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50th Annual Meeting of the Australian Mathematical Society

Plenary Talk in Macquarie Theatre

Thursday 28 September 2006 at 10:00


Christopher Skinner (University of Michigan)


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.