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Modular forms of varying weight. III. (English) Zbl 0588.10021

Let \(S(r,\lambda)\) be the space of real analytic cusp forms for the full modular group with weight \(r\in {\mathbb{R}}\), eigenvalue \(\lambda\in {\mathbb{R}}\) and the same multiplier system as the \((2r)\)-th power of the Dedekind eta-function \(\eta\). In part I [Math. Z. 190, 477-495 (1985; Zbl 0553.10021)] the existence was shown of a collection \(\{\phi_ j:\) \(j\geq 1\}\) of families of cusp forms, analytic on (0,12), with \(\phi_ j(r)\in S(r,\mu_ j(r))\) and \(\mu_ j\) also analytic on (0,12). The \(\phi_ j(r)\) form an orthonormal basis of the corresponding Hilbert space. Let \(\phi_ 1\) be the family corresponding to \(\eta^{2r}.\)
The present paper shows: (1) The analytic functions \(\mu_ j\) on (0,12) are all different. (2) \(\mu_ j>1/4\) on (0,12) for all \(j\geq 2\). (3) For \(j\geq 2\) the family \(\phi_ j\) and the eigenvalue \(\mu_ j\) have an analytic extension to a maximal interval \((-12n_ j, 12m_ j),\) with \(m_ j\geq 1\), \(n_ j\geq 0\) integers. (4) For \(j\geq 2\) the limit of \(\mu_ j\) at \(-12n_ j\) and \(12m_ j\) exists and is equal to 1/4. (5) For \(j\geq 2\) and \(n_ j\geq 1\) there is a unique \(k\geq 2\) such that \(\mu_ j(r-12)=\mu_ k(r);\) \(\phi_ j(r-12)\) and \(\phi_ k(r)\) are related as well.
(6) Let \(C\) be the coefficient occurring in the Fourier term of order zero of the meromorphic continuation in two variables of the Eisenstein series, as studied in part II [Math. Z. 192, 297-328 (1986; Zbl 0569.10011)]. For \(t>0:\) C is not holomorphic at \((0,it)\) if and only if \(S(0,1/4+t^ 2)\) is not spanned by the \(\phi_ j(0)\) with \(j\geq 2\), \(n_ j\geq 1\), \(\mu_ j(0)=1/4+t^ 2\).

MSC:

11F11 Holomorphic modular forms of integral weight
35P15 Estimates of eigenvalues in context of PDEs
47A55 Perturbation theory of linear operators
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