Bogomolov conjecture

The conjecture was proven by Emmanuel Ullmo and Shou-Wu Zhang in 1998 using Arakelov theory.

Let C be an algebraic curve of genus g at least two defined over a number field K, let

denote the algebraic closure of K, fix an embedding of C into its Jacobian variety J, and let

denote the Néron-Tate height on J associated to an ample symmetric divisor.

The original Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using Arakelov theory in 1998.

[1][2] In 1998, Zhang proved the following generalization:[2] Let A be an abelian variety defined over K, and let