Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups

Valentijn Karemaker ; Gunther Cornelissen

Documenta mathematica, Tome 22 (2017), pp. 851-871

Résumé

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​)≅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​)≅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.

Détail

DOI : 10.4171/dm/580

Classification : 11F70, 11R56, 14L10, 20C08, 20G35
Mots-clés : algebraic groups, Hecke algebras, adeles, arithmetic equivalence