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

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.

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.

In order to state our results precisely, let $\tau = u + iv \in {\mathfrak {h}}$ be the variable in Poincaré’s upper-half plane, with $u, v\in {\mathbb {R}}$, let $\Gamma _0(N)$ be the standard congruence subgroup of $\mathrm {SL}_2({\mathbb {Z}})$ of level 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

$$\begin{aligned} F = F^{+} + F^{-} \end{aligned}$$

into a holomorphic part $F^{+}$ with poles supported at the cusps and a nonholomorphic part $F^{-}$. After Zwegers’ work [15] (see also [14] for an influential overview), the function $F^{+}:{\mathfrak {h}}\rightarrow {\mathbb {C}}$ is called a 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.

As shown in [3], harmonic Maass forms map into classical modular forms via differential operators. Denote by $M^{!}_k(\Gamma _0(N),\chi )$ (resp. $S_k(\Gamma _0(N),\chi )$) the space of weakly holomorphic modular forms (resp. cusp forms) of weight k, level N, and character $\chi$. If for any $w\in {\mathbb {Z}}$, we let

$$\begin{aligned} \xi _w := 2iv^w\overline{\frac{\partial }{\partial {\overline{\tau }}}}, \end{aligned}$$

(1)

then $f:=\xi _{2-k}(F)=\xi _{2-k}(F^-)$ is a cusp form in $S_k(\Gamma _0(N),\chi )$ for all $F\in {\mathcal {H}}_{2-k}(\Gamma _0(N),{\overline{\chi }})$. We say that 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

A harmonic Maass form $F\in {\mathcal {H}}_{2-k}(\Gamma _1(N))$ is 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.

Suppose that $f\in S_k(\Gamma _1(N),K)$ is a (normalized) newform defined over K, let F be a harmonic Maass form that is good for f, and write

$$\begin{aligned} F^{+} = \sum _{n\gg -\infty } c^{+}(n)q^n \end{aligned}$$

for the holomorphic part of 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

$$\begin{aligned} {\mathcal {F}}_{\alpha }:= F^{+} - \alpha E_f =\sum _{n\gg -\infty }c_\alpha (n)q^n \end{aligned}$$

also lie in K. In particular, this applies of course to $\alpha =c^+(1)$.

Now fix a prime $p\not \mid N$, and a choice of complex and 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

$$\begin{aligned} c^+(1)+{\mathbb {C}}_p:=\{c^+(1)+\gamma \;:\;\gamma \in {\mathbb {C}}_p\}, \end{aligned}$$

(2)

the 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.

For the statement, let $\beta$ and $\beta '$ be the roots of the Hecke polynomial of f at p:

$$\begin{aligned} T^2 - a_pT + \chi (p)p^{k-1}=(T-\beta )(T-\beta '), \end{aligned}$$

ordered so that $v_p(\beta )\leqslant v_p(\beta ')$. Let V be the operator acting as $q\mapsto q^p$ on q-expansions.

Theorem 1.2

With the above notations and hypotheses, suppose $v_p(\beta )< v_p(\beta ')$ and $v_p(\beta ')<k-1$, and set ${\mathcal {F}}_\alpha ^*:={\mathcal {F}}_\alpha -p^{1-k}\beta 'V({\mathcal {F}}_\alpha )$. Then, among all values $\alpha \in c^+(1)+{\mathbb {C}}_p$, the value

$$\begin{aligned} \alpha =c^+(1)+(\beta -\beta ')\lim _{w\rightarrow +\infty }\frac{c_{c^+(1)}(p^w)}{\beta ^{w+1}} \end{aligned}$$

is the unique one such that ${\mathcal {F}}_\alpha ^*$ is an overconvergent modular form of weight $2-k$.

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.

