Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups

Abstract

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. We establish conditions on the group $G$, related to the structure of its Borel groups, under which the existence of a group isomorphism $G({\mathbf{A}}_{K,f})\cong G({\mathbf{A}}_{L,f})$ over the finite adeles implies that $K$ and $L$ have isomorphic adele rings. Furthermore, if $G$ satisfies these conditions, $K$ or $L$ is a Galois extension of $\mathbf{Q}$, and $G({\mathbf{A}}_{K,f})\cong G({\mathbf{A}}_{L,f})$, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$ that are Galois over $\mathbf{Q}$, the finite Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n\ge 2$) are isomorphic by an isometry for the $L^1$-norm, then the fields $K$ and $L$ are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field, if it is Galois over $\mathbf{Q}$.