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.
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.