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. Assumethat there is a group isomorphism G(AK,f ) ∼= G(AL,f ) of points on G over the finite adeles ofK and L, respectively. We establish conditions on the group G, related to the structure of its Borelgroups, under which K and L have isomorphic adele rings. Under these conditions, if K or L is aGalois 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, thefinite 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 ofautomorphic representations of the theorem of Neukirch that the absolute Galois group of a numberfield determines the field, if it is Galois over Q.