# A geometric perspective on *p*-adic properties of mock modular forms

- Luca Candelori
^{1}Email author and - Francesc Castella
^{2}

**4**:5

**DOI: **10.1186/s40687-017-0095-z

© The Author(s) 2017

**Received: **31 October 2016

**Accepted: **5 January 2017

**Published: **3 March 2017

## Abstract

Bringmann et al. (Trans Am Math Soc 364(5):2393–2410, 2012) showed how to ‘regularize’ mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain *p*-adic modular forms in the sense of Serre. In this paper, we give a new proof of a refined form of their results (for good primes *p*) by employing the geometric theory of harmonic Maass forms developed by Candelori (Math Ann 360(1–2):489–517, 2014) and the theory of overconvergent modular forms due to Katz and Coleman. In particular, our main results imply that the *p*-adic modular forms in Bringmann et al.
(2012) are overconvergent.

### Mathematics Subject Classification

11F33 11F23## 1 Background

Over the past decade, there has been a renewed interest in Ramanujan’s *mock modular forms* and related objects, such as *harmonic (weak) Maass forms*, whose Fourier coefficients have been found in many instances to encode interesting arithmetic data, similarly as in the classical theory of modular forms. In this paper, we introduce a new perspective on the *p*-adic properties of Fourier coefficients of mock modular forms, based on the algebro-geometric theory of *p*-adic modular forms due Katz [12] and Coleman [8]. Such *p*-adic properties were originally discovered by Guerzhoy–Kent–Ono [11] and Bringmann–Guerzhoy–Kane [1], but we believe that our methods offer a most natural approach to such results.

*N*, and let \(\chi \) be a Dirichlet character modulo

*N*. Denote by \({\mathcal {H}}_k(\Gamma _0(N),\chi )\) the space of harmonic Maass forms on \(\Gamma _0(N)\) of integral weight

*k*and character \(\chi \) (as defined in [1, §2]). Any harmonic Maass form

*F*has a decomposition

*mock modular form*; in general, it does not transform like a modular form, but (as first discovered by Ramanujan) the properties of its Fourier coefficients resemble those of a classical modular form.

*k*, level

*N*, and character \(\chi \). If for any \(w\in {\mathbb {Z}}\), we let

*f*is the

*shadow*of

*F*, and a fundamental question in the subject is to relate the coefficients of a mock modular form \(F^+\) to the coefficients of its shadow.

However, with the differential operator (1) having an infinite-dimensional kernel, to obtain results in this direction it becomes necessary to work with a refined notion of harmonic Maass forms lifting a given *f*. For any congruence subgroup \(\Gamma \) of \(\mathrm {SL}_2({\mathbb {Z}})\), let \(S_k(\Gamma , K)\) (resp. \(M^{!}_k(\Gamma , K)\)) be the space of cusp forms (weakly homomorphic modular forms) of weight *k* and level \(\Gamma \) whose *q*-expansion coefficients all lie in \(K\subseteq {\mathbb {C}}\).

### Definition 1.1

*good*for \(f\in S_k(\Gamma _1(N),K)\) if:

- (i)
The principal parts of

*F*at all cusps are defined over*K*. - (ii)
We have \(\xi _{2-k}(F) = f/\Vert f\Vert ^2\), where \(\Vert f\Vert \) is the Petersson norm of

*f*.

*K*, let

*F*be a harmonic Maass form that is good for

*f*, and write

*F*. Let \(E_f = \sum _{n=1}^\infty n^{1-k}a_nq^n\) be the so-called Eichler integral of

*f*, so that \(D^{k-1}(E_f)=f\) for the differential operator \(D^{k-1}\) acting as \((q d/dq)^{k-1}\) on

*q*-expansions. It is shown in [11] (and in Theorem 4.1 below by different methods) that for any \(\alpha \in {\mathbb {C}}\) such that \(\alpha - c^{+}(1) \in K\), the coefficients of

*K*. In particular, this applies of course to \(\alpha =c^+(1)\).

