Rank one elliptic curves and rank stability

TL;DR

This paper proves the existence of infinitely many elliptic curves of rank one over any number field, using a novel specialization approach and rank computations, extending previous results related to Hilbert's tenth problem.

Contribution

It introduces a new method to construct rank one elliptic curves over number fields via specialization of a nonisotrivial family, generalizing prior theorems.

Findings

Infinitely many rank 1 elliptic curves over any number field

Construction of elliptic curves with controlled rank over quadratic extensions

Extension of results related to Hilbert's tenth problem

Abstract

For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic curves of rank $1$ over any number field. This result generalizes theorems of Koymans-Pagano and Alp\"oge-Bhargava-Ho-Shnidman which were used to independently show that Hilbert's tenth problem over the ring of integers of any number field has a negative answer. Our approach differs since we are obtaining our elliptic curves by specializing a nonisotrivial rank $1$ family of elliptic curves and we compute all the ranks involved.