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SUMMARY:Adelic representations of elliptic type - Professor David Rohrlich
  (Boston)
DTSTART:20080311T170000Z
DTEND:20080311T180000Z
UID:TALK8788@talks.cam.ac.uk
CONTACT:Helen Innes
DESCRIPTION:In axiomatizing their study of Frobenius distributions in GL(2
 )-extensions of the rationals\, Lang and Trotter introduce the notion of a
 n adelic Galois representation of “elliptic type\,” and they ask in pa
 ssing whether every such representation arises from an elliptic curve.  Ro
 ughly speaking\, the question is whether certain conditions that are neces
 sary for a representation to come from an elliptic curve over the rational
 s 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 intro
 duction to the ring of adelic integers and the notion of a Galois represen
 tation over this ring (essentially the same thing as a strictly compatible
  family of l-adic Galois representations).  We shall then look more closel
 y at the problem posed by Lang and Trotter.  In spite of the extraordinary
  advances that have been made in Galois representation theory during the i
 ntervening decades\, the problem of Lang and Trotter may not yet be ripe f
 or a solution\, but the obstacles that remain appear to be interesting que
 stions in their own right. 
LOCATION:Wolfson Room (MR 2) Centre for Mathematical Sciences\, Wilberforc
 e Road\, Cambridge
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