Shafarevich予想について

Abstract

The Shafarevich conjecture, known as a geometric analogue of the Hermite-Minkowski theorem, states the finiteness of certain varieties over a fixed number field admitting good reduction away from a fixed finite set of finite places. In the abelian varieties of a fixed dimension case, this conjecture was proved by Faltings-Zarhin and applied to the Mordell conjecture. Moreover, the author proved a certain generalization of this conjecture in the K3 surfaces case. In the first part of this survey, we will sketch the proof of Faltings-Zarhin and its relation to the Mordell conjecture, and the second part we will present the result of the author.