We have well-known canonical isomorphisms

$$\begin{aligned} M^{!}_k(\Gamma _1(N), K)\simeq H^0(Y_K, {\underline{\omega }}^k), \quad S_k(\Gamma _1(N), K) \simeq H^0(X_K, {\underline{\omega }}^k\otimes {\mathcal {I}}_C), \end{aligned}$$

where a modular form 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

$$\begin{aligned} {\mathcal {H}}_r:= \mathrm {Sym}^r({\mathcal {H}}^1_{\mathrm {dR}}). \end{aligned}$$

The Gauss–Manin connection of $\pi :{\mathcal {E}}\rightarrow Y$ extends to a connection with logarithmic poles $\nabla :{\mathcal {H}}^1_{\mathrm {dR}}\rightarrow {\mathcal {H}}^1_{\mathrm {dR}}\otimes \Omega ^1_X(\log C)$ over X, and we let

$$\begin{aligned} \nabla _r: {\mathcal {H}}_r \longrightarrow {\mathcal {H}}_r\otimes \Omega ^1_X(\log C) \end{aligned}$$

denote its r-th symmetric power. Define

$$\begin{aligned} {\mathbb {H}}^1_{\mathrm{{par}}}(X,{\mathcal {H}}_r):= {\mathbb {H}}^1(X,{\mathcal {H}}_r\otimes {\mathcal {I}}_C \mathop {\longrightarrow }\limits ^{\nabla _r}{\mathcal {H}}_r\otimes \Omega ^1_X), \end{aligned}$$

(3)

where ${\mathbb {H}}^{\bullet }$ denotes hypercohomology. The formation of ${\mathbb {H}}^1_{\mathrm{{par}}}(X,{\mathcal {H}}_r)$ is compatible with base-change under field extensions $K/{\mathbb {Q}}$, and over ${\mathbb {C}}$ it is canonically isomorphic to the parabolic cohomology group attached to the space of cusp forms of weight $r+2$ and level $\Gamma _1(N)$. In particular, by the Shimura isomorphism (see, e.g., [9, Thm. 2.10], [2]) ${\mathbb {H}}^1_{\mathrm{{par}}}(X_{{\mathbb {C}}},{\mathcal {H}}_r)$ is canonically isomorphic to the direct sum of $S_{r+2}(\Gamma _1(N))$ and its complex conjugate.

More generally, the following second description of ${\mathbb {H}}^1_{\mathrm{{par}}}(X,{\mathcal {H}}_r)$ in terms of modular forms will play an important role here. Recall that for all $k\geqslant 2$ there is a differential operator

$$\begin{aligned} D^{k-1}: M_{2-k}^{!}(\Gamma _1(N)) \longrightarrow M_{k}^{!}(\Gamma _1(N)) \end{aligned}$$

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

Theorem 2.1

([4, Thm. 6]) Let 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:

$$\begin{aligned} {\mathbb {H}}_{\mathrm{{par}}}^1(X_K,{\mathcal {H}}_{k-2}) \simeq \frac{S_{k}^{!}(\Gamma _1(N),K)}{D^{k-1}M_{2-k}^{!}(\Gamma _1(N),K)}. \end{aligned}$$

The spaces ${\mathbb {H}}^1_{\mathrm{{par}}}(X_K,{\mathcal {H}}_{k-2})$ are endowed with an action of the Hecke operators $T_\ell$ for all primes $\ell \not \mid N$, and if $f \in S_{k}(\Gamma _1(N),K)$ is a newform, we let

$$\begin{aligned} M_{\mathrm{dR}}(f):={\mathbb {H}}^1_{\mathrm{{par}}}(X_K,{\mathcal {H}}_{k-2})^f \end{aligned}$$

denote the 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 $M_{\mathrm{dR}}(f)\otimes _{K}{\mathbb {C}} \simeq {\mathbb {C}}[f] \oplus {\mathbb {C}}[{\bar{f}}]$.

