Geodesic
Reidemeister torsion for vector bundles on $\mathbb{P}^1_\mathbb{Z}$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 156-169
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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
Mots-clés : torsion. 
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V. M. Polyakov. Reidemeister torsion for vector bundles on $\mathbb{P}^1_\mathbb{Z}$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 156-169. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a11/

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