On Arthur's class expansion of the relative trace formula
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.
scientific article; zbMATH DE number 24262