A class of nonholomorphic modular forms III: real analytic cusp forms for \(\mathrm {SL}_2(\mathbb {Z})\)
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Abstract
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasiperiods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit ‘weak harmonic lifts’ for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebrogeometric and have Fourier coefficients proportional to \(n^{1k}(a_n^{\prime } + \rho a_n)\) for \(n\ne 0\), where \(\rho \) is the normalised permanent of the period matrix of the corresponding motive, and \(a_n, a_n^{\prime }\) are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.
1 Introduction
In this paper, we shall construct real analytic cusp forms \(\mathcal {H}(f)_{r,s}\) which are canonically associated with any Hecke cusp form, and satisfy an analogous system of differential equations. It is clear from their construction that they are ‘motivic’, in that their coefficients only involve the periods of pure motives associated with cusp forms [21]. The functions \(\mathcal {H}(f)_{r,s}\) generate \(\mathcal {MI}^!_1\), and furthermore, they generate the subspace of \(\mathcal {HM}^! \subset \mathcal {M}^!\) of eigenfunctions of the Laplacian. In other words, the overlap between the space \(\mathcal {M}^!\) and the set of Maass waveforms is exactly described by the functions studied in this paper.
1.1 Real Frobenius
The essential ingredient in our construction is the real Frobenius, also known as complex conjugation. For all \(n\in {\mathbb {Z}}\) let \(M_{n}^!\) denote the space of weakly holomorphic modular forms of weight n.
1.2 Summary of results
Theorem 1.1
The theorem can be rephrased as follows. Consider the real analytic vectorvalued function \(\mathcal {H}(f) : \mathfrak {H} \rightarrow {\mathbb {C}}[X,Y]\) defined by
Theorem 1.2
The constant term \(\alpha _f\) can be computed (Sect. 6.6) from the Fourier coefficients of f and \(\mathbf {s}(f)\) in the case when f is cuspidal, and is given by an odd zeta value in the case when f is an Eisenstein series. It is a pure period in the cuspidal case; and a mixed period in the Eisenstein case. This dichotomy is due to the fact that the Tate twists of the Tate motive have nontrivial extensions, but the Tate twists of the motive of a cusp form do not (in the relevant range). When f is holomorphic, the constant \(\alpha _f\) is proportional to the Petersson norm of f.
 (1)
If f is a Hecke eigenfunction with eigenvalues \(\lambda _m\), then the functions \(\mathcal {H}(f)_{r,s}\) satisfy an inhomogeneous Hecke eigenvalue equation with eigenvalues \(m^{1} \lambda _m\). See Sect. 6.5 for precise statements.
 (2)
The action of \(\mathrm {Gal}({\overline{{\mathbb {Q}}}}/{\mathbb {Q}})\) on Hecke eigenfunctions extends to an action on the functions \(\mathcal {H}(f)_{r,s}\), for every r, s. In fact, this action extends to an action of a ‘motivic’ Galois group on a larger class of modular forms which acts on the coefficients in the expansion (1.2). This will be discussed elsewhere.
1.3 Weak harmonic lifts and mock modular forms of integral weight
Corollary 1.3
The problem of constructing weak harmonic lifts has a long history, but an explicit construction has remained elusive. The existence of weak harmonic lifts in a much more general setting was proved in [9]. Having established existence, the general shape of the Fourier expansion is easily deduced—the only issue is to determine the unknown Fourier coefficients. On the other hand a direct, but highly transcendental, construction using Poincaré series was given in [8, 20], involving complicated special functions. This procedure is potentially illdefined: when the space of cusp forms has dimension greater than one, it involves choices, since there are relations between Poincaré series. The question of whether weak harmonic lifts have irrational coefficients or not has been raised in [10, 11, 20]. Our results imply that these functions, despite appearances, are in fact of geometric, and indeed, motivic, origin.
Since the space of cusp forms of weight 12 is onedimensional, the method of Poincaré series [8] also yields in this case an explicit expression for this mock modular form in terms of special functions. Comparing the Fourier coefficients of the two gives:
Corollary 1.4
Since modular forms of level one do not have complex multiplication, Grothendieck’s period conjecture, applied to the motives of cusp forms, would imply that its Fourier coefficients are transcendental. The reader will easily be able to generalise the results of this paper to the case of a general congruence subgroup using the results of [23].^{1} In an “Appendix”, we explain how the existence of a complex multiplication on the motive of a cusp form implies an algebraicity constraint on the singlevalued involution. This explains the phenomena studied in the recent paper [10] which observed algebraicity of the Fourier coefficients of suitably normalised Maass waveforms associated with modular forms with complex multiplication.
