Abstract
If \(\rho \) denotes a finite-dimensional complex representation of \(\mathbf {SL}_{2}(\mathbf {Z})\), then it is known that the module \(M(\rho )\) of vector-valued modular forms for \(\rho \) is free and of finite rank over the ring M of scalar modular forms of level one. This paper initiates a general study of the structure of \(M(\rho )\). Among our results are absolute upper and lower bounds, depending only on the dimension of \(\rho \), on the weights of generators for \(M(\rho )\), as well as upper bounds on the multiplicities of weights of generators of \(M(\rho )\). We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free basis for \(M(\rho )\) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series \(E_6\) of weight six.
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Notes
Nevertheless, it is worth remarking that the term \(h(-p)\) in the Euler characteristic arises via the exponents of \(\rho _i(T)\) through Dirichlet’s analytic class number formula.
The transpose is missing from the formula in [1].
Since we do not have explicit equations for the character variety of \(\mathbf {SL}_2(\mathbf {Z})\) in dimensions six or greater, we do not know that there in fact exist representations \(\rho \) of \(\mathbf {SL}_2(\mathbf {Z})\) having all of the possible prescribed values for \(\mathbf {Tr}(\rho (R))\), \(\mathbf {Tr}(\rho (S))\), and \(\mathbf {Tr}(L)\) satisfying the obvious constraints. Also, in ranks six or greater, we do not, in general, know how to compute dimensions of spaces of forms of weight one or less.
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Franc, C., Mason, G. On the structure of modules of vector-valued modular forms. Ramanujan J 47, 117–139 (2018). https://doi.org/10.1007/s11139-017-9889-2
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DOI: https://doi.org/10.1007/s11139-017-9889-2