Abstract
We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things, we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.
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Acknowledgements
I would like to thank Peter Bantay for a fruitful collaboration, Chris Marks, Geoff Mason, and Arturo Pianzola for conversations, and my students Jitendra Bajpai and Tim Graves. My research is supported in part by NSERC.
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Gannon, T. (2014). The Theory of Vector-Valued Modular Forms for the Modular Group. In: Kohnen, W., Weissauer, R. (eds) Conformal Field Theory, Automorphic Forms and Related Topics. Contributions in Mathematical and Computational Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43831-2_9
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