*p*-adic embeddings \(\overline{{\mathbb {Q}}}\hookrightarrow {\mathbb {C}}\) and \(\overline{{\mathbb {Q}}}\hookrightarrow {\mathbb {C}}_p\), and let \(v_p\) be the resulting

*p*-adic valuation on \(\overline{{\mathbb {Q}}}\) normalized so that \(v_p(p)=1\). Thus, for any value of \(\alpha \) in the set

*q*-expansion of \({\mathcal {F}}_\alpha \) lies in \({\mathbb {C}}_p[[q]][q^{-1}]\), and it becomes meaningful to ask about the

*p*-adic properties of its coefficients; in particular, whether the resulting

*q*-expansion corresponds to a

*p*-adic modular form. In general, the coefficients \(c_\alpha (n)\) of \({\mathcal {F}}_\alpha \) will have unbounded

*p*-adic valuation (see, e.g., [1, p. 2396]), but the following special case of our main result shows that, for a specific value of \(\alpha \), a certain regularization of \({\mathcal {F}}_\alpha \) indeed gives rise to a

*p*-adic modular form.

*f*at

*p*:

*V*be the operator acting as \(q\mapsto q^p\) on

*q*-expansions.

### Theorem 1.2

We refer the reader to Definition 3.1 for the precise notion of overconvergent modular forms to which Theorem 1.2 applies, but suffice it to say that they bear a relation to Coleman’s overconvergent modular forms [8] analogous to that of *p*-adic modular forms in the sense of [1] to Serre’s *p*-adic modular forms [13]. In particular, our results in Sect. 5 (of which Theorem 1.2 is a special case) yield a new proof of a refined form of the main results obtained by Bringmann–Guerzhoy–Kane in [1], showing that the *p*-adic modular forms constructed in *loc.cit.* are overconvergent.

We conclude this Introduction by briefly mentioning some key ideas behind our proof of Theorem 1.2. Let \(f_\beta \) and \(f_{\beta '}\) be the *p*-stabilizations of *f*, which are modular forms of level *Np* that are eigenvectors for the *U*-operator with eigenvalues \(\beta \) and \(\beta '\), respectively. In Theorem 4.3 we show that, for all but one value of \(\alpha \), the *p*-stabilized shadow \(f_\beta \) can be recovered from an iterated application of *U* on \(D^{k-1}({\mathcal {F}}_\alpha )\); the exceptional value of \(\alpha \) yields the precise value in Theorem 1.2. The forms \(f_{\beta }\) and \(f_{\beta '}\) define classes in the *f*-isotypical component of a certain parabolic cohomology group, and in Proposition 3.4 we show that under the assumptions of Theorem 1.2 they form a basis for this space. Writing the class of \(D^{k-1}({\mathcal {F}}_\alpha )\) in terms of this basis, our proof of Theorem 4.3 then follows from an analysis of the action of *U* on cohomology.

## 2 Harmonic Maass forms: the geometric point of view

We begin by briefly recalling the geometric interpretation of harmonic Maass forms given in [4]. For \(N>4\), consider the moduli functor \({\mathscr {M}}_1(N)\) of generalized elliptic curves with a point of order *N*, which is represented by a smooth and proper scheme over \({\mathbb {Z}}[1/N]\). Let \({\mathcal {E}}^\mathrm{{gen}}\rightarrow {\mathscr {M}}_1(N)\) be the universal generalized elliptic curve, and let \({\underline{\omega }}\) be its relative dualizing sheaf. Let \(X:= {\mathscr {M}}_1(N)\times _{{\mathbb {Z}}[1/N]} {\mathbb {Q}}\) and \(Y:= X\smallsetminus C\), where *C* is the cuspidal subscheme, whose ideal sheaf we denote by \({\mathcal {I}}_C\). For any extension \(K/{\mathbb {Q}}\), we denote by \(X_K, Y_K\) the base-change to *K*.

*f*of weight

*k*is identified with the differential \(f(dq/q)^k\). Let \(\pi :{\mathcal {E}}\rightarrow Y\) be the universal elliptic curve with \(\Gamma _1(N)\)-level structure. The relative de Rham cohomology of \(\pi :{\mathcal {E}}\rightarrow Y\) canonically extends to a rank two vector bundle \({\mathcal {H}}^1_{\mathrm{{dR}}}\) over

*X*. Let

*X*, and we let

*r*-th symmetric power. Define

*q*-expansion as \((qd/dq)^{k-1}\). In particular, \(D^{k-1}\) preserves fields of definition.

