# A class of non-holomorphic modular forms I

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## 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 single-valued 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.

*r*,

*s*. They

*do not*satisfy a simple condition involving the Laplacian. The

*raison d’être*for this class of functions is two-fold:

- (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 non-holomorphic 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 single-valued 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

*G*be a connected graph with no self-edges. It is permitted to have a number of half-edges. Denote its set of vertices by \(V_G\) and number its edges (including the half-edges) \(1,\ldots , r\). Choose an orientation of

*G*. The associated modular graph function is defined, when it converges, by the sum [10] (3.12):

*z*is a variable in the upper half plane \(\mathfrak {H} \), the prime over a summation symbol denotes a sum over \((m,n) \in \mathbb {Z}^2 \backslash (0,0)\), and for every vertex \(v \in V_G\),

*i*is not incident to the vertex

*v*, \(+1\) if

*i*is oriented towards the vertex

*v*, and \(-1\) if it is oriented away from

*v*.

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 half-edges 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 ‘single-valued’ multiple zeta values. The quantity$$\begin{aligned} \zeta (n_1,\ldots , n_r) = \sum _{1\le k_1< \cdots < k_r} {1 \over n_1^{k_1} \ldots n_r^{k_r}}, \end{aligned}$$
*r*is called the depth. The ‘single-valued’ 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$$\begin{aligned} \zeta _{\mathrm {sv}} (3,5, 3): = 2 \zeta (3,5,3) - 2 \zeta (3) \zeta (3,5) -10 \zeta (3)^2 \zeta (5)\ . \end{aligned}$$ - (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 finite-dimensional space of modular-invariant 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 | Single-valued 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 ‘single-valued’ 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 \({\mathcal {M}}\)

*z*will denote a variable in the upper half plane

*q*-disc. The latter normalisation (i.e. 2\(\mathbb {L}\) rather than \(\mathbb {L}\)), simplifies some formulae and may be preferred.

### 2.1 First definitions

## Definition 2.1

*modular of weights*(

*r*,

*s*), where \(r,s\in \mathbb {Z}\), if for all \(\gamma \in \mathrm {SL}_2(\mathbb {Z})\) of the form (2.1) it satisfies

*f*vanishes [put \(\gamma = - \mathrm {id} \) in (2.3)]. Let \({\mathcal {M}}_{r,s}\) denote the space of real analytic functions of modular weights (

*r*,

*s*) which admit an expansion of the form

*R*. Define a bigraded vector space

*w*the total weight, and let

*w*,

*h*be even.

### 2.2 *q*-Expansions 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

*q*and \(\overline{q}\) are invariant under translations \(z\mapsto z+1\), it suffices to show that

*T*denotes analytic continuation of

*q*around a loop around 0 in the punctured

*q*-disc. We have \(T \log q = \log q + 2 i \pi \) and \(T \log \overline{q} = \log \overline{q} - 2 i \pi \). It is a simple exercise in invariant theory to show that every

*T*-invariant polynomial in \(\log q\) and \(\log \overline{q}\) is a polynomial in \(2 \log |q| = \log q + \log \overline{q}\). \(\square \)

*q*-expansion of the form (2.5) for some

*N*. This expansion is unique. Define the

*constant part*of

*f*to be

## Example 2.3

*M*is the ring of holomorphic modular forms, is called the ring of almost holomorphic modular forms. By the previous example, it is contained in \({\mathcal {M}}\).

### 2.3 Differential operators (Maass)

## Definition 2.4

*r*,

*s*and \(f, g: \mathfrak {H} \rightarrow \mathbb {C}\), and in addition the formula

*r*,

*k*. Both formulae (2.12) and (2.13) remain true on replacing \(\partial \) by \(\overline{\partial }\) and are verified by straightforward computation. Finally, one checks that

## 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

*k*,

*m*,

*n*by

## 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

*s*, and similarly, \(\overline{\partial }\) acts on \({\mathcal {M}}_{r,s}\) via \(\overline{\partial }_s\) for any

*r*.

