Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero.
Abstract
Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having
analytic rank zero, i.e., such that the $L$-function $L_E(s)$ of $E$ does not
vanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of
the same conductor $N$ whose Mordell-Weil rank is greater than zero 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 product of the order of the Shafarevich-Tate group
of $E$ and the orders of the arithmetic component groups of $E$. We extract an
explicit integer factor from the the Birch and Swinnerton-Dyer conjectural
formula for the product mentioned above, and under some hypotheses similar to
the ones made in the situation above, we show that $r$ divides this integer
factor. This provides theoretical evidence for the second part of the Birch and
Swinnerton-Dyer conjecture in the analytic rank zero case.
Вход в систему
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