Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-10-04T16:27:48.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Hecke structure of spaces of half-integral weight cusp forms

Published online by Cambridge University Press:  22 January 2016

Sharon M. Frechette
Sharon M. Frechette*
Affiliation:
Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481-8203, U.S.A., sfrechette@wellesley.edu
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.

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Keywords

11F3711F3211F11
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 159 , 2000 , pp. 53 - 85
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[1] Atkin, A. O. L. and Lehner, J., Hecke operators on Γ0(m), Math. Ann., 185 (1970), 134–160.CrossRefGoogle Scholar
[2] Atkin, A. O. L. and Li, W., Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math., 48 (1978), 222–243.CrossRefGoogle Scholar
[3] Birch, B. J. and Swinnerton-Dyer, H. P. F., Notes on elliptic curves I and II, J. reine angew. Math., 212 (1963), 7–25 and 218 (1965), 79–108.CrossRefGoogle Scholar
[4] Cremona, J. E., Algorithms for modular elliptic curves, Cambridge University Press, 1992.Google Scholar
[5] Flicker, Y., Automorphic forms on covering groups of GL(2), Inventiones Math., 57 (1980), 119–182.CrossRefGoogle Scholar
[6] Hijikata, H., Pizer, A. and Shemanske, T., Twists of newforms, J. Num. Theory, 35, no. 3 (1990), 287–324.CrossRefGoogle Scholar
[7] Kohnen, W., Newforms of half-integral weight, J. reine angew. Math., 333 (1982), 32–72.Google Scholar
[8] Li, W., Newforms and functional equations, Math. Ann., 212 (1975), 285–315.CrossRefGoogle Scholar
[9] Pizer, A., Theta series and modular forms of level p2 M, Compositio Math., 40 (1980), 177–241.Google Scholar
[10] Ross, S., A Simplified trace formula for hecke operators for Γ0(N), Trans. Amer. Math. Soc, 331 (1992), 425–447.CrossRefGoogle Scholar
[11] Saito, H. and Yamauchi, M., Trace formula of certain hecke operators for Γ0(q), Nagoya Math. J., 76 (1979), 1–33.CrossRefGoogle Scholar
[12] Shimura, G., On modular forms of half-integral weight, Annals of Math., 97 (1973), 440–481.CrossRefGoogle Scholar
[13] Shintani, T., On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J., 58 (1975), 83–126.CrossRefGoogle Scholar
[14] Ueda, M., The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators, J. Math. Kyoto Univ., 28 (1988), 505–555.Google Scholar
[15] Ueda, M., On twisting operators and newforms of half-integral weight II — complete theory of newforms for Kohnen space, Nagoya Math. J., 149 (1998), 117–172.Google Scholar
[16] Ueda, M., Supplement to the decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators, J. Math. Kyoto Univ., 31 (1991), 307–309.Google Scholar
[17] Waldspurger, J. L., Sur Les Coefficients de Fourier des Formes Modulaires de Poids Demi-Entier, J. Math. Pures et Appl., 60 (1981), 375–484.Google Scholar
[18] Yamauchi, M., On the traces of Hecke operators for a normalizer of Γ0(N), J. Math. Kyoto Univ., 13 (1973), 403–411.Google Scholar