*k*and all \(f\in {\mathcal {M}}\), and similarly for \(\overline{\partial }\). This is equivalent to (2.13). We can rewrite the previous equation in the form

### 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.

*M*is generated by \(\mathbb {G} _4\) and \(\mathbb {G} _6\), we conclude that \(M[ \mathbb {L}, \mathfrak {m} ]\) is indeed closed under the action of \(\partial \). These formulae are equivalent to a computation due to Ramanujan. In general, for any \(f\in M_{n}\) we have

*f*[34] (53). The previous formula is compatible with the commutation relation \(h = [\partial , \overline{\partial }]\), as the reader may wish to check.

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

*r*,

*s*, consider the Laplace operator

**176**and (9). It follows from the previous computation (2.15) that \(\Delta _{r,s}\) acts via:

## 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

*f*we have

### 2.7 Real analytic Petersson inner product

*r*,

*s*, let

*f*whose constant part \(f^0\) vanishes. If \({\mathcal {S}}= \bigoplus _{r,s} {\mathcal {S}}_{r,s}\), there is an exact sequence

## Definition 2.13

*n*, consider the pairing

*y*as \(y\rightarrow \infty \), but via \( q = \exp (2 \pi i x) \exp (-2 \pi y)\) and \(\overline{q} = \exp (-2 \pi i x) \exp (-2 \pi y)\) it tends to zero in absolute value exponentially fast in

*y*at the cusp since \(g(z)\in {\mathcal {S}}\).

Two spaces \({\mathcal {M}}_{r,s}\) and \({\mathcal {S}}_{r',s'}\) can be paired via (2.24) if and only if \(r-s= r'-s'\). Equivalently, \(\langle f,g\rangle \) exists whenever \(h(f) = h(g)\), where *h* was defined in (2.6).

### 2.8 Holomorphic projections [32]

*r*and

*s*, this defines a linear map

### 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

*F*. We exhibit three obstructions for the existence of modular primitives: the first is combinatorial, the second relates to modularity, and the third is arithmetic.

### 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 anti-holomorphic 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

*n*. In the case \(r> s\) i.e. ‘below the diagonal’,

*F*vanishes. In the case \(r=s\), we have

## 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 \((0,s-r)\) and transforms like a modular form of weight \(s-r\), i.e. \(g( \gamma (z)) = (cz +d )^{s-r} g(z) \) for all \(\gamma \in \mathrm {SL}_2(\mathbb {Z})\) of the form (2.1). Thus, \(g \in M_{s-r}\). For the last part, use the well-known 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 anti-holomorphic projection \(p^a(F')\) vanishes. It is unique up to addition by a multiple of \( \mathbb {L}^{-r} \overline{\mathbb {G} }_{s-r}\) for \(s-r\ge 4\), where \(\mathbb {G} _{n}\) is the Eisenstein series (2.8).

## Proof

Since the Petersson inner product is non-degenerate, there exists a unique cusp form \(g\in S_{s-r}\) 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 \(S_{s-r}\) in \(M_{s-r}\) 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

*f*satisfies the condition

## Proof

*f*by \(a^{(k)}_{m,n}\). By assumption, they vanish for all \(k\le -r\) and \(k\ge N\) for some \(N\ge -r\). Denote the coefficients of

*F*by \(b^{(k)}_{m,n}\). Equation (2.15) is equivalent to the set of equations

*m*,

*n*. Fix an

*n*and an \(m\ge 1\). Then, if we set \(b_{m,n}^{(k)} =0\) for all \(k\ge N\), (3.5) holds for all \(k\ge N+1\). For \(k=N\), we can solve it by setting

*k*. In the case \(m=0\), the Eqs. (3.5) can be solved trivially, provided that (3.3) holds. This is certainly implied by (3.4). \(\square \)

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

*f*has a \(\partial \)-primitive in \( {\mathcal {M}}\), then

*f*is in the kernel of the holomorphic projection (2.28).

## Proof

*i*and from

*i*to \(-\overline{\rho }\) cancel due to \(\omega (-z^{-1}) = \omega (z)\); and finally the contribution along a path from \(i\infty \) to \(i \infty +1\), which corresponds to a small loop in the

*q*-disc, also gives zero because

*g*is cuspidal. \(\square \)

## 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

*M*(resp. \(\overline{M}\)) denotes the ring of holomorphic (anti-holomorphic) modular forms. The conditions (3.7) are stable under the operation of taking sums of vector spaces, and therefore a largest such space exists and is unique.

