2009/06/29 by Gunther Cornelissen, Oliver Lorscheid, Cornelissen, Gunther +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Automorphic form #Converse theorem #Cusp (singularity) #Cusp form #Dirichlet distribution #Dirichlet series #Eisenstein series #Field (mathematics) #Fourier series #Geometry #Hecke operator #Mathematical analysis #Mathematics #Modular form #Physics #Pure mathematics #Quadratic equation #Quadratic field #Quadratic function #Quantum mechanics #Series (stratigraphy) #Space (punctuation) #Toroid #Torus #math.NT #msc:11F12 #msc:11M06 #msc:11M41 #msc:11R42
paper · pdf · doi:10.48550/arxiv.0906.5284
published in arXiv (Cornell University) (Cornell University) · 14 pages
openalex publication_date 2009/06/29 · arxiv created 2011/08/15 · arxiv updated 2011/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space spanned by all derivatives up to order n-1 of an Eisenstein series of weight s and class group character omega precisely if s is a zero of order n of the L-series corresponding to omega at s, and a space consisting of exactly those cusp forms the central value of whose L-series is zero. The proofs are based on an identity of Hecke for toroidal integrals of Eisenstein series and a result of Waldspurger about toroidal integrals of cusp forms combined with non-vanishing results for twists of L-series proven by the method of double Dirichlet series.