A class of nonholomorphic modular forms I
 Francis Brown^{1}Email author
https://doi.org/10.1007/s4068701801308
© The Author(s) 2018
Received: 6 November 2017
Accepted: 18 December 2017
Published: 6 February 2018
Abstract
This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms and are modular analogues of singlevalued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms) and to the modular graph functions arising in genus one string perturbation theory.
 (1)
Holomorphic modular forms f with rational Fourier coefficients correspond to certain pure motives \(M_f\) over \(\mathbb {Q}\). Using iterated integrals, we can construct nonholomorphic modular forms which are associated with iterated extensions of the pure motives \(M_f\). Their coefficients are periods.
 (2)
In genus one closed string perturbation theory, one assigns a lattice sum to a graph [16], which defines a real analytic function on the upper half plane invariant under \(\mathrm {SL}_2(\mathbb {Z})\). It is an open problem to give a complete description of this class of functions and prove their conjectured properties.
This paper is based on a talk at a conference in honour of Don Zagier’s birthday, and connects with his work in several ways: through his work on modular graph functions [10], on singlevalued polylogarithms [33], on period polynomials [22], on periods [21], on multiple zeta values [15], on double Eisenstein series [19], and doubtless many others.
It is a great pleasure to dedicate it to him on his 65th birthday.
1 Modular graph functions
For motivation, we briefly recall the definition of modular graph functions.
Definition 1.1
The function \(I_G\) depends neither on the edge numbering, nor on the choice of orientation of G. It defines a function \(I_G \) on the upper half plane which is real analytic and invariant under the action of \(\mathrm {SL}_2(\mathbb {Z})\) (Fig. 1).
Examples 1.2
Consider the graph with 3 halfedges depicted on the left:
1.1 Properties
 (1)Zerbini [36] has shown that in all known examples, the ‘zeroth modes’ of modular graph functions involve a certain class of multiple zeta valueswhere \(n_1,\ldots , n_r \in \mathbb {N}\) and \(n_r \ge 2\), which are called ‘singlevalued’ multiple zeta values. The quantity r is called the depth. The ‘singlevalued’ subclass is generated in depth one by odd zeta values \(\zeta (2n+1)\) for \(n\ge 1\), in depth two by products \(\zeta (2m+1)\zeta (2n+1)\), but starting from depth three includes the following combination of triple zeta values$$\begin{aligned} \zeta (n_1,\ldots , n_r) = \sum _{1\le k_1< \cdots < k_r} {1 \over n_1^{k_1} \ldots n_r^{k_r}}, \end{aligned}$$$$\begin{aligned} \zeta _{\mathrm {sv}} (3,5, 3): = 2 \zeta (3,5,3)  2 \zeta (3) \zeta (3,5) 10 \zeta (3)^2 \zeta (5)\ . \end{aligned}$$
 (2)The \(I_{G}\) satisfy some mysterious inhomogeneous Laplace eigenvalue equations. A simple example of this is the Eq. [13] (1.4)where \(\Delta \) is the Laplace–Beltrami operator. The function \(C_{2,1,1}\) corresponds to the modular graph function of the graph with four edges and two vertices depicted above on the right. As an illustration of our methods, we shall solve this Laplace eigenvalue equation in Sect. 9.3 using a new family of functions constructed here and determine its kernel. Note that the operator \(\Delta \) in the physics literature has the opposite sign from the usual convention (2.21).$$\begin{aligned} (\Delta +2)\, C_{2,1,1}(z) = 16 \, \mathbb {L}^2 \, {\mathcal {E}}_{1,1}^2  \textstyle {2\over 5} \,\mathbb {L}^3 \, {\mathcal {E}}_{3,3}, \end{aligned}$$(1.1)
 (3)
Modular graph functions satisfy many relations [11], which suggests that they should lie in a finitedimensional space of modularinvariant functions.
 (4)
The zeroth modes of modular graph functions are homogeneous [10], Sect. 6.1, for a grading called the weight, in which rational numbers have weight 0, and multiple zeta values have weight \(n_1+\cdots +n_r\). The weight of \({\mathrm {Im}}(z)\) is zero.
1.2 Landscape
Open string  Closed string  

Genus 0  Multiple polylogs  Singlevalued polylogs 
Genus 1  Multiple elliptic polylogs  Equivariant iterated Eisenstein integrals 
The open genus zero amplitudes are integrals on the moduli spaces of curves of genus 0 with n marked points \({\mathfrak {M}}_{0,n}\). They involve multiple polylogarithms, whose values are multiple zeta values. The genus one string amplitudes are integrals on the moduli space \({\mathfrak {M}}_{1,n}\) and are expressible [5] in terms of multiple elliptic polylogarithms [3]. Viewed as a function of the modular parameter, the latter are given by certain products of iterated integrals of Eisenstein series. The passage from the open to the closed string involves a ‘singlevalued’ construction [31]. The closed superstring amplitudes in genus one are thus linear combinations of products of iterated of Eisenstein series and their complex conjugates which are modular. This is the definition of the space \(\mathcal {MI}^E\). A rigorous proof of the relation between closed superstring amplitudes and our class \(\mathcal {MI}^E\) might go along the broad lines of the author’s thesis, generalised to genus one using [3].
