Modular forms of real weights and generalised Dedekind symbols

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

The Dedekind eta function is a holomorphic function of the complex variable z with positive imaginary part given by

$$\begin{aligned} \eta (z) = e^{\pi iz/24} \prod _{k=1}^{\infty } (1-e^{2\pi i kz}) . \end{aligned}$$

It is a cusp form of weight 1 / 2, and from $PSL (2,{\mathbf {Z}})$-invariance of $\eta (z)^{24} dz^6$ it follows that for any fractional linear transformation $\gamma \in PSL (2,{\mathbf {Z}})$,

$$\begin{aligned} \gamma z := \frac{az+b}{cz+d} \end{aligned}$$

with $c> 0$ we can obtain a rational number

$$\begin{aligned} s(a,c):=- \frac{1}{\pi i} \left[ \mathrm{log}\,\eta (\gamma z) - \mathrm{log} \eta (z) - \frac{1}{2} \mathrm{log} ((cz+d)/i)\right] + \frac{a+d}{12c} \end{aligned}$$

called the (classical) Dedekind symbol.

It satisfies the reciprocity relation (which can be easily extended to all $c\in {\mathbf {Z}}$)

$$\begin{aligned} s(a,c)+ s(c,a) =a/c+c/a +1/ac -3\, \mathrm{sgn} (a) . \end{aligned}$$

1.2 Generalised Dedekind symbols with values in an abelian group

Using a slightly different normalisation and terminology of [6], one can define the generalised Dedekind symbol 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

$$\begin{aligned}&d(p,q)=d(p,q+p), \quad d(p,-q)= -d(p,q), \end{aligned}$$

(1.1)

$$\begin{aligned}&d(p,q)-d(q,-p) = \frac{p^2+q^2-3pq+1}{12pq}. \end{aligned}$$

(1.2)

Studying other $PSL(2,{\mathbf {Z}})$-modular forms in place of $\eta$, one arrives to 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.

In particular, let 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 $t\in \mathbf {C}$:

$$\begin{aligned} P_F (t):= \int _0^{i\infty } F (z) (z-t)^k\mathrm{d}z. \end{aligned}$$

(1.3)

Fukuhara has shown that (slightly normalised) values of period polynomials at rational points form a reciprocity function

$$\begin{aligned} f_F (p, q):= p^k P_F (q/p) = \int _0^{i\infty } F (z) (pz-q)^k \mathrm{d}z, \end{aligned}$$

(1.4)

whose respective Dedekind symbol is

$$\begin{aligned} d_F (p,q) = \int _{p/q}^{i\infty } F (z) (pz-q)^k \mathrm{d}z. \end{aligned}$$

(1.5)

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

A $\mathbf {G}$-valued reciprocity function is a map $f: W\rightarrow {\mathbf {G}}$ satisfying the following conditions:

$$\begin{aligned}&f(p,-q)=f(-p,q), \end{aligned}$$

(1.6)

$$\begin{aligned}&f(p,q)f(-q,p)=1_{{\mathbf {G}}}, \end{aligned}$$

(1.7)

$$\begin{aligned}&f(p,p+q)f(p+q,q)=f(p,q). \end{aligned}$$

(1.8)

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.

A generalised ${\mathbf {G}}$-valued Dedekind symbol $\mathcal{D}$ with reciprocity function f is a map

$$\begin{aligned} \mathcal{D}:\,W\rightarrow {\mathbf {G}}:\ (p,q)\mapsto \mathcal{D}(p,q) \end{aligned}$$

satisfying the following conditions (1.9)–(1.11):

$$\begin{aligned} \mathcal{D}(p,q)= & {} \mathcal{D}(p, q+p), \end{aligned}$$

(1.9)

$$\begin{aligned} \mathcal{D}(p,-q)= & {} \mathcal{D}(-p,q), \end{aligned}$$

(1.10)

so that $\mathcal{D}(-p,-q)= \mathcal{D}(p,q)$. Finally,

$$\begin{aligned} \mathcal{D}(p,q)\mathcal{D}(q,-p)^{-1}=f(p,q). \end{aligned}$$

(1.11)

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.

