A converse to a theorem of Gross-Zagier-Kolyvagin

A converse to a theorem of Gross-Zagier-Kolyvagin

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Christopher Skinner , Princeton University and IAS
IAS Room S-101

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite. This talk will describe an approach to the converse. The methods are mostly p-adic.