A Belyi-Type Criterion for Vector Bundles on Curves Defined over a Number Field
Abstract
Let X0 be an irreducible smooth projective curve defined over Q‾, and let f0:X0⟶PQ‾1 be a nonconstant morphism whose branch locus is contained in the subset {0,1,∞}⊂PQ‾1. For any vector bundle E on X=X0×SpecQ‾SpecC, consider the direct image f∗E on PC1, where f=(f0)C. It decomposes into a direct sum of line bundles, and also it has a natural parabolic structure. We prove that E is the base change, to C, of a vector bundle on X0 if and only if there is an isomorphism f∗E⟶ ∼⨁ i=1rOPC1(mi), where r=rank(f∗E), that takes the parabolic structure on f∗E to a parabolic structure on ⨁ i=1rOPC1(mi) defined over Q‾.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
advanced mathematical theories
Publication details
- Journal / source
- The Michigan Mathematical Journal
- Publisher
- Michigan Mathematical Journal
- Volume
- -1
- Issue
- -1
- Pages
- 1-12
- Publication date
- 1 January 2026
- Language
- EN