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 $G(\AK,f) \cong G(\AL,f)$ 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 $G(\AK,f) \cong G(\AL,f)$, 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 $L1$-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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