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).