An algebraic characterization of the Kronecker function

We characterize the generating series of extended period polynomials of normalized Hecke eigenforms for $\operatorname{PSL}_2(\mathbb Z)$ studied by Zagier 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' to the Fay identity.

The main object of this paper is the two-variable meromorphic function F τ defined by where θ τ is the classical odd Jacobi theta function. It first appeared in work of Kronecker, [12,Ch. VIII] and was rediscovered by Zagier, [13], who demonstrated its central role in the theory of modular forms for the full modular group Γ := PSL 2 (Z). The function F τ also naturally arises in the study of configuration spaces of points on a complex elliptic curve, especially in relation with the universal elliptic Knizhnik-Zamolodchikov-Bernard (KZB) connection, [2,3,9]. Following [2,11], we call F τ the Kronecker function.
One of the key properties of F τ is the Fay identity 1 , [2], which is the elliptic curve case of an identity for the theta functions of general compact Riemann surfaces, known as Fay's trisecant identity, [6, p.34, eqn. (45)]. A geometric interpretation of this identity was given by Mumford [10,Ch. IIIb.2]. As an application, the Fay identity gives quadratic relations in the de Rham cohomology of configuration spaces of points on a complex elliptic curve, [2], providing an elliptic analog of Arnold's relations in the case of configuration spaces of points on the complex plane, [1]. Related to this, the Fay identity also plays a key role in the proof of flatness of the universal elliptic KZB connection, [3, Proposition 1.2], [9, Proposition 3.2.2]. 2 The Fay identity also has a modular interpretation. Namely, it is equivalent to the period relations, equation (1.2), which are related to the group cohomology of Γ and were found in this form in [13].
1.2. Main results. The goal of this paper is to characterize the Kronecker function among all meromorphic functions using the Fay identity, equation (1.1). To put our result into context, we mention that there already exists at least one characterization of the Kronecker function. Namely, it is the unique meromorphic function f : C 2 → C which has simple poles for u or v in 2πi(Z + Zτ ) and simple zeros if u + v ∈ 2πi(Z + Zτ ), which is quasiperiodic, i.e. f (u+2πi(nτ +s), v+2πi(mτ +r)) = e −2πimnτ −mu−nv F τ (u, v), with m, n, r, s ∈ Z, and whose residue along u = 0 is equal to 1 (cf. [13,Theorem 3]). This characterization ultimately relies on a version of Liouville's theorem and is therefore of a global, complexanalytic nature.
The characterization of the Kronecker function we give in this paper is entirely different. Instead of a condition on all zeros and poles, we only require that in a small neighborhood of (0, 0) ∈ C 2 the function f has poles at most along the hyperplanes u = 0 and v = 0. The Weierstrass preparation theorem (see for example [7, p.8]) then implies that f has a Laurent expansion at (0, 0), In fact, as we will only work with the Laurent expansion of f , our characterization will more generally distinguish the Kronecker function among all formal Laurent series. Our main result is then the following theorem which in particular answers a question of Ecalle, Using the known expansion of [13,Theorem 3], where q = e 2πiτ , we can impose additional conditions on the coefficients to fix the parameters α = β = δ = 1 and γ = 0, thereby obtaining an algebraic characterization of the Kronecker function. Also, note that F τ (u, v)| τ =i∞ = (coth(u/2) + coth(v/2))/2 in which case the Fay identity follows from the well-known addition formula for the hyperbolic cotangent.
The idea of the proof of Theorem 1.1 is that the Fay identity puts recursive conditions on the Laurent coefficients a m,n . More precisely, if a −1,0 a 0,−1 = 0, then f is uniquely determined by its coefficients a 0,−1 and a m,0 for m = −1, 1, 3, 5 (see Theorem 3.6). A similar result holds if a −1,0 a 0,−1 = 0 (see Propositions 3.1 and 3.3). Conversely, one can verify directly that given any complex numbers a 0,−1 and a m,0 , for m = −1, 1, 3, 5, there exists a choice of parameters α, β, γ, δ and τ such that the Laurent coefficients of one of the functions in Theorem 1.1 match with the a m,n in the five cases given above, and this implies that they match for all m, n ∈ Z by Proposition 3.3 and Theorem 3.6.
Our second main result is an application of Theorem 1.1. Let Zagier, [13], proved that C is a generating series of the extended period polynomials of normalized Hecke eigenforms for Γ (divided by their Petersson norm). As a consequence C satisfies the period relations As observed in this paper, the period relations for C are equivalent to the Fay identity for F τ (see Proposition 2.5 and Remark 2.6). Combining this with Theorem 1.1, we obtain a characterization of C. More precisely, given f ∈ C((u, v)), define