We put

$$\begin{aligned} H^+ := \{z\in {\mathbf {C}}\,|\, \mathrm{Im}\,z>0 \} ,\quad H^- := \{t\in {\mathbf {C}}\,|\, \mathrm{Im}\,t<0 \}. \end{aligned}$$

As in [4], we identify holomorphic functions on $H^+$ and $H^-$ using 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.

Cusps ${\mathbf {P}}^1({\mathbf {Q}})$ are rational points on the common boundary of $H^+$ and $H^-$ (including infinite point). Denote by ${\mathcal{P}}^+$, resp. ${\mathcal{P}}^-$ the space of functions F(z) holomorphic in $H^+$, resp. $H^-$ satisfying for some constants $K, A>0$ and all $z\in H^{\pm }$ inequality

$$\begin{aligned} |F(z)|< K(|z|^A+ |\mathrm{Im} z|^{-A} ). \end{aligned}$$

(2.1)

This is called the 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

The standard left action of $PSL(2,{\mathbf {Z}})$ upon ${\mathbf {P}}^1({\mathbf {C}})$ by linear fractional transformations of z

$$\begin{aligned} z\mapsto \gamma (z)= \frac{az+b}{cz+d} \end{aligned}$$

defines the right action upon holomorphic functions in $H^{\pm }$. In the theory of automorphic forms of even integral weight, this natural right action is considered first upon tensor powers of 1-forms $F(z) (dz)^{k/2}$ and then transported back to holomorphic functions by dividing the result by $(dz)^{k/2}$. Equivalently, the last action on functions can be defined using integral powers of $j(\gamma , z):=cz+d$:

$$\begin{aligned} (F|_k^1\gamma )(z)= j(\gamma ,z)^{-k} F(\gamma z) \end{aligned}$$

(2.2)

and similarly in t-coordinate.

In the theory of automorphic forms of general real weight k, the relevant generalisation requires two additional conventions. First, we define $(cz+d)^k$ using the following choice of arguments:

$$\begin{aligned} \mathrm{arg}\,(cz+d)\in & {} (-\pi ,\pi ]\ \quad \mathrm{for}\ z\in H^+, \\ \mathrm{arg}\,(ct+d)\in & {} [-\pi ,\pi )\ \quad \mathrm{for}\ t\in H^-. \end{aligned}$$

Second, multiplication by $(cz+d)^k$ is completed by an additional complex factor depending on $\gamma$.

Definition 2.1

A unitary multiplier system 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

$$\begin{aligned} j_{v,k}(\gamma , z):= v(\gamma ) (cz+d )^k. \end{aligned}$$

Then we have

$$\begin{aligned} j_{v,k}(\gamma \delta ,z)= j_{v,k}(\gamma , \delta z)\cdot j_{v,k}(\delta ,z) \end{aligned}$$

(2.3)

and

$$\begin{aligned} j_{v,k}(-\gamma , z)= j_{v,k}(\gamma , z). \end{aligned}$$

(2.4)

Identities (2.3) and (2.4) imply that the formula

$$\begin{aligned} F(z) \mapsto v(\gamma ) j(\gamma ,z)^{k} F(\gamma z)=j_{v,k}(\gamma ,z) F(\gamma z) \end{aligned}$$

(2.5)

defines a right action of $SL(2,{\mathbf {Z}})/(\pm \mathrm{id})=PSL(2,{\mathbf {Z}})$ upon functions holomorphic in $H^+$. Functions invariant with respect to this action and having exponential decay at cusps (in terms of geodesic distance, cf. [12] and [3]) are called 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.

Moreover, behaviour of period functions of modular forms with respect to modular transformations involves the action

$$\begin{aligned} (P|_{-k}^v\gamma )(t) := v(\gamma ) ^{-1}j(\gamma ,t)^{k} P(\gamma t)=j_{v,k}(\gamma ,t)P(\gamma t), \end{aligned}$$

(2.6)

which is the right action of $PSL(2,{\mathbf {Z}})$ upon functions holomorphic in $H^-$.

