Bounded ranks and Diophantine error terms

TL;DR

This paper demonstrates that Lang's conjecture on Diophantine error terms implies Honda's conjecture on elliptic curve ranks, providing evidence for boundedness of ranks without probabilistic assumptions.

Contribution

It establishes a link between Lang's conjecture and Honda's conjecture, showing that error term conjectures imply boundedness of elliptic curve ranks over number fields.

Findings

Lang's conjecture implies Honda's conjecture on ranks

Weak Lang's error term conjecture suggests bounded ranks for quadratic twists

Provides evidence for rank boundedness independent of probabilistic heuristics

Abstract

We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of Lang's error term conjecture would be enough to deduce boundedness of ranks for quadratic twists of elliptic curves over number fields. This can be seen as evidence for boundedness of ranks not relying on probabilistic heuristics on elliptic curves.