*n*, 0) with \(n\ge 0\) must satisfy

*r*,

*s*) with \(r\ge s\), the first equation of (3.7) determines

*F*in terms of previously determined functions by increasing induction on

*s*. The functions in the region \(r<s\) are deduced by complex conjugation [or by using the second equation of (3.7), starting from weights (0,

*n*)].

## 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

*M*of holomorphic modular forms in the Eq. (3.7) with another space of modular forms \(M'\) to define a class of functions \(\mathcal {MI}(M')\). Some examples:

- (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

*k*. It is stable under multiplication by \(\mathbb {L}\). Write \({\mathcal {M}}^D= {\mathcal {M}}^{D+0}\). Define an operator

*i*. For example, the one-form of weight zero

## Lemma 3.12

*F*, i.e. \(F\in {\mathcal {M}}^D\).

## Proof

*F*is modular of weights (

*r*,

*r*). \(\square \)

## 4 Real analytic Eisenstein series

We consider in some detail the simplest possible family of non-holomorphic functions in \({\mathcal {M}}\) as a concrete illustration.

### 4.1 Modular primitives of Eisenstein series

## Proposition 4.1

## Proof

*r*,

*s*. We must verify (4.1) and (4.2). These follow from the following identity, which holds for any integers

*r*,

*s*:

*r*and

*s*. It follows from the definition of the holomorphic Eisenstein series as a sum:

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 anti-diagonals 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 \(\Delta f = - \partial _{n-1} \overline{\partial }_0 f = 0\). The same is true of \(\mathfrak {m} \) defined in Sect. 2.5. More generally, \(\mathbb {L}^k f\) is an eigenfunction with eigenvalue \((n-k-1)k\). 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 anti-holomorphic) modular form lies in \({\mathcal {HM}}\).

### 5.1 Proof of Theorem 5.1

## Lemma 5.2

*F*is of the form

## 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 \(x^2+ x(w-1) + \lambda =0\) has two integral solutions \(k_0\) and \(1-w-{k_0}\), 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 \(\lambda a^{(k+1)}_{m,0} = 2 m(k+s) a^{(k)}_{m,n} - k (k+w-1) a^{(k+1)}_{m,0}\), 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 \(\lambda a^{(k)}_{0,0} = - k (k+w-1) a^{(k)}_{0,0}\), so by the above \(a^{(k)}_{0,0}\) is non-zero only for \(k\in \{ k_0, 1-w-{k_0}\} \). \(\square \)

## 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 \((r+1, s-1)\). Applying \(\partial _{r-1}\) kills the term \(\mathbb {L}^{k} g_{k}(\overline{q})\) for with \(k=1-r\). 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

*V*. By the above, we can assume that \(\partial F\) is a linear combination of

*f*is a cusp form. Since these elements have distinct \(\mathsf {h}\)-degrees, we can treat each case in turn, by linearity. But we showed in corollary 3.5 that \(\mathfrak {m} \mathbb {L}^k\) has no \(\partial \)-primitive in \({\mathcal {M}}\), and likewise, in corollary 3.8 that cusp forms have no primitives either. The elements \({\mathcal {E}}_{0,2n}\) (and hence \(\mathbb {L}^k {\mathcal {E}}_{0,2n}\)) have no modular primitives by Lemma 3.4, since the coefficient of \(\mathbb {L}\) in \({\mathcal {E}}^0_{0,2n}\) is nonzero by (4.5). Therefore, none of these cases can arise, and we conclude that if \(\partial F \in V\), so too is \(F\in V\). \(\square \)

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

*Df*)

*g*of mixed weights, where

The properties of the brackets (6.1) are well-known. For instance, the bracket is anti-symmetric and satisfies the Jacobi identity [34] §5.2.