Now let $[\phi ]$ be a class in $M_{\mathrm{dR}}(f)$ represented by an element $\phi \in S_{k}^{!}(\Gamma _1(N),K)$ using Theorem 2.1. Extending scalars to ${\mathbb {C}}$, we may write

$$\begin{aligned} {[}\phi ] = s_1[f] + s_2[{\bar{f}}], \end{aligned}$$

(4)

for some $s_1,s_2 \in {\mathbb {C}}$. Let $C^{\infty }_Y$ (resp. ${\mathcal {A}}^1_Y$) be the sheaf of smooth functions (resp. smooth differential forms) on $Y_{{\mathbb {C}}}$. The differential $\phi - s_1f - s_2{\bar{f}}$ is smooth over $Y_{{\mathbb {C}}}$, and it defines a class in

$$\begin{aligned} {\mathbb {H}}^1({\mathcal {H}}_{k-2}\otimes C_Y^{\infty }\xrightarrow {\nabla _{{k-2}}}{\mathcal {H}}_{k-2}\otimes {\mathcal {A}}_Y^{1}) \simeq \frac{H^0(Y_{{\mathbb {C}}},{\mathcal {H}}_{k-2}\otimes {\mathcal {A}}^1_Y)}{ \nabla _{{k-2}}H^0(Y_{{\mathbb {C}}},{\mathcal {H}}_{k-2}\otimes C_Y^{\infty })} \end{aligned}$$

which is trivial by construction. Therefore, there exists a smooth ${\mathcal {H}}_{k-2}$-valued section ${\mathbf {F}}$ such that

$$\begin{aligned} \nabla _{k-2}({\mathbf {F}}) = \phi - s_1f - s_2{\bar{f}}. \end{aligned}$$

The vector bundle ${\mathcal {H}}_{k-2}$ decomposes into line bundles as

$$\begin{aligned} {\mathcal {H}}_{k-2} \simeq {\underline{\omega }}^{2-k}\oplus {\underline{\omega }}^{4-k}\oplus \cdots \oplus {\underline{\omega }}^{k-2}, \end{aligned}$$

and we let $F \in {\underline{\omega }}^{2-k}$ be the projection of ${\mathbf {F}}$ to the first factor. With this construction, it is shown in [4, Prop. 4] that F is a harmonic Maass form of weight $2-k$ satisfying

$$\begin{aligned} D^{k-1}(F^{+}) = \phi - s_1 f, \quad \frac{2i v^{2-k}}{(-4\pi )^{k-1}}\frac{\partial }{\partial {\overline{\tau }}}(F^{-}) = s_2 {\bar{f}}. \end{aligned}$$

Carrying out the above construction of 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

$$\begin{aligned} s_2 = 1/\langle f,{\bar{f}} \rangle = 1/(-4\pi )^{k-1}\Vert f\Vert ^2, \end{aligned}$$

which shows that $\xi _{2-k}(F) = f/\Vert f\Vert ^2$ and F is good for f in the sense of Definition 1.1.

3 Overconvergent modular forms

Let $p\geqslant 5$ be a prime and let ${\mathbb {C}}_p$ be the completion of an algebraic closure of ${\mathbb {Q}}_p$. We fix a valuation $v_p$ on ${\mathbb {C}}_p$ such that $v_p(p)=1$ and an absolute value $|\cdot |$ on ${\mathbb {C}}_p$ compatible with $v_p$. Let $K_p$ be a complete discretely valued subfield of ${\mathbb {C}}_p$ and let $R_p$ be its ring of integers. Suppose $(p,N)=1$, and let ${\mathscr {X}}:= {\mathscr {M}}_1(N)\times _{{\mathbb {Z}}[1/N]} R_p$ be the base-change to $R_p$. Let $E_{p-1} \in H^0({\mathscr {X}}\times _{R_p} K_p,{\underline{\omega }}^{p-1})$ be the global section given by the Eisenstein series of weight $p-1$ and level 1, normalized so that its constant coefficient is 1. As in [8, §1], for any $\epsilon \in |R_p|$ there are rigid analytic spaces $X_{(\epsilon )}$ characterized by

