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**Trace formula and trace identity of twisted Hecke operators on the spaces of cusp forms of weight \(k+1/2\) and level \(32M\).**
*(English)*
Zbl 1238.11054

In a sequence of papers [J. Math. Kyoto Univ. 28, No. 3, 505–555 (1988; Zbl 0673.10021) and Jap. J. Math., New Ser. 17, No. 1, 83–135 (1991; Zbl 0742.11031)] among others, the author studied trace formulae for Hecke operators on spaces of cusp forms of half-integral weight \(k + \frac12\), level \(N\) and character \(\chi\). In the present note a case is discussed which was not treated before: For level \(N = 32 M\) with \(M\) odd, these spaces are not generally closed under twisted Hecke operators \(R_{\psi} \tilde{T}(n^{2})\). However, they are closed under certain conditions on the conductors of \(\chi\) and \(\psi\), and the author presents explicit trace formulae for these cases. The details and proofs are announced to appear elsewhere.

Reviewer: Günter Köhler (Würzburg)

### MSC:

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11F25 | Hecke-Petersson operators, differential operators (one variable) |

### Keywords:

trace formula; twisting operator; half-integral weight; trace identity; Hecke operator; cusp form
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\textit{M. Ueda}, Proc. Japan Acad., Ser. A 85, No. 2, 11--15 (2009; Zbl 1238.11054)

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### References:

[1] | T. Miyake, Modular forms , Translated from the Japanese by Yoshitaka Maeda, Springer, Berlin, 1989. · Zbl 0701.11014 |

[2] | G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440-481. · Zbl 0266.10022 |

[3] | 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), no. 3, 505-555. · Zbl 0673.10021 |

[4] | M. Ueda, The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and some trace relations, Japan. J. Math. (N.S.) 17 (1991), no. 1, 83-135. · Zbl 0742.11031 |

[5] | M. Ueda, Trace formula of twisting operators of half-integral weight in the case of even conductors, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 4, 85-88. · Zbl 1053.11044 |

[6] | M. Ueda, Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weight, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 131-135. · Zbl 1078.11030 |

[7] | M. Ueda, The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and trace identities II. (in preparation). · Zbl 0742.11031 |

[8] | M. Ueda, Newforms of half-integral weight in the case of level \(2^{\text{m}}\). (Preprint). http://www.math.nara-wu.ac.jp/personal/ueda/index_e.html · Zbl 0956.65003 |

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