Illinois Journal of Mathematics

Irreducible vector-valued modular forms of dimension less than six

Christopher Marks

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An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, $T$-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert–Poincaré series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert–Poincaré series.

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Illinois J. Math., Volume 55, Number 4 (2011), 1267-1297.

First available in Project Euclid: 12 July 2013

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Zentralblatt MATH identifier

Primary: 11F03: Modular and automorphic functions 11F99: None of the above, but in this section


Marks, Christopher. Irreducible vector-valued modular forms of dimension less than six. Illinois J. Math. 55 (2011), no. 4, 1267--1297. doi:10.1215/ijm/1373636684.

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