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While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group Γ and an arbitrary representation R of Γ in GL(n, C), their existence, as far as we know, has been established in t...he literature only when restrictions are imposed on Γor R. In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group Γ and any n-dimensional complex representation R of Γ. To this end, we realize these automorphic forms as global sections of a special rank n vector bundle built using solutions to the Riemann-Hilbert problem over various non-compact Riemann surfaces and Kodaira’s vanishing theorem.続きを見る
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