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Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one | | Amod Agashe
; | | Date: |
14 Oct 2008 | | Abstract: | 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. | | Source: | arXiv, 0810.2487 | | Services: | Forum | Review | PDF | Favorites | |
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