2014/09/04 by Gunther Cornelissen, Cornelissen, Gunther, Valentijn Karemaker +1
Mathematics · #11F70 #11R56 #14L10 #20C08 #20G35 #22D20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11F70 #msc:11R56 #msc:14L10 #msc:20C08 #msc:20G35 #msc:22D20
paper · pdf · doi:10.48550/arxiv.1409.1385
19 pages - completely rewritten
arxiv created 2015/08/04 · arxiv updated 2015/08/05
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,f) and G(AL,f) 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.