Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T10:43:12.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular forms of half-integral weights on SL(2, ℤ)

Published online by Cambridge University Press:  11 January 2016

Yifan Yang
Yifan Yang*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University and National Center for Theoretical Sciences, Hsinchu 300, Taiwan, yfyang@math.nctu.edu.tw
Rights & Permissions [Opens in a new window]

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 this paper, we prove that, for an integer r with (r, 6) = 1 and 0 < r < 24 and a nonnegative even integer s, the set

is isomorphic to

as Hecke modules under the Shimura correspondence. Here Ms(1) denotes the space of modular forms of weight is the space of newforms of weight 2k on Γ0 (6) that are eigenfunctions with eigenvalues €2 and €3 for Atkin-Lehner involutions W2 and W3, respectively, and the notation ⊕(12/.) means the twist by the quadratic character (12/-). There is also an analogous result for the cases (r, 6) = 3.

Keywords

11F3711F11
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 215 , September 2014 , pp. 1 - 66
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Ahlgren, S., Distribution of the partition function modulo composite integers M, Math. Ann. 318 (2000), 795–803. MR 1802511. DOI 10.1007/s002080000142.Google Scholar
[2] Ahlgren, S. and Ono, K., Congruence properties for the partition function, Proc. Natl. Acad. Sci. USA 98 (2001), 12882–12884. MR 1862931. DOI 10.1073/pnas.191488598.CrossRefGoogle ScholarPubMed
[3] Atkin, A. O. L. and Lehner, J., Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160. MR 0268123.Google Scholar
[4] Cox, D. A., Primes of the Form x2 +ny2: Fermat, Class Field Theory and Complex Multiplication, Wiley, New York, 1989. MR 1028322.Google Scholar
[5] Eichler, M., “The basis problem for modular forms and the traces of the Hecke operators” in Modular Functions of One Variable, I (Antwerp, 1972), Lecture Notes in Math. 320, Springer, Berlin, 1973, 75–151. MR 0485698.Google Scholar
[6] Flicker, Y. Z., Automorphic forms on covering groups of GL(2), Invent. Math. 57 (1980), 119–182. MR 0567194. DOI 10.1007/BF01390092.Google Scholar
[7] Frechette, S. M., Hecke structure of spaces of half-integral weight cusp forms, Nagoya Math. J. 159 (2000), 53–85. MR 1783564.Google Scholar
[8] Frechette, S. M., A classical characterization of newforms with equivalent eigenforms in Sk+1/2(4N,χ), J. Lond. Math. Soc. (2) 68 (2003), 563–578. MR 2009437. DOI 10.1112/S0024610703004666.CrossRefGoogle Scholar
[9] Garvan, F. G., Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank, Int. J. Number Theory 6 (2010), 281–309. MR 2646759. DOI 10.1142/S179304211000296X.Google Scholar
[10] Gelbart, S. S., Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Math. 530, Springer, Berlin, 1976. MR 0424695.Google Scholar
[11] Guo, L. and Ono, K., The partition function and the arithmetic of certain modular L-functions, Int. Math. Res. Not. IMRN 1999, no. 21, 1179–1197. MR 1728677. DOI 10.1155/S1073792899000641.Google Scholar
[12] Hijikata, H., Explicit formula of the traces of Hecke operators for Γ0(N), J. Math. Soc. Japan 26 (1974), 56–82. MR 0337783.Google Scholar
[13] Kohnen, W., Modular forms of half-integral weight on Γ0 (4), Math. Ann. 248 (1980), 249–266. MR 0575942. DOI 10.1007/BF01420529.Google Scholar
[14] Kohnen, W., Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. MR 0660784. DOI 10.1515/crll.1982.333.32.Google Scholar
[15] Kohnen, W. and Zagier, D., Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175–198. MR 0629468. DOI 10.1007/BF01389166.Google Scholar
[16] Niwa, S., Modular forms of half-integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161. MR 0364106.Google Scholar
[17] Ogg, A. P., “Real points on Shimura curves” in Arithmetic and Geometry, I, Progr. Math. 35, Birkhäuser, Boston, 1983, 277–307. MR 0717598.Google Scholar
[18] Ono, K., Distribution of the partition function modulo m, Ann. of Math. (2) 151 (2000), 293–307. MR 1745012. DOI 10.2307/121118.Google Scholar
[19] Shimura, G., On modular forms of half-integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 0332663.Google Scholar
[20] Shimura, G., On the trace formula for Hecke operators, Acta Math. 132 (1974), 245–281. MR 0562508.Google Scholar
[21] Shintani, T., On construction of holomorphic cusp forms of half-integral weight, Nagoya Math. J. 58 (1975), 83–126. MR 0389772.Google Scholar
[22] Szmidt, J., Urbanowicz, J., and Zagier, D., Congruences among generalized Bernoulli numbers, Acta Arith. 71 (1995), 273–278. MR 1339132.Google Scholar
[23] Ueda, M. and Yamana, S., On newforms for Kohnen plus spaces, Math. Z. 264 (2010), 1–13. MR 2564929. DOI 10.1007/s00209-008-0449-8.CrossRefGoogle Scholar
[24] Waldspurger, J.-L., Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), 1–132. MR 0577010.Google Scholar
[25] Waldspurger, J.-L., Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), 375–484. MR 0646366.Google Scholar
[26] Weber, H., Lehrbuch der Algebra, III, 3rd ed., Chelsea, New York, 1961.Google Scholar
[27] Yamauchi, M., On the traces of Hecke operators for a normalizer of Γ0(N), J. Math. Kyoto Univ. 13 (1973), 403–411. MR 0340179.Google Scholar
[28] Yang, Y., Congruences of the partition function, Int. Math. Res. Not. IMRN 2011, no. 14, 3261–3288. MR 2817679. DOI 10.1093/imrn/rnq194.Google Scholar