## Example 6.2

*f*and

*g*and is an element of \({\mathcal {M}}_{r_1+r_2, s_1+s_2}\). It can be written more elegantly as a composition of operators, as follows:

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

*f*is equivariant:

*r*,

*s*). Suppose now that the coefficients (7.2) of

*f*admit expansions of the form

*f*is equivariant if and only if \(f_{r,s} \in P^{-r-s} {\mathcal {M}}_{r,s}.\)

## Proof

*z*in the usual manner and on the right of \(V_{2n}\), like a modular function of weights \((-r,-s)\). The coefficient of \((X-zY)^r (X-\overline{z} Y)^s\) for

*f*equivariant is therefore modular of weights (

*r*,

*s*). For the last statement, the assumption on the Fourier expansions of \(f^{r,s}\) implies that the coefficients \((z-\overline{z})^{r+s} f_{r,s}\) admit expansions in the ring \(\mathbb {C}[[q, \overline{q}]][z, \overline{z}]\) by (7.5). By Lemma 2.2, the \(f_{r,s}\) have expansions of the form (2.5). \(\square \)

We construct equivariant functions *f* from iterated integrals. These only involve non-negative 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 Vector-valued differential equations

The operators \(\partial , \overline{\partial }\) of Definition 2.4 admit the following interpretation.

## Proposition 7.2

## Proof

*Y*using the second line of (7.5), the right-hand side becomes

## Lemma 7.3

## Proof

*r*,

*s*) with \((r-2m+k,s+k)\). \(\square \)

### 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 single-valued periods as we presently explain in some simple examples.

### 8.1 Single-valued functions

*z*on \(\mathbb {C}\). The fundamental group of \(\mathbb {C}^{\times }\) at the point 1 is isomorphic to \(\mathbb {Z}\) and is generated by a simple loop around the origin. Analytic continuation around this loop creates a discontinuity

*D*(

*z*).

*r*,

*s*). The equivariance

*f*(

*z*) on the orbifold quotient of \(\mathfrak {H} \) by the action of \(\mathrm {SL}_2(\mathbb {Z})\).

### 8.2 Notation

*f*a holomorphic modular form of weight

*n*, let us denote by

*z*and

*X*,

*Y*. We shall write

### 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

*f*. By the Eichler–Shimura theorem, the classes of \(C^+\) and \( C^-\) are independent in group cohomology \(H^1(\mathrm {SL}_2(\mathbb {Z}), \mathbb {C}[X,Y])\), and so there is no way, by taking real and imaginary parts, that we can kill the right-hand side of (8.4) to obtain a single-valued function. Therefore, a single iterated integral, or primitive, of a cusp form yields nothing new. Indeed, by Proposition 7.2, such a function, if it existed, would provide a solution to the equation \(\partial F = f\), in \({\mathcal {M}}\), which would contradict 3.8. This obstruction can be circumvented by introducing poles; cusp forms do have primitives in \({\mathcal {M}}^!\) (see §11).

#### 8.4.2 Eisenstein series

*C*satisfies ([1], §7):

*C*replaced with \({\mathrm {Re}}\, C\). The key point is that the real part of \(C_{\gamma }\) only involves the first term in the previous equation, which is a coboundary for \(\mathrm {SL}_2(\mathbb {Z})\). Therefore, the function \({\mathrm {Re}}\, F(\tau )\) can be modified in the following manner to define a vector-valued real analytic function

## 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$$\begin{aligned} e_{2k}^0(S)= & {} {(2k-2)! \over 2} \, \sum _{i=1}^{k-1} {B_{2i} \over (2i)!}{B_{2k-2i} \over (2k-2i)!} X^{2i-1} Y^{2k-2i-1} \ , \\ e_{2k}^0(T)= & {} {(2k-2)! \over 2} {B_{2k} \over (2k)!} \Big ( { (X+Y)^{2k-1} - X^{2k-1} \over Y}\Big )\ . \end{aligned}$$
*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. -
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 Haberland-Petersson 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

*f*is a holomorphic modular form, and

*g*a cusp form, both of weight \(2a+2b-2-2k\), which is modular equivariant:

*f*,

*g*,

*c*and \(M^{(k)}_{2a,2b}\), except in the case when \(2a+2b-4-2k=0\) since we can add an arbitrary constant \(c\in \mathbb {C}\). Extracting the coefficients of \(M^{(k)}_{2a,2b}\) via (7.3) yields a class of functions in \({\mathcal {M}}\).

## Theorem 9.3

*f*is the unique cusp form of weight \(2w+2\) satisfying

*m*,

*n*are negative.

## Proof

*f*is a cusp form. It is uniquely determined by Theorem 3.7, which gives Eq. (9.6).

