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

Ueda, Masaru

Proc. Japan Acad., Ser. A 80, No. 7, 131-135 (2004).

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.

Reviewer: Shoyu Nagaoka (Osaka)

MSC:

11F37	Forms of half-integer weight; nonholomorphic modular forms
11F25	Hecke-Petersson operators, differential operators (one variable)

References:

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