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CATEGORIES:Kuwait Foundation Lectures
SUMMARY:Adelic representations of elliptic type - Professo
r David Rohrlich (Boston)
DTSTART;TZID=Europe/London:20080311T170000
DTEND;TZID=Europe/London:20080311T180000
UID:TALK8788AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/8788
DESCRIPTION:In axiomatizing their study of Frobenius distribut
ions in GL(2)-extensions of the rationals\, Lang a
nd Trotter introduce the notion of an adelic Galoi
s representation of “elliptic type\,” and they ask
in passing whether every such representation aris
es from an elliptic curve. Roughly speaking\, the
question is whether certain conditions that are n
ecessary for a representation to come from an elli
ptic curve over the rationals without complex mult
iplication - cyclotomic determinant\, Weil bound f
or the trace of Frobenius elements\, openness of
the image in GL(2) of the adelic integers - are al
so sufficient. This talk will begin with an intro
duction to the ring of adelic integers and the not
ion of a Galois representation over this ring (ess
entially the same thing as a strictly compatible f
amily of l-adic Galois representations). We shall
then look more closely at the problem posed by La
ng and Trotter. In spite of the extraordinary adv
ances that have been made in Galois representation
theory during the intervening decades\, the probl
em of Lang and Trotter may not yet be ripe for a s
olution\, but the obstacles that remain appear to
be interesting questions in their own right.
LOCATION:Wolfson Room (MR 2) Centre for Mathematical Scienc
es\, Wilberforce Road\, Cambridge
CONTACT:Helen Innes
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