On the structure of modules of vector-valued modular forms

If ρ denotes a finite-dimensional complex representation of SL 2 ( Z ) , then it is known that the module M ( ρ ) of vector-valued modular forms for ρ 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 ( ρ ) . A...

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Bibliographic Details
Published in:The Ramanujan journal 2018-10, Vol.47 (1), p.117-139
Main Authors: Franc, Cameron, Mason, Geoffrey
Format: Article
Language:English
Subjects:
Analytic functions
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Generators
Lower bounds
Mathematics
Mathematics and Statistics
Modular structures
Number Theory
Upper bounds
Weight
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Summary:If ρ denotes a finite-dimensional complex representation of SL 2 ( Z ) , then it is known that the module M ( ρ ) of vector-valued modular forms for ρ 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 ( ρ ) . Among our results are absolute upper and lower bounds, depending only on the dimension of ρ , on the weights of generators for M ( ρ ) , as well as upper bounds on the multiplicities of weights of generators of M ( ρ ) . 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 ( ρ ) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series E 6 of weight six.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-017-9889-2