$$\begin{aligned} X_{(\epsilon )}^\mathrm{cl} = \{ x\in ({\mathscr {X}}\times _{R_p} K_p)^\mathrm{{cl}}: |E_{p-1}(x)| > \epsilon \}, \end{aligned}$$

where the superscript ‘cl’ denotes the set of closed points. In the terminology of [6], the spaces $X_{(\epsilon )}$ for $0<\epsilon <1$ are wide-open neighborhoods of the ordinary locus $X^{\mathrm {ord}}$ of X, which is the rigid analytic space characterized by

$$\begin{aligned} (X^{\mathrm {ord}})^\mathrm{cl} = \{ x\in ({\mathscr {X}}\times _{R_p} K_p)^\mathrm{{cl}}: |E_{p-1}(x)| \geqslant 1 \}. \end{aligned}$$

Since $|E_{p-1}(c)| = 1$ for all $c\in C$, we have $C\subseteq X_{(\epsilon )}$ for all $\epsilon \in |R_p|$, and we let

$$\begin{aligned} Y^{{{\mathrm{ord}}}}:= X^\mathrm{{ord}}\smallsetminus C, \quad Y_{(\epsilon )}:= X_{(\epsilon )} \smallsetminus C \end{aligned}$$

be the rigid analytic spaces obtained by removing the cusps. The invertible sheaves ${\underline{\omega }}^k$ restrict to rigid analytic line bundles on these spaces denoted in the same manner.

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].

For any wide-open neighborhood W of $X^\mathrm{{ord}}$, set $W^\circ :=W\smallsetminus C$ and define

$$\begin{aligned} {\mathbb {H}}^1(W^\circ ,{\mathcal {H}}_r):={\mathbb {H}}^1(W^\circ ,{\mathcal {H}}_r\mathop {\longrightarrow }\limits ^{\nabla _{r}}{\mathcal {H}}_r\otimes \Omega ^1_{X}) \simeq \frac{H^0(W^\circ ,{\mathcal {H}}_r\otimes \Omega _X^1)}{\nabla _rH^0(W^\circ ,{\mathcal {H}}_{r})}, \end{aligned}$$

where the isomorphism follows from the fact that $H^q(W^\circ ,{\mathcal {H}})=0$ for $q>0$ and any coherent sheaf ${\mathcal {H}}$ on $W^\circ$. The next two results will play an important role in the proofs of our main results.

Theorem 3.3

(Coleman) For every $r\geqslant 0$, there is linear map

$$\begin{aligned} \theta ^{r+1}:H^0(W^\circ ,{\underline{\omega }}^{-r})\longrightarrow H^0(W^\circ ,{\underline{\omega }}^{r+2}) \end{aligned}$$

whose action on q-expansions is $(qd/dq)^{r+1}$, and the natural injection

$$\begin{aligned} H^0(W^\circ ,{\underline{\omega }}^{r+2})\simeq H^0(W^\circ ,{\underline{\omega }}^r\otimes \Omega _X^1) \hookrightarrow H^0(W^\circ ,{\mathcal {H}}_r\otimes \Omega _X^1) \end{aligned}$$

induces an isomorphism

$$\begin{aligned} {\mathbb {H}}^1(W^\circ ,{\mathcal {H}}_r) \simeq \frac{H^0(W^\circ ,{\underline{\omega }}^{r+2})}{\theta ^{r+1} H^0(W^\circ ,{\underline{\omega }}^{-r})}. \end{aligned}$$

Proof

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

Consider now the wide-open neighborhoods of $X^{\mathrm {ord}}$ given by $W_1 := X_{(p^{-p/p+1})}$ and $W_2 := X_{(p^{-1/p+1})}\subseteq W_1$, and let

be the operators defined in [8, §§2, 3] and whose action on q-expansions is given by the usual formulas

Let $f=\sum _{n=1}^\infty a_nq^n\in S_{k}(\Gamma _0(N),\chi )$ be a newform defined over a number field 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

$$\begin{aligned} a_p f = U(f) + \chi (p)p^{k-1}V(f)\in H^0(W_2,{\underline{\omega }}^{k}), \end{aligned}$$

from which it follows easily that the p-stabilizations

$$\begin{aligned} f_{\beta } := f - \beta ' V(f)\quad f_{\beta '} := f - \beta V(f) \end{aligned}$$

(5)

are 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.

