Elliptic curves of rank one over number fields

TL;DR

This work settles a longstanding question by proving that every number field $K$ supports infinitely many elliptic curves $E/K$ with rank $1$. The authors blend $2$-descent with additive combinatorics, introducing a pre-twisting step and leveraging Kai’s theorem to produce auxiliary twists that force the $2$-Selmer rank to drop to the level needed for rank one, all without relying on archimedean inputs. Key innovations include a non-archimedean analogue to real-place methods via split multiplicative reduction and a Markov-chain view of Selmer ranks, enabling uniformly applicable results across all number fields. The paper also highlights substantial applications, including a corollary on rank-one curves and implications for definability and Hilbert’s tenth problem over big rings, situating the findings within broader conjectures about rank distribution in natural families. Overall, the work provides a robust framework to generate rank-one elliptic curves over arbitrary number fields and connects arithmetic statistics with explicit descent techniques in a novel way.

Abstract

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

Theorems & Definitions (35)

• Corollary 1.1
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4