We consider vector bundles of rank 2 with a 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 a 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 Spec$(\mathbb{Z})$, we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.
Keywords: vector bundle, arithmetic surface, projective line, torsion
Received November 29, 2023
Revised December 19, 2023
Accepted December 25, 2023
Funding Agency: This work was supported by the Ministry of Science and Higher Education of the Russian Federation (grant for the creation and development of the Leonhard Euler International Mathematical Institute, agreement no. 075-15-2022-289).
Vladimir Mikhailovich Polyakov, doctoral student, St. Petersburg Department of V.A.Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, 191023 Russia, e-mail: polyakov@pdmi.ras.ru
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Cite this article as: V.M. Polyakov. Reidemeister torsion for vector bundles on $\mathbb{P}^1_\mathbb{Z}$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 156–169. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S155–S167.
