**Amod Agashe**

University of Florida, USA

December 16, 2010

**Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one**:
Let E be an (optimal) elliptic curve over the rational numbers such
that the L-function of E vanishes to order one at s=1. Then by work of Gross and Zagier, there is a point on E defined over a suitable quadratic imaginaryfield K, called a Heegner point, that has infinite order. Furthermore,they showed that the second part of the Birch and Swinnerton-Dyer conjecture then predicts that the index of the cyclic group generated by the Heegner point in the group of K-rational points on E is the product of the order of the Shafarevich-Tate group of E over K and certain other integer invariants of E. In our talk, we will extract a factor from the index mentioned above, and use
the theory of visibility to show that if an odd prime divides this factor, then it divides the order of the Shafarevich-Tate group (as predicted), under certain hypotheses, the most serious of which is the first part of the Birch and Swinnerton-Dyer conjecture.