Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups

Let $G$ denote a linear algebraic group over $\Q$ 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 $G(\A_{K,f})
\cong G(\A_{L,f})$ 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 $\Q$, and $G(\A_{K,f}) \cong G(\A_{L,f})$,
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 $\Q$, the
finite Hecke algebras for $\GL(n)$ (for fixed $n ≥ 2$) 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 $\Q$.

2010 Mathematics Subject Classification: 11F70, 11R56, 14L10, 20C08, 20G35, 22D20

Keywords and Phrases: algebraic groups, adeles, Hecke algebras, arithmetic equivalence

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