On Arthur's class expansion of the relative trace formula
type: Item
instance of: scholarly article
title: On Arthur's class expansion of the relative trace formula
zbMATH Open document ID: 0738.11039
DOI: 10.1215/S0012-7094-91-06405-7
published in: Duke Mathematical Journal
publication date: 1992-06-26T00:00:00Z
review text: Let \(F\) be an algebraic number field, \(E\) a finite Galois extension of \(F\), \(\mathbb{G}\) a connected reductive algebraic group defined over \(F\), \(G\) its group of \(F\)-rational points, and \(^ 0G\) the intersection of the kernels of all continuous homomorphisms \(G\to \mathbb{R}^*_ +\). The author applies Arthur's technique of class expansion of the trace formula [\textit{J. Arthur}, Duke Math. J. 45, 915--952 (1978; Zbl 0499.10032)] to the relative trace formula \(\iint K(x,y)\,dx\,dy\) where \(K\) is the Selberg kernel function on \(\mathbb{G}(E)\) and \(x,y\in G\backslash^ 0G_{\mathbb{A}}\), and proves the convergence theorem of the integration of the modified kernel. The author claims that this might provide an alternative route to study the base change problem, and in particular to prove Tate's conjecture on algebraic cycles for a family of compact Shimura surfaces.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 24262
zbMATH Keywords: connected reductive algebraic group
zbMATH Keywords: class expansion
zbMATH Keywords: relative trace formula
zbMATH Keywords: convergence theorem
zbMATH Keywords: base change problem
zbMATH Keywords: Tate's conjecture on algebraic cycles
zbMATH Keywords: compact Shimura surfaces
author: King-Fai Lai
reviewed by: Yangbo Ye
MaRDI profile type: Publication
cites work: A trace formula for reductive groups. I: Terms associated to classes in G(Q)
cites work: Relative Trace Formula and Simple Algebras
cites work: Q3714151
cites work: Sur un résultat de Waldspurger
cites work: Algebraic cycles on compact Shimura surface
cites work: Kloosterman integrals and base change for GL (2).
description: scientific article; zbMATH DE number 24262