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

TL;DR

This paper proves the existence of infinitely many elliptic curves over the rationals with rank exactly 2, providing explicit models and using a 2-descent method, building on a theorem by Tao and Ziegler.

Contribution

It establishes the infinitude of elliptic curves over a0Q with rank 2, including explicit models and a novel application of Tao and Ziegler's theorem.

Findings

Infinitely many elliptic curves over a0Q have rank exactly 2.

Explicit models of such elliptic curves are provided.

Rank is verified via 2-descent method.

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.