1.4 Contents
In Sect. 2 we review the theory of weakly holomorphic modular forms. Much of this material is standard, but many aspects are not widely known and may be of independent interest. In Sect. 3 we review some properties of the space \(\mathcal {M}^!\) of real analytic modular forms from [2], and its subspaces \(\mathcal {HM}^!\) (Sect. 4) of Laplace eigenfunctions and \(\mathcal {MI}^!\) (Sect. 5) of modular integrals. In Sect. 6 we describe the action of Hecke operators on \(\mathcal {HM}^!\). Much of this material is well known. In Sect. 7 we prove the existence of weak modular lifts, and in Sect. 8 we discuss Ramanujan’s function \(\Delta \).
2 Background on weakly holomorphic modular forms
2.1 Weakly holomorphic modular forms
Theorem 2.1
This theorem has a number of consequences that we shall spell out below. Many of these have been known for some time, others apparently not.
2.1.1 Hecke operators
The isomorphism (2.3) is equivariant with respect to the action of Hecke operators \(T_m\), for \(m\ge 1\), which act via the formula (6.2) (which we shall rederive, in a more general context, in Sect. 6). If a formal power series (2.1) has a pole of order p at the cusp, then \(T_m f\) has a pole of order mp at the cusp.
Remark 2.2
One could fix a ‘canonical’ basis of \(H^{dR}_{\underline{\lambda }}\) either by assuming that the Fourier coefficient \(a_1\) of \(f'\) is equal to 1, or by demanding that \( \{f'_{\underline{\lambda }} , f'_{\underline{\lambda }} \} = 0\) (note that \(\{f_{\underline{\lambda }} , f_{\underline{\lambda }} \} =0\) holds automatically). This will not be required in this paper. The latter condition holds for the basis chosen in Sect. 8.
2.1.2 Group cohomology and cocycles
2.1.3 Eichler–Shimura isomorphism
The following corollary is a consequence of a mild extension [6] of Grothendieck’s algebraic de Rham theorem.
Corollary 2.3
In particular, the comparison isomorphism respects the decomposition into Eisenstein and cuspidal parts. It can be computed as follows. Fix a point \(z_0 \in \mathfrak {H} \).
Definition 2.4
2.1.4 Period matrix
Since the comparison isomorphism is Hecke equivariant, it respects the decomposition into Hecke eigenspaces.
Definition 2.5
2.1.5 Hodge theory
The de Rham cohomology group \(H^1_{dR}(\mathcal {M}_{1,1}; \mathcal {V}_n)\) admits an increasing weight filtration W and a decreasing Hodge filtration F by \({\mathbb {Q}}\)vector spaces. The basis (2.7) is compatible with the Hodge filtration.
Similarly, \(H^1(\Gamma ; V_n)\) is equipped with an increasing filtration W compatible with the weight filtration on de Rham cohomology via the comparison isomorphism.
2.1.6 Real Frobenius and singlevalued map
3 The space \(\mathcal {M}^!\) of nonholomorphic modular forms
3.1 Differential operators
Lemma 3.1
Since there exist weakly holomorphic modular forms of negative weight, it follows that primitives in \(\mathcal {M}^!_{r,s}\), unlike the space \(\mathcal {M}_{r,s}\), are never unique.
Lemma 3.2
The equations \([\partial , \mathbb {L} ] = [{\overline{\partial }}, \mathbb {L} ]\) imply that \(\mathbb {L} \) is constant for the differential operators \(\partial , {\overline{\partial }}\), and justify calling \(f^0\) the ‘constant’ part.
3.2 Bol’s operator
Lemma 3.3
Proof
3.3 Vectorvalued modular forms
3.4 Some useful lemmas
Lemma 3.4
Proof
Corollary 3.5
Proof
4 The space \(\mathcal {HM}^!\) of harmonic functions
Definition 4.1
Lemma 4.2
Proof
The first equation follows since \([\nabla , \partial ]=[\nabla , {\overline{\partial }}]=0\) by Lemma 3.2. For the second, \([\mathbb {L} , \Delta ] = w \mathbb {L} \) implies that if \(\Delta F = \lambda F\), then \(\Delta (\mathbb {L} F) = (\lambda  w) \mathbb {L} F\). \(\square \)
The lemma remains true on replacing \(\mathcal {HM}^!(\lambda )\) by \(\mathcal {HM}(\lambda ) = \mathcal {HM}^!(\lambda ) \cap \mathcal {M}\).