2.3 Modular forms

Let $F\in {\mathcal{P}}^+$ be a holomorphic function of polynomial growth in $H^+$ (see 2.2) satisfying the $SL(2,{\mathbf {Z}})$-invariance condition:

$$\begin{aligned} (F |_{k+2}^v\gamma )(z) = F (z)\ \quad \mathrm{for\ all}\quad \gamma \in SL(2,{\mathbf {Z}}). \end{aligned}$$

(2.7)

It is called 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

For a cusp form $F\in C^0(\Gamma ,k,v)$ and points 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 $t\in H^-$:

$$\begin{aligned} I_a^b (\omega _F;t) :=\int _a^b \omega _F (z;t) . \end{aligned}$$

If 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.

More generally, for a finite sequence of cusp forms $F_j\in C^0(\Gamma ,k_j+2,v_j)$ and $\omega _j(z) = \omega _j(z;t) := F_j(z)(z-t)^{k_j}dz$, $j=1,\ldots ,n$, where t is considered as parameter, we put

$$\begin{aligned} I_a^b (\omega _1,\ldots ,\omega _n; t) :=\int _a^b \omega _1(z_1) \int _a^{z_1} \omega _2(z_2) \ldots \int _a^{z_{n-1}} \omega _n(z_n). \end{aligned}$$

(2.8)

These functions of 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

$$\begin{aligned} \mathbf {v} (\gamma ):= v_1(\gamma ) v_2(\gamma ) \ldots v_n(\gamma ), \quad \mathbf {k} = k_1+k_2 +\cdots + k_n. \end{aligned}$$

The key role in our constructions is played by the following result ([4], Lemma 3.2):

Lemma 2.1

The iterated period integral (2.8) as a function of $t \in H$ satisfies for all $\gamma \in SL(2,{\mathbf {Z}})$ the following functional equations:

$$\begin{aligned} (I_a^b (\omega _1,\ldots ,\omega _n; \cdot )|_{-\mathbf {k}}^{\mathbf {v}}\gamma )(t)= I_{\gamma ^{-1}a}^{\gamma ^{-1}b} (\omega _1,\ldots ,\omega _n; t) \end{aligned}$$

(2.9)

(see 2.6).

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.

As in [15], we produce the generating series of all integrals of the type (2.8):

$$\begin{aligned} J_a^b(\Omega ;t) := 1+ \sum _{n\ge 1} \sum _{1\le m_1,\ldots ,m_n\le l} I_a^b (\omega _{m_1},\ldots ,\omega _{m_n};t) A_{m_1}\ldots A_{m_n}. \end{aligned}$$

Consider the multiplicative group $\mathbf {G}$ of the formal series in $(A_j)$ with coefficients in functions of 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:

$$\begin{aligned} (J_a^b(\Omega ;\cdot )|_{-\mathbf {k}}^{\mathbf {v}}\gamma )(t)= := 1+ \sum _{n\ge 1} \sum _{1\le m_1,\ldots ,m_n\le l} (I_a^b (\omega _{m_1},\ldots ,\omega _{m_n};\cdot )|_{-\mathbf {k}}^{\mathbf {v}}\gamma )(t)A_{m_1}\ldots A_{m_n}. \end{aligned}$$

Moreover, the linear action of the group $GL (l,{\mathbf {C}})$ upon $(A_1,\ldots ,A_l)$ extends to its action upon the group of formal series ${\mathbf {G}}$. We will need here only the action of the subgroup of diagonal matrices.

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

The generalised reciprocity function associated with the family $\omega := (\omega _j)$ is the map of the set of coprime pairs of positive integers $(p,q)\mapsto f_{\omega }(p,q)\in \mathbf {G}$ defined by

$$\begin{aligned} \begin{aligned} f_{\omega }(p,q)&= f(p,q) \\&:= 1+ \sum _{n\ge 1} \sum _{ 1\le m_1,\ldots ,m_n\le l} p^{k_{m_1}+\cdots +k_{m_n}} I_0^{\infty } (\omega _{m_1},\ldots , \omega _{m_n}; qp^{-1})A_{m_1}\ldots A_{m_n}. \end{aligned} \end{aligned}$$