Let $K_p$ be the completion of 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

$$\begin{aligned} {\mathbb {H}}_{\mathrm{{par}}}^1(X_{K_p},{\mathcal {H}}_{k-2})\longrightarrow {\mathbb {H}}^1(W^\circ ,{\mathcal {H}}_{k-2}) \end{aligned}$$

(6)

is injective. (See [6, Thm. 4.2] for the case $k=2$ and [7, Prop. 10.3] for higher weights.) The image of this map can be described in terms of 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

Let $f = \sum _{n=1}^\infty a_nq^n\in S_{k}(\Gamma _0(N),\chi )$ be a newform of weight $k\geqslant 2$, and let $\beta$ and $\beta '$ be the roots of $T^2 -a_pT + \chi (p)p^{k-1}$, ordered so that $v_p(\beta )\leqslant v_p(\beta ')$. Assume that the following two conditions hold:

• (i) $\beta \ne \beta '$.

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

Then $\{[f], [V(f)]\}$ is a basis for $M_{\mathrm{dR},p}(f)$.

Proof

Since $M_{\mathrm{dR},p}(f):=M_{\mathrm{dR}}(f)\otimes _{K}K_p$ is 2-dimensional (see remark after Thm. 2.1), it suffices to show that the classes [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

$$\begin{aligned} \begin{bmatrix} f\\ V(f) \end{bmatrix} =\frac{1}{\beta -\beta '} \begin{bmatrix} \beta&1\\beta'&-1 \end{bmatrix} \begin{bmatrix} f_\beta \\ f_{\beta '} \end{bmatrix}, \end{aligned}$$

and since $\det \left[ {\begin{matrix}\beta &{}1\\ beta'&{}-1\end{matrix}}\right] =\beta '-\beta \ne 0$, the result follows. $\square$

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

Let $f=\sum _{n=1}^{\infty } a_nq^n$ be a normalized newform and let 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

$$\begin{aligned} D^{k-1}(F)=D^{k-1}(F^{+}) = \phi - s_1 f \end{aligned}$$

(7)

for some $\phi \in S_k^{!}(\Gamma _1(N),K)$ and $s_1\in {\mathbb {C}}$.

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

Let $\alpha \in {\mathbb {C}}$ be such that $\alpha -c^+(1)\in K$. Then, the coefficients of

$$\begin{aligned} {\mathcal {F}}_\alpha :=F^+-\alpha E_f:=\sum _{n\gg -\infty }c^+(n)q^n-\alpha \sum _{n=1}^\infty a_nn^{1-k}q^n \end{aligned}$$

are all in K.

Proof

Write $\phi = \sum _{n\gg -\infty } d(n)q^n$, with $d(n)\in K$. By (7), we have the formula

$$\begin{aligned} c^+(n)=\left( \frac{d(n) - s_1 a_n}{n^{k-1}}\right) \end{aligned}$$

(8)

where $a_n:=0$ for $n\leqslant 0$. The result is thus clear for $n\leqslant 0$. Now let $n\geqslant 1$, and write $\alpha =c^+(1)+\gamma$ with $\gamma \in K$, or equivalently, $\alpha =d(1) - s_1 +\gamma$. Using (8), an immediate calculation reveals that the coefficient of $q^n$ in ${\mathcal {F}}_\alpha$ is given by $(d(n)-d(1)-\gamma )n^{1-k}$. $\square$

Since one can always take $\alpha =c^+(1)$ in Theorem 4.1, the coefficients of ${\mathcal {F}}_{c^+(1)}$ are all in K. Writing

$$\begin{aligned} D^{k-1}({\mathcal {F}}_{c^+(1)})= \sum _{n\gg -\infty } c_{c^+(1)}(n)q^n, \end{aligned}$$

we may thus view the coefficients $c_{c^+(1)}(n)$ inside ${\mathbb {C}}_p$ via our fixed embedding $\overline{{\mathbb {Q}}}\hookrightarrow {\mathbb {C}}_p$. Our next result is a special case of [11, Thm. 1.2(i)], but the ideas in the proof of the general case (see Theorem 4.3 below) already appear here.

