Global Weierstrass equations of hyperelliptic curves

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathcal{C}$ of $C$ over the ring of integers ${\mathcal O}_K$(i\mathrm{.}e\mathrm{.}thehyperellipticinvolutionof$C$extendsto$\mathscr{C}$ and the quotient is a smooth model of ${\mathbb{P}}_K^1$ over ${\mathcal{O}}_K$), we give necessary and sometimes sufficient conditions for $\mathcal{C}$ to be defined by a global Weierstrass equation. In particular, if $C$ has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number $h_K$ is prime to $2(2g+1)$, confirming a conjecture of M. Sadek.