2 A class of functions \({\mathcal {M}}\)
2.1 First definitions
Definition 2.1
2.2 qExpansions and pole filtration
Lemma 2.2
Suppose that \(f: \mathfrak {H} \rightarrow \mathbb {C}\) satisfies Eq. (2.3) and admits an expansion in the ring \( \mathbb {C}[[q, \overline{q}]] [ \log q , \log \overline{q} ]\). Then, \(f \in \mathbb {C}[[q, \overline{q}]] [ \mathbb {L}].\)
Proof
Example 2.3
2.3 Differential operators (Maass)
Definition 2.4
Lemma 2.5
Proof
Direct computation. \(\square \)
See Sect. 7 for another interpretation of \(\partial _r, \overline{\partial }_s\) in terms of sections of vector bundles.
Lemma 2.6
Proof
The first part follows immediately from the formulae (2.15), which are easily derived from the definitions. The second line follows by complex conjugation. \(\square \)
Corollary 2.7
Definition 2.8
2.4 Action of \(\mathfrak {sl}_2\)
Proposition 2.9
Proof
Straightforward computation. \(\square \)
2.5 Almost holomorphic modular forms
The subspace \({\widetilde{M}}[\mathbb {L}^{\pm }]\) of almost holomorphic modular forms inherits an \(\mathfrak {sl}_2\) module structure which is not to be confused with another \(\mathfrak {sl}_2\) module structure [34] Sect. 5.3, which involves multiplication by \(\mathbb {G} _2\). For the convenience of the reader, we describe the differential structure here.
For example, the Hecke normalised cusp form \(\Delta \) of weight 12 satisfies \(\vartheta (\Delta ) = 0\). It follows that \( \partial (\Delta ) = 12 \mathfrak {m} \Delta \), which gives another interpretation of \(\mathfrak {m} \).
2.6 Bigraded Laplace operator
By taking polynomials in \(\mathbb {L}, \partial \) and \(\overline{\partial }\) one can define any number of operators acting on the space \({\mathcal {M}}\). Examples include the Laplace operator, Rankin–Cohen brackets Sect. 6, and the Bol operator (see [4]).
Definition 2.10
Corollary 2.11
Let \(\Delta : {\mathcal {M}} \rightarrow {\mathcal {M}}\) denote the linear operator which acts by \(\Delta _{r,s}\) on \({\mathcal {M}}_{r,s}\). Let \(\mathsf {w}: {\mathcal {M}} \rightarrow {\mathcal {M}}\) be the linear map which acts by multiplication by \(w=r+s\) on \({\mathcal {M}}_{r,s}\).
Lemma 2.12
Proof
2.7 Real analytic Petersson inner product
Definition 2.13
Two spaces \({\mathcal {M}}_{r,s}\) and \({\mathcal {S}}_{r',s'}\) can be paired via (2.24) if and only if \(rs= r's'\). Equivalently, \(\langle f,g\rangle \) exists whenever \(h(f) = h(g)\), where h was defined in (2.6).
2.8 Holomorphic projections [32]
2.9 A picture of \({\mathcal {M}}\)
The bigraded algebra \({\mathcal {M}}\) can be depicted as follows.
The dashed arrows represent the action of \(\mathbb {L}, \mathbb {L}^{1}, \partial , \overline{\partial }\). Each solid circle represents a copy of \({\mathcal {M}}_{r,s}\) for \(r+s\) even. Some examples of modular forms are indicated in red.
3 Primitives and obstructions
3.1 Constants
Lemma 3.1
Proof
Since \(\partial _r \mathbb {L}^k f = \mathbb {L}^k \partial _{r+k} f\) (2.13), we can assume, by multiplying by \(\mathbb {L}^{r}\), that \(r=0\). The kernel of \(\partial _0 = (z\overline{z}){ \partial \over \partial z} \) consists of antiholomorphic functions. The second formula in (3.2) is the complex conjugate of the first. \(\square \)
We now consider the kernel of the operator \(\partial \) acting on the space \({\mathcal {M}}\).
