Rank $2$ vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ and quadratic forms
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 121-134
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We study the action of the group $\mathrm{SL}_2(\mathbb{Z})$ on $\mathrm{Ext}^1(\mathcal{O}(2),\mathcal{O}(-2))$ and on isomorphism classes of vector bundles on $\mathbb {P}^1_{\mathbb{Z}}$ of rank $2$ with a trivial generic fiber and simple jumps. It is proved that such bundles are equivariant under the action of this group. The concept of a rigged bundle is introduced and studied. It is shown that the group of isomorphism classes of rigged bundles of rank $2$ with a trivial generic fiber and simple jumps is isomorphic to the $2$-torsion quotient of the class group of binary quadratic forms of the corresponding discriminant up to a $\mathbb{Z}/2$ factor.
@article{ZNSL_2023_523_a6,
author = {V. M. Polyakov},
title = {Rank $2$ vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ and quadratic forms},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {121--134},
publisher = {mathdoc},
volume = {523},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a6/}
}
V. M. Polyakov. Rank $2$ vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ and quadratic forms. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 6, Tome 523 (2023), pp. 121-134. http://geodesic.mathdoc.fr/item/ZNSL_2023_523_a6/