From sparsity of rational points on curves to the generic positivity of Beilinson-Bloch height (Ziyang Gao, UCLA)

It is a fundamental question to find rational solutions to a given system of polynomials, and in modern language this translates into finding rational points in algebraic varieties.  It is already very deep for algebraic curves defined over Q.  An intrinsic natural number associated with the curve, called its genus, plays an important role in studying rational points on curves.  In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points.  Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness.  An answer to each grade requires a better understanding of the distribution of the rational points.

In my talk, I will explain the historical and recent developments of this problem according to the different grades.  I will also mention a recent work (joint with Shouwu Zhang) about a generic positivity property and a Northcott property of the Beilison-Bloch height of the Gross-Schoen cycles and the Ceresa cycles.