Gael Rémond - Theta heights and Kodaira construction

We prove that in Kodaira construction a height on the base curve is described by the theta height of the jacobians of the fibres. We make use of theta heights as introduced by Bost and David who prove a comparison with Faltings' height. Therefore our result shows that a suitable effective version of Shafarevich conjecture (a bound for Faltings' height) implies an effective Mordell.