Number Theory Seminar

Tuesday 11 March 2008, MR2, 5pm

Adelic representations of elliptic type (Kuwait lecture)
David Rohrlich (Boston)


     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 l-adic 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.

Last modified on 29/1/2008.

Comments/corrections to Tim Dokchitser