A note on Bremner's conjecture and uniformity

Abstract

In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of different rational points whose $x$-coordinates are in arithmetic progression, have large rank. This conjecture was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang conjecture of Gao--Ge--Kühne. In particular, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a more direct proof of this last statement, which only uses the uniform Mordell--Lang conjecture for curves (due to Dimitrov--Gao--Habegger) and avoids the technicalities of our original argument with Nevanlinna theory.

Theorems & Definitions (5)

• Theorem 1.1: Strong form of Bremner's conjecture
• Theorem 1.3
• Conjecture 2.1: Mordell
• Theorem 2.2: Height-uniform Mordell
• proof : Proof of Theorem \ref{['ThmUniform']}