Comparison of spectra (Q1291969)

Let \(\mathbb{H}\) denote the upper half-plane. A famous conjecture of A. Selberg says that every non-zero eigenvalues \(\lambda\) of \(-\Delta\) on \(L^2(\Gamma\setminus \mathbb{H})\) satisfies \(\lambda\geq \frac{1}{4}\) whenever \(\Gamma\) is a congruence subgroup of \(SL_2 (\mathbb{Z})\). It is the author's aim to prove this conjecture. (Reviewer's remark: In the English version of the article it is only said that ``in the present paper, we deal with the Selberg conjecture'' whereas the Russian version claims that ``the Selberg conjecture will be proved in the present work''. A similar discrepancy occurs in the English and Russian version of the author's abstract.) The author's approach is based on two spectral expansions of the same double convolution. One of these is given in terms of the spectral data for \(\Gamma_0(q)\) whereas the second is given by the data for \(SL_2(\mathbb{Z})\). Since the Selberg conjecture is known to be true for \(SL_2(\mathbb{Z})\) it is the author's idea to reduce the Selberg conjecture for \(\Gamma_0(q)\) to its validity of \(SL_2(\mathbb{Z})\). The work under review is written in a very sketchy manner leaving a lot of work to the reader and relies heavily on two rather long and technical papers. Hence the reviewer feels unable to form an opinion on the correctness of the author's alleged proof. In view of the extraordinary importance of the Selberg conjecture the author should publish a detailed version of his argument. Reference [4] is available in English (not in Russian as stated in the English translation); the full quotation is: \textit{V. Bykovsky, N. Kuznetsov} and \textit{A. Vinogradov}, Generalized summation formula for inhomogeneous convolution. In: International Conference Automorphic Functions and Their Applications, Khabarovsk, June 27--July 4, 1988, 18-63 (1990; Zbl 0758.11028). The author has just published a continuation of the paper under review: \textit{A. I. Vinogradov}: Comparison of spectra. II, J. Math. Sci., New York 95, No. 2, 2074-2084 (1999); translation from Zap. Nauchn. Semin. POMI 236, 50-67 (1997). In this work the author tackles the Selberg conjecture for the group \(\Gamma_0(q)\) with the quadratic character \(\chi= \chi_q\) of modulus \(q\).