# Institute of Mathematics

## 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

References:
[1] E. Bannai, M. Koike, A. Munemasa, J. Sekiguchi: Some results on modular forms-subgroups of the modular group whose ring of modular forms is a polynomial ring. Groups and Combinatorics - In Memory of Michio Suzuki. Advanced Studies in Pure Mathematics 32. Mathematical Society Japan, Tokyo (2001), 245-254. DOI 10.2969/aspm/03210245 | MR 1893493 | Zbl 1029.11012
[2] D. J. Benson: Polynomial Invariants of Finite Groups. London Mathematical Society Lecture Note Series 190. Cambridge University Press, Cambridge (1993). DOI 10.1017/CBO9780511565809 | MR 1249931 | Zbl 0864.13001
[3] L. Candelori, C. Franc: Vector-valued modular forms and the modular orbifold of elliptic curves. Int. J. Number Theory 13 (2017), 39-63. DOI 10.1142/S179304211750004X | MR 3573412 | Zbl 1419.11076
[4] L. Candelori, C. Franc: Vector bundles and modular forms for Fuchsian groups of genus zero. Commun. Number Theory Phys. 13 (2019), 487-528. DOI 10.18576/amis/130321 | MR 4013728 | Zbl 07124990
[5] C. Franc, G. Mason: Fourier coefficients of vector-valued modular forms of dimension 2. Can. Math. Bull. 57 (2014), 485-494. DOI 10.4153/CMB-2014-007-3 | MR 3239110 | Zbl 1302.11027
[6] C. Franc, G. Mason: Hypergeometric series, modular linear differential equations and vector-valued modular forms. Ramanujan J. 41 (2016), 233-267. DOI 10.1007/s11139-014-9644-x | MR 3574630 | Zbl 1418.11064
[7] T. Gannon: The theory of vector-valued modular forms for the modular group. Conformal Field Theory, Automorphic Forms and Related Topics. Contributions in Mathematical and Computational Sciences 8. Springer, Berlin (2014), 247-286. DOI 10.1007/978-3-662-43831-2_9 | MR 3559207 | Zbl 1377.11055
[8] R. Gottesman: The arithmetic of vector-valued modular forms on $\Gamma_0(2)$. Int. J. Number Theory 16 (2020), 241-289. DOI 10.1142/S1793042120500141 | MR 4077422 | Zbl 07182420
[9] C. Marks: Fourier coefficients of three-dimensional vector-valued modular forms. Commun. Number Theory Phys. 9 (2015), 387-411. DOI 10.4310/CNTP.2015.v9.n2.a5 | MR 3361298 | Zbl 1381.11038
[10] C. Marks, G. Mason: Structure of the module of vector-valued modular forms. J. Lond. Math. Soc., II. Ser. 82 (2010), 32-48. DOI 10.1112/jlms/jdq020 | MR 2669639 | Zbl 1221.11134
[11] G. Mason: On the Fourier coefficients of 2-dimensional vector-valued modular forms. Proc. Am. Math. Soc. 140 (2012), 1921-1930. DOI 10.1090/S0002-9939-2011-11098-0 | MR 2888179 | Zbl 1276.11064
[12] A. Selberg: On the estimation of Fourier coefficients of modular forms. Proc. Sympos. Pure Math. 8. American Mathematical Society, Providence (1965), 1-15. MR 0182610 | Zbl 0142.33903

Affiliations:   Richard Gottesman, Queen's University, 48 University Avenue, Kingston, Ontario, Canada, e-mail: richard.b.gottesman@gmail.com

PDF available at: