An algebraic characterization of the Kronecker function

We characterize Zagier’s generating series of extended period polynomials of normalized Hecke eigenforms for PSL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{PSL}\,}}_2(\mathbb {Z})$$\end{document} in terms of the period relations and existence of a suitable factorization. For this, we prove a characterization of the Kronecker function as the “fundamental solution” of the Fay identity.


Zagier's generating series of extended period polynomials
To every cusp form 1 f , one can attach its period polynomial r f (X). This is an important arithmetic invariant whose coefficients are, up to elementary factors, the critical values of the completed L-function of f . Period polynomials were studied extensively by Eichler, Shimura and Manin (see, for example, [7, Chapter V] for a textbook account), and their definition was extended to all modular forms by Zagier [12]. Nowadays, extended period polynomials are well-studied objects in number theory which keep attracting attention. For example, they are related to zeta functions of real quadratic fields [6] and multiple zeta values [4]. More recently, an analogue of the Riemann hypothesis has been proved for them [2,5].
In [12], Zagier introduced a generating series C(X, Y ; τ ; T ) of extended period polynomials of normalized Hecke eigenforms (see Sect. 4.1 for its definition) and related it to the Kronecker function [10,Chapter VIII], Here θ τ (u) denotes Jacobi's odd theta function (see Sect. 2.1 for the definition). More precisely, Zagier's main result is as follows.
As two applications, we mention the algorithmic computation (up to scalars) of the period polynomials of all cuspidal Hecke eigenforms, as well as a new proof of the Eichler-Selberg trace formula for PSL 2 (Z) [11].

Period relations and the main result
The goal of this paper is to give another application of Zagier's result, namely a purely algebraic characterization of C in terms of the period relations for extended period polynomials where S = 0 −1 1 0 , U = 1 −1 1 0 and | denotes the slash operator. Indeed, since C is defined as a generating series of extended period polynomials it also satisfies a suitable version of the period relations; see Eq. (4.1).
It turns out that the period relations alone do not suffice to characterize C, as for any modular form f of weight k the series C + r f (X)T k−2 also satisfies the period relations. However, if we demand in addition the existence of a factorization as in Theorem 1.1, then the situation is much better and our main result is as follows.

Theorem 1.2 For a formal Laurent series f
satisfies the period relations (Eq. (4.1)). Then either: Moreover, if α , β , δ , τ are a different choice of parameters as above, then there exists a matrix a b c d ∈ SL 2 (Z) such that The basic idea is that, via Theorem 1. (i) If (a 3,0 , a 5,0 ) = (0, 0), then there exist unique α, β, γ ∈ C such that Moreover, if α , β , γ , δ , τ are a different choice of parameters as above, then there exists a b c d ∈ SL 2 (Z) such that The idea of proof is that the Fay identity implies recurrence relations for the coefficients of f such that every solution is uniquely determined by at most five of its coefficients (see Proposition 3.3 and Theorem 3.6 for the precise statements). Varying the parameters α, β, γ , δ, τ suitably, we can arrange that f is equal to one of the functions in Theorem 1.3.
After a previous version of this manuscript had been submitted, the author was informed that the existence part of Theorem 1.3 is equivalent to a result of Polishchuk [8,Theorem 5]. To be precise, Polishchuk works with the (scalar) associative Yang-Baxter equation which is equivalent to the Fay identity [9] and classifies solutions in the space of meromorphic functions defined in a neighborhood of (0, 0) instead of formal Laurent series. The two proofs are quite similar, but ours may still be of independent interest as it yields slightly more information about the coefficients of solutions to the Fay identity (see Proposition 3.8). This in turn sheds some light on how solutions to the associative Yang-Baxter equation can be constructed algorithmically.

Content
Section 2 is preliminary; we recall the definition of the Kronecker function F τ (u, v) and state the Fay identity. This identity is then studied in detail in Sect. 3 which culminates in the proof of Theorem 1.3. In Sect. 4, we explain the relation between Fay identity and period relations, building on Zagier's fundamental result, and finish by giving a proof of Theorem 1.2.
Notation Given variables X 1 , . . . , X n and a ring R, we will denote by R((X 1 , . . . , X n )) the R-algebra of formal Laurent series of the form

The Kronecker function and its functional equations
The reference for this section is [12].

Definition 2.1 The Kronecker function is the meromorphic function
From the properties of θ τ mentioned above, it follows that , for m, n, r, s ∈ Z. Also, at the cusp i∞, the Kronecker function degenerates to a trigonometric function: Remark 2.2 The above version of the Kronecker function is the one given in [12]. Other sources such as [1,8] use the function 2πiF τ (2πiu, 2πiv) instead.
We are interested in functional equations satisfied by it satisfies the following three-term functional equation [1], Proposition 5.(iii). 2

Proposition 2.3 The Kronecker function satisfies the Fay identity,
Proof We give a proof for convenience of the reader. Writing out the definition of F τ (u, v) and multiplying by the common denominator of the left-hand side of (2.1), we see that (2.1) is equivalent to where we also used θ τ (−u) = −θ τ (u). Now substituting and again using antisymmetry of θ τ , we can write (2.2) in the more symmetric form and this is precisely [12,Proposition 5].
Slightly more generally, we have the following result.

Corollary 2.4
The functions both satisfy the Fay identity, for all α, β, γ ∈ C in the first case and for all α, β ∈ C × , γ , δ ∈ C and τ ∈ H ∪ {i∞} in the second. Proof The key observation is that the product e γ u 1 v 1 e γ u 2 v 2 is invariant under the following two linear transformations which occur in the Fay identity: The corollary then follows from partial fractions in the first case and from Proposition 2.3 in the second.

Algebraic structure of the Fay identity
In this section, we always let f (u, v) = m,n> >−∞ a m,n u m v n ∈ C((u, v)) be a formal Laurent series. The goal of this section is to derive constraints on the coefficients a m,n imposed by the Fay identity. Our main results (Theorem 3.6 and Proposition 3.3) show that if f satisfies the Fay identity, it is uniquely determined by at most five of its coefficients.

Some basic implications of the Fay identity
We begin by showing how the Fay identity implies the vanishing of many of the coefficients a m,n .
The first equation implies f (u, 0) = a −1,0 u −1 , as can be seen by comparing coefficients.
In particular, if f does not have a pole at u = 0, we must have f (u, 0) = 0, but then the second equation implies f (u, v) = 0. Now assume that f = 0 and let M, N ∈ Z be the largest integers such that a −M,n = 0 for some n ∈ Z and a m,−N = 0 for some m ∈ Z. Since f has a pole at either u = 0 or v = 0, we have M ≥ 1 or N ≥ 1. Define Both g and h are well defined by construction of M and N . Now, if f has a pole at u = 0, we multiply the Fay identity by (u 1 u 2 ) M and then set u = u 1 = u 2 = 0 to get and it is straightforward to verify that this implies M = 1 and that g(v) = a −1,0 . Likewise, if f (u, v) has a pole at v = 0 (i.e., N ≥ 1), then a similar argument yields N = 1 and h(u) = a 0,−1 . This shows that It remains to prove that a m,n = 0 if m + n is even which is equivalent to antisymmetry . For this, we may clearly assume that f = 0. Taking the residues of the Fay identity at u 1 = 0, respectively, at v 1 = 0, gives and since (a −1,0 , a 0,−1 ) = (0, 0), the result follows.