Put now

$$\begin{aligned} \sigma = \left( \begin{matrix} 0&{} -1\\ 1&{} 0 \end{matrix} \right) ,\quad \theta = \left( \begin{matrix} 1&{} 1\\ 0&{} 1 \end{matrix} \right) . \end{aligned}$$

Theorem 3.1

The function f satisfies the following functional equation generalising (1.8):

$$\begin{aligned} f(p,q) = (\mathbf {v}(\theta )* f(p,q+p)) \cdot (\mathbf {v}(\theta \sigma \theta )* f(q+p,q) ), \end{aligned}$$

(3.1)

It can be extended to the function defined on coprime pairs $(p,q), p>0, q<0,$ using the functional equation generalising (1.7)

$$\begin{aligned} f(p,q):= (\mathbf {v}(\sigma )*f(-q,p))^{-1} \end{aligned}$$

(3.2)

and defining the right hand side via Definition 3.1.

Remark

According to [3], (2.12), we have

$$\begin{aligned} v_m(\sigma )= e^{-\pi i (k_m+2)/2} ,\quad v_m(\theta )= e^{\pi i (k_m+2)/6}. \end{aligned}$$

Hence, $v_m(\theta \sigma \theta )= e^{-\pi i (k_m+2)/6}$.

Proof

The direct computation shows that

$$\begin{aligned} \theta ^{-1}(0)=-1,\ \theta ^{-1}(\infty )= \infty ,\quad (\theta \sigma \theta )^{-1}(0)= 0,\ (\theta \sigma \theta )^{-1}(\infty )= -1. \end{aligned}$$

Therefore (cf. Eq. (1.5) of [15])

$$\begin{aligned} J_0^{\infty }(\Omega ;t) = J_{-1}^{\infty }(\Omega ;t) J_0^{-1 }(\Omega ;t) = J_{\theta ^{-1}(0)}^{\theta ^{-1}(\infty ) }(\Omega ;t)\cdot J_{(\theta \sigma \theta )^{-1}(0)}^{(\theta \sigma \theta )^{-1}(\infty ) }(\Omega ;t) . \end{aligned}$$

(3.3)

We now apply (2.9) to both factors in the right hand side of (3.3) term wise. To avoid cumbersome notations, we will restrict ourselves to simple integrals, that is, to the coefficients of terms linear in $(A_j)$. The general case easily follows from this one.

We have then from (2.6):

$$\begin{aligned} I_{\theta ^{-1}(0)}^{\theta ^{-1}(\infty ) }(\omega _j (z;t))= & {} v(\theta )^{-1} j(\theta ,t)^{k_j}\cdot I_0^{\infty } (\omega _j (z; \theta ( t))) \nonumber \\= & {} v(\theta )^{-1} \cdot I_0^{\infty } (\omega _j (z; t+1)) \end{aligned}$$

(3.4)

and similarly

$$\begin{aligned} I_{(\theta \sigma \theta )^{-1}(0)}^{(\theta \sigma \theta )^{-1}(\infty ) }(\omega _j (z;t))= & {} v(\theta \sigma \theta )^{-1} j(\theta \sigma \theta ,t)^{k_j}\cdot I_0^{\infty } (\omega _j (z; \theta ( t))) \nonumber \\= & {} v(\theta \sigma \theta )^{-1}\cdot (t+1)^{k_j}\cdot I_0^{\infty } \left( \omega _j \left( z; \frac{t}{t+1}\right) \right) . \end{aligned}$$

(3.5)

Thus, from (3.3) we get the following identity for $A_j$-coefficients:

$$\begin{aligned} I_0^{\infty }(\omega _j(z;t))= & {} v(\theta )^{-1} I_0^{\infty }(\omega _j(z;t+1)) \nonumber \\&+\, v(\theta \sigma \theta )^{-1}\cdot (t+1)^{k_j}\cdot I_0^{\infty } \left( \omega _j \left( z; \frac{t}{t+1}\right) \right) . \end{aligned}$$

(3.6)

