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On Selberg’s small eigenvalue conjecture and residual eigenvalues. (English) Zbl 1237.11024

Let \(\mathcal H\) be the complex upper half-plane. Many years ago A. Selberg [Proc. Sympos. Pure Math. 8, 1–15 (1965; Zbl 0142.33903)] conjectured that the automorphic Laplacian on \(L^2(\Gamma_0(q)\setminus \mathcal H)\), \(q\in\mathbb N\), has no eigenvalues in the interval \((0, 1/4)\). This conjecture has not been proved so far. The author formulates a new conjecture, asserting that given an interval \([a, b]\subseteq(0, 1/4)\), \(q\in\mathbb N\), a Dirichlet character \(\chi_0\) modulo \(q\), and \(f\in M_2(\Gamma_0(q))\) with \[ \int^1_0 f(x+ iy)\,dx= 0\quad\text{for }y> 0, \] there is an \(\varepsilon_0> 0\) such that the automorphic Laplacian on \(L^2(\Gamma_0(q),\chi_\varepsilon)\) has no residual eigenvalues in \([a,b]\) for \(|\varepsilon|\leq \varepsilon_0\). Here \(\chi_\varepsilon: \Gamma_0(q)\to S^1\) is defined by \[ \chi_\varepsilon(\gamma)= \chi_0(\gamma)\exp\Biggl(2\pi i\varepsilon\text{\,Re\,}\int^{\gamma z_0}_{z_0} f(z)\,dz\Biggr) \] for any \(\gamma\) in \(\Gamma_0(q)\) and some \(z_0\in\mathcal H\), and \[ L^2(\Gamma,\chi):= \{f: H\to\mathbb C\mid f\in L^2(\Gamma\setminus \mathcal H),\;f(\gamma z)= \chi(\gamma)f(z),\;\gamma\in\Gamma,\;z\in \mathcal H\} \] for a homomorphism \(\chi: \Gamma\to S^1\) and \(\Gamma\in \{\Gamma_0(q)\mid q\in\mathbb N\}\). Making use of Kato’s perturbation theory, cf. T. Kato [Perturbation theory for linear operators. Reprint. Berlin: Springer-Verlag (1995; Zbl 0836.47009)], Selberg’s trace formula with characters, and a non-vanishing theorem of D. E. Rohrlich [Invent. Math. 97, No. 2, 381–403 (1989; Zbl 0677.10020)], the author proves that his conjecture is equivalent to Selberg’s conjecture.
Reviewer: B. Z. Moroz (Bonn)

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
47A55 Perturbation theory of linear operators
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