# Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero - Mathematics > Number Theory

Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ havinganalytic rank zero, i.e., such that the $L$-function $L Es$ of $E$ does notvanish at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ ofthe same conductor $N$ whose Mordell-Weil rank is greater than zero and whoseassociated newform is congruent to the newform associated to $E$ modulo aninteger $r$. The theory of visibility then shows that under certain additionalhypotheses, $r$ divides the product of the order of the Shafarevich-Tate groupof $E$ and the orders of the arithmetic component groups of $E$. We extract anexplicit integer factor from the the Birch and Swinnerton-Dyer conjecturalformula for the product mentioned above, and under some hypotheses similar tothe ones made in the situation above, we show that $r$ divides this integerfactor. This provides theoretical evidence for the second part of the Birch andSwinnerton-Dyer conjecture in the analytic rank zero case.

Author: ** Amod Agashe**

Source: https://arxiv.org/