Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Mock modular period functions and $ L$-functions of cusp forms in higher level cases


Authors: SoYoung Choi and Chang Heon Kim
Journal: Proc. Amer. Math. Soc. 142 (2014), 3369-3386
MSC (2010): Primary 11F11, 11F67; Secondary 11F37
DOI: https://doi.org/10.1090/S0002-9939-2014-12073-9
Published electronically: June 19, 2014
MathSciNet review: 3238414
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing the results of Bringmann, Guerzhoy, Kent and Ono, we investigate mock modular period polynomials associated to harmonic Maass forms for $ \Gamma _0^+(p)$. In particular, using period relations generated from the period polynomials, we derive congruence relations involving the critical values of modular $ L$-functions and show that these congruence relations are indeed equalities by using integral representations of nonholomorphic parts of harmonic Maass forms.


References [Enhancements On Off] (What's this?)

  • [1] G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), no. nos. 3-4, 1-28 (German). MR 0033411 (11,437a)
  • [2] Kathrin Bringmann, Pavel Guerzhoy, Zachary Kent, and Ken Ono, Eichler-Shimura theory for mock modular forms, Math. Ann. 355 (2013), no. 3, 1085-1121. MR 3020155, https://doi.org/10.1007/s00208-012-0816-y
  • [3] K. Bringmann, B. Kane and R. C. Rhoades, Duality and differential operators for harmonic weak Maass forms, Dev. Math., 28, Springer, New York, 2013. MR 2986955
  • [4] Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45-90. MR 2097357 (2005m:11089), https://doi.org/10.1215/S0012-7094-04-12513-8
  • [5] B. Cho, S. Choi and C. H. Kim, Harmonic weak Maass-modular grids in higher level cases, Acta Arith. 160 (2013), no. 2, 129-141.MR 3105331
  • [6] SoYoung Choi and Chang Heon Kim, Congruences for Hecke eigenvalues in higher level cases, J. Number Theory 131 (2011), no. 11, 2023-2036. MR 2825109 (2012j:11101), https://doi.org/10.1016/j.jnt.2011.04.013
  • [7] SoYoung Choi and Chang Heon Kim, Basis for the space of weakly holomorphic modular forms in higher level cases, J. Number Theory 133 (2013), no. 4, 1300-1311. MR 3004001, https://doi.org/10.1016/j.jnt.2012.09.015
  • [8] P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3051-3059. MR 2407067 (2009e:11089), https://doi.org/10.1090/S0002-9939-08-09277-0
  • [9] Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)
  • [10] Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 347-454. MR 2555930 (2010m:11060)
  • [11] Junichi Shigezumi, On the zeros of certain Poincaré series for $ \Gamma _0^*(2)$ and $ \Gamma _0^*(3)$, Osaka J. Math. 47 (2010), no. 2, 487-505. MR 2722370 (2011k:11058)
  • [12] J. Shigezumi, On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels, arXiv:0882.1307v2.
  • [13] W. Szpankowski , Mellin Transform and Its Applications, http://www.cs.purdue.edu/homes/spa/papers/chap9.ps

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F11, 11F67, 11F37

Retrieve articles in all journals with MSC (2010): 11F11, 11F67, 11F37


Additional Information

SoYoung Choi
Affiliation: Department of Mathematics Education, Dongguk University-Gyeongju, 123 Dongdae-ro, Gyeongju, Gyeongbuk, 780-714, Republic of Korea
Email: young@dongguk.ac.kr

Chang Heon Kim
Affiliation: Department of Mathematics and Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
Address at time of publication: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Email: chhkim@skku.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12073-9
Keywords: Weakly holomorphic modular forms, mock modular forms, period polynomials
Received by editor(s): September 5, 2012
Received by editor(s) in revised form: September 19, 2012, and October 20, 2012
Published electronically: June 19, 2014
Additional Notes: The first author was supported by the Dongguk University Research fund of 2013 and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A3011711)
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2A10004632)
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society