Trace formula of twisting operators of half-integral weight in the case of even conductors. (English) Zbl 1053.11044

Ueda, Masaru

Proc. Japan Acad., Ser. A 79, No. 4, 85-88 (2003).

From the text: Let \(k\) and \(N\) be positive integers with \(4\mid N\). Let \(\chi\) be an even Dirichlet character defined modulo \(N\) with \(\chi^2= 1\). Denote the space of cusp forms of weight \(k+1/2\), level \(N\), and character \(\chi\) by \(S(k+1/2,N,\chi)\).
In the previous paper M. Ueda [The trace formulae of twisting operators on the spaces of cup forms of half-integral weight and some trace relations. Jap. J. Math., New Ser. 17, 83–135 (1991; Zbl 0742.11031)], the author calculated an explicit trace formula of the twisting operator \(R_\psi\) on \(S(k+1/2, N,\chi)\) for a quadratic primitive character \(\psi\) of odd conductor. Here the author now gives an explicit trace formula of the twisting operator \(R_\psi\) for a quadratic primitive character \(\psi\) of even conductor. Details will appear in Part II of the above cited paper.

MSC:

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

Citations:

Zbl 0742.11031

References:

[1]	Miyake, T.: Modular Forms. Springer, Berlin (1989). · Zbl 0701.11014
[2]	Shimura, G.: On modular forms of half integral weight. Ann. of Math., 97 , 440-481 (1973). · Zbl 0266.10022
[3]	Ueda, M.: The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and some trace relations. Japan. J. of Math., 17 , 83-135 (1991). · Zbl 0742.11031
[4]	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
[5]	Ueda, M.: Some trace relations of twisting operators on the spaces of cusp forms of half-integral weight. Proc. Japan Acad., 66A , 169-172 (1990). · Zbl 0726.11027