### Theorem 2.1

*K*be a subfield of \({\mathbb {C}}\) and let \(S_{k}^{!}(\Gamma _1(N),K)\) be the subspace of those modular forms in \(M_k^{!}(\Gamma _1(N),K)\) with vanishing constant coefficient in their

*q*-expansions at the cusps. Then, for all \(k\geqslant 2\) there is a canonical isomorphism:

*f*-isotypical component for this action. Note that this is a 2-dimensional

*K*-vector space. This can be seen by extending scalars to \({\mathbb {C}}\) and then noting that the Shimura isomorphism is compatible under the action of Hecke operators, thus \(M_{\mathrm{dR}}(f)\otimes _{K}{\mathbb {C}} \simeq {\mathbb {C}}[f] \oplus {\mathbb {C}}[{\bar{f}}]\).

*F*is a harmonic Maass form of weight \(2-k\) satisfying

*F*with a class \([\phi ]\) normalized so that \(\langle f,\phi \rangle =1\) under the cup product, one then finds that the constant \(s_2\) in (4) is given

*F*is good for

*f*in the sense of Definition 1.1.

## 3 Overconvergent modular forms

*wide-open neighborhoods*of the

*ordinary locus*\(X^{\mathrm {ord}}\) of

*X*, which is the rigid analytic space characterized by

### Definition 3.1

An *overconvergent modular form* of integral weight *k* is a rigid analytic section of \({\underline{\omega }}^k\) on \(Y_{(\epsilon )}\) for some \(\epsilon <1\).

### Remark 3.2

As shown by Katz [12], sections of \({\underline{\omega }}^k\) over \(X^\mathrm{{ord}}\) are the same as Serre’s *p*-adic modular forms [13] of integral weight *k*, and therefore elements in \(H^0(Y^\mathrm{ord},{\underline{\omega }}^k)\) correspond to *p*-adic modular forms in the sense considered in [1]. As explained in [*loc.cit.*, p. 2394], the latter give rise to Serre’s *p*-adic modular forms upon multiplication by an appropriate power of the modular discriminant \(\Delta \in S_{12}(\mathrm {SL}_2({\mathbb {Z}}))\), and the same argument shows that overconvergent modular forms in the sense of Definition 3.1 give rise to overconvergent modular forms in the sense of Coleman [8].

*W*of \(X^\mathrm{{ord}}\), set \(W^\circ :=W\smallsetminus C\) and define

### Theorem 3.3

*q*-expansions is \((qd/dq)^{r+1}\), and the natural injection

### Proof

See [8, Prop. 4.3] for the construction of \(\theta ^{r+1}\) and [*loc.cit.*, Thm. 5.4] for the last isomorphism. \(\square \)

*q*-expansions is given by the usual formulas

*K*and with \(T_p\)-eigenvalue \(a_p\). This is a section of \({\underline{\omega }}^k\) defined over

*X*, and thus by restriction it gives a section of \({\underline{\omega }}^k\) over \(W_2\) as well. The relation \(T_p=U+\chi (p)p^{k-1}V\) trivially implies that

*p*-

*stabilizations*

*U*-eigenvectors with eigenvalues \(\beta \) and \(\beta '\), respectively. After replacing

*K*by a quadratic extension if necessary, we assume from now on that both \(\beta \) and \(\beta '\) lie in

*K*.

*K*at the prime above

*p*induced by our fixed embedding \(\overline{{\mathbb {Q}}}\hookrightarrow {{\mathbb {C}}}_p\), and set \(M_{\mathrm{dR},p}(f):=M_{\mathrm{dR}}(f)\otimes _{K}K_p\). For any wide-open neighborhood

*W*of \(X^\mathrm{ord}\), the natural restriction

*p*-adic residues, and as a result for any newform

*f*as above, the classes \([f_{\beta }], [f_{\beta '}]\in {\mathbb {H}}^1(W_2^\circ ,{\mathcal {H}}_{k-2})\) naturally lie in \({\mathbb {H}}_{\mathrm{{par}}}^1(X_{K_p},{\mathcal {H}}_{k-2})\). In fact, similarly as \({\mathbb {H}}_{\mathrm{{par}}}^1(X_{K_p},{\mathcal {H}}_{k-2})\), the spaces \({\mathbb {H}}^1(W^\circ ,{\mathcal {H}}_{k-2})\) are endowed with an action of the Hecke operators \(T_\ell \) for \(\ell \not \mid Np\) (see [7, §8]), and the restriction map (6) is equivariant for these actions. Therefore, the classes \([f_{\beta }], [f_{\beta '}]\) naturally lie in \(M_{\mathrm{dR},p}(f)\).

