New York Journal of Mathematics
Volume 16 (2010) 1-12


Joseph H. Silverman

Lang's height conjecture and Szpiro's conjecture

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Published: April 22, 2010
Keywords: elliptic curve, canonical height, Szpiro conjecture, Lang conjecture
Subject: Primary: 11G05; Secondary: 11G50, 11J97, 14H52

It is known that Szpiro's conjecture, or equivalently the ABC-conjecture, implies Lang's conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly weaker version of Szpiro's conjecture, which we call "prime-depleted,'' suffices to prove Lang's conjecture.


The author's research partially supported by NSF grants DMS-0650017 and DMS-0854755.

Author information

Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA