Holomorphic modular forms and cocycles
Abstract
This is a slightly expanded version of my lecture at the conference Modular forms are everywhere at Bonn, May 2017, taking into account remarks by Don Zagier and Shaul Zemel, and suggestions of the referees.
1 Introduction
 Classical

For modular cusp forms of even weight the Eichler integral determines for a given modular cusp form of even positive weight a cohomology class with values in a module of polynomial functions. The names to mention are Eichler [5] and Shimura [10].
 Real weight
 There are interesting holomorphic modular cusp forms of positive real weight. For instance the Dedekind eta functionwith weight \(\frac{1}{2}\). One can assign cohomology classes to such cusp forms. The values are in a much larger module. The name to mention is Knopp [6].$$\begin{aligned} \eta (\tau ) = \mathrm{e}^{\pi i \tau /12} \prod _{n\ge 1} \bigl (1\mathrm{e}^{2\pi i n \tau }\bigr ), \end{aligned}$$(1.1)
 Maass cusp forms

Modular Maass wave forms are functions on the upper halfplane that are invariant under the transformation of the modular group. They are not holomorphic, but satisfy a secondorder linear differential equation: There are infinitely many linearly independent Maass cusp forms, although none of them can be given explicitly for the full modular group.
There is a way to associate cohomology classes to Maass cusp form, with values in a large module of functions on the real projective line \(\mathbb {P}^1_{{\mathbb {R}}}\). This can be generalized to cuspidal automorphic forms on larger Lie groups than \({\mathrm {SL}}_2({\mathbb {R}})\). I mention Bunke and Ohlbrich [3, 4].
Don Zagier, John Lewis and I looked at the relation between Maass forms and cohomology. We could associate cocycles to Maass forms with large growth at the cusps and established a quite satisfactory theory [1]. In the Eichler–Shimura theory the cohomology groups have finite dimension. So a linear map from infinitedimensional spaces of modular forms (with unrestricted growth at the cusp) has a huge kernel. In the context of Maass forms we obtained an injective map to cohomology and could describe the image of various spaces of Maass forms.
Later YoungJu Choie, Nikos Diamantis and I tried to apply the ideas that worked for Maass forms to holomorphic modular forms. For classical modular forms this does not work well. However, in the case of holomorphic forms of real weight not in \({\mathbb {Z}}_{\ge 2}\) the ideas in [1] turned out to be applicable. We get an injective map to a cohomology group and characterize the image. The module for the cohomology has still infinite dimension, but is much smaller than the module in Knopp’s approach.
Many details in [2] differ from those in [1], but on the whole I would put holomorphic forms of real weight not in \({\mathbb {Z}}_{\ge 2}\) and Maass forms in one group, and the classical case in another group, as far as it concerns the relation between modular forms and cohomology. This in contrast with what one might think at first. Holomorphic modular forms of real weight look like a reasonable generalization of classical modular forms, and those strange Maass forms seem to form a quite different class.
Here I discuss results from [2] for the full modular group \(\Gamma ={\mathrm {SL}}_2({\mathbb {Z}})\), to stay close to the theme of the conference. Figure 5 summarizes the results, which in fact are valid for general cofinite discrete groups with cusps.
2 Eichler cocycles
3 Knopp cocycles
Functions with a transformation like that in (3.2) correspond to functions on the universal covering group \({\tilde{G}}\) of \({\mathrm {SL}}_2({\mathbb {R}})\) that transform on the left according to a character of a discrete subgroup of \({\tilde{G}}\). See, e.g., [2, Appendix], of for halfintegral weight [11, Section 2]. Conceptually this approach has advantages. In this overview I prefer to use the classical point of view.
The formula in (2.1) for the Eichler cocycle causes problems if \(r\not \in {\mathbb {Z}}\): The factor \((\tau z)^{r2}\) is multivalued in \(\tau \) and z, branching where \(\tau =z\). This difficulty does not occur for the conjugate cocycle in (2.5). Now \({\bar{\tau }}\) is in the lower halfplane and z in the upper halfplane. We choose a branch of \(({\bar{\tau }}z)^{r2}\) and get a cocycle \(\psi ^c\) associated with a cusp form f of real weight.
Knopp used a simpler notation for the module of functions with at most polynomial growth. Here I use a notation that comes from representation theory, and was useful in [1, 2].
The Eichler cocycle in (2.1) leads for real weights to holomorphic functions on the lower halfplane \({{\mathfrak {H}}}^\) with at most polynomial growth at the boundary. That is what we did in [2]. The plus in the notation of the modules indicates that here we work with functions on the upper halfplane. Both approaches are equivalent. The operator \((\textit{JF})(z) = \overline{F({\bar{z}})}\) gives for real weights an \({\mathbb {R}}\)linear isomorphism \({}^+{\mathcal {D}}_{{\bar{v}},2r}^{\infty } \leftrightarrow {}{\mathcal {D}}_{v,2r}^{\infty }\).
The theorem of Knopp and Mawi is valid in the classical situation \(r\in 2{\mathbb {Z}}_{\ge 1}\). The cocycles defined by the holomorphic Eichler integral become coboundaries when we enlarge the module from \(V_{r2}\) to \({}^+{\mathcal {D}}_{1,2r}^{\infty }\).
4 Smaller modules
5 Base point in the upper halfplane
The image of \(A_r(\Gamma ,v)\) in \(H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega })\) can be described as a mixed parabolic cohomology group. If \(V\subset W\) is an inclusion of \(\Gamma \)modules, the mixed parabolic cohomology group \(H^1_p(\Gamma ;V,W)\) consists of those cohomology classes \([\psi ]\) in \(H^1(\Gamma ;V)\) for which \(\psi _T \in W(T1)\). (For groups with more than one cuspidal orbit the definition is slightly more complicated; one has to take into account all cuspidal orbits.) The image of \(S_{\!r}(\Gamma ,v)\) is equal to the mixed parabolic cohomology group \(H^1_p(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega }, {}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega ^0,\infty })\).
Figure 5 summarizes the results we discussed. Following the arrows we get an \({\mathbb {R}}\)linear map \(A_r(\Gamma ,v)\rightarrow H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\infty })\). This map has a huge kernel since this cohomology group is isomorphic to \(S_{\!r}(\Gamma ,v)\). The kernel of the composition \(H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega })\rightarrow H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\infty })\) on the right in Fig. 5 is even larger, since the mixed parabolic cohomology group \(H^1_p(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega }, {}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega ^0,{\mathrm {exc}}})\) has infinite codimension in \(H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega })\). See Part i)c) of Theorem E in [2].
On top in Fig. 5 I have added the cohomology group \(H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega })\). In the classical case \(r\in {\mathbb {Z}}_{\ge 2}\) it is known to be zero [8, Theorem 5]. I do not know whether this cohomology group vanishes for general real weights r. Each element of \(A_r(\Gamma ,v)\) has an image in \(H^1(\Gamma ;{}^+{\mathcal {D}}_{{\bar{v}},2r}^{\omega })\) via the bottom and the righthand side of the diagram in Fig. 5. Theorem C in [2] relates the vanishing of this image and the existence of harmonic lifts of f.
I end with the conclusion that for real weights \(r\not \in {\mathbb {Z}}_{\ge 2}\) the theory of cocycles attached to modular form is more complicated than the classical theory. It is more similar to what is discussed in [1] for Maass forms.
Notes
References
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