2009/07/03 by Gilles Lachaud, Lachaud, Gilles
Mathematics · #11F03 #11F12 #11M26 #11M41 #47A10 #58B34 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F03 #msc:11F12 #msc:11M26 #msc:11M41 #msc:47A10 #msc:58B34
paper · pdf · doi:10.48550/arxiv.0907.0536
35 pages
arxiv created 2009/07/03 · openalex publication_date 2009/07/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An algebraic number field K defines a maximal torus T of the linear group G = GLn. Let χ be a character of the idele class group of K, satisfying suitable assumptions. The χ-toroidal forms are the functions defined on G(Q) Z(A) \backslash G(A) such that the Fourier coefficient corresponding to χ with respect to the subgroup induced by T is zero. The Riemann hypothesis is equivalent to certain conditions concerning some spaces of toroidal forms, constructed from Eisenstein series. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of zeroes of L(s, χ) on the critical line.