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The first eigenvalue problem and tensor products of zeta functions. (English) Zbl 1065.11035

The paper under review is concerned with Selberg’s eigenvalue conjecture for the lowest positive eigenvalue \(\lambda_1\) of the Laplacian on a congruence subgroup of \(\text{SL}_2(\mathbb{Z})\) or \(\text{SL}_2({\mathcal O}_\alpha)\), where \({\mathcal O}_\alpha\) is the ring of integers of an imaginary quadratic number field of discriminant \(d<0\). The author introduces the following assumption on the integer \(n\geq 2\) and a cuspidal automorphic representation \(\pi\) of \(\text{GL}_2(\mathbb{A}_k)\) with a number field \(k\):
There is a cuspidal automorphic representation \(\pi_n=\text{Sym}^{n-1}(\pi)\) of \(\text{GL}_n(\mathbb{A}_k)\) whose \(L\)-function is equal to \(L(s,\text{Sym}^{n-1}(\pi))\).
This assumption is known to hold for \(n\geq 5\). If this assumption is true for all \(n\) then Selberg’s conjecture holds true. Moreover, if this assumption holds for some integer \(n\geq 2\) the author gives lower bounds for \(\lambda_1\) depending on \(n\) which give the best known lower bounds in the two-dimensional case (for \(n=4\) or 5, respectively). For \(n=5\) this bound yields in the three-dimensional case (i.e. for congruence subgroups of \(\text{SL}_2({\mathcal O}_d)): \lambda_1\geq \frac{160}{169}=0.946\dots\) which is remarkably close to Selberg’s conjecture \(\lambda_1\geq 1\).
In the last section the author comments on the general principle that tensor products of zeta-functions will be useful for solving problems of the type of the Riemann hypothesis, and he introduces a possibly useful candidate.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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References:

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