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

TL;DR

This paper provides new theoretical evidence supporting the Birch and Swinnerton-Dyer conjecture for rank one elliptic curves over quadratic imaginary fields by linking visibility theory and divisibility properties of the Shafarevich-Tate group.

Contribution

It demonstrates that under certain hypotheses, the divisibility of the Shafarevich-Tate group order by an integer r can be inferred from congruences between modular forms, supporting the BSD conjecture.

Findings

r divides the Shafarevich-Tate group order of E over K

r divides the BSD conjectural order of the Shafarevich-Tate group

Provides new evidence for BSD in rank one case

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.