New York Journal of Mathematics
Volume 16 (2010) 525-537


Sonal Jain

A new record for the canonical height on an elliptic curve over C(t)

view    print

Published: November 22, 2010
Keywords: Elliptic surface, canonical height, elliptic curve, Szpiro conjecture, Lang conjecture, integral points
Subject: Primary 11; Secondary 14

We exhibit an elliptic curve E/C(t) of discriminant degree 84 with a nontorsion point P of canonical height 2987/120120 (a new record). We also prove that if (E,P) has Szpiro ratio σ ≦ 4, then \hat{h}(P) must exceed this value, providing some evidence that our example may yield the smallest height possible over C(t). Using the same strategy, we find other E/C(t) with nontorsion points of small canonical height, including Elkies' previous record.


The author's work was partially supported by the NSF RTG grant DMS-0739380

Author information

Departent of Mathematics, New York University Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012