Modular forms of real weights and generalized Dedekind symbols

In a previous paper, I have defined non--commutative generalized Dedekind symbols for classical $PSL(2,Z)$--cusp forms using iterated period polynomials. Here I generalize 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.


Introduction: generalized 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. [KiMel], [Mel]). Fukuhara in [Fu1], [Fu2], and others ( [Ap], [ChZ], gave an abstract definition of generalized Dedekind symbols with values in an arbitrary commutative group and produced such symbols from period polynomials of P SL(2, Z)-modular forms of any even weight.
In the note [Ma5], I have given an abstract definition of generalised Dedekind symbols for the full modular group P SL(2, Z) taking values in arbitrary nonnecessarily commutative group and constructed such symbols from iterated versions of period integrals of modular forms of integral weights considered earlier in [Ma3], [Ma4].
In this article, I extend these constructions to cusp forms of real weights, studied in particular in [Kn], [KnMa], [BrChDi]. The essential ingredient here is furnished by the introduction of iterated versions of their period integrals following [BrCh] . (1 − e 2πikz ).
1 It is a cusp form of weight 1/2, and from P SL(2, Z)-invariance of η(z) 24 dz 6 it follows that for any fractional linear transformation γ ∈ P SL(2, Z), It satisfies the reciprocity relation (which can be easily extended to all c ∈ Z) s(a, c) + s(c, a) = a/c + c/a + 1/ac − 3 sgn(a).
1.2. Generalised Dedekind symbols with values in an abelian group. Using a slightly different normalisation and terminology of [Fu1], one can define the generalised Dedekind symbol d(p, q) as a function d : W → G where W is the set of pairs of co-prime integers (p, q), and G an abelian group. It can be uniquely reconstructed from the functional equations Studying other P SL(2, Z)-modular forms in place of η, 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 [Fu1], [Fu2] and 1.3.2 below.
In particular, let F (z) be a cusp form of even integral weight k + 2 for Γ := P SL(2, Z). Its period polynomial is the following function of t ∈ C: Fukuhara has shown that (slightly normalized) values of period polynomials at rational points form a reciprocity function 1.3. Non-commutative generalised Dedekind symbols. In [Ma5], I introduced non-commutative generalised Dedekind symbols with values in a nonnecessarily abelian group G by the following definition. (1.8) Applying (1.8) to p = 1, q = 0, we get f (1, 1) = 1 G where 1 G is the identity. From (1.7) we then get f (−1, 1) = 1 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 (1.11) Clearly, knowing D one can uniquely reconstruct its reciprocity function f . Conversely, any reciprocity function uniquely defines the respective generalised Dedekind symbol ([Ma5], Theorem 1.8).
In [Ma5] 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 ( [Ma1], [Ma2] and many other works); they can be used in order to produce "local zeta-factors" in the mythical algebraic geometry of characteristic 1 ( [Ma6]); they describe relations between certain inner derivations of a free Lie algebra ( [Po], [Hai], [HaiMo], [BaSch]), 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 ∈ 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 M 1,w , w ∈ R." Notice that certain p-adic interpolations appeared already long time ago in the theory of p-adic L-functions.
Acknowledgements. This note was strongly motivated and inspired by the recent preprint [BrCh] due to R. Bruggeman and Y. Choie. R. Bruggeman kindly answered my questions, and clarified for me many issues regarding modular forms of noninteger weights. Together with Y. Choie, he carefully read a preliminary version of this note. I am very grateful to them.

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 [Kn], [KnMa], and [BrCh]. I adopt conventions of [BrCh], where modular forms of real weights are holomorphic functions on the upper complex halfplane, whereas their period integrals, analogs 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 P 1 (C) be the set of C-points of the projective line endowed with a fixed projective coordinate z. This coordinate identifies the complex plane C with the maximal subset of P 1 (C) where z is holomorphic.
We put As in [BrCh], we identify holomorphic functions on H + and H − using antiholomorphic involution: F (z) → F (z). In the future holomorphic functions on H − will often be written using coordinate t = z. On the other hand, the standard hyperbolic metric of curvature −1 on H + ∪ H − , ds 2 = |dz| 2 /(Im z) 2 looks identically in both coordinates. Cusps P 1 (Q) are rational points on the common boundary of H + and H − (including infinite point). Denote by P + , resp. P − the space of functions F (z) holomorphic in H + , resp. H − satisfying for some constants K, A > 0 and all z ∈ H ± inequality |F (z)| < K(|z| A + |Imz| −A ).
(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 P SL(2, Z) upon P 1 (C) by linear fractional transformations of z z → γ(z) = az + b cz + d defines the right action upon holomorphic functions in H ± . 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(γ, z) := cz + d: 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: Second, multiplication by (cz + d) k is completed by an additional complex factor depending on γ.
2.2.1. Definition. A unitary multiplier system v of weight k ∈ R (for the group SL(2, Z)) is a map v : SL(2, Z) → C, |v(γ)| = 1, satisfying the following conditions. Put Then we have

Identities (2.3) and (2.4) imply that the formula
defines a right action of SL(2, Z)/(±id) = P SL(2, 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. [KnMa] and BrChDi]) 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 which is the right action of P SL(2, Z) upon functions holomorphic in H − .

