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Small eigenvalues of Laplacian for \(\Gamma_ 0(N)\). (English) Zbl 0702.11034
Let \(0=\lambda_ 0<\lambda_ 1\leq \lambda_ 2\leq...\), \(\lambda_ n\to \infty\) (n\(\to \infty)\) be the eigenvalues of the Laplace-Beltrami operator \(-\Delta\) for the hyperbolic metric on the upper half-plane H, where \(-\Delta\) acts on its domain in \(L^ 2(\Gamma_ 0(N)\setminus H)\). The work under review is a contribution to Selberg’s conjecture which claims that \(\lambda_ 1\geq 1/4\) for all congruence groups. The first idea is to investigate the distribution of the points \(s_ j\) \((\lambda_ j=s_ j(1-s_ j))\) in the interval \(],1[\) from a statistical point of view. A quantitative version of this idea is the author’s “density theorem” which says that \[ \sum_{<s_ j<1}V^{c(s_ j-1/2)}\quad \ll \quad V^{1+\epsilon}, \] where \(V=vol(\Gamma_ 0(N)\setminus H)\), \(\epsilon\) is any positive number and the constant implied in \(\ll\) depends only on \(\epsilon\) (but not on N). Improving remarkably on his earlier results, the author proves in the present work that the density theorem holds true with exponent \(c=4\). This result is deduced from a sharp estimate for the Fourier coefficients of Maaß cusp forms (see Theorem 2).
Second, of crucial importance for the work under review is a sharp estimate for bilinear forms whose coefficients are certain Kloosterman sums (see Theorem 3). The latter estimate is applied to investigate the exceptional spectrum of \(-\Delta\) by means of a version of the Kuznetsov summation formula. Considering the eigenvalue problem of \(-\Delta\) on \(L^ 2(\Gamma_ 0(p)\setminus H,\chi_ p)\) with a Dirichlet character \(\chi_ p mod p\), the author improves on Selberg’s bound \(\lambda_ 1\geq 3/16\) for almost all groups \(\Gamma_ 0(p).\)
This is indeed a remarkable achievement.
Reviewer: J.Elstrodt

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F11 Holomorphic modular forms of integral weight
11L05 Gauss and Kloosterman sums; generalizations
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