Vector-valued modular forms and the modular orbifold of elliptic curves

Abstract

ABSTRACT. This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL 2 (Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line P(4, 6).

On the structure of modules of vector-valued modular forms

“…If ρ is a finite-dimensional complex representation of SL 2 (Z) of dimension d, then the module M(ρ) of vector valued modular forms for ρ is known [8], [2] to be free of rank d over the ring M of classical scalar modular forms of level one. A basic problem about M(ρ) is then to determine the weights of a generating set of modular forms in this module.…”

“…These are invariants of the isomorphism class of ρ. In [2] it was observed that this question is tantamount to determining the decomposition of a certain vector bundle V 0 (ρ) on the moduli stack of elliptic curves into line bundles. In dimension less than six, some results on this questions have been obtained by Marks [7], but otherwise very little has been proved about the general situation.…”

“…We explored the idea of using the decomposition (1) and the results of [2] to study the dimensions of spaces of modular forms of weight one on Γ(p) for a prime p. We were pleased to observe that when p ≡ 3 (mod 4), and if ρ 1 and ρ 2 denote the irreducible representations of SL 2 (F p ) obtained as certain constituents of reducible principal series representations, then the Euler characteristics of the corresponding vector bundles V 0 (ρ i ) equal 1 2 (1 ± h(−p)), where h(−p) denotes the class number of Q( √ −p). Using this, it is not too hard to show that dim M 1 (ρ i ) ≥ (1 + h(−p)).…”

“…This elementary argument detects the dihedral theta series of weight one without writing them down explicitly, and is presumably well-known to experts 1 . Unfortunately, the elementary arguments that [2] enables do not, by themselves, shed any new light on the question of dimensions of spaces of modular forms of weight one. However, the question of the module structure of M(ρ) seems to be a richer one, and a methodical 1 Nevertheless, it is worth remarking that the term h(−p) in the Euler characteristic arises via the exponents of ρ i (T ) through Dirichlet's analytic class number formula.…”

“…Recall that since det e M = e Tr(M ) , the quantity 12 Tr(L) is an integer for any choice of exponents L for ρ. Let V k,L (ρ) denote the vector bundle introduced in [2]. If L has eigenvalues with real part in [0, 1) then we write simply…”