A visible factor for analytic rank one

TL;DR

This paper investigates the relationship between certain integer factors derived from the index of Heegner points on optimal elliptic curves and the divisibility of the Shafarevich-Tate group or component groups, providing evidence for the Birch and Swinnerton-Dyer conjecture.

Contribution

It introduces a method to extract an integer factor from the Heegner point index and relates it to congruences of modular forms, linking divisibility properties to the BSD conjecture.

Findings

Prime divisors of the extracted factor relate to the order of the Shafarevich-Tate group.

Prime divisors also relate to the order of arithmetic component groups.

Supports the BSD conjecture under certain hypotheses.

Abstract

Let $E$ be an optimal elliptic curve of conductor $N$, such that the $L$-function 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 $K$ also vanishes to order one at $s=1$. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in $E(K)$ of the subgroup generated by the Heegner point is equal to the product of the Manin constant of $E$, the Tamagawa numbers of $E$, and the square root of the order of the Shafarevich-Tate group of $E$ (over $K$). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to $E$ with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides the order of the Shafarevich-Tate group or the order of an arithmetic component group of $E$, as predicted by the second part of the Birch and Swinnerton-Dyer conjecture.