Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

TL;DR

This paper generalizes Birch's lemma to produce infinite families of quadratic twists of elliptic curves with known ranks, constructs rational points explicitly, and verifies the 2-part of the BSD conjecture for these families.

Contribution

It extends Birch's classical lemma to a broader class of elliptic curves, providing explicit constructions and verifying the 2-part of BSD for these cases.

Findings

Constructed infinite families of quadratic twists with ranks 0 and 1.

Explicitly constructed rational points of infinite order on rank 1 twists.

Verified the 2-part of the BSD conjecture for these families.

Abstract

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove the existence of explicit infinite families of quadratic twists with analytic ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank $1$. In addition, we show that these families of quadratic twists satisfy the $2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.