On the infinitude of elliptic curves over a number field with prescribed small rank

TL;DR

This paper proves that for any number field and rank between 0 and 4, there are infinitely many elliptic curves over that field with exactly that rank, constructed via specialization of nonisotrivial elliptic curves.

Contribution

It establishes the infinitude of elliptic curves with prescribed small ranks over any number field using specialization techniques and advanced descent methods.

Findings

Infinitely many elliptic curves of rank 0 over any number field.

Infinitely many elliptic curves of rank 4 over any number field.

Control over bad primes enables rank computation via 2-descent.

Abstract

For any number field $K$ and integer $0\leq r \leq 4$, we prove that there are infinitely many elliptic curves over $K$ of rank $r$. Our elliptic curves are obtained by specializing well-chosen nonisotrivial elliptic curves over the function field $K(T)$. We use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to choose our specializations so that we have control over the bad primes and can perform a $2$-descent to compute ranks.