Vector-valued automorphic forms and vector bundles

Hicham Saber; Abdellah Sebbar

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arXiv · 1312.2992

Vector-valued automorphic forms and vector bundles

Hicham Saber·Abdellah Sebbar

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Abstract

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $\Gamma$ and an arbitrary representation $R$ of $\Gamma$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $\Gamma$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $\Gamma$ and any $n$-dimensional complex representation $R$ of $\Gamma$. 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 noncompact Riemann surfaces and Kodaira's vanishing theorem.

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BibTeX RIS

Hicham Saber, Abdellah Sebbar. 2013-12-10. Vector-valued automorphic forms and vector bundles. https://arxiv.org/abs/1312.2992

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