Proposition 3.2
Proof
By Lemma 3.1, we can write \( \mathbb {L} ^r F = \overline{g}\) where \( g: \mathfrak {H} \rightarrow \mathbb {C}\) is a holomorphic function. Since f (respectively \(\mathbb {L}^r\)) has weights (r, s) (respectively \((r,r)\)), it follows that \( \overline{g}\) has weights \((0,sr)\) and transforms like a modular form of weight \(sr\), i.e. \(g( \gamma (z)) = (cz +d )^{sr} g(z) \) for all \(\gamma \in \mathrm {SL}_2(\mathbb {Z})\) of the form (2.1). Thus, \(g \in M_{sr}\). For the last part, use the wellknown fact that there are no nonzero holomorphic modular forms of negative weight. \(\square \)
Thus, if Eq. (3.1) has a solution, it is unique up to addition by an element of \(\mathbb {C}\mathbb {L} ^{r}\) if \(h(F)=0\), and is unique if \(h(F)>0\).
Corollary 3.3
Let \(F \in {\mathcal {M}}_{r,s}\) and let \(f= \partial F \). There is a solution \(F' \in {\mathcal {M}}_{r,s}\) to (3.1) whose antiholomorphic projection \(p^a(F')\) vanishes. It is unique up to addition by a multiple of \( \mathbb {L}^{r} \overline{\mathbb {G} }_{sr}\) for \(sr\ge 4\), where \(\mathbb {G} _{n}\) is the Eisenstein series (2.8).
Proof
Since the Petersson inner product is nondegenerate, there exists a unique cusp form \(g\in S_{sr}\) such that \(p^{a} (\overline{g}) = p^{a}(F)\). Then, \(F' = F (2\pi )^r \mathbb {L}^{r} \overline{g}\) has the required properties. The second part follows since the orthogonal complement of \(S_{sr}\) in \(M_{sr}\) is exactly the vector space generated by the Eisenstein series. \(\square \)
3.2 Combinatorial obstructions
The maps \(\partial _r, \overline{\partial }_s\) are far from surjective.
Lemma 3.4
Proof
Follows immediately from Lemma 2.6. \(\square \)
This is not the only constraint: for every \(m,n \ge 0\), there is a condition on the \(a^{(k)}_{m,n}\), for varying k, in order for f to lie in the image of the map \(\partial _r\). Nonetheless, (3.3) is already sufficient to rule out the existence of primitives in many interesting cases.
Corollary 3.5
There exists no element \(F \in \mathbb {C}[[q, \overline{q}]][\mathbb {L} ^{\pm }]\) satisfying \(\partial _0 F = \mathbb {L}\mathbb {G} ^*_2\).
3.3 A condition involving the pole filtration
Lemma 3.6
Proof
The commutation relation \(\mathsf {h}= [\partial , \overline{\partial }]\) implies that \( h\, f + \overline{\partial } \partial f\) is in the image of \(\partial \) for all \(f \in {\mathcal {M}}\). This remark, combined with (3.4), enables one to prove the existence of combinatorial primitives in many cases of interest.
3.4 Obstructions from the Petersson inner product
Another obstruction comes from the fact that a formal power series solution to (3.1) is not necessarily modular.
Theorem 3.7
Proof
Corollary 3.8
For every nonzero cusp form \(f \in S_{n}\), and every \(k\in \mathbb {Z}\), the equation \(\partial F = \mathbb {L}^k f\) has no solution in \({\mathcal {M}}\).
Proof
By (2.13), we can assume that \(k=0\). If F were to exist, the previous theorem with \(g=f\) would imply that \(0 = \langle f, f \rangle \). But this contradicts the fact that the Petersson inner product is positive definite. \(\square \)
Primitives of cusp forms do exist if one allows poles at the cusp (Sect. 11 and [4]).
3.5 Arithmetic obstructions
Although this is largely irrelevant here, since we work mostly over the complex numbers, the equation \(\partial F = f\) involves some subtle questions regarding the field of definition of the coefficients \(a^{(k)}_{m,n}\). Fundamentally, complex conjugation is not rationally defined on algebraic de Rham cohomology.
For example, \(\partial F = \mathbb {G} _{4} \mathbb {L} \) has a unique solution given by a real analytic Eisenstein series \({\mathcal {E}}_{2, 0} \in {\mathcal {M}}_{2,0}\), to be defined in Sect. 4, but it has no solution with rational coefficients. This is because \({\mathcal {E}}_{2,0}\) involves the value of the Riemann zeta function \(\zeta (3)\), which is irrational as shown by Apéry. The examples of functions in \({\mathcal {M}}\) constructed in this paper arise from iterated integrals of modular forms, and their coefficients \(a^{(k)}_{m,n}\) are, in a certain sense, periods. The period conjecture suggests that they are transcendental.
3.6 A class of modular iterated primitives
The functions studied in this paper lie in a special subclass of functions inside \({\mathcal {M}}\).
Definition 3.9
Lemma 3.10
\(\mathcal {MI}_0 = \mathbb {C}[\mathbb {L}^{1}]\).
