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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2024 Volume 30, Number 1, Pages 156–169 (Mi timm2069)

Reidemeister torsion for vector bundles on $\mathbb{P}^1_\mathbb{Z}$

V. M. Polyakov

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: 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.

UDC: 512.75

MSC: 14G40

Received: 29.11.2023
Revised: 19.12.2023
Accepted: 25.12.2023

DOI: 10.21538/0134-4889-2024-30-1-156-169


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2024, 325, suppl. 1, S155–S167

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© Steklov Math. Inst. of RAS, 2025