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**Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weights.**
*(English)*
Zbl 1078.11030

The author shows that certain identities between traces of Hecke operators of half-integral weight and those of integral weight.

Let \(R_\psi\) be a twisting operator for a quadratic primitive character \(\psi\) and \(\widetilde T(n^2)\) the \(n^2\)th Hecke operator of half-integral weight. In the case that \(\psi\) has an odd conductor, the author already found trace identities between twisted Hecke operators \(R_\psi\widetilde T(n^2)\) of half-integral weight and certain Hecke operators of integral weight for almost all cases.

In this paper, he proves that the restriction on the conductor is removed. Namely, he gives similar trace identities for every quadratic primitive character \(\psi\), including the case that \(\psi\) has an even conductor.

At the end of this paper, he remarks that a theory of newforms in the case of level \(2^m\) is established.

Let \(R_\psi\) be a twisting operator for a quadratic primitive character \(\psi\) and \(\widetilde T(n^2)\) the \(n^2\)th Hecke operator of half-integral weight. In the case that \(\psi\) has an odd conductor, the author already found trace identities between twisted Hecke operators \(R_\psi\widetilde T(n^2)\) of half-integral weight and certain Hecke operators of integral weight for almost all cases.

In this paper, he proves that the restriction on the conductor is removed. Namely, he gives similar trace identities for every quadratic primitive character \(\psi\), including the case that \(\psi\) has an even conductor.

At the end of this paper, he remarks that a theory of newforms in the case of level \(2^m\) is established.

Reviewer: Shoyu Nagaoka (Osaka)

### MSC:

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

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

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\textit{M. Ueda}, Proc. Japan Acad., Ser. A 80, No. 7, 131--135 (2004; Zbl 1078.11030)

### References:

[1] | Kohnen, W.: Newforms of half-integral weight. J. Reine und Angew. Math., 333 , 32-72 (1982). · Zbl 0475.10025 |

[2] | Miyake, T.: Modular Forms. Springer, Berlin (1989). · Zbl 0701.11014 |

[3] | Niwa, S.: On Shimura’s trace formula. Nagoya Math. J., 66 , 183-202 (1977). · Zbl 0351.10018 |

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

[5] | 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 , 505-555 (1988). · Zbl 0673.10021 |

[6] | Ueda, M.: The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and some trace relations. Japan J. Math., 17 , 83-135 (1991). · Zbl 0742.11031 |

[7] | 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 , 307-309 (1991). · Zbl 0726.11029 |

[8] | Ueda, M.: Trace formula of twisting operators of half-integral weight in the case of even conductors. Proc. Japan Acad., 79A , 85-88 (2003). · Zbl 1053.11044 |

[9] | Ueda, M.: 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 |

[10] | Ueda, M.: Newforms of half-integral weight in the case of level \(2^m\). (Preprint). (Please see my web page URL: http//euler.math.nara-wu.ac.jp/ ueda/ index_e.html). |

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