In 1983, Faltings proved the Mordell conjecture: let $K$ be a number field and let $C$ be a curve over $K$ of genus $g \ge 2$, then $\#C(K)$ is finite. One wishes to look for a uniform bound for $\#C(K)$ (conjectured by Caporaso-Harris-Mazur). After a series of work by David-Philippon and R\'{e}mond, it is known that $\#C(K)$ is bounded above in terms of $g$, $[K:\mathbb{Q}]$, $\mathrm{rank}J(C)(K)$ and the Faltings height of $J(C)$ (denoted by $h_F(J(C))$).

The goal of this series of lectures is to present an idea to remove the dependence on $h_F(J(C))$ and prove that $\#C(K) \le c(g,[K:\mathbb{Q}])^{\mathrm{rank}J(C)(K)}$ for some constant $c(g,[K:\mathbb{Q}]) > 0$. The crucial point is to prove a height inequality.

Talk 1: State the desired height inequality. Explain how it implies the desired bound. We will focus on $1$-parameter families.

Talk 2: Overview on functional transcendence. Introduction to the universal abelian variety and the Betti map. State and explain the Ax-Schanuel theorem for the universal abelian varieties.

Talk 3: Use Ax-Schanuel to study the maximal rank of the Betti map.

Talk 4: Explain how to prove the height inequality on a given subvariety when the Betti map is a homeomorphism on a small neighborhood of the subvariety.