Proof
Since \(\mathcal {MI}_{1}=0\), any \(F\in \mathcal {MI}_{0}\) of weights (n, 0) satisfies \(\partial F=0\) by (3.8). If \(n>0\), then F vanishes by Proposition 3.2. Continuing in this manner, we see that any \(F\in \mathcal {MI}_{0}\) of weights (r, s) for \(r>s\) must also vanish, and in the case \(r=s\), it must be of the form \(F \in \mathbb {C}\mathbb {L}^{r}\). Therefore, \(\mathcal {MI}_0 \subset \mathbb {C}[\mathbb {L}^{1}]\). Since \(\partial \mathbb {L}= \overline{\partial } \mathbb {L}=0\), the ring \(\mathbb {C}[\mathbb {L}^{1}]\) indeed satisfies the conditions (3.7) and hence \(\mathcal {MI}_0 = \mathbb {C}[\mathbb {L}^{1}]\). \(\square \)
Remark 3.11
 (1)
Replace M with S, the space of cusp forms. Since cusp forms do not admit modular primitives, one deduces by induction that \(\mathcal {MI}(S) = \mathbb {C}[\mathbb {L}^{1}]\).
 (2)Replace M withthe \(\mathbb {Q}\)vector space generated by Eisenstein series. We obtain a space$$\begin{aligned} E = \bigoplus _{n\ge 2} \mathbb {G} _{2n} \mathbb {Q}\end{aligned}$$(3.9)In the sequel to this paper, we construct a subspace \(\mathcal {MI}^E\otimes \mathbb {C}\subset \mathcal {MI}(E)\) (Sect. 10) and hope that equality holds, which would have deep consequences. We shall show below that \(\mathcal {MI}(E)_k = \mathcal {MI}_k\) for \(k=0,1\) but not for \(k=2\).$$\begin{aligned} \mathcal {MI}(E) \subset \mathcal {MI}. \end{aligned}$$
The class of functions \(\mathcal {MI}\) has an interesting \(\mathfrak {sl}_2\)module structure which could profitably be reformulated in the language of [6].
3.7 Homological interpretations
Lemma 3.12
Proof
4 Real analytic Eisenstein series
We consider in some detail the simplest possible family of nonholomorphic functions in \({\mathcal {M}}\) as a concrete illustration.
4.1 Modular primitives of Eisenstein series
Proposition 4.1
Proof
We immediately deduce the following properties:
Corollary 4.2
Proof
The compatibility with complex conjugation follows by symmetry of (4.1) and (4.2) and uniqueness. The Laplace equation follows from (2.20), (4.1) and (4.2). The last equation follows from Theorem 3.7 since \({\mathcal {E}}_{r,s}\) is in the image of either \(\partial \) or \(\overline{\partial }\). \(\square \)
Proposition 4.3
Proof
The statement is well known for \(r=s=w\), since it reduces to the Fourier expansion of the real analytic Eisenstein series \(E(z,w+1)\). The remaining cases are deduced by applying \(\partial \) via (4.1) and by \({\mathcal {E}}_{r,s} = \overline{{\mathcal {E}}}_{s,r}\). An alternative way to prove this theorem is to use the expression for \({\mathcal {E}}_{r,s}\) as the real part of the single iterated integral of holomorphic Eisenstein series [1] §8, and use the computation of the cocycle of the latter [1], Lemma 7.1, to write down the constant terms directly. See Sect. 8.4.2. \(\square \)
4.2 Explicit formulae
4.3 Description of \(\mathcal {MI}_1\)
We already showed that \(\mathcal {MI}_0 = \mathbb {C}[\mathbb {L}^{1}]\).
Corollary 4.4
Proof
Let \(F\in \mathcal {MI}_{1}\) of weights (n, 0). By (3.8), it satisfies \(\partial F \in M \mathbb {L}\). Since \(\partial F\) is orthogonal to cusp forms by Theorem 3.7, it must satisfy \(\partial F \in \mathbb {C}\mathbb {G} _{n+2} \mathbb {L}\). This equation has the unique family of solutions \(F \in \mathbb {C}{\mathcal {E}}_{n,0}\). By eq. (3.7), the elements \(F\in \mathcal {MI}_{1}\) of weights (r, s) with \(r>s\) are iterated primitives of real analytic Eisenstein series and modular forms \(M[\mathbb {L}]\), and hence also real analytic Eisenstein series, by a similar argument. We conclude that \(\mathcal {MI}_1\) is contained in the \(\mathbb {C}[\mathbb {L}^{1}]\)module generated by the \({\mathcal {E}}_{r,s}\). Since the latter satisfy (3.7), this proves equality. \(\square \)
4.4 Picture of the real analytic Eisenstein series
Based on the previous picture of \({\mathcal {M}}\), the real analytic Eisenstein series can be viewed as follows:
The dashed arrows going up and down the antidiagonals are \(\partial \) and \(\overline{\partial }\). The classical real analytic Eisenstein series are the functions \({\mathcal {E}}_{n,n}\) lying along the diagonal \(r=s\).