Theorem 4.2

Assume that $v_p(\beta )<v_p(\beta ')$ and that $f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})$. Then

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{U^wD^{k-1}({\mathcal {F}}_{c^+(1)})}{c_{c^+(1)}(p^w)} = f_{\beta }. \end{aligned}$$

Proof

First note that by Eqs. (7) and (8), we have

$$\begin{aligned} D^{k-1}({\mathcal {F}}_{c^+(1)}) = \phi - d(1) f, \end{aligned}$$

which is a weakly holomorphic cusp form of weight 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

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{c^+(1)})] = t_1 [f_{\beta }] + t_2[f_{\beta '}] \end{aligned}$$

for some constants $t_1, t_2\in K_p$. By restriction, the differential $D^{k-1}({\mathcal {F}}_{c^+(1)}) - t_1 f_{\beta } - t_2f_{\beta '}$ defines a class in ${\mathbb {H}}^1(W_2^\circ ,{\mathcal {H}}_{k-2})\simeq H^0(W_2^\circ ,{\underline{\omega }}^k)/\theta ^{k-1}H^0(W_2^\circ ,{\underline{\omega }}^{2-k})$ which is trivial by construction. Thus, we may write

$$\begin{aligned} D^{k-1}({\mathcal {F}}_{c^+(1)}) = t_1f_{\beta } + t_2f_{\beta '} + \theta ^{k-1}h \end{aligned}$$

for some $h \in H^0(W_2^\circ ,{\underline{\omega }}^{2-k})$. Applying U to both sides of the equation gives

$$\begin{aligned} UD^{k-1}({\mathcal {F}}_{c^+(1)}) = t_1\beta f_{\beta } + t_2 \beta ' f_{\beta '} + U(\theta ^{k-1}h); \end{aligned}$$

and more generally, for any power $w\geqslant 1$, we obtain

$$\begin{aligned} U^wD^{k-1}({\mathcal {F}}_{c^+(1)}) = t_1 \beta ^w f_{\beta } + t_2\beta '^w f_{\beta '} + U^w(\theta ^{k-1}h). \end{aligned}$$

(9)

Dividing by $\beta ^{w}$, we get

$$\begin{aligned} \beta ^{-w} U^wD^{k-1}({\mathcal {F}}_{c^+(1)}) = t_1 f_{\beta } + t_2 \left( \frac{\beta '}{\beta }\right) ^w f_{\beta '} + \beta ^{-w}U^w(\theta ^{k-1}h) \end{aligned}$$

and taking the limit as $w\rightarrow +\infty$ we arrive at

$$\begin{aligned} \lim _{w\rightarrow +\infty } \beta ^{-w} U^wD^{k-1}({\mathcal {F}}_{c^+(1)}) = t_1 f_{\beta }. \end{aligned}$$

Here we used the hypothesis $v_p(\beta '/\beta )>0$, and that fact that (since the coefficients of h have bounded denominators) the differential $U^w(\theta ^{k-1}h)$ has coefficients with arbitrarily high valuation as $w\rightarrow +\infty$.