### Proposition 3.4

- (i)
\(\beta \ne \beta '\).

- (ii)
\(f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})\).

### Proof

*f*] and [

*V*(

*f*)] are linearly independent. Since clearly \(v_p(\beta )<k-1\), by [8, Lem. 6.3] we have \([f_\beta ]\ne 0\). Thus, by conditions (i) and (ii) the classes \([f_{\beta }]\) and \([f_{\beta '}]\) are linearly independent. On the other hand, from the definitions (5) we see that

### Remark 3.5

By results of Coleman–Edixhoven [5], condition (i) in Proposition 3.4 holds if \(k=2\), and for \(k>2\) it is a consequence of the semi-simplicity of crystalline Frobenius, which remains an open conjecture. On the other hand, by [8, Prop. 7.1] condition (ii) fails if *f* has CM by an imaginary quadratic field in which *p* splits, and the ‘*p*-adic variational Hodge conjecture’ of Emerton–Mazur (see [10]) predicts that these are the *only* cases where it fails.

## 4 Recovering the shadow

*F*be a harmonic Maass form which is good for

*f*in the sense Definition 1.1. By the construction in Sect. 2, we may assume that

*F*satisfies

In [11], Guerzhoy, Kent, and Ono showed that one of the *p*-stabilizations of *f* can be recovered *p*-adically from an iterated application of *U* to a certain ‘regularization’ of \(D^{k-1}(F^+)\). In this section, we give a new proof of this result using the *p*-adic techniques developed above. We begin by giving a new proof of [*loc.cit.*, Thm. 1.1].

### Theorem 4.1

*K*.

### Proof

*K*. Writing

### Theorem 4.2

### Proof

*k*with

*q*-expansion coefficients in

*K*, hence defining a class in \(M_{\mathrm{dR}}(f)\) (see Theorem 2.1). Our assumptions clearly imply conditions (i) and (ii) of Proposition 3.4, and so (as shown in the proof) the \(K_p\)-vector space \(M_{\mathrm{dR},p}(f)\) has a basis \(\{ [f_{\beta }], [f_{\beta '}] \}\) of eigenvectors for

*U*. In particular, we can write

*U*to both sides of the equation gives

*h*have bounded denominators) the differential \(U^w(\theta ^{k-1}h)\) has coefficients with arbitrarily high valuation as \(w\rightarrow +\infty \).

*n*-th Fourier coefficients in a

*q*-expansion

*g*, and we used the fact that both \(f_\beta \) and \(f_{\beta '}\) are normalized, so that \(a_1(f_{\beta })=a_1(f_{\beta '})=1\). Thus, taking the limit as \(w\rightarrow +\infty \) we obtain

*n*-th coefficient in the expansion

### Theorem 4.3

### Proof

*except*in the case where

## 5 Mock modular forms as overconvergent modular forms

*p*-stabilization. Recall that we let \(\beta \) and \(\beta '\) be the roots of the

*p*-th Hecke polynomial of

*f*, ordered so that \(v_p(\beta )\leqslant v_p(\beta ')\).

### Definition 5.1

Our first result shows that, similarly as in Theorem 4.3 for \({\mathcal {F}}_\alpha \), the *p*-stabilization \(f_{\beta }\) of the shadow of \(F^+\) can be recovered *p*-adically from \({\mathcal {F}}_\alpha ^*\).

### Theorem 5.2

### Proof

*except*in the case where

Considering the exceptional value of \(\alpha \) arising in the proof of Theorem 5.2, we recover a refined form of [1, Thm. 1.1].

