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Мemorial conference dedicated to the memory of Ivan Matveevich Vinogradov
March 28, 2019 14:00–14:25, Moscow, Steklov Mathematical Institute, Conference hall
 


An additive problem with coefficients of automorphic forms and exceptional eigenvalues of the Laplacian

I. S. Rezvyakova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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I. S. Rezvyakova
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Abstract: We shall talk about the formulae which can be obtained by means of the spectral theory for the additive problem with the coefficients of automorphic forms for the Hecke congruence-group and for the problem of estimating the sum of Kloosterman sums. In these formulae the error terms depend on the exceptional eigenvalues of the Laplacian. By the methods of analytic number theory (i.e., the circle method, the estimation of exponential sums) we provide an estimate for the additive problem, which can be exploit to give a lower bound for the exceptional eigenvalues of the discrete spectrum of the Laplacian for the congruence-group due to the existence of the formulae from the spectral theory. The so obtained estimate for the exceptional eigenvalues is not good enough in accordance with the current known result, but is proved with rather simple tools.

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