Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
| dc.creator | Agashe, Amod | |
| dc.date | 2008-10-14 | |
| dc.date.accessioned | 2026-07-07T10:09:58Z | |
| dc.date.available | 2026-07-07T10:09:58Z | |
| dc.description | Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ also divides the Birch and Swinnerton-Dyer {\em conjectural} order of the Shafarevich-Tate group of $E$ over $K$, which provides new theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank one case. | |
| dc.identifier | https://arxiv.org/abs/0810.2487 | |
| dc.identifier | http://arxiv.org/abs/0810.2487 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/171486 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G40 | |
| dc.title | Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one | |
| dc.type | text |