On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp
The sum formula of \textit{N. V. Kuznetsov} [Mat. Sb., Nov. Ser. 111, 334- 383 (1980; Zbl 0427.10016)] relates an infinite sum \(\sum^{\infty}_{c=1}c^{-1}f(4\pi \sqrt{| nm|}/c)S(n,m;c)\), with f a test function and S(n,n;c) the classical Kloosterman sum, to an integral and a sum over the automorphic spectrum for \(PSL_ 2({\mathbb{Z}})\). The spectral side of the sum formula contains Fourier coefficients of Eisenstein series and of cuspidal Maass forms. The test function on this side is related to f by an explicit Bessel transformation. This sum formula has been generalized in several ways. Here it is discussed for automorphic forms on \(PSL_ 2({\mathcal O})\setminus {\mathcal H}^ 2\), with \({\mathcal H}\) the upper halfplane and \({\mathcal O}\) the ring of integers in \({\mathbb{Q}}(\sqrt{D})\), D square-free, \(D\equiv 2,3 mod 4\), with the condition that \({\mathcal O}\) has class number one. The proof of the generalization of Kuznetsov's formula to this situation follows a method due to D. Zagier. This method is published here for the first time. The map \((z_ 1,z_ 2)\mapsto (z_ 2,z_ 1)\) in \({\mathcal H}^ 2\) leads to involutions in the spaces of automorphic forms. Composition with these involutions gives a twisted version of the sum formula. The author did not obtain a direct description of the relation between the test functions in both sides of the twisted version of the sum formula. In the final section various other ideas presented in Kuznetsov's paper are worked out for the real quadratic situation.
scientific article; zbMATH DE number 4148183