Finiteness of the Number of Classes of Vector Bundles on \({\mathbb{P}}_{\mathbb{Z}}^{1}\) with Jumps of Height 2

Vector bundles of rank 2 with jumps of heights 1 and 2 and trivial generic fiber on the arithmetic surface \({\mathbb{P}}_{\mathbb{Z}}^{1}\) ${\mathbb{P}}_{\mathbb{Z}}^1$ are considered. The finiteness of the number of isomorphism classes of such vector bundles with fixed discriminant and, as a consequence, with fixed genus, is obtained.