# Modular forms of real weights and generalised Dedekind symbols

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## Abstract

In a previous paper, I have defined non-commutative generalised Dedekind symbols for classical \(PSL(2,\mathbf {Z})\)-cusp forms using iterated period polynomials. Here I generalise this construction to forms of real weights using their iterated period functions introduced and studied in a recent article by R. Bruggeman and Y. Choie.

## 1 Introduction: generalised Dedekind symbols

The classical Dedekind symbol encodes an essential part of modular properties of the Dedekind eta function and appears in many contexts seemingly unrelated to modular forms (cf. [10, 19]). Fukuhara in [6, 7], and others ([1, 5], gave an abstract definition of generalised Dedekind symbols with values in an arbitrary commutative group and produced such symbols from period polynomials of \(PSL (2,{\mathbf {Z}})\)-modular forms of any even weight.

In the note [17], I have given an abstract definition of generalised Dedekind symbols for the full modular group \(PSL (2,{\mathbf {Z}})\) taking values in arbitrary non-necessarily commutative group and constructed such symbols from iterated versions of period integrals of modular forms of integral weights considered earlier in [15, 16].

In this article, I extend these constructions to cusp forms of real weights, studied in particular in [3, 11, 12]. The essential ingredient here is furnished by the introduction of iterated versions of their period integrals following [4].

### 1.1 Dedekind symbols

*z*with positive imaginary part given by

### 1.2 Generalised Dedekind symbols with values in an abelian group

*d*(

*p*,

*q*) as a function \(d:\,W\rightarrow \mathbf {G}\) where

*W*is the set of pairs of coprime integers (

*p*,

*q*), and \(\mathbf {G}\) an abelian group. It can be uniquely reconstructed from the functional equations

*generalised Dedekind symbols*, satisfying similar functional equations, in which the right hand side of (1.2) is replaced by a different

*reciprocity function*, which in turn satisfies simpler functional equations and which uniquely defines the respective Dedekind symbol: see [6, 7] and 1.1 below.

*F*(

*z*) be a cusp form of even integral weight \(k+2\) for \(\Gamma := PSL(2,\mathbf {Z})\). Its period polynomial is the following function of \(t\in \mathbf {C}\):

### 1.3 Non-commutative generalised Dedekind symbols

In [17], I introduced *non-commutative generalised Dedekind symbols* with values in a non-necessarily abelian group \({\mathbf {G}}\) by the following definition.

## Definition 1.1

Applying (1.8) to \(p=1, q=0\), we get \(f(1,1)=1_{\mathbf {G}}\) where \(1_{\mathbf {G}}\) is the identity. From (1.7), we then get \(f(-1,1)=1_{\mathbf {G}}.\) Moreover, \(f(-p,-q)=f(p,q)\) so that *f*(*p*, *q*) depends only on *q* / *p* which obviously can now be an arbitrary point in \(\mathbf {P}^1(\mathbf {Q})\) including \(\infty \). (Of course, \(i\infty \) in integrals like (1.3)–(1.5) coincides with \(\infty \) of the real projective line).

The function (1.4) taking values in the additive group of complex numbers satisfies equations (1.6)–(1.8) (written additively).

## Definition 1.2

Let *f* be a \(\mathbf {G}\)-valued reciprocity function.

*f*is a map

Clearly, knowing \(\mathcal{D}\) one can uniquely reconstruct its reciprocity function *f*. Conversely, any reciprocity function uniquely defines the respective generalised Dedekind symbol ([17], Theorem 1.8).

In [17] I constructed such reciprocity functions using iterated integrals of cusp forms of integral weights. In the main part of this note, I will (partly) generalise this construction to cusp forms of real weights.

Period polynomials of cusp forms of integer weights appear in many interesting contexts. Their coefficients are values of certain *L*-functions in integral points of the critical strip ([13, 14] and many other works); they can be used in order to produce “local zeta–factors” in the mythical algebraic geometry of characteristic 1 ([18]); they describe relations between certain inner derivations of a free Lie algebra ([2, 8, 9, 20]), essentially because iterated period polynomials define representations of unipotent completion of basic fundamental modular groupoids.

Iterated period polynomials of real weights can be compared to various other constructions where interpolation from integer values to real values occurs, e. g. Deligne’s theory of “ symmetric groups \(S_w\), \(w\in \mathbf {R}\)” using a categorification. It would be very interesting to find similar categorifying constructions also in the case of modular forms of real weights. One can expect perhaps appearance of “modular spaces \(\overline{M}_{1,w}, w\in \mathbf {R}.\)” Notice that certain *p*-adic interpolations appeared already long time ago in the theory of *p*-adic *L*-functions.

## 2 Modular forms of real weight and their period integrals

In this section, I fix notation and give a brief survey of relevant definitions and results from [11, 12], and [4]. I adopt conventions of [4], where modular forms of real weights are holomorphic functions on the upper complex half-plane, whereas their period integrals, analogues of (1.3), are holomorphic functions on the lower half-plane.

### 2.1 Growth conditions for holomorphic functions in upper/lower complex half-planes

Let \({\mathbf {P}}^1({\mathbf {C}})\) be the set of \({\mathbf {C}}\)-points of the projective line endowed with a fixed projective coordinate *z*. This coordinate identifies the complex plane \({\mathbf {C}}\) with the maximal subset of \({\mathbf {P}}^1({\mathbf {C}})\) where *z* is holomorphic.

*antiholomorphic*involution: \(F(z)\mapsto \overline{F(\overline{z})}\). In the future, holomorphic functions on \(H^-\) will often be written using coordinate \(t=\overline{z}\). On the other hand, the standard hyperbolic metric of curvature \(-1\) on \(H^+\cup \, H^-\), \(ds^2=|dz|^2/(\mathrm{Im}\,z)^2\) looks identically in both coordinates.

*F*(

*z*) holomorphic in \(H^+\), resp. \(H^-\) satisfying for some constants \(K, A>0\) and all \(z\in H^{\pm }\) inequality

*polynomial growth*condition.

Cusp forms and their iterated period functions with which we will be working actually satisfy stronger growth conditions near the boundary: see 2.4 below.

### 2.2 Actions of modular group

*z*

*t*-coordinate.

*k*, the relevant generalisation requires two additional conventions. First, we define \((cz+d)^k\) using the following choice of arguments:

## Definition 2.1

*v*of weight \(k\in \mathbf {R}\) (for the group \(SL(2,{\mathbf {Z}})\)) is a map \(v:\, SL(2,{\mathbf {Z}}) \rightarrow {\mathbf {C}}\), \(|v(\gamma )|=1\), satisfying the following conditions. Put

*cusp forms*for the full modular group of weight

*k*with multiplier system

*v*.

For such a form *F*(*z*), one can define its *period function* \(P_F(t)\) by the formula similar to (1.3). Generally, it is defined only on \(H^-\) and satisfies the polynomial growth condition near the boundary.

### 2.3 Modular forms

*a modular form of weight*\(k+2\)

*and multiplier system*

*v*. Such a modular form is called

*a cusp form*if in addition its Fourier series at all cusps contain only positive powers of the relevant exponential function (cf. [12]).

The space of all such forms is denoted \(C^0(\Gamma ,k+2,v)\). It can be non-trivial only if \(k>0\).

### 2.4 Period integrals

*a*,

*b*in \(H^+\cup \{ cusps\}\) we put \(\omega _F(z;t):=F (z)(z-t)^kdz\) and define its integral as a function of \(t\in H^-\):

*a*and/or

*b*is a cusp, then the integration path near it must follow a segment of geodesic connecting

*a*and

*b*. We may and will assume that in our (iterated) integrals the integration path is always the segment of geodesic connecting limits of integration.

*t*is considered as parameter, we put

*t*are holomorphic on \(H^-\) and extend holomorphically to a neighbourhood of \(\mathbf {P}^1(\mathbf {R}){\setminus } \{a,b\}\) (we assume here that

*a*,

*b*are cusps). More precisely, they belong to the \(PSL(2,\mathbf {R})\)-module \({\mathcal{D}}_{\mathbf {v},-\mathbf {k}}^{\omega ^0,\infty }\) defined in Sec. 1.6 of [3], where \(\mathbf {v}, \mathbf {k}\) are defined by

## Lemma 2.1

## 3 Generalised reciprocity functions from iterated period integrals

### 3.1 Non-commutative generating series

Fix a finite family of cusp forms as in 2.4 and the respective family of 1-forms \(\omega := (\omega _j(z;t))\). Let \((A_j), j=1,\ldots ,l\), be independent associative but non-commuting formal variables.

*t*and lower term 1 (\(A_j\) commute with coefficients). The right action of \(SL(2,{\mathbf {Z}})\) upon this group is defined coefficientwise. In particular, the action upon \(J_a^b(\Omega ;t)\) is given by:

Let \(\gamma \in PSL (2,\mathbf {Z})\) and \((p,q)\in W.\) Then, we will denote by \(\mathbf {v}(\gamma )*\) the automorphism of such a ring sending \(A_m\) to \(v_m(\gamma )^{-1}A_m\).

