Bogomolov conjecture

In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov , in arithmetic geometry about algebraic curves that generalizes the Manin–Mumford conjecture in arithmetic geometry.

The conjecture was proven by Emmanuel Ullmo in 1998 and was subsequently generalized by Shou-Wu Zhang to subvarieties of general abelian varieties in 1998. Both Ullmo's and Zhang's proofs rely on Arakelov theory. The idea of using Arakelov theory to attack the Bogomolov conjecture is due to Lucien Szpiro.

In recent years, generalizations and variants of the Bogomolov conjecture have become a central area of research in both arithmetic dynamics and diophantine geometry, motivating many developments in both fields.

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