An effective proof of the hyperelliptic Shafarevich conjecture

Abstract: Let \(C\) be a hyperelliptic curve of genus \(g\geq 1\) 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 \(g\geq 1\) one can in principle determine the set of \(K-\)isomorphism classes of hyperelliptic curves over \(K\) of genus \(g\) with good reduction outside \(S\).