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

We consider vector bundles of rank \(2\) $2$ with trivial generic fiber on the projective line over \(\mathbb{Z}\) $\mathbb{Z}$ . 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 \(\mathcal{O}^{2}\) $\mathrm{superscript}(𝒪, 2)$ in the fiber over \(\mathbb{Q}\) $\mathbb{Q}$ and are isomorphic to \(\mathcal{O}^{2}\) $\mathrm{superscript}(𝒪, 2)$ or \(\mathcal{O}(-1)\oplus\mathcal{O}(1)\) $\mathrm{direct-sum}(𝒪-1, 𝒪\times 1)$ over each closed point of Spec \((\mathbb{Z})\) $\mathbb{Z}$ , we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.