Lemma 4.3
Proof
This was proved for the space \(\mathcal {HM}\) in [2, lemma 5.2]. The proof is more or less identical for \(\mathcal {M}^!\). \(\square \)
5 The space \(\mathcal {MI}_1^!\) of weak modular primitives
The subspace \(\mathcal {MI}^! \subset \mathcal {M}^!\) of modular iterated integrals was defined in [2].
Definition 5.1
We call the increasing filtration \(\mathcal {MI}^!_k \subset \mathcal {MI}^!\) the length. In this paper we shall focus only on length \(\le 1\). We first dispense with the subspace of length 0.
Proposition 5.2
\(\mathcal {MI}^!_0 = {\mathbb {C}}[\mathbb {L} ^{1} ]\).
Proof
Firstly, the space \({\mathbb {C}}[\mathbb {L} ^{1}]\) satisfies the conditions of the definition since \([\partial , \mathbb {L} ] = [{\overline{\partial }}, \mathbb {L} ]=0\), and so \({\mathbb {C}}[\mathbb {L} ^{1}] \subset \mathcal {MI}^!_0\). Now let \(F \in \mathcal {MI}^!_0\) be of modular weights (n, 0), where \(n\ge 0\). Since \(\partial F\) has weights \((n+1,1)\), which lies outside the positive quadrant, we must by (5.1) and \(\mathcal {MI}^!_{1}=0\) have \(\partial F =0\). Similarly, the element \(F'={\overline{\partial }}^n F\) has weights (0, n) and so \({\overline{\partial }} F'=0\) since it also lies outside the positive quadrant. By Corollary 3.5, F vanishes if \(n>0\) and \(F\in {\mathbb {C}}\) if \(n=0\). By complex conjugation, it follows that \(\mathcal {MI}^!_0\) vanishes in modular weights (0, n) and (n, 0) for all \(n\ge 1\) and is contained in \({\mathbb {C}}\) in weights (0, 0). We can now repeat the argument for any \(F\in \mathcal {MI}_0^!\) of modular weights (n, 1) by replacing F with \(\mathbb {L} F\) and arguing as above. We deduce that \(\mathcal {MI}_0^!\) vanishes in all weights (n, 1) and (1, n) for \(n\ge 2\) and is contained in \({\mathbb {C}}\mathbb {L} ^{1}\) in weights (1, 1). Continuing in this manner, we conclude that \(\mathcal {MI}_0^! \subset {\mathbb {C}}[\mathbb {L} ^{1}]\). \(\square \)
5.1 Modular iterated integrals of length one
Remark 5.3
As a consequence, \(\mathbb {L} ^{1} F\) satisfies \(\partial \mathbb {L} ^{1} F = f\) and \(\Delta \mathbb {L} ^{1} F =0\). It is therefore what is known as a weak harmonic lift of f.
Proposition 5.4
Proof
Remark 5.5
We now turn to uniqueness.
Lemma 5.6
Proof
By Lemma 3.1, \(X'_{n,0} X_{n,0} \in \mathbb {L} ^{n} \overline{M^!}_{n}\). Apply (5.4) and (3.6) to conclude. \(\square \)
Corollary 5.7
5.2 Harmonic functions and structure of \(\mathcal {MI}^!_1\)
We show that the modular primitives of Proposition 5.4 generate \(\mathcal {MI}^!_1\) under multiplication by \(\mathbb {L} ^{1}\). This section can be skipped and is not required for the rest of the paper.
Proposition 5.8
In the case \(s\le r\), we can take \(F_k = \mathbb {L} ^{k} \partial ^{rk} \overline{X}_k\), with \(X_k\) a modular primitive of \(\mathbb {L} g_k\), where \(g_k \in M_{r+s+22k}^!\) is weakly holomorphic.
Proof

an element \(F\in \mathcal {MI}^!_1\) of modular weights (n, 0) is necessarily an eigenfunction of the Laplacian with eigenvalue \(n\).

an element \(F\in \mathcal {MI}^!_1\) of modular weights \((n1,1)\) is a linear combination of two eigenfunctions of the Laplacian with possible eigenvalues \(\{n, 0\}\).
 an element \(F\in \mathcal {MI}^!_1\) of total weight w can have eigenvalues in the set$$\begin{aligned} \{w \ , \ 0 \ , \ w2 \ , \ 2(w3) \ , \ 3(w4) \ , \ \ldots \ , \ \textstyle {w \over 2}(1 {w\over 2}) \}. \end{aligned}$$
Remark 5.9
Elements in \(\mathcal {MI}^!_k\) for \(k\ge 2\) are no longer harmonic and satisfy a more complicated structure with respect to the Laplace operator. See, for example, [3, Sect. 11.34].
