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

Abstract

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.