Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups
Gunther CornelissenMathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland
Valentijn KaremakerDepartment of Mathematics, University of Pennsylvania, David Rittenhouse Labs, 209 South 33rd Street, Philadelphia, PA 19104-6395, U.S.A.
Let denote a linear algebraic group over and and two number fields. We establish conditions on the group , related to the structure of its Borel groups, under which the existence of a group isomorphism over the finite adeles implies that and have isomorphic adele rings. Furthermore, if satisfies these conditions, or is a Galois extension of , and , then and are isomorphic as fields. We use this result to show that if for two number fields and that are Galois over , the finite Hecke algebras for (for fixed ) are isomorphic by an isometry for the -norm, then the fields and 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 .
Cite this article
Gunther Cornelissen, Valentijn Karemaker, Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups. Doc. Math. 22 (2017), pp. 851–871DOI 10.4171/DM/580