An effective proof of the hyperelliptic Shafarevich conjecture

Let C be a hyperelliptic curve of genus g >= 1 over a number field K with good reduction outside a finite set of places S of K. We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g, S and K. In particular, we obtain that for any given number field K, finite set of places S of K and integer g >= 1 one can in principle determine the set of K-isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside S.