What Do
We Know about the Birch-Swinnerton-Dyer (and Related) Conjectures?
The Birch-Swinnerton-Dyer conjecture
(BSD) predicts that the order of vanishing at s=1 of the
L-function L(E,s) of an elliptic curve
E is equal to the rank of the group of rational points
on E and that the first non-zero Taylor series coefficient
around s=1 can be expressed in terms of various number-theoretic
information about E. Today, this is just one conjecture
in a vast array of conjectures about special values of L-functions
(with the names Deligne, Beilinson, Bloch-Kato, ... attached).
This talk will be a survey of some of the things we know about
BSD and related conjectures for modular forms, focusing on recent
results making use of Shimura varieties for groups other than
GL2.
