Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 1211-1218, 2020

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman

Received October 29, 2019.   Published online October 27, 2020.

Abstract:  Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote the collection of modular forms on $H$. Then $M(\rho)$ is a $\mathbb{Z}$-graded $M(H)$-module. It has been proven that $M(\rho)$ may not be projective as a $M(H)$-module. We prove that $M(\rho)$ is Cohen-Macaulay as a $M(H)$-module. We also explain how to apply this result to prove that if $M(H)$ is a polynomial ring, then $M(\rho)$ is a free $M(H)$-module of rank $\dim\rho.$
Keywords:  vector-valued modular form; Cohen-Macaulay module
Classification MSC:  11F03, 13C14
DOI:  10.21136/CMJ.2020.0476-19

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Affiliations:   Richard Gottesman, Queen's University, 48 University Avenue, Kingston, Ontario, Canada, e-mail:

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