In view of growth estimates of period functions in [3, 4], we may put here $t=q/p$, where q, p are coprime integers, $p > 0$. We get

$$\begin{aligned} I_0^{\infty }(\omega _j(z;qp^{-1}))= & {} v(\theta )^{-1} I_0^{\infty }(\omega _j(z;(q+p)p^{-1})) \nonumber \\&+ \,v(\theta \sigma \theta )^{-1}\cdot ((q+p)p^{-1})^{k_j}\cdot I_0^{\infty } (\omega _j (z; q(q+p)^{-1})) \end{aligned}$$

(3.7)

that is,

$$\begin{aligned} p^{k_j}I_0^{\infty }(\omega _j(z;qp^{-1}))= & {} v(\theta )^{-1} p^{k_j} I_0^{\infty }(\omega _j(z;(q+p)p^{-1})) \nonumber \\&+ \,v(\theta \sigma \theta )^{-1}\cdot (q+p)^{k_j}\cdot I_0^{\infty } (\omega _j (z; q(q+p)^{-1})). \end{aligned}$$

(3.8)

Thus, if we put, interpolating formula (1.4) (established for even integer weights)

$$\begin{aligned} f_{0,j}(p,q):=p^{k_j}I_0^{\infty }(\omega _j(z;qp^{-1})), \end{aligned}$$

(3.9)

then we get from (3.7) functional equations

$$\begin{aligned} f_{0,j}(p,q) = v(\theta )^{-1} f_{0,j}(p,q+p) + v(\theta \sigma \theta )^{-1} f_{0,j}(q+p,q) . \end{aligned}$$

(3.10)

This is the $A_j$-linear part of (3.1). The general case is obtained in the same way.

Similarly, looking at the $A_j$-coefficients in the identity $J_0^{\infty }(\Omega ;t)J_{\sigma (0)}^{\sigma ( \infty )} (\Omega ; t)=1$ and applying (2.6) for $t=-pq^{-1}$ we get

$$\begin{aligned} f_{0,j}(p,q) + v(\sigma ) f_{0,j}(-q,p)=0. \end{aligned}$$

The reasoning as above completes the proof. $\square$

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 $\Gamma := PSL (2,\mathbf {Z})$.

More precisely, let ${\mathbf {G}}$ be a possibly non-commutative left $\Gamma$-module. It is known that $PSL(2,\mathbf {Z})$ is the free product of its two subgroups $\mathbf {Z}_2$ and $\mathbf {Z}_3$ generated, respectively, by

$$\begin{aligned} \sigma = \left( \begin{matrix} 0&{} -1\\ 1&{} 0 \end{matrix} \right) ,\quad \tau = \left( \begin{matrix} 0&{} -1\\ 1&{} -1\end{matrix} \right) . \end{aligned}$$

Restriction to $(\sigma ,\tau )$ of any cocycle in $Z^1(PSL(2,\mathbf {Z}), {\mathbf {G}})$ belongs to the set

$$\begin{aligned} \{\, (X,Y) \in {\mathbf {G}}\times {\mathbf {G}}\,|\, X\cdot \sigma X=1,\, Y\cdot \tau Y\cdot \tau ^2Y=1\,\}. \end{aligned}$$

(3.11)

In fact, this defines a bijection between cocycles and pairs (3.11).

An element (X, Y) of (3.11) is called (the representative of) the Dedekind cocycle, iff it satisfies the relation

$$\begin{aligned} Y=\tau X. \end{aligned}$$

(3.12)

Let now ${\mathbf {G}}_0$ be a group. Denote by ${\mathbf {G}}$ the group of functions $f:\, {\mathbf {P}}^1({\mathbf {Q}})\rightarrow {\mathbf {G}}_0$ with pointwise multiplication. Define the left action of $\Gamma$ upon $\mathbf {G}$ by

$$\begin{aligned} (\gamma f)(x) = f(\gamma ^{-1}x);\quad f\in {\mathbf {G}},\ x\in {\mathbf {P}}^1({\mathbf {Q}}),\ \gamma \in \Gamma . \end{aligned}$$

(3.13)