Theorem 1.2.
Assume that C f (X, Y, T ) satisfies the period relations. Then either: Note that the exponential prefactor e γuv cancels in Theorem 1.2. Also, if αβ = 0, then As before, we could impose further conditions on the coefficients of C f (X, Y, T ), ensuring that α = β = δ = 1 and thereby characterizing the series C(X, Y ; τ ; T ) uniquely. However, the existence of a factorization (1.3) is essential; given any modular form f of weight k for Γ with extended period polynomial r f (X), the series C(X, Y ; τ ; T ) + r f (X)T k−2 satisfies equation (1.2) as well. In other words, the period relations alone do not suffice to characterize C(X, Y ; τ ; T ) among all formal Laurent series in C((X, Y, T )).

Content.
In Section 2, we recall the definition of the Kronecker function F τ (u, v) and of Zagier's generating series C(X, Y ; τ ; T ) as well as the functional equations they satisfy. In Section 3, we state results on the Laurent expansion of a general solution f (u, v) to the Fay identity. Our key results are Theorem 3.6 and Proposition 3.3 which are used in Section 4 to prove Theorems 1.1 and 1.2.
Acknowledgements: This work was done while the author was a JSPS postdoctoral fellow, partly supported by JSPS KAKENHI Grant No. 17F17020.
Notation: Unless otherwise stated, by τ we will always denote a fixed point in the upper half-plane H := {z ∈ C | Im(z) > 0} and q := e 2πiτ . Given variables X 1 , . . . , X n , we will denote by C((X 1 , . . . , X n )) the C-algebra of formal Laurent series of the form

The Kronecker function and its functional equations
We introduce the Kronecker function F τ (u, v) and the series C(X, Y ; τ ; T ) constructed from it, basically following [13]. These satisfy certain functional equations (antisymmetry and Fay identity, respectively the period relations) which turn out to be equivalent.

Definition 2.1. The Kronecker function is the meromorphic function
From the properties of θ τ mentioned above, it follows that F τ (u, v) has simple poles if u or v is in 2πi(Z + Zτ ), simple zeros for u + v ∈ 2πi(Z + Zτ ) and transforms as Remark 2.2. The above version of the Kronecker function is the one given in [13]. Other sources such as [2,11] work with the function 2πiF τ (2πiu, 2πiv) instead.  45)]). We give a proof for the convenience of the reader.

Proposition 2.3. The Kronecker function satisfies the Fay identity
Proof. 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 Here, we have also used that θ τ (−u) = −θ τ (u). Now substituting we can write (2.2) in the more symmetric form and this is precisely [13,Proposition 5].
Slightly more generally, we have the following result.
Proof. The main observation is that e γ(u 1 v 1 +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.

Comparison with the period relations.
In [13], Zagier considers the series and shows that it satisfies the period relations, equation (1.2). 4 Using the "slash operator" are the standard generators of SL 2 (Z) and | is extended linearly to the group ring of SL 2 (Z).
The period relations are closely related to the Fay identity. To see this, given a formal Laurent series f (u, v) ∈ C((u, v)), we define The proof of the following proposition is straightforward.
Proof. We first show that if f = 0, then f must have a pole at either u = 0 or v = 0.
Indeed, if f does not have a pole at v = 0, then the Fay identity implies The first equation implies f (u, 0) = a −1,0 u −1 , as can be seen by comparing coefficients. In particular, if furthermore 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 set 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 f (−u, −v) = −f (u, v). 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   Note that γ is well-defined in both cases. Indeed, since f = 0 we must have a −1,0 = 0 in the first case and a 0,−1 = 0 in the second one, by Proposition 3.1.
Proof. We only prove (i), the proof of (ii) is analogous. Since a 0,−1 = 0, Proposition 3.1 implies that f (u, v) does not have a pole at v = 0 and the Fay identity with v := v 1 = v 2 and u := u 1 = u 2 yields Writing out the Laurent expansion of f and using that f (u, 0) = a −1,0 u −1 (see the proof of Proposition 3.1), equation This implies a m,n = 0 by induction on n, if m = n − 1, and more generally 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 γuv 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 for some c m 1 ,m 2 ,n 1 ,n 2 ∈ A.
Theorem 3.6. The map ι 0 is an isomorphism of algebras.
More concretely, using the 1-1 correspondence f ↔ ϕ f described above, Theorem 3.6 says that every solution f (u, v) ∈ C ((u, v)) to the Fay identity with a −1,0 a 0,−1 = 0 is uniquely determined by its coefficients a 0,−1 and a m,0 , for m = −1, 1, 3, 5. 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.

Proof of the main results
Using the results established in the last section, we can now prove our main results.