*g*is uniquely determined from the anti-holomorphic projection \(p^{a}( \overline{\partial } M_{0,2w})=0\). \(\square \)

## 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 *L*-functions and constant terms

All expansion coefficients (2.5) of an element \(f \in \mathcal {MI}_k\) are uniquely determined by those of functions in \(\mathcal {MI}_{k-1}\) 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 *L*-function [27].

## Definition 9.6

*L*-function is

## Theorem 9.7

*f*by analytic continuation. Indeed, we have

*L*-function. Similarly, one can assign (a family of)

*L*-functions to universal mixed elliptic motives [18]. This will be discussed elsewhere.

### 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

*X*is dual to the relations between double zeta values in weight 12.

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 single-valued 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

*M*-degree, where \(\mathbb {L}\) has*M*-filtration 2, and the \({\mathcal {E}}_{r,s}\) have*M*-filtration 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*M*-filtration \(\le m\) is finite-dimensional for every*m*,*w*. - (5)Every element of \(\mathcal {MI}^E\) admits an expansion in the ringi.e. its coefficients are single-valued multiple zeta values. An element of total modular weight$$\begin{aligned} \mathcal {Z}^{\mathrm {sv}} [[q, \overline{q}]][\mathbb {L}^{\pm }]\ , \end{aligned}$$
*w*has poles in \(\mathbb {L}\) of order at most*w*. An element of*M*-filtration 2*m*has terms in \(\mathbb {L}^k\) for \(k \le m\). - (6)The space \(\mathcal {MI}^E\) has the following differential structure:where$$\begin{aligned} \partial \big ( \mathcal {MI}_k^E\big )\subset & {} \mathcal {MI}^E_{k} + E[\mathbb {L}] \times \mathcal {MI}_{k-1}^E \\ \overline{\partial } \big ( \mathcal {MI}_k^E \big )\subset & {} \mathcal {MI}^E_{k} + \overline{E}[\mathbb {L}] \times \mathcal {MI}_{k-1}^E \ \end{aligned}$$
*E*is (3.9). The operators \(\partial , \overline{\partial }\) respect the*M*-filtration, i.e. \(\deg _M \partial = \deg _M \overline{\partial }=0\), where the generators \(\mathbb {G} _{2n+2}\) of*E*are placed in*M*-degree 0. - (7)Every element \(F \in \mathcal {MI}_k^E\) of total modular weight
*w*satisfies an inhomogeneous Laplace equation of the form:$$\begin{aligned} (\Delta + w )\, F \in (E+ \overline{E})[\mathbb {L}] \times \mathcal {MI}^E_{k-1} + E \overline{E} [\mathbb {L}] \times \mathcal {MI}^E_{k-2} \ . \end{aligned}$$

## 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 \(\mathcal {MI}^E[\mathbb {L}^{\pm }]\).

*w*,

*w*). Then, \(\mathbb {L}^{w} f\) is modular invariant, and by (7) and repeated application of (2.23) it satisfies an inhomogenous Laplace eigenvalue equation with eigenvalue

*s*is a positive integer, and the same statement was proved in [14] for two-loop modular graphs functions using the representation theory of \(\mathrm {SO}(2,1)\).

*M*-filtration can be made more precise. If \(F \in \mathcal {MI}^E\) of

*M*-filtration \(\le 2 m\), then the coefficient of \(\mathbb {L}^{-k}\) in the constant part \(F^0\) of

*F*is a single-valued multiple zeta value of weight \( \le k +m.\) If one assumes (for example, by replacing multiple zeta values with their motivic versions) that multiple zeta values are graded, rather than filtered, by weight, then this filtration would also be a grading. For example, the elements \({\mathcal {E}}_{r,s}\) have constant parts

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

*r*,

*s*) with \(r,s\ge 0\)), such that

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

*f*. The \({\mathcal {H}}(f)_{r,s}\) are eigenfunctions of the Laplacian with eigenvalue \(-n\).

## 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 \({\mathcal {M}}_{r,s}\)

*N*(in fact, \(N= \dim S_n\) will do). In particular,

### 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.

## Notes

## 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 well-known results in the theory of modular forms. I apologise in advance if I have failed to provide attributions in every case.

## Ethics approval and consent to participate

Not applicable.

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