Modular forms.
Let F ∈ P + be a holomorphic function of polynomial growth in H + (see 2.1) satisfying the SL(2, Z)-invariance condition: 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. [KnMa]). The space of all such forms is denoted C 0 (Γ, k + 2, v). It can be non-trivial only if k > 0.

Period integrals.
For a cusp form F ∈ C 0 (Γ, k, v) and points a, b in H + ∪{cusps} we put ω F (z; t) := F (z)(z −t) k dz and define its integral as a function of t ∈ H − : 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.

Generalized 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 ω := (ω j (z; t)). Let (A j ), j = 1, . . . , l, be independent associative but non-commuting formal variables.
Consider the multiplicative group 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, Z) upon this group is defined coefficientwise. In particular, the action upon J b a (Ω; t) is given by: Moreover, the linear action of the group GL(l, C) upon (A 1 , . . . , A l ) extends to its action upon the group of formal series G. We will need here only the action of the subgroup of diagonal matrices.
Let γ ∈ P SL(2, Z) and (p, q) ∈ W. Then we will denote by v(γ) * the automorphism of such a ring sending A m to v m (γ) −1 A m .
3.1.1. Definition. The generalised reciprocity function associated to the family ω := (ω j ) is the map of the set of coprime pairs of positive integers (p, q) → f ω (p, q) ∈ G defined by Put now
Proof. The direct computation shows that Therefore 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): and similarly Thus, from (3.3) we get the following identity for A j -coefficients: In view of growth estimates of period functions in [BrCh] and [BrChDi], we may put here t = q/p, where q, p are coprime integers, p > 0. We get Thus, if we put, interpolating formula (1.4) (established for even integer weights) f 0,j (p, q) := p k j I ∞ 0 (ω j (z; qp −1 )), (3.9) then we get from (3.7) functional equations This is the A j -linear part of (3.1). The general case is obtained in the same way.
Remark. In this statement 3.2, 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 · ((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 ∞ should also be treated separately already in the definition of f (p, q). We leave this as an exercise for the reader.
3.3. Dedekind cocycles. In the last section of [Ma5], equations for reciprocity functions of even integer weights (cf. Definition 1.3.1 above) were interpreted as defining a special class of 1-cocycles of Γ := P SL(2, Z).
More precisely, let G be a possibly noncommutative left Γ-module. It is known that P SL(2, Z) is the free product of its two subgroups Z 2 and Z 3 generated respectively by Restriction to (σ, τ ) of any cocycle in Z 1 (P SL(2, Z), G) belongs to the set 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 Y = τ X. (3.12) Let now G 0 be a group. Denote by G the group of functions f : P 1 (Q) → G 0 with pointwise multiplication. Define the left action of Γ upon G by (3.13) Let f : W → G 0 be a G 0 -valued reciprocity function, as in Definition 1.3.1. Define elements X f , Y f ∈ G as the following functions P 1 (Q) → G 0 : (3.14) Then the map f → (X f , Y f ) establishes a bijection between the set of G 0valued reciprocity functions and the set of (representatives of) Dedekind cocycles from Z 1 (Γ, G) ( [Ma5], 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 Γmodule of coefficients will not be of the form (3.13).
3.4. A digression: left vs. 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.
This construction establishes a bijection between the sets of structures of left/right Γ-modules on G.
In this way, Dedekind right cocycles defined by the additional condition V = U |σ bijectively correspond to the Dedekind left cocycles defined in [Ma5] by the condition Y = τ X. This can be checked by straightforward computations.
3.5. Dedekind cocycles from cusp forms of real weights. In this subsection, we take for G the subgroup of non-commutative series in A i (with coefficients in functions on H − ) generated by all series of the form J b a (Ω; t), a, b ∈ P 1 (Q) and fixed Ω (cf. 3.1).
In order to construct our Dedekind cocycle, we combine the σ-component of λ ∞ with τ -component of λ 0 : is (the representative of ) a left Dedekind cocycle.