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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

##### MSC:
 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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