An effective proof of the hyperelliptic Shafarevich conjecture

Let $K$ be a field, $S$ be a finite set of places of $K$ and let $g\ge 1$ be an integer. The Shafarevich conjecture says that there are only finitely many $K$-isomorphism classes of curves over $K$ of genus $g$ with good reduction outside $S$. This was proved by Faltings in 1983. An effective version of the conjecture would imply inter alia the effective Mordell and the abc conjecture. In the talk we give an effective version of the Shafarevich conjecture for hyperelliptic curves and discuss some applications.