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.
Assume that there is a group isomorphism of points on G over the adeles of
K and L, respectively. We establish conditions on the group G, related to
the structure of its Borel groups, under which K and L have isomorphic
adele rings. Under these conditions, if K or L is a Galois extension of Q
and G(A_K) and G(A_L) are isomorphic, then K and L are isomorphic as
fields. As a corollary, we show that an isomorphism of Hecke algebras for
GL(n) (for fixed n > 1), which is an isometry in the L^1 norm over two
number fields K and L that are Galois over Q, implies that 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.