## Definition 3.1

## Theorem 3.1

*f*satisfies the following functional equation generalising (1.8):

## Remark

## Proof

*q*,

*p*are coprime integers, \(p > 0\). We get

## Remark

In this Theorem 3.1, we avoided the simultaneous direct treatment of all cusps \(qp^{-1}\) because in our context we have to use the identities of the type \(p^k\cdot ((p+q)p^{-1})^k= (p+q)^k\) which require a separate treatment depending on the signs of integers involved, and the cusps 0 and \(\infty \) should also be treated separately already in the definition of *f*(*p*, *q*). We leave this as an exercise for the reader.

### 3.2 Dedekind cocycles

In the last section of [17], equations for reciprocity functions of even integer weights (cf. Definition 1.1 above) were interpreted as defining a special class of 1-cocycles of \(\Gamma := PSL (2,\mathbf {Z})\).

*X*,

*Y*) of (3.11) is called (the representative of)

*the Dedekind cocycle*, iff it satisfies the relation

We will now show that generalised Dedekind cocycles also can be constructed from iterated integrals of cusp forms of real weights, although the respective \(\Gamma \)-module of coefficients will not be of the form (3.13).

### 3.3 A digression: left versus right

Treating cocycles with non-commutative coefficients, we may prefer to work with left or right modules of coefficients, depending on the concrete environment. We will give a description of Dedekind cocycles with coefficients in a right module.

## Lemma 3.1

- (a)
If the map \({\mathbf {G}}\times \Gamma \rightarrow {\mathbf {G}}\): \((g, \gamma )\mapsto g| \gamma \) defines on \({\mathbf {G}}\) the structure of right \(\Gamma \)-module, then the map \(( \gamma , g)\mapsto \gamma g:= g| \gamma ^{-1} \) defines on \({\mathbf {G}}\) the structure of left \(\Gamma \)-module.

This construction establishes a bijection between the sets of structures of left/right \(\Gamma \)-modules on \({\mathbf {G}}\).

- (b)For such a pair of left/right structures and an element \(\lambda \in Z^1_l(PSL(2,\mathbf {Z}), {\mathbf {G}})\), defineThis establishes a bijection of the respective set of 1-cocycles \(Z^1_l(\Gamma ,{\mathbf {G}})\) and \(Z^1_r(\Gamma ,{\mathbf {G}}).\)$$\begin{aligned} \rho :\Gamma \rightarrow {\mathbf {G}}:\ \rho (\gamma ) = (\lambda (\gamma ^{-1}))^{-1}. \end{aligned}$$(3.18)
- (c)
Assume now that \(\Gamma =PSL(2,\mathbf {Z})\). If \((\lambda , \rho )\) is a pair of cocycles connected by (3.18), put as above \(U:=\rho (\sigma ), V:=\rho (\tau )\), and as in [17], sec. 3, \(X:=\lambda (\sigma ), Y:=\lambda (\tau )\).

In this way, Dedekind right cocycles defined by the additional condition \(V=U|\sigma \) bijectively correspond to the Dedekind left cocycles defined in [17] by the condition \(Y=\tau X\).

This can be checked by straightforward computations.

### 3.4 Dedekind cocycles from cusp forms of real weights

In this subsection, we take for \({\mathbf {G}}\) the subgroup of non-commutative series in \(A_i\) (with coefficients in functions on \(H^-\)) generated by all series of the form \(J_a^b(\Omega ;t)\), \(a,b\in \mathbf {P}^1({\mathbf {Q}})\) and fixed \(\Omega \) (cf. Sect. 3.1).

*a*defines the left cocycle \(\lambda _a: \,\Gamma \rightarrow {\mathbf {G}}\):

## Theorem 3.2

## Proof

## Acknowledgements

Open access funding provided by Max Planck Society. This note was strongly motivated and inspired by the recent preprint [4] due to R. Bruggeman and Y. Choie. R. Bruggeman kindly answered my questions and clarified for me many issues regarding modular forms of non-integer weights. Together with Y. Choie, he carefully read a preliminary version of this note. I am very grateful to them.

## Competing interests

The author declares that he has no competing interests.

## Ethics approval and consent to participate

The author declares that this study does not involve human subjects, human material and human data.

## Notes

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