Geodesic
Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 137-160

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We consider vector bundles of rank $2$ with jumps of heights $1$ and $2$ and a trivial generic fiber on the arithmetic surface $\mathbb{P}^1_{\mathbb{Z}}$. The finiteness of the number of isomorphism classes of such vector bundles with a fixed discriminant and, as a consequence, with a fixed genus is obtained. 
@article{ZNSL_2022_511_a4,
     author = {V. M. Polyakov},
     title = {Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--160},
     publisher = {mathdoc},
     volume = {511},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/}
}
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V. M. Polyakov. Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 137-160. http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/