Henri Darmon - Recent progress in the theory of elliptic curves

After Wiles' proof of the Shimura-Taniyama conjecture, the Birch and Swinnerton-Dyer conjecture has become the outstanding open problem in the arithmetic theory of elliptic curves. In the late 80's, the work of Kolyvagin led to an almost complete proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves of (analytic) rank at most one. The higher rank case remains shrouded in mystery. Recently Bertolini and I have been able to prove part of a p-adic variant of the Birch and Swinnerton Dyer conjecture which applies to curves of higher rank. The proof combines ideas that arose in the work of Wiles and earlier work of Ribet on Fermat's Last Theorem with the methods of Thaine and Kolyvagin.