A coarse Jacquet–Zagier trace formula for GL(n), with applications

We establish a coarse Jacquet-Zagier trace identity for GL.n/ over a global field.We prove the absolute convergence in Re.s/ > 1, and obtain holomorphic continuation under almost all character twists.In particular, we compute all P -regular orbital integrals in the geometric side, and the contributions from constant terms and nondegenerate terms of Eisenstein series in the continuous spectrum, and further derive meromorphic continuations of them.As an application, we give a conditional proof of the Dedekind conjecture, which asserts that given any finite extension E=F of number fields, the ratio E .s/=F .s/ of the Dedekind zeta functions is entire, which is only known for ŒE W F Ä 4. Some nonvanishing results are also obtained.1. Introduction 862 1A.The Rankin-Selberg-Zagier integral R K 0 .x;x/E.x; s/ dx 862 1B.Statement of the main results 863 2. Idea of proof and structure of the paper 865 2A.Basic notation 866 2B.Decomposition of the kernel function 867 2C.Decomposition of I ' 0 .s;/ 868 2D.Structure of the paper 869 2E.Some remarks on reading this paper 870 3. Fourier expansion and decomposition of I 0 .s;/ 870 3A.Mirabolic Fourier expansions of automorphic functions 870 3B.Decomposition of I 0 .s;/ 871 4. I Geo;Reg .s;/ as Dedekind zeta functions 873 4A.Structure of G.F /-conjugacy classes 873 4B.Decomposition of G.F / 874 4C.Algebraic expansion: P -regular conjugacy classes 883 5.I P;Reg .s;/ as intertwining operators 892 5A.P .F /-conjugacy classes 892 5B.Holomorphic continuation 896 6. Convergence of the spectral side 900 6A.Reduction to the relative trace formula of Kuznetsov type 901 6B.Spectral decomposition of the kernel function 905 6C.Spectral expansion of I Whi .s;/