Fiber products of hyperelliptic curves

Abstract

Let $k$ be a number field, and let $S$ be a finite set of maximal ideals of the ring of integers of $k$. In his 1962 ICM address, Shafarevich asked if there are only finitely many $k$-isomorphism classes of algebraic curves of a fixed genus $g\ge 1$ with good reduction outside $S$. He verified this for $g=1$ by reducing the problem to Siegel's theorem. Parshin extended this argument to all hyperelliptic curves (cf. also the work of Oort). The general case was settled by Faltings' celebrated work. In this note we give a short proof of Shafarevich's conjecture for hyperelliptic curves, by reducing the problem to the case $g=1$ using the Theorem of de Franchis plus standard facts about discriminants of hyperelliptic equations.