Let $f:\, W\rightarrow {\mathbf {G}}_0$ be a ${\mathbf {G}}_0$-valued reciprocity function, as in Definition 1.1. Define elements $X_f, Y_f\in {\mathbf {G}}$ as the following functions ${\mathbf {P}}^1({\mathbf {Q}})\rightarrow {\mathbf {G}}_0$:

$$\begin{aligned}&X_f(qp^{-1}):= f(p,q), \end{aligned}$$

(3.14)

$$\begin{aligned}&Y_f(qp^{-1}):= (\tau X_f)(qp^{-1})= X_f(\tau ^{-1}(qp^{-1}))= f(q, q-p). \end{aligned}$$

(3.15)

Then, the map $f\mapsto (X_f,Y_f)$ establishes a bijection between the set of ${\mathbf {G}}_0$-valued reciprocity functions and the set of (representatives of) Dedekind cocycles from $Z^1(\Gamma ,{\mathbf {G}})$ ([17], Theorem 3.6).

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.

The sets of left/right 1-cocycles with coefficients in a left/right module ${\mathbf {G}}$ are defined by

where $*|\gamma$ denotes the right action of $\gamma$.

Let $\rho \in Z^1_r(PSL(2,\mathbf {Z}), {\mathbf {G}})$ and set $U:=\rho (\sigma ), V:=\rho (\tau )$. From $\sigma ^2=\tau ^3 =1$ and from ($3.16_r$) it follows that

$$\begin{aligned} (U|\sigma )\cdot U=1,\, (V|\tau ^2)\cdot (V|\tau )\cdot V=1. \end{aligned}$$

(3.17)

Moreover, any pair $(U,V)\in {\mathbf {G}}\times {\mathbf {G}}$ satisfying (3.17) comes from a unique cocycle $\rho \in Z^1_r(PSL(2,\mathbf {Z}), {\mathbf {G}})$.

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 $\lambda \in Z^1_l(PSL(2,\mathbf {Z}), {\mathbf {G}})$, define

  $$\begin{aligned} \rho :\Gamma \rightarrow {\mathbf {G}}:\ \rho (\gamma ) = (\lambda (\gamma ^{-1}))^{-1}. \end{aligned}$$

  (3.18)

  This establishes a bijection of the respective set of 1-cocycles $Z^1_l(\Gamma ,{\mathbf {G}})$ and $Z^1_r(\Gamma ,{\mathbf {G}}).$

• (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).

The left action of $\Gamma = PSL(2,\mathbf {Z})$ upon ${\mathbf {G}}$ is defined by

$$\begin{aligned} \gamma J_a^b(\Omega ;t) := J_{\gamma a}^{\gamma b}(\Omega ;t). \end{aligned}$$

Every cusp a defines the left cocycle $\lambda _a: \,\Gamma \rightarrow {\mathbf {G}}$:

$$\begin{aligned} \lambda _a(\gamma ):= J^a_{\gamma a} (\Omega ;t) \end{aligned}$$

because

$$\begin{aligned} J^a_{\gamma _1\gamma _2 a} = J^a_{\gamma _1 a} \cdot J^{\gamma _1 a}_{\gamma _1\gamma _2 a}= J^a_{ \gamma _1 a} \cdot \gamma _1 J^a_{\gamma _2 a}. \end{aligned}$$

In order to construct our Dedekind cocycle, we combine the $\sigma$-component of $\lambda _{\infty }$ with $\tau$-component of $\lambda _0$:

Theorem 3.2

The pair

$$\begin{aligned} X:=J_0^{\infty }(\Omega ;t), \quad Y:=J_1^0(\Omega ;t) \end{aligned}$$

(3.19)

is (the representative of) a left Dedekind cocycle.

Proof

We have

$$\begin{aligned} X\cdot \sigma X= & {} J_0^{\infty }(\Omega ;t)\cdot J^0_{\infty }(\Omega ;t) =1, \\ Y\cdot \tau Y\cdot \tau ^2Y= & {} J_1^0(\Omega ;t)\cdot J^1_{\infty }(\Omega ;t) \cdot J_0^{\infty }(\Omega ;t) =1, \end{aligned}$$

and finally

$$\begin{aligned} \tau X= J_{\tau 0}^{\tau \infty }(\Omega ;t) = J_1^0(\Omega ;t) =Y. \end{aligned}$$

$\square$

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.