5 Eigenfunctions of the Laplacian
This section is not needed for the rest of the paper. We show that the space \({\mathcal {M}}\) has very limited overlap with the theory of Maass waveforms [23], and determine to what extent the solutions to a Laplace eigenvalue equation are not unique.
Call \(F\in {\mathcal {M}}\) an eigenfunction of \(\Delta \) if there exists \(\lambda \in \mathbb {C}\), the eigenvalue, such that \(\Delta F = \lambda F\). It decomposes into a sum of terms \(F_{r,s} \in {\mathcal {M}}_{r,s}\) satisfying \(\Delta _{r,s} F = \lambda F\).
Theorem 5.1
Let F be an eigenfunction of the Laplacian. Then, its eigenvalue is an integer, and F is a linear combination over \(\mathbb {C}[\mathbb {L}^{\pm }]\) of real analytic Eisenstein series \({\mathcal {E}}_{r,s}\), almost holomorphic modular forms and their complex conjugates.
Let us write \({\mathcal {HM}} \subset {\mathcal {M}}\) to denote the space of Laplace eigenfunctions. It follows from Lemma 2.12 that it is stable under the action of \(\mathcal {O}= \mathbb {Q}[\mathbb {L}^{\pm }][\partial , \overline{\partial }]\). Furthermore, the subspace \({\mathcal {HM}}(n)\) of eigenfunctions with eigenvalue n is stable under the action of the Lie algebra \(\mathfrak {sl}_2\) generated by \(\partial , \overline{\partial }\).
Every holomorphic modular form \(f\in M_{n}\) lies in \({\mathcal {HM}}(0)\) since \(\Delta f =  \partial _{n1} \overline{\partial }_0 f = 0\). The same is true of \(\mathfrak {m} \) defined in Sect. 2.5. More generally, \(\mathbb {L}^k f\) is an eigenfunction with eigenvalue \((nk1)k\). Since the ring of almost holomorphic modular forms is generated by holomorphic modular forms and \(\mathfrak {m} \) by the action of \(\partial \), it follows that any almost holomorphic (or antiholomorphic) modular form lies in \({\mathcal {HM}}\).
5.1 Proof of Theorem 5.1
Lemma 5.2
Proof
Assume that F is nonzero and denote the coefficients in its expansion (2.5) by \(a_{m,n}^{(k)}\). We first show that \(a^{(k)}_{m,n}=0\) if \(mn\ne 0\). Fix m, n such that \(a^{(k)}_{m,n}\ne 0 \) for some k. Choose k maximal with this property. Taking the coefficient of \(\mathbb {L}^{k+2} q^m \overline{q}^n\) in the equation \(\Delta _{r,s} F = \lambda F\) implies, via (2.22), that \( \lambda a^{(k+2)}_{m,n} =  4mn\, a^{(k)}_{m,n} \), which implies that \(mn =0\). Therefore, all \(a^{(k)}_{m,n}\) vanish for \(mn\ne 0\). Now, for any m, n, choose k minimal such that \(a^{(k)}_{m,n}\) is nonzero. Equation (2.22) implies that \(\lambda a^{(k)}_{m,n} =  k (k + w 1) a^{(k)}_{m,n}\), which proves the first part of the lemma. The equation \(x^2+ x(w1) + \lambda =0\) has two integral solutions \(k_0\) and \(1w{k_0}\), which are distinct since w is even. The assumption that \(k_0\) is the smaller of the two implies that \(a^{(k)}_{m,n}\) vanishes for all \(k< k_0\).
Now consider a nonzero coefficient of the form \(a^{(k)}_{m,0}\) with \(m\ne 0\). Let k be maximal. Equation (2.22) implies that \(\lambda a^{(k+1)}_{m,0} = 2 m(k+s) a^{(k)}_{m,n}  k (k+w1) a^{(k+1)}_{m,0}\), which implies that \(m(k+s)=0\) since \(a^{(k+1)}_{m,0} =0\). Therefore, \(k=s\). A similar computation with terms of the form \(a^{(k)}_{0,n}\) shows that they all vanish if \(k> r\). It remains to determine the constant terms \(a^{(k)}_{0,0}\). Equation (2.22) implies that \(\lambda a^{(k)}_{0,0} =  k (k+w1) a^{(k)}_{0,0}\), so by the above \(a^{(k)}_{0,0}\) is nonzero only for \(k\in \{ k_0, 1w{k_0}\} \). \(\square \)
Lemma 5.3
Let \(F\in {\mathcal {M}}_{r,s}\) be an eigenfunction of the Laplacian. Then, there exist integers \(M, N\ge 0 \) such that \(\overline{\partial }^M \partial ^N F \in \mathbb {C}[\mathbb {L}^{\pm }]\).
