Mathematical Research Letters

Volume 26 (2019)

Number 5

Bounded ranks and Diophantine error terms

Pages: 1559 – 1570



Hector Pasten (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.; and Dept. de Matemáticas, Facultad de Matemáticas,Pontificia Universidad Católica de Chile, Macul, RM, Chile)


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.

Received 29 June 2018

Accepted 14 December 2018

Published 27 November 2019