On twisting operators and newforms of half-integral weight. (English) Zbl 0778.11027
The theory of newforms is very important and useful for arithmetical study of modular forms of integral weight. It is natural to try to extend this theory into the case of modular forms of half-integral weight. Until now, several authors have attempted to find a theory of newforms of half- integral weight. But complete results have not been obtained yet. The purpose of this paper is to complete W. Kohnen’s work in [J. Reine Angew. Math. 333, 32-72 (1982; Zbl 0475.10025)], and to establish a theory of newforms for (what is called) Kohnen space of arbitrary level.
Let us explain the contents of this paper, precisely. Let \(k\) be a positive integer, \(N=4\times\) (odd positive integer), and \(\chi\) an even character 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 paper cited above, Kohnen defined a canonical subspace (what is called Kohnen space) \(S(k+1/2,N,\chi)_ K\) as follows: \[ S\left( k+{1\over 2},N,\chi\right)_ K=\left\{ \begin{matrix} S(k+{1\over 2},N,\chi)\ni f(z)=\sum^ \infty_{n=1} a(n)e(nz);\\ a(n)=0 \text{ for }\chi_ 2 (- 1)(-1)^ k n\equiv 2, 3\pmod 4\end{matrix}\right\}, \] where \(e(z):= \exp(2\pi\sqrt {-1}z)\) \((z\in {\mathfrak H})\) and \(\chi_ 2\) is the 2- primary component of \(\chi\). Moreover when \(N/4\) is (odd) squarefree, he also established the theory of newforms and a strong multiplicity one theorem for this subspace.
Unfortunately, when \(N/4\) is not squarefree, Kohnen’s theory does not work. In fact, there exists a case such that all common eigen subspaces of \(S(k+1/2,N,\chi)_ K\) for Hecke operators have dimension \(\geq 2\) and hence a strong “multiplicity one theorem” does not hold good [cf. the author, J. Math. Kyoto Univ. 28, 505-555 (1988; Zbl 0673.10021), Proposition 3(3)]. This difficulty can be resolved in this paper by decomposing \(S(k+1/2,N,\chi)_ K\) into eigen subspaces of twisting operators. We define canonical subspaces \(S^{\emptyset,\kappa}\) of \(S(k+1/2,N,\chi)_ K\) by using twisting operators. Moreover we establish a theory of newforms and a strong multiplicity one theorem for \(S^{\emptyset,\kappa}\) in §3 of this paper (cf. Theorems (3.10- 11)).
Finally, we have some comments. It seems likely that the results in §3 and §4 of this paper can be generalized to the full space \(S(k+1/2,N,\chi)\). We remark that Kohnen’s results on the operator \(w^ N_{p,k+1/2,\chi}\) (loc. cit., Proposition 4, Theorem 1) contain a mistake. The factor \(\bigl({{N/p} \over p}\bigr)\) falls out. Our operator \(w_ p\) in §3 is a correction of \(w^ N_{p,k+1/2,\chi}\).
Let us explain the contents of this paper, precisely. Let \(k\) be a positive integer, \(N=4\times\) (odd positive integer), and \(\chi\) an even character 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 paper cited above, Kohnen defined a canonical subspace (what is called Kohnen space) \(S(k+1/2,N,\chi)_ K\) as follows: \[ S\left( k+{1\over 2},N,\chi\right)_ K=\left\{ \begin{matrix} S(k+{1\over 2},N,\chi)\ni f(z)=\sum^ \infty_{n=1} a(n)e(nz);\\ a(n)=0 \text{ for }\chi_ 2 (- 1)(-1)^ k n\equiv 2, 3\pmod 4\end{matrix}\right\}, \] where \(e(z):= \exp(2\pi\sqrt {-1}z)\) \((z\in {\mathfrak H})\) and \(\chi_ 2\) is the 2- primary component of \(\chi\). Moreover when \(N/4\) is (odd) squarefree, he also established the theory of newforms and a strong multiplicity one theorem for this subspace.
Unfortunately, when \(N/4\) is not squarefree, Kohnen’s theory does not work. In fact, there exists a case such that all common eigen subspaces of \(S(k+1/2,N,\chi)_ K\) for Hecke operators have dimension \(\geq 2\) and hence a strong “multiplicity one theorem” does not hold good [cf. the author, J. Math. Kyoto Univ. 28, 505-555 (1988; Zbl 0673.10021), Proposition 3(3)]. This difficulty can be resolved in this paper by decomposing \(S(k+1/2,N,\chi)_ K\) into eigen subspaces of twisting operators. We define canonical subspaces \(S^{\emptyset,\kappa}\) of \(S(k+1/2,N,\chi)_ K\) by using twisting operators. Moreover we establish a theory of newforms and a strong multiplicity one theorem for \(S^{\emptyset,\kappa}\) in §3 of this paper (cf. Theorems (3.10- 11)).
Finally, we have some comments. It seems likely that the results in §3 and §4 of this paper can be generalized to the full space \(S(k+1/2,N,\chi)\). We remark that Kohnen’s results on the operator \(w^ N_{p,k+1/2,\chi}\) (loc. cit., Proposition 4, Theorem 1) contain a mistake. The factor \(\bigl({{N/p} \over p}\bigr)\) falls out. Our operator \(w_ p\) in §3 is a correction of \(w^ N_{p,k+1/2,\chi}\).
Reviewer: M.Ueda (Nara)
MSC:
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
References:
| [1] | Table for modular forms of weight 3/2 (1987) |
| [3] | Japanese J. Math. 17 pp 83– (1991) |
| [4] | J. Math. Kyoto Univ. 28 pp 505– (1988) · Zbl 0673.10021 · doi:10.1215/kjm/1250520402 |
| [5] | DOI: 10.1016/0022-314X(82)90070-1 · Zbl 0485.10018 · doi:10.1016/0022-314X(82)90070-1 |
| [6] | DOI: 10.2307/1970831 · Zbl 0266.10022 · doi:10.2307/1970831 |
| [7] | Springer Lee. Notes in Math. pp 27– (1977) |
| [8] | Nagoya Math. J. 80 pp 129– (1980) · Zbl 0455.10015 · doi:10.1017/S0027763000019097 |
| [9] | Pacific J. of Math |
| [10] | Nagoya Math. J. 66 pp 183– (1977) · Zbl 0351.10018 · doi:10.1017/S0027763000017803 |
| [11] | J. reine angew. Math. 328 pp 58– (1981) |
| [12] | DOI: 10.1016/0022-314X(90)90151-G · Zbl 0704.11013 · doi:10.1016/0022-314X(90)90151-G |
| [13] | Publ. Math. Soc. Japan 11 (1971) |
| [14] | Modular Forms (1989) |
| [15] | DOI: 10.2969/jmsj/03340649 · Zbl 0494.10018 · doi:10.2969/jmsj/03340649 |
| [16] | J. reine und angew. Math. 333 pp 32– (1982) |
| [17] | DOI: 10.2969/jmsj/02610056 · Zbl 0266.12009 · doi:10.2969/jmsj/02610056 |
| [18] | Nagoya Math. J. 91 pp 49– (1983) · Zbl 0523.10014 · doi:10.1017/S0027763000020468 |
| [19] | DOI: 10.1007/BF01390245 · Zbl 0369.10016 · doi:10.1007/BF01390245 |
| [20] | Proc. Japan Acad. 66 pp 173– (1990) |
| [21] | On operators of Atkin-Lehner type for modular forms of half-integral weight and theta series (1985) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