Proof
Apply \(\partial _r\) to the expansion (5.1). By Lemma 3.1, this annihilates the term \(\mathbb {L}^{k} g_{k}(\overline{q})\) for \(k=r\). The terms of the form \(\mathbb {L}^{k} g_{k}(\overline{q})\) are simply multiplied by \(k+r\). Its action on terms of the form \(\mathbb {L}^k f_k(q)\) increases the degree in \(\mathbb {L}\) by at most one, by (2.15). Therefore, \(\partial _r F \) has a similar expansion to (5.1), with (r, s) replaced by \((r+1, s1)\). Applying \(\partial _{r1}\) kills the term \(\mathbb {L}^{k} g_{k}(\overline{q})\) for with \(k=1r\). Proceeding in this manner, every term of the form \(\mathbb {L}^{k} g_{k}(\overline{q})\) is eventually annihilated (this also follows directly from Lemma 5.2 since \(\partial ^m F\) are eigenfunctions of the Laplacian with the same eigenvalue \(\lambda \) as F). Now, by a similar argument, repeated application of \(\overline{\partial }\) annihilates all the terms of the form \(\mathbb {L}^k f_k(q)\). \(\square \)
Lemma 5.4
The maps \(\overline{\partial }: {\widetilde{M}} \rightarrow {\widetilde{M}}\) and \(\partial : \widetilde{\overline{M}} \rightarrow \widetilde{\overline{M}}\) are surjective.
Proof
Since \(\overline{\partial } \,\mathfrak {m} =1\), any element \( f \mathfrak {m} ^i\), where \(i\ge 0\) and \(f\in M[\mathbb {L}^{\pm }]\), is the \(\overline{\partial }\)image of \((i+1)^{1} f \mathfrak {m} ^{i+1}\). The second statement follows by complex conjugation. \(\square \)
Lemma 5.5
Proof
Corollary 5.6
Let \(V \subset {\mathcal {M}}\) denote the \(\mathbb {C}[\mathbb {L}^{\pm }]\)module generated by the real analytic Eisenstein series \({\mathcal {E}}_{r,s}\), \({\widetilde{M}}\) and \(\widetilde{\overline{M}}\). If \(F \in {\mathcal {M}}\) satisfies \(\partial F \in V\), then \(F \in V\). By complex conjugation, the same statement holds with \(\partial \) replaced with \(\overline{\partial }\).
Proof
An eigenfunction of the Laplacian F satisfies \(\overline{\partial }^M \partial ^N F \in \mathbb {C}[\mathbb {L}^{\pm }] \subset V\). It follows from the previous corollary and induction on N that \(F \in V\). This completes the proof.
Remark 5.7
In passing, we have shown that the ring of almost holomorphic modular forms \(M[\mathfrak {m} , \mathbb {L} ^{\pm }]\) is the subspace of functions \(f\in {\mathcal {M}}\) such that \(a_{m,n}^{(k)}(f)=0\) for all \(n>0\), or equivalently, which satisfy \(\overline{\partial }^N f=0\) for sufficiently large N.
6 Mixed Rankin–Cohen brackets
Example 6.1
The properties of the brackets (6.1) are wellknown. For instance, the bracket is antisymmetric and satisfies the Jacobi identity [34] §5.2.
Example 6.2
Interesting operators of order two in the ring \(\mathcal {O}\otimes \mathcal {O}\) therefore include: the Laplace operators \(\Delta \otimes \mathrm {id} \) and \(\mathrm {id} \otimes \Delta \), the Rankin–Cohen bracket \([f,g]_2\) and its conjugate, and a symmetric product \((f,g)_2\). All this is part of the general study of differential operators on \({\mathcal {M}}\), which we shall not pursue any further here.
7 Modular forms and equivariant sections
In this section, all tensor products are over \(\mathbb {Q}\).
7.1 Reminders on representations of \(\mathrm {SL}_2\)
7.2 A characterisation of functions in \({\mathcal {M}}\)
See also [35], Proposition 2.1.
Proposition 7.1
Proof
We construct equivariant functions f from iterated integrals. These only involve nonnegative powers of \(\log q\). Their coefficients \(f_{ij}\) will have poles in \(\mathbb {L}\) of degree at most the total weight, and their modular weights will naturally be located in the first quadrant \(r,s\ge 0\).
7.3 Vectorvalued differential equations
The operators \(\partial , \overline{\partial }\) of Definition 2.4 admit the following interpretation.
Proposition 7.2
Proof
Lemma 7.3
Proof
7.4 Example: real analytic Eisenstein series
8 Modular forms from equivariant iterated integrals
The main idea behind our construction of functions in \({\mathcal {M}}\) is a modification of the theory of singlevalued periods as we presently explain in some simple examples.
8.1 Singlevalued functions
8.2 Notation
8.3 The modular function \({\mathrm {Im}}(z)\)
8.4 Primitives of holomorphic modular forms
Now, we construct, or fail to construct, equivariant versions of classical Eichler integrals in the same vein.
