Nagoya Mathematical Journal

Hecke structure of spaces of half-integral weight cusp forms

Sharon M. Frechette

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Abstract

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [4], we obtain structure theorems for spaces of half-integral weight cusp forms $S_{k/2}{4N}{\chi}$ where $k$ and $N$ are odd nonnegative integers with $k \geq 3$, and $\chi$ 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 $\chi$, by giving conditions under which a newform has an equivalent cusp form in $S_{k/2}{4N}{\chi}$. 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 $\chi$. 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.

Article information

Source
Nagoya Math. J., Volume 159 (2000), 53-85.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631452

Mathematical Reviews number (MathSciNet)
MR1783564

Zentralblatt MATH identifier
1018.11022

Subjects
Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms
Secondary: 11F11: Holomorphic modular forms of integral weight 11F32: Modular correspondences, etc.

Citation

Frechette, Sharon M. Hecke structure of spaces of half-integral weight cusp forms. Nagoya Math. J. 159 (2000), 53--85. https://projecteuclid.org/euclid.nmj/1114631452


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References

  • A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma_0(m)$ , Math. Ann., 185 (1970), 134–160.
  • A. O. L. Atkin and W. Li, Twists of newforms and pseudo-eigenvalues of $W$-operators , Invent. Math., 48 (1978), 222–243.
  • B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves I and II , J. reine angew. Math., 212 (1963), 7–25 and 218 (1965), 79–108.
  • J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press (1992).
  • Y. Flicker, Automorphic forms on covering groups of $GL(2)$ , Inventiones Math., 57 (1980), 119–182.
  • H. Hijikata, A. Pizer and T. Shemanske, Twists of newforms , J. Num. Theory, 35, no. 3 (1990), 287–324.
  • W. Kohnen, Newforms of half-integral weight , J. reine angew. Math., 333 (1982), 32–72.
  • W. Li, Newforms and functional equations , Math. Ann., 212 (1975), 285–315.
  • A. Pizer, Theta series and modular forms of level $p^2M$ , Compositio Math., 40 (1980), 177–241.
  • S. Ross, A Simplified trace formula for hecke operators for $\Gamma_0(N)$ , Trans. Amer. Math. Soc., 331 (1992), 425–447.
  • H. Saito and M. Yamauchi, Trace formula of certain hecke operators for $\Gamma_0(q^\nu)$ , Nagoya Math. J., 76 (1979), 1–33.
  • G. Shimura, On modular forms of half-integral weight , Annals of Math., 97 (1973), 440–481.
  • T. Shintani, On construction of holomorphic cusp forms of half integral weight , Nagoya Math. J., 58 (1975), 83–126.
  • M. Ueda, 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.
  • ––––, On twisting operators and newforms of half-integral weight II –- complete theory of newforms for Kohnen space , Nagoya Math. J., 149 (1998), 117–172.
  • ––––, 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.
  • J. L. Waldspurger, Sur Les Coefficients de Fourier des Formes Modulaires de Poids Demi-Entier , J. Math. Pures et Appl., 60 (1981), 375–484.
  • M. Yamauchi, On the traces of Hecke operators for a normalizer of $\Gamma_0(N)$, J. Math. Kyoto Univ., 13 (1973), 403–411.