Let X be a curve of genus g≥ 2 over a number field F of degree d=[F: Q]. The conjectural
existence of a uniform bound N (g, d) on the number# X (F) of F-rational points of X is an …

Consider a smooth, geometrically irreducible, projective curve of genus g≥2 defined over a
number field of degree d≥1. It has at most finitely many rational points by the Mordell …

We show that there is a bound depending only on g, r and [K: Q] for the number of K-rational
points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil …

In 1983, G. Faltings proved the famous Mordell conjecture, which states that a curve of
genus bigger than 1 defined over a number field K has only finitely many K-rational points …

Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of
results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion …

1. Introduction. Let E be an elliptic curve over a number field K. A classical conjecture in the
theory of elliptic curves is the uniform boundedness conjecture that there is a bound BK on …

We present seven theorems on the structure of prime order torsion points on CM elliptic
curves defined over number fields. The first three results refine bounds of Silverberg and …

We give an introductory account of two recent approaches towards an effective proof of the
Mordell conjecture, due to Lawrence–Venkatesh and Kim. The latter method, which is …

We introduce a general strategy for proving quantitative and uniform bounds on the number
of common points of height zero for a pair of inequivalent height functions on P^1(Q). We …

We provide new upper bounds on N_q (g), the maximum number of rational points on a
smooth absolutely irreducible genus-g curve over F_q, for many values of q and g. Among …