8.4.1 Cusp forms
8.4.2 Eisenstein series
Remark 8.1

The period polynomial of the Eisenstein series is equivalent to formulae which must have been known to Ramanujan and are given by [1], §9:However, I could not find this precise formulation elsewhere. The literature tends to focus on period polynomials (value of a cocycle on S) which only determine the cocycle in the cuspidal case. Zagier’s approach is to introduce poles in X, Y to force the Eisenstein cocycle to be cuspidal.$$\begin{aligned} e_{2k}^0(S)= & {} {(2k2)! \over 2} \, \sum _{i=1}^{k1} {B_{2i} \over (2i)!}{B_{2k2i} \over (2k2i)!} X^{2i1} Y^{2k2i1} \ , \\ e_{2k}^0(T)= & {} {(2k2)! \over 2} {B_{2k} \over (2k)!} \Big ( { (X+Y)^{2k1}  X^{2k1} \over Y}\Big )\ . \end{aligned}$$

It is often stated that \(X^{2n}  Y^{2n}\) is the period polynomial of an Eisenstein series, but is in fact the value of the cuspidal coboundary cocycle at S and vanishes in cohomology. It is, however, nonzero in relative cohomology and is dual to the Eisenstein cocycle under the Petersson inner product (which pairs cocycles and compactly supported cocycles). This is discussed in [1] §9.

The ‘extra’ relation satisfied by period polynomials of cusp forms [21] expresses the orthogonality of the cocycle of a cusp form to the Eisenstein cocycle with respect to the HaberlandPetersson inner product.
9 Equivariant double iterated integrals
We now define equivariant versions of double Eisenstein integrals, which are modular analogues of the Bloch–Wigner function D(z).
9.1 Double Eisenstein integrals
Lemma 9.1
Proof
Definition 9.2
9.2 Equivariant versions of double Eisenstein integrals
Theorem 9.3
Proof
Remark 9.4
The antisymmetrization of \(K^{(k)}_{2a,2b}\) is related to the function \(I^{(k)}_{2a,2b}\) defined in [1], and its holomorphic projection is related to the double Eisenstein series of [9].
The Eqs. (9.5) and (9.7) uniquely determine \(F_{r,s}\) when \(r+s>0\), and determine it up to a constant when \(r=s=0\). We can show that the functions \(F_{r,s}\) are linearly independent for distinct values of a, b and k.
9.3 Example
Remark 9.5
The coefficients in the expansion (2.5) of these functions are easily determined from the formulae for the action (2.15) of \(\partial , \overline{\partial }\) and the above differential equations, up to the sole exception of a constant term \(\alpha \mathbb {L} ^{w}\). When \(w>0\), it is uniquely determined by modularity. If \(w=0\), we can assume this coefficient is zero.
9.4 Lfunctions and constant terms
All expansion coefficients (2.5) of an element \(f \in \mathcal {MI}_k\) are uniquely determined by those of functions in \(\mathcal {MI}_{k1}\) of lower length by the defining Eq. (3.7) and Lemma 2.6 except for a single constant term of the form \(\alpha \mathbb {L} ^{w}\), where \(\alpha \) is typically transcendental. This missing constant (when \(w>0\)) can be determined from the others by analytic continuation using an Lfunction [27].
Definition 9.6
Theorem 9.7
9.5 Orthogonality conditions
We now wish to consider the problem of finding linear combinations of equivariant iterated integrals which only involve Eisenstein series, i.e. in which all integrals of cusp forms cancel out. This is equivalent to finding linear combinations of the \(M_{2a,2b}^{(k)}\) which are orthogonal to all cusp forms under the Petersson inner product. Since this problem is discussed in [1], §22 in an essentially equivalent form, we illustrate with a simple example.
Example 9.8
Viewed in this manner, it might seem hopeless to find iterated integrals of Eisenstein series of higher lengths which are equivariant. Already in length three, the Rankin–Selberg method can no longer be applied in any obvious manner to find the necessary linear combinations of triple Eisenstein integrals. Fortunately, using the theory of the motivic fundamental group of the Tate curve, we can find an infinite class and, conjecturally all, solutions to this problem. This is summarised below.
10 A space of equivariant Eisenstein integrals
Recall that E is the graded \(\mathbb {Q}\) vector space generated by Eisenstein series (3.9). Let \(\mathcal {Z}^{\mathrm {sv}}\) denote the ring of singlevalued multiple zeta values.
Theorem 10.1
 (1)
It is the \(\mathcal {Z}^{\mathrm {sv}}\)vector space generated by certain (computable) linear combinations of real and imaginary parts of regularised iterated integrals of Eisenstein series.
 (2)
The space \(\mathcal {MI}^E[\mathbb {L}^{\pm }]\) is stable under multiplication and complex conjugation.