5.3 Ansatz for primitives
Proposition 5.10
Proof
Lemma 5.11
Proof
Finally, if E is modular equivariant, \(E_{r,s} \in {\mathbb {C}}\mathbb {L} ^{n}\) is modular of weights (r, s) with \(r+s=n>0\). But \(\mathbb {L} ^{n}\) is modular of weights (n, n), which implies that \(E_{r,s}= 0\). \(\square \)
Corollary 5.12
Proof
We shall determine the unknown coefficient \(\alpha \) using Hecke operators. Another way to prove the corollary is to use the fact that \(X_{r,s}\) are eigenfunctions of the Laplacian (Proposition 5.8) and the explicit shape (4.3) and (4.4) for the latter. We chose the approach above since it explains the origin of the indeterminate coefficient \(\alpha \), and since functions in \(\mathcal {MI}\) are not harmonic in general.
Corollary 5.13
5.4 Example: real analytic Eisenstein series
6 Hecke operators
6.1 Definition
Lemma 6.1
Proof
6.2 Properties
Lemma 6.2
Proof
6.3 qexpansions
Corollary 6.3
Corollary 6.4
Proof
Condition (6.5) holds in particular if \(a^{(k)}_{m,n}=0\) for all \(mn\ne 0\).
Corollary 6.5
The Hecke algebra acts on \(\mathcal {HM}^!\).
If \(f = f^a + f^0 + f^h\) as in (4.2) then \((T_N f )^{\bullet }= T_N (f^{\bullet })\) for \(\bullet \in \{a, 0, h\}\). It follows from the formula that if \(f^{\bullet }\) has a pole of order at most p at the cusp, then \(T_N f^{\bullet }\) has a pole of order at most Np at the cusp, for \(\bullet = a, h\).
Corollary 6.6
Corollary 6.7
The space of almost weakly holomorphic modular forms \(M^![\mathbb {G} ^*_2, \mathbb {L} ^{\pm }]\) consists of harmonic functions. It is preserved by the Hecke operators.
Example 6.8
Remark 6.9
The quotient \(\mathcal {HM}^! / \partial (\mathcal {HM}^!)\) also admits an action of the Hecke algebra.
6.4 Hecke operators on weakly holomorphic modular forms
6.5 Hecke action on modular primitives
Let \(f, g, X_{r,s}\) be as in Proposition 5.4.
Proposition 6.10
Proof
Remark 6.11
6.6 Determination of the coefficient of \(\mathbb {L} ^{n}\)
Corollary 6.12
Proof
The \(\lambda _m\) are the eigenvalues of a normalised holomorphic Hecke eigenform \(g \in S_{n+2}\). Then \(\lambda _m = a_m(g)\) and an elementary estimate [16, Lemma 2], implies that \(a_m(g)\) grows at most like \(m^{n/2+1}\). Since \(\sigma _{n+1}(m)\ge m^{n+1}\), it follows that \(( \sigma _{n+1}(m) \lambda _m)\) is nonzero for sufficiently large m. \(\square \)
The consistency of equations (6.12) for different values of m follows from (6.8). Equation (6.13) would have poles for every n if f were an Eisenstein series by (2.5).
7 Existence of modular primitives
Having determined the form of modular primitives, we now turn to their existence.
7.1 Cocycles and periods
Lemma 7.1
Proof
See [4, Lemma 7.3]. \(\square \)
We can assume that \(P^+, P^ \in Z_{\mathrm {cusp}}^1(\Gamma ; V_n \otimes K)\) are the unique Heckeequivariant lifts of the cohomology classes chosen earlier. They satisfy \(P^{\pm }_T=0\).
Corollary 7.2
7.2 Real and imaginary analytic cusp forms
We shall construct explicit modular primitives of cusp forms in two steps.
Recall that the integrals \(F_{f}(z)\) were defined in (2.11) relative to the basepoint \(z_0 \in \mathfrak {H} \).
Definition 7.3
Theorem 7.4
The functions \(\mathcal {I}_f(z)\) and \(\mathcal {R}_f(z)\) are well defined (independent of the choice of basepoint \(z_0\)), and \(\Gamma \)equivariant.
