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Disappearance of cusp forms in special families. (English) Zbl 0826.11024

Real analytic cusp forms and Eisenstein series of weight zero are the automorphic forms occurring in the spectral decomposition of the Laplace operator on quotients of the upper half plane \({\mathcal H}\) by cofinite discrete groups. These quotients are Riemann surfaces with a hyperbolic structure. The present paper gives new evidence that cusp forms are rare, unless there are symmetry reasons for their presence.
The main theorems all have the same structure: The author forms a family \(l\mapsto {\mathcal R}_l\) of Riemann surfaces with hyperbolic structure. For \(l> 0\) the topological structure of \({\mathcal R}_l\) is constant; there are some cusps, and one or more special closed geodesics with varying length \(l\). At \(l=0\) each of these special geodesics degenerates into a pair of cusps. The surface \({\mathcal R}_0\) is not necessarily connected. The surfaces \({\mathcal R}_l\) have one or two reflections fixing cusps. The statement is that for generic values of \(l\) the surface \({\mathcal R}_l\) has at most a finite number of linearly independent cusp forms that are invariant under the reflections. The restriction to reflection-even functions is essential. To span the orthogonal complement, infinitely many linearly independent cusp forms may be needed.
The statements are proved under assumption of a hypothesis for the arithmetic cases \(\text{PSL}_2 (\mathbb{Z}) \setminus {\mathcal H}\), \(\Gamma_0 (2) \setminus {\mathcal H}\), and \(\Gamma_0 (4) \setminus {\mathcal H}\): The dimension of each space of cusp forms of weight zero is not larger than newform oldform considerations force it to be.
The three arithmetic cases form the starting point for the proof. These surfaces occur as components of \({\mathcal R}_0\) for five special families. For \(l>0\) the curves in these families have types \((g,n)= (0,4)\), \((1,1)\), \((1,2)\), \((1,4)\), and \((0,6)\) (\(g\) denotes the genus and \(n\) the number of cusps). Generic elements of the special families are used as components of \({\mathcal R}_0\) for other families. In this way the author arrives at families with type \((g,n)\) with \(g\geq 2\) and \(n\geq 1\), and with \(g=0\) and \(n\) even. By a slightly different method the case \(g=1\) is handled.
Compact hyperbolic surfaces have no cusps at all. For each \(g\geq 2\) a family is given for which the eigenvalues larger than \(1/4\) are not constant. These results are proved by a study of the perturbation from \({\mathcal R}_0\) to \({\mathcal R}_l\), in which a pair of cusps becomes a geodesic of length \(l\). This perturbation is more complicated than the perturbation of surfaces that has been studied in the papers by R. S. Phillips and P. Sarnak [Invent. Math. 80, 339-364 (1985; Zbl 0558.10017) and Commun. Pure Appl. Math. 38, 853-866 (1985; Zbl 0614.10027)]. There analytic perturbation theory of linear operators suffices, here asymptotic perturbation theory is needed. Infinitesimally, the perturbation is not described by a holomorphic cusp form on \({\mathcal R}_0\) but by a holomorphic Eisenstein series of weight 4. That makes the nonvanishing of the relevant special value of an \(L\)-function easier to handle than in the case of Phillips and Sarnak.
The introduction of the paper contains a detailed overview of the proof (see pp. 241-244).

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F11 Holomorphic modular forms of integral weight
58J37 Perturbations of PDEs on manifolds; asymptotics
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