3:00 p.m., Friday (March 28th)
Math Annex 1100
Department of Mathematics
University of California, Berkeley
Galois groups arising from modular forms
My talk will focus on the Galois representations that are
attached to cusp forms on congruence subgroups of SL(2,Z).
The existence of these representations was conjectured
in the late 1960s by J-P. Serre, who saw that they would
provide a structural explanation for congruences that had
been found by Ramanjuan, Hardy and others. Soon after Serre
saw that the representations should exist, Deligne sketched
their construction in a Bourbaki seminar. (A full account
of Deligne's construction is contained in a forthcoming
Cambridge University Press volume by Brian Conrad.) Serre
and Swinnerton-Dyer then exploited Deligne's representations
to show that there were no unknown congruences for the
Ramanujan ``tau" function: the relevant representations were
shown to have ``maximum possible" images in all cases where
known congruences constrained the sizes of the images.
In the work of Serre/Swinnerton-Dyer, and also in the analogous
work of Serre on elliptic curves, the representations are integral
in an obvious sense; for example the congruences mentioned above
are congruences between ordinary integers. One can consider also
families of representations that are defined over rings of integers
of number fields. In that case, a tiny bit of group theory is
needed to prove analogues of the theorems of Serre and Swinnerton-Dyer. I will describe a surprising group-theoretic argument of H. W. Lenstra, Jr. that allows one to study images of representations in certain ``ramified" cases that could not be treated before.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).