On spaces of modular forms spanned by eta-quotients

Abstract

An eta-quotient of level N is a modular form of the shape f (z) = δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ 0 (N ) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f (z) is a holomorphic modular form that is nonvanishing on the upper half plane and has integer Fourier coefficients at infinity, then f (z) is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of J 0 (2 k )(Q).

Zagier duality for level p weakly holomorphic modular forms

“…Bringmann and Ono extended Zagier's results by proving duality theorems for harmonic Maass forms and Poincaré series of level 4 and half-integral weight [4]. Likewise, Rouse [21], Choi [6], and Zhang [24] showed that duality holds for certain Hilbert modular forms and forms with quadratic character.…”

Finiteness of irreducible holomorphic eta quotients of a given level

“…Since for all t ∈ D N , * The invertibility of the order matrix (and hence, the existence of such eta quotients) has been known classically. For example, see Satz 8 in [15], Proposition 3.2 in [12], the proof of Theorem 3 in [13] or the proof of Theorem 2 in [19].…”

“…From (3.6), it follows readily that η p /η p is holomorphic for each prime p. In particular, so is the rightmost eta quotient in (1.9). Also, from Newman's criteria (see [16,17] or [19]), it follows that the multiplier systems of both of the eta quotients on the right hand side of (1.9) are trivial. For k ∈ 2N, we define the normalized Eisenstein series E k by…”

A proof of the Thompson moonshine conjecture

“…Let f ∈ S Then f · ϑ is a modular form of weight 2 with the same multiplier and level (respectively, trivial multiplier and level N [g] ). Using programs 3 written by Rouse and Webb [34], one can verify that the algebra of modular forms of level N [g] , so in particular the space M 2 (N [g] ), is generated by eta quotients. One can also compute a generating system consisting of eta quotients for all remaining N [g] that still need to be considered.…”

“…In order to compute the necessary Fourier coefficients exactly without relying on the rather slow convergence of the Fourier coefficients of the Rademacher series, one can construct linear combinations of weight 1 2 weakly holomorphic eta quotients again using the programs 4 written by Rouse and Webb [34] which have the same principal part at ∞ (and the related cusp [35] is easily seen to be 0. 5 The largest bound up to which coefficients need to be checked turns out to be 384 for [g] = 24CD.…”