Abstract
We give a method for constructing vector-valued modular forms from singular scalar-valued ones of a suitable type. As an application, we prove that two remarkable spaces of vector-valued modular forms, which seem to be unrelated at a first look since they are constructed in two very different ways, are the same. More precisely, let \(V_{grad}\) be the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and let \(V_\Theta \) be the one generated by vector-valued modular forms arising from second order theta constants with our construction. We will prove that \(V_{grad}=V_\Theta \).
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Acknowledgements
The author is grateful to Professor E. Freitag for reading a first version of the manuscript. Also, special thanks are due to F. Dalla Piazza, A. Fiorentino, S. Grushevsky for stimulating discussions.
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Perna, S. Heat equation for theta functions and vector-valued modular forms. manuscripta math. 157, 81–99 (2018). https://doi.org/10.1007/s00229-017-0980-1
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DOI: https://doi.org/10.1007/s00229-017-0980-1