There are infinitely many elliptic curves over the rationals of rank 2

TL;DR

The authors construct an explicit infinite family of elliptic curves over $\mathbb Q$ with rank exactly $2$ by a careful 2-descent using a 2-isogeny. Infinitude is established via a Tao–Ziegler result guaranteeing infinitely many admissible parameter pairs $(m,n)$ with primality and congruence conditions. A detailed Selmer-group analysis together with explicit rational points shows $E(\mathbb Q)\cong \mathbb Z/2\mathbb Z \times \mathbb Z^2$ for these curves, confirming rank $2$. They also provide an explicit formula for the $j$-invariant and prove that distinct parameter choices yield non-isomorphic curves over $\overline{\mathbb Q}$, hence infinitely many such curves exist up to isomorphism. The work combines arithmetic of explicit models, isogeny descent, and a density argument to settle the infinitude of rank-$2$ curves over $\mathbb Q$.

Abstract

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit models and their rank is shown to be $2$ via a $2$-descent. That there are infinitely many such elliptic curves makes use of a theorem of Tao and Ziegler.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof