In axiomatizing their study of Frobenius distributions in
GL(2)extensions of the rationals, Lang and Trotter introduce the
notion of an adelic Galois representation of "elliptic type," and they
ask in passing whether every such representation arises from an
elliptic curve. Roughly speaking, the question is whether certain
conditions that are necessary for a representation to come from an
elliptic curve over the rationals without complex multiplication 
cyclotomic determinant, Weil bound for the trace of Frobenius
elements, openness of the image in GL(2) of the adelic integers  are
also sufficient. This talk will begin with an introduction to the ring
of adelic integers and the notion of a Galois representation over this
ring (essentially the same thing as a strictly compatible family of
ladic Galois representations). We shall then look more closely at the
problem posed by Lang and Trotter. In spite of the extraordinary
advances that have been made in Galois representation theory during
the intervening decades, the problem of Lang and Trotter may not yet
be ripe for a solution, but the obstacles that remain appear to be
interesting questions in their own right.
