Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero

TL;DR

This paper provides theoretical evidence supporting the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero by linking visibility theory, congruences, and explicit factors in the BSD formula.

Contribution

It derives an explicit integer factor from the BSD formula and shows that this factor is divisible by the congruence integer under certain hypotheses, supporting BSD conjecture in rank zero.

Findings

r divides the explicit BSD-related integer factor

Visibility theory links congruences to Shafarevich-Tate group order

Supports BSD conjecture in the analytic rank zero case

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.