Hecke algebra isomorphisms and adelic points on algebraic groups. (English) Zbl 1418.11093
Summary: Let \(G\) denote a linear algebraic group over \(\mathbb{Q}\) 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 \(G(\mathbb{A}_{K,f}) \cong G(\mathbb{A}_{L,f})\) 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 \(\mathbb{Q}\), and \(G(\mathbb{A}_{K,f}) \cong G(\mathbb{A}_{L,f})\), 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 \(\mathbb{Q}\), the finite Hecke algebras for \(\mathrm{GL}(n)\) (for fixed \(n \geq 2\)) 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 \(\mathbb{Q}\).
We use this result to show that if for two number fields \(K\) and \(L\) that are Galois over \(\mathbb{Q}\), the finite Hecke algebras for \(\mathrm{GL}(n)\) (for fixed \(n \geq 2\)) 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 \(\mathbb{Q}\).
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11R56 | Adèle rings and groups |
| 20C08 | Hecke algebras and their representations |
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
| 22D20 | Representations of group algebras |
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