A visible factor for analytic rank one
| dc.creator | Agashe, Amod | |
| dc.date | 2008-10-29 | |
| dc.date.accessioned | 2026-07-07T10:13:50Z | |
| dc.date.available | 2026-07-07T10:13:50Z | |
| dc.description | Let $E$ be an optimal elliptic curve of conductor $N$, such that the $L$-function 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 $K$ also vanishes to order one at $s=1$. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in $E(K)$ of the subgroup generated by the Heegner point is equal to the product of the Manin constant of $E$, the Tamagawa numbers of $E$, and the square root of the order of the Shafarevich-Tate group of $E$ (over $K$). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to $E$ with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides the order of the Shafarevich-Tate group or the order of an arithmetic component group of $E$, as predicted by the second part of the Birch and Swinnerton-Dyer conjecture. | |
| dc.identifier | https://arxiv.org/abs/0810.5177 | |
| dc.identifier | http://arxiv.org/abs/0810.5177 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/172673 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G40; 14G99 | |
| dc.title | A visible factor for analytic rank one | |
| dc.type | text |