### Theorem 5.3

### Proof

Next we consider a second modification of \({\mathcal {F}}_\alpha =\sum _{n\gg -\infty }a_{{\mathcal {F}}_\alpha }(n)q^n\).

### Definition 5.4

Our next result determines the values of \(\alpha \) and \(\delta \) for which \({\mathcal {F}}_{\alpha ,\delta }\) is an overconvergent modular form, recovering a refined form of [1, Thm 1.2(2)].

### Theorem 5.5

### Proof

## 6 The CM case

In this section, we treat the case in which *f* has CM. This case is of special interest, since then one can choose a good harmonic Maass form *F* for *f* as in Section 2 with \(F^+\) having algebraic coefficients.

Thus, assume that \(f=\sum _{n=1}^\infty a_nq^n\in S_k(\Gamma _1(N),K)\) has CM by an imaginary quadratic field *M* of discriminant prime to *p*, and let \(F=F^++F^-\) be a good harmonic Maass form attached to *f*. We also assume (upon enlarging *K* if necessary) that *K* contains a primitive *m*-th root of unity, where \(m=N\cdot \mathrm{disc}(M)\). Then, by [3, Thm. 1.3], \(F^+\) has coefficients in *K*, and so \(D^{k-1}(F^+)\) defines a class in \(M_{\mathrm{dR}}(f)\).

We first treat the case in which *p* is inert in *M*. In this case, \(a_p=\beta +\beta '=0\), and so by the proof of Proposition 3.4, the space \(M_{\mathrm{dR},p}(f)\) admits a basis given by the classes \([f_\beta ]\) and \([f_{\beta '}]\).

### Lemma 6.1

*p*is inert in

*M*, and write \([D^{k-1}(F^+)]=t_1[f_\beta ]+t_2[f_{\beta '}]\) with \(t_1, t_2\in K_p\). Then

### Proof

The proof will be obtained by arguments similar to the proof of Theorem 4.2, but some adjustments are necessary due to the fact that condition \(v_p(\beta )\ne v_p(\beta ')\) clearly does not hold in this case. Instead, we shall exploit the extra symmetry \(\beta '=-\beta \).

### Definition 6.2

Armed with Lemma 6.1, in Corollary 6.4 below we will determine the values of \(\alpha \) for which \(\widetilde{{\mathcal {F}}}_{\alpha }\) is an overconvergent modular form, thus recovering a refined form of [1, Thm. 1.3]. This will be an immediate consequence of the following result.

### Theorem 6.3

*M*, and for any \({\widetilde{\alpha }}\in {\mathbb {C}}_p\) define

### Proof

*f*] and [

*V*(

*f*)] form a basis for \(M_{\mathrm{dR}}(f)\), and rewriting (25) in terms of them we arrive at

*p*-adic limit in the statement. \(\square \)

### Corollary 6.4

*M*. Then, there exists a unique value of \(\alpha \) such that \(\widetilde{{\mathcal {F}}}_\alpha \) is an overconvergent modular form of weight \(2-k\), and it is given by

### Proof

*f*has CM by an imaginary quadratic field

*M*in which

*p*splits, characterizing the values of \(\alpha \in {\mathbb {C}}_p\) for which \({\mathcal {F}}_{\alpha }^*\) is an overconvergent modular form. As noted in Remark 3.5, the class \([f_{\beta '}]\) vanishes in this case, and so the proofs of Theorems 5.2 and 5.3 break down. However, based on the observation that (using the algebraicity of \(c^+(1)\) to set \(\alpha =\gamma \))

### Theorem 6.5

Assume that \(p\not \mid N\) is split in *M*. Then, among all values of \(\alpha \in {\mathbb {C}}_p\), the value \(\alpha =0\) is the unique one for which \({\mathcal {F}}_\alpha ^*\) is an overconvergent modular form of weight \(2-k\).

### Proof

### Acknowledgements

We would like to sincerely thank our Ph.D. advisor Henri Darmon, who generously shared with one of us his ideas on mock modular forms. We would also like to thank Matt Boylan and Pavel Guerzhoy for their comments on an earlier version of this paper, and the anonymous referee for a very careful reading of our manuscript and a number of suggestions that led to significant improvements in the exposition.

## Declarations

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## Authors’ Affiliations

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