Proof
Corollary 7.5
Proof
This is a straightforward application of (3.9) to the previous discussion. \(\square \)
7.3 Modular primitives of cusp forms
Since the period isomorphism is invertible, we can change basis, to deduce the existence of modular primitives for all cusp forms.
Definition 7.6
Write \(\mathcal {H}(f)= \sum _{r+s=n} \mathcal {H}(f)_{r,s} (XzY)^r(X\overline{z}Y)^s\) as usual.
Theorem 7.7
Proof
It follows from uniqueness (lemma 5.11) that \(\mathcal {H}(f)_{r,s}\) is well defined (only depends on f and not the choice of basis \(f, f'\)), since it only depends on f and its image under the singlevalued involution \(\mathbf {s}(f)\), which is canonical.
Corollary 7.8
Corollary 7.9
Proof
This follows from (6.13): the term \(\psi _m\) vanishes since f has no pole, and the sole contribution to \(\alpha \) comes, via \(\phi _m\), from the action of Hecke operators on the \(f'\) term in \(g= \mathbf {s}(f) \in {\mathbb {C}}f \oplus {\mathbb {C}}f'\). But the coefficient of \(f'\) in \(\mathbf {s}(f)\) is proportional to \(\{f,\mathbf {s}(f)\}\), since \(\{f,f\}=0\) and \(\{f, f'\} \in K_{\underline{\lambda }}\). The quantity \(\{f,\mathbf {s}(f)\}\) can be interpreted as the Petersson norm via (2.14) and the comments which follow. \(\square \)
Corollary 7.10
Every modular form admits a modular primitive in \(\mathcal {M}^!\).
Proof
Every modular form of integral weight is a linear combination of Eisenstein series and cuspidal Hecke eigenforms. \(\square \)
7.4 Vanishing constant term
An element \(f \in (H^{dR}\otimes {\mathbb {C}})^+\) satisfies \(\mathbf {s}(f) = f\), and hence, \(\mathcal {H}(f)_{r,s}\) is proportional to the ‘real’ function \(\mathcal {R}(f)_{r,s}\).
An element \(f \in (H^{dR}\otimes {\mathbb {C}})^\) satisfies \(\mathbf {s}(f) =  f\), and hence, \(\mathcal {H}(f)_{r,s}\) is proportional to the ‘imaginary’ function \(\mathcal {I}(f)_{r,s}\). The latter satisfies \(\mathcal {I}_{r,s}^0 =0 \) since by Corollary 7.5 and (6.13), the constant term \(\alpha \) is real and hence vanishes since \(\mathcal {I}(f)_{r,r} =  \overline{\mathcal {I}}(f)_{r,r}\). It is therefore cuspidal: \(\mathcal {I}(f)_{r,s } \in \mathcal {S}^!\).
8 Example: Real analytic version of Ramanujan’s function \(\Delta \)
8.1 Weakly holomorphic cusp forms in weight 12
8.2 Cocycles
8.3 Periods
8.4 Singlevalued involution
8.5 The constant term
8.6 Real analytic cusp forms
The real analytic cusp forms \(\mathcal {H}(\Delta )_{r,s}\) for \(r+s =10\) can be written down explicitly from the formulae given in Theorem 1.2.
8.7 The mock modular form \(M_{\Delta }\)
Footnotes
 1.
After we had written this paper, K. Ono and N. Diamantis kindly pointed out the recent work of Candelori [11], which is closely related to our construction and applies for modular forms of level \(\ge 5\). His formula (48) for the Fourier coefficients in the case \(n\ne 0\) is very similar to (1.9). The case \(n=0\) requires an additional argument, which we provide in this paper using Hecke operators.
 2.The reason for the factor \(m^{1}\) is that f is a function; the usual formula for Hecke operators involves oneforms: for \(\alpha \) as in (6.1),and the \(\det (\alpha )\) accounts for an extra multiple of m in the formula for \(T_m\).$$\begin{aligned} \mathrm{d} (\alpha z) = { \det (\alpha ) \over (c_{\alpha } z+\mathrm{d}_{\alpha })^2} \mathrm{d}z\ , \end{aligned}$$
Notes
Author's contributions
Acknowlegements
The author is partially supported by ERC Grant GALOP 724638. Many thanks to the IHES, where this paper was written, for hospitality. Many thanks to Larry Rolen for informing me of the problem of finding weak harmonic lifts during the conference ‘modular forms are everywhere’ in honour of Zagier’s 65th birthday, and to Luca Candelori for comments and corrections.
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