Transition: Eisenstein series on adele groups

[1] Despite contrary assertions in the literature, rewriting Eisenstein series, as opposed to more general automorphic forms, on adele groups does not use Strong Approximation. Strong Approximation does make precise the relation between general automorphic forms on adele groups and automorphic forms on SL2 and even on SLn, but rewriting these Eisenstein series does not need this comparison. Indeed, Strong Approximation does not hold in the simplest form for general semi-simple or reductive groups, but this does harm anything. Strong approximation does show that there can be no other extension of the Eisenstein series to the adele group beyond that described here.