Remark 3.2
We have already mentioned that our version of the Fay identity is slightly different from the one in [1,Proposition 5]. In particular, the latter does not imply antisymmetry. Indeed, for every α = 0 the function f (u, v) := α(coth(αu) + 1) satisfies the Fay identity as given in [1] (with u corresponding to ξ ), but does not satisfy Eq. (2.1). On the other hand, antisymmetry together with the version of the Fay identity given in loc.cit. are equivalent to (2.1). Therefore, our version of the Fay identity subsumes antisymmetry as well which is the reason why we prefer to work with it.
for every n, showing that a n−1,n is recursively determined by a −1,0 and a 0,1 , and therefore, f itself is uniquely determined by a −1,0 and a 0,1 . On the other hand, by Corollary 2.4 there exists a solution to the Fay identity for any given values of a −1,0 ∈ C × and a 0,1 ∈ C, namely αe γ uv u −1 with α = a −1,0 and γ = a 0,1 /a −1,0 , and this ends the proof.

The ideal of Fay relations
To study the Fay identity in more detail, it will be convenient to replace the coefficients a m,n ∈ C by symbols A m,n and accordingly to study a formal version of the Fay identity. More precisely, consider the polynomial C-algebra (the restriction on the indices (m, n) is justified by Proposition 3.1) and let be the generic element where the sum is over all (m, n) as in (3.2). By definition, it satisfies Proof By definition, (u, v) has simple poles exactly along u = 0 and v = 0 with residues A −1,0 and A 0,−1 , respectively. It is therefore sufficient to check that all residues of F along u 1 = 0, u 2 = 0, u 1 + u 2 = 0, v 1 = 0, v 2 = 0 and v 1 − v 2 = 0 vanish, which is straightforward and essentially only uses that (−u, −v) = − (u, v). Extending scalars to C[Ā −1 −1,0 ,Ā −1 0,−1 ], we get an induced map which is clearly injective.

Theorem 3.6
The map ι 0 is an isomorphism of algebras.
Before we prove Theorem 3.6, we need to introduce some more notation. Let p ⊂ A be the ideal generated by all A m,n with m, n ≥ 0. This is a homogeneous prime ideal of A, the grading being defined by giving A m,n degree m + n. Moreover, since the ideal J of Fay relations is homogeneous, this grading descends to the quotientĀ. In general, given a homogeneous ideal I of either A orĀ, we will denote by I k its component of degree k.
The following proposition gives explicit formulas for some of the coefficients c m 1 ,m 2 ,n 1 ,n 2 and will be the key for proving Theorem 3.6. Proposition 3. 8 We have the following formulas for the coefficients of F: and for k ≥ 2 even, (3.8) Finally, for 0 < m < k with k as above, we have Proof Since the Fay identity is homogeneous, for every monomial u r Proof of Theorem 3.6 It is clearly enough to show thatĀ m,n ∈Ā 0 for all m, n. We prove this by induction on the degree d = m + n ofĀ m,n . For d = 1, we haveĀ 1,0 ∈Ā by definition and it follows from (3.4) thatĀ 0,1 =Ā −1 −1,0Ā 0,−1Ā1,0 ∈Ā 0 . Note that this also implies that (p 2 0 ) 2 ⊂Ā 0 wherep 2 0 denotes the ideal generated by the image of p 2 inĀ 0 . Now we use induction on d to show thatĀ m,n ∈Ā 0 for all m, n with d = m + n. (This shows in particular that (p 2 0 ) d−1 ⊂Ā 0 .) For d = 3 or d = 5, sinceĀ d,0 ∈Ā in that case, we see from (3.6) together with (p 2 0 ) d−1 ⊂Ā 0 (which follows from the induction hypothesis) thatĀ d−1,1 ∈Ā 0 . Repeating the same argument, using (3.9) and finally (3.5), we obtainĀ m,n ∈Ā 0 for all m + n = d, if d = 3 or d = 5.

Proof of Theorem 1.3
Let f (u, v) = m,n> >−∞ a m,n u m v n be a solution to the Fay identity. By Proposition 3.1, we have a m,n u m v n , with a m,n = 0, if m + n ∈ 2Z.