013) no. 8, pp. 2567-2587 corrigendum in ibid. 159 (2016), p. 434-435 | DOI | MR | Zbl

[18] Zachary Kent $p$ -adic analysis and mock modular forms, Ph. D. Thesis, University of Hawaii (USA) (2010) (https://scholarspace.manoa.hawaii.edu/handle/10125/25934) | MR

[19] Neal Koblitz Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, 97, Springer, 1993 | MR | Zbl

[20] Hayato Kohama; Yoshinori Mizuno Kernel functions of the twisted symmetric square of elliptic modular forms, Mathematika, Volume 64 (2018) no. 1, pp. 184-210 corrigendum in ibid. 65 (2019), no. 1, p. 130-131 | DOI | MR | Zbl

[21] Winfried Kohnen Modular forms of half-integral weight of $Γ_{0} (4)$ , Math. Ann., Volume 28 (1980), pp. 249-266 | DOI | Zbl

[22] Winfried Kohnen Fourier coefficients and modular forms of half-integral weight, Math. Ann., Volume 271 (1985), pp. 237-268 | DOI | MR | Zbl

[23] Winfried Kohnen; Don Zagier Values of $L$ -series of modular forms at the center of the critical strip, Invent. Math., Volume 64 (1981), pp. 175-198 | DOI | MR | Zbl

[24] Murugesan Manickam; Balakrishnan Ramakrishnan; T. C. Vasudevan On Saito–Kurokawa descent for congruence subgroups, Manuscr. Math., Volume 81 (1993) no. 1-2, pp. 161-182 | DOI | MR | Zbl

[25] Toshitsune Miyake Modular forms, Springer Monographs in Mathematics, Springer, 2006 | MR | Zbl

[26] Shin Mizumoto Congruences for Fourier coefficients of lifted Siegel modular forms. I: Eisenstein lifts, Abh. Math. Semin. Univ. Hamb., Volume 75 (2005), pp. 97-120 | DOI | MR | Zbl

[27] Yoshinori Mizuno The Rankin–Selberg convolution for Cohen’s Eisenstein series of half integral weight, Abh. Math. Semin. Univ. Hamb., Volume 75 (2005), pp. 1-20 | DOI | MR | Zbl

[28] Yoshinori Mizuno Rankin–Selberg convolutions of noncuspidal half-integral weight Maass forms in the plus space, Nagoya Math. J., Volume 237 (2020), pp. 127-165 | DOI | MR | Zbl

[29] NIST handbook of mathematical functions (Frank W. J. Olver; Daniel W. Lozier; Ronald F. Boisvert; Charles W. Clark, eds.), Cambridge University Press, 2010 | Zbl

[30] Vicentiu Pasol; Alexandru A. Popa On the Petersson scalar product of arbitrary modular forms, Proc. Am. Math. Soc., Volume 142 (2014) no. 3, pp. 753-760 | DOI | MR | Zbl

[31] Goro Shimura On modular forms of half integral weight, Ann. Math., Volume 97 (1973), pp. 440-481 | DOI | MR | Zbl

[32] Goro Shimura On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc., Volume 31 (1975), pp. 79-98 | DOI | MR | Zbl

[33] Goro Shimura Elementary Dirichlet series and modular forms, Springer Monographs in Mathematics, Springer, 2007 | Zbl

[34] Shigeaki Tsuyumine Petersson scalar products and $L$ -functions arising from modular forms, Ramanujan J., Volume 52 (2020) no. 1, pp. 1-40 | DOI | MR | Zbl

[35] Ian Wagner Harmonic Maass form eigencurves, Res. Math. Sci., Volume 5 (2018) no. 2, 24, 16 pages | MR | Zbl

[36] Xueli Wang; Dingyi Pei Modular forms with integral and half-integral weights, Springer, 2012 | Zbl

[37] Don Zagier Nombres de classes et formes modulaires de poids 3/2, C. R. Math. Acad. Sci. Paris, Volume 281 (1975), pp. 883-886 | MR | Zbl

[38] Don Zagier Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable VI (Bonn, 1976) (Lecture Notes in Mathematics), Volume 627, Springer, 1976, pp. 105-169 | DOI | Zbl

[39] Don Zagier The Rankin–Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 28 (1981), pp. 415-437 | MR | Zbl

[40] Don Zagier Periods of modular forms and Jacobi theta functions, Invent. Math., Volume 104 (1991) no. 3, pp. 449-465 | DOI | MR | Zbl

[41] Don Zagier Introduction to modular forms, From number theory to physics (Les Houches, 1989), Springer, 1992, pp. 238-291 | DOI | Zbl

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