#### DOCUMENTA MATHEMATICA, Vol. 22 (2017), 851-871

Gunther Cornelissen and Valentijn Karemaker

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

Full text: dvi.gz 42 k, dvi 98 k, ps.gz 333 k, pdf 237 k.