The number of rational points on a curve of genus at least two

Abstract

The Mordell Conjecture states that a smooth projective curve of genus at least $2$ defined over number field $F$ admits only finitely many $F$-rational points. It was proved by Faltings in the 1980s and again using a different strategy by Vojta. Despite there being two different proofs of the Mordell Conjecture, many important questions regarding the set of $F$-rational points remain open. This survey concerns recent developments towards upper bounds on the number of rational points in connection with a question of Mazur.