Mock modular period functions and $L$-functions of cusp forms in higher level cases
HTML articles powered by AMS MathViewer
- by SoYoung Choi and Chang Heon Kim PDF
- Proc. Amer. Math. Soc. 142 (2014), 3369-3386 Request permission
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
- G. Bol, Invarianten linearer differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), no. nos. 3-4, 1–28 (German). MR 33411, DOI 10.1007/BF03343515
- 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, DOI 10.1007/s00208-012-0816-y
- Kathrin Bringmann, Ben Kane, and Robert C. Rhoades, Duality and differential operators for harmonic Maass forms, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 85–106. MR 2986955, DOI 10.1007/978-1-4614-4075-8_{6}
- Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, DOI 10.1215/S0012-7094-04-12513-8
- Bumkyu Cho, SoYoung Choi, and Chang Heon Kim, Harmonic weak Maass-modular grids in higher level cases, Acta Arith. 160 (2013), no. 2, 129–141. MR 3105331, DOI 10.4064/aa160-2-3
- 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, DOI 10.1016/j.jnt.2011.04.013
- 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, DOI 10.1016/j.jnt.2012.09.015
- P. Guerzhoy, Hecke operators for weakly holomorphic modular forms and supersingular congruences, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3051–3059. MR 2407067, DOI 10.1090/S0002-9939-08-09277-0
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- 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
- 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
- J. Shigezumi, On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels, arXiv:0882.1307v2.
- W. Szpankowski , Mellin Transform and Its Applications, http://www.cs.purdue.edu/homes/spa/papers/chap9.ps
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
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3238414