Reidemeister Torsion for Vector Bundles on $$\mathbb{P}^{1}_{\mathbb{Z}}$$

We consider vector bundles of rank $$2$$ with trivial generic fiber on the projective line over $$\mathbb{Z}$$ . For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to $$\mathcal{O}^{2}$$ in the fiber over $$\mathbb{Q}$$ and are isomorphic to $$\mathcal{O}^{2}$$ or $$\mathcal{O}(-1)\oplus\mathcal{O}(1)$$ over each closed point of Spec $$(\mathbb{Z})$$ , we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.