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 finite 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(AK,fβ) and
G(AL,fβ) are isomorphic, 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>1) 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.Comment: 19 pages - completely rewritte