 (3)
It carries an even filtration (conjecturally a grading) by Mdegree, where \(\mathbb {L}\) has Mfiltration 2, and the \({\mathcal {E}}_{r,s}\) have Mfiltration 2. It is also filtered by the length (number of iterated integrals), which we denote by \(\mathcal {MI}^E_k \subset \mathcal {MI}^E\).
 (4)
The subspace of elements of \(\mathcal {MI}^E\) of total modular weight w and Mfiltration \(\le m\) is finitedimensional for every m, w.
 (5)Every element of \(\mathcal {MI}^E\) admits an expansion in the ringi.e. its coefficients are singlevalued multiple zeta values. An element of total modular weight w has poles in \(\mathbb {L}\) of order at most w. An element of Mfiltration 2m has terms in \(\mathbb {L}^k\) for \(k \le m\).$$\begin{aligned} \mathcal {Z}^{\mathrm {sv}} [[q, \overline{q}]][\mathbb {L}^{\pm }]\ , \end{aligned}$$
 (6)The space \(\mathcal {MI}^E\) has the following differential structure:where E is (3.9). The operators \(\partial , \overline{\partial }\) respect the Mfiltration, i.e. \(\deg _M \partial = \deg _M \overline{\partial }=0\), where the generators \(\mathbb {G} _{2n+2}\) of E are placed in Mdegree 0.$$\begin{aligned} \partial \big ( \mathcal {MI}_k^E\big )\subset & {} \mathcal {MI}^E_{k} + E[\mathbb {L}] \times \mathcal {MI}_{k1}^E \\ \overline{\partial } \big ( \mathcal {MI}_k^E \big )\subset & {} \mathcal {MI}^E_{k} + \overline{E}[\mathbb {L}] \times \mathcal {MI}_{k1}^E \ \end{aligned}$$
 (7)Every element \(F \in \mathcal {MI}_k^E\) of total modular weight w satisfies an inhomogeneous Laplace equation of the form:$$\begin{aligned} (\Delta + w )\, F \in (E+ \overline{E})[\mathbb {L}] \times \mathcal {MI}^E_{k1} + E \overline{E} [\mathbb {L}] \times \mathcal {MI}^E_{k2} \ . \end{aligned}$$
Remark 10.2
A more precise statement about the Laplace equation (7) can be derived from the differential equations (6). In fact, the differential equations with respect to \(\partial , \overline{\partial }\) are the more fundamental structure. This simplicity is obscured when looking only at the Laplace operator. Recently, a generalisation of modular graph functions called modular graph forms were introduced in [11]. These define functions in \({\mathcal {M}}\) of more general modular weights (r, s), and, up to scaling by \(\mathbb {L}^{\pm }\), are closed under the action of \(\partial , \overline{\partial }\). It suggests that one should try to find systems of differential equations, with respect to \(\partial , \overline{\partial }\), satisfied by modular graph forms using partial fraction identities (see [11], (2.30)), and match their solutions with elements in \(\mathcal {MI}^E[\mathbb {L}^{\pm }]\).
This theorem and further properties of \(\mathcal {MI}^E\) will be proved in the sequel.
11 Meromorphic primitives of cusp forms
We revisit the problem of finding primitives of cusp forms. If we allow poles at the cusp, then we can indeed construct modular equivariant versions of cusp forms [4].
11.1 Weakly analytic variant of \({\mathcal {M}}\)
Definition 11.1
We now give some examples of elements in \(\mathcal {MI}^!_k\) for \(k\le 2\).
11.2 Primitives of cusp forms
The following theorem is proved in [4].
Theorem 11.2
Theorem 11.3
\(\mathcal {MI}^!_1\) is the free \(\mathbb {C}[\mathbb {L} ^{1}]\)module generated by the \({\mathcal {H}}(f)_{r,s}\) .
11.3 New elements in \({\mathcal {M}}_{r,s}\)
11.4 Double integrals
As above, by multiplying by sufficiently large powers of \(\Delta (z)\overline{\Delta }(z)\), we can clear poles in the denominators to obtain yet more elements in \({\mathcal {M}}\), and so on.
Declarations
Acknowledgements
Many thanks to Michael Green, Eric d’Hoker, Pierre Vanhove, Don Zagier and Federico Zerbini for explaining properties of modular graph functions. Many thanks also to Martin Raum for pointing out connections with the literature on mock modular forms. This work was partially supported by ERC grant 724638. Our original construction [1], §19 is technical and obscures a very elementary underlying theory which we wished to describe here. It diverges from the classical theory of modular forms which emphasises eigenfunctions of the Laplacian, which our functions are not. For these reasons, I strived to make the exposition in this paper as accessible and elementary as possible (perhaps overly so). As a result, there is considerable overlap with some classical constructions and wellknown results in the theory of modular forms. I apologise in advance if I have failed to provide attributions in every case.
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