A SEMI-ADELIC KUZNETSOV FORMULA OVER NUMBER FIELDS

In this paper, we prove a semi-adelic version of the Kuznetsov formula over number fields. This formula matches a weighted sum made of Fourier coefficients of cusp forms and Eisenstein series with a weighted sum of Kloosterman sums, the latter weight function is a kind of Bessel transform of the former one. We obtain a variant which is valid over all number fields. The admissible weight functions are important in applications, they depend on the archimedean parameters of the representations and show exponential decay. The automorphic vectors are not necessarily spherical in the archimedean aspect. Such formulas are proven to be useful in analytic number theory, e.g., in the estimate of L-functions on the critical line.