On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp
type: Item
instance of: scholarly article
title: On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp
zbMATH Open document ID: 0701.11020
DOI: 10.1007/BF02570723
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174114
review text: 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.
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 4148183
zbMATH Keywords: Hilbert modular forms
zbMATH Keywords: real quadratic field
zbMATH Keywords: sum formula of Kuznetsov
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: automorphic spectrum
author: David Joyner
reviewed by: Roelof Bruggeman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4148183