Elliptic curves of rank one over number fields

TL;DR

This paper proves that for any number field, there are infinitely many elliptic curves with rank exactly one, expanding understanding of elliptic curve ranks over various fields.

Contribution

It establishes the existence of infinitely many elliptic curves of rank one over any number field, a significant advancement in the study of elliptic curves.

Findings

Infinitely many elliptic curves of rank one over any number field exist.

The result applies universally to all number fields.

It advances the understanding of the distribution of ranks of elliptic curves.

Abstract

We prove that for every number field $K$, there exist infinitely many elliptic curves $E$ over $K$ with rank exactly equal to 1.