Rank $2$ vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ and quadratic forms

We study the action of the group $\mathrm{SL}_2(\mathbb{Z})$ on $\mathrm{Ext}^1(\mathcal{O}(2),\mathcal{O}(-2))$ and on isomorphism classes of vector bundles on $\mathbb {P}^1_{\mathbb{Z}}$ of rank $2$ with a trivial generic fiber and simple jumps. It is proved that such bundles are equivariant under the action of this group. The concept of a rigged bundle is introduced and studied. It is shown that the group of isomorphism classes of rigged bundles of rank $2$ with a trivial generic fiber and simple jumps is isomorphic to the $2$-torsion quotient of the class group of binary quadratic forms of the corresponding discriminant up to a $\mathbb{Z}/2$ factor.