Reidemeister Torsion for Vector Bundles on subscriptsuperscriptℙ1ℤ - Polyakov, V. M. (Proceedings of the Steklov Institute of Mathematics, 2024)
We consider vector bundles of rank 2 with trivial generic fiber on the projective line over 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 superscript 𝒪 2 in the fiber over ℚ and are isomorphic to superscript 𝒪 2 or direct-sum 𝒪 1 𝒪 1 over each closed point of Spec ℤ , we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.