To determine the value of $t_1$, consider the coefficient of $q^{p^w}$ in (9), which is given by

$$\begin{aligned} c_{c^+(1)}(p^w)=a_{p^w}(D^{k-1}({\mathcal {F}}_{c^+(1)}))&=a_1(U^wD^{k-1}({\mathcal {F}}_{c^+(1)}))\\&=t_1\beta ^w+t_2\beta '^w+O(p^{w(k-1)}), \end{aligned}$$

where we let $a_n(g)$ denote the 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

$$\begin{aligned} \lim _{w\rightarrow +\infty }\beta ^{-w}c_{c^+(1)}(p^w) = t_1 \end{aligned}$$

(10)

which gives the result. $\square$

Now for any $\alpha$ with $\alpha -c^+(1)\in K$, define

$$\begin{aligned} {\mathcal {F}}_\alpha :=F^+-\alpha E_f \end{aligned}$$

and let $c_\alpha (n)$ denote the n-th coefficient in the expansion

$$\begin{aligned} D^{k-1}({\mathcal {F}}_\alpha )=\sum _{n\gg -\infty }c_\alpha (n)q^n. \end{aligned}$$

The following is the content of [11, Thm. 1.2(i)] for primes $p\not \mid N$.

Theorem 4.3

Assume that $v_p(\beta )< v_p(\beta ')$ and that $f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})$. Then for all but at most one choice of $\alpha$ with $\alpha -c^+(1)\in K$, we have

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{U^wD^{k-1}({\mathcal {F}}_\alpha )}{c_\alpha (p^w)}=f_{\beta }. \end{aligned}$$

Proof

As in the proof of Theorem 4.2, we can write

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{c^+(1)})]=t_1[f_\beta ]+t_2[f_{\beta '}] \end{aligned}$$

(11)

in $M_{\mathrm{dR},p}(f)$ with the value of $t_1$ given by (10). Let $\gamma \in K$ be such that $\alpha =c^+(1)+\gamma$, so that ${\mathcal {F}}_\alpha ={\mathcal {F}}_{c^+(1)}-\gamma E_f$ by definition. Noting that

$$\begin{aligned} f=\frac{\beta f_{\beta }-\beta 'f_{\beta '}}{\beta -\beta '}, \end{aligned}$$

(12)

and substituting into the expression (11) with ${\mathcal {F}}_\alpha$ in place of ${\mathcal {F}}_{c^+(1)}$, we obtain

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_\alpha )]=\left( t_1-\gamma \frac{\beta }{\beta -\beta '}\right) [f_\beta ] +\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) [f_{\beta '}], \end{aligned}$$

and hence we have the equality

$$\begin{aligned} D^{k-1}({\mathcal {F}}_\alpha )=\left( t_1-\gamma \frac{\beta }{\beta -\beta '}\right) f_\beta +\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) f_{\beta '}+\theta ^{k-1}h \end{aligned}$$

(13)

as sections in $H^0(W_2^\circ ,{\underline{\omega }}^k)$, for some $h\in H^0(W_2^\circ ,{\underline{\omega }}^{2-k})$. Applying $U^w$ to both sides of this equation and letting $w\rightarrow +\infty$, we deduce that

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{U^wD^{k-1}({\mathcal {F}}_\alpha )}{\beta ^w}=\left( t_1-\gamma \frac{\beta }{\beta -\beta '}\right) f_{\beta } \end{aligned}$$

(14)

as in the proof of Theorem 4.2. On the other hand, arguing again as in Theorem 4.2, we find that the $p^w$-th coefficient of $D^{k-1}({\mathcal {F}}_\alpha )$ is given by

$$\begin{aligned} c_\alpha (p^w)=\left( t_1-\gamma \frac{\beta }{\beta -\beta '}\right) \beta ^w +\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) \beta '^w+O(p^{w(k-1)}), \end{aligned}$$

and hence

$$\begin{aligned} \left( t_1-\gamma \frac{\beta }{\beta -\beta '}\right) =\lim _{w\rightarrow +\infty }\frac{c_\alpha (p^w)}{\beta ^w}. \end{aligned}$$

(15)

Therefore, except in the case where

$$\begin{aligned} \gamma =\frac{t_1(\beta -\beta ')}{\beta }=(\beta -\beta ')\lim _{w\rightarrow +\infty }\frac{c_{c^+(1)}(p^w)}{\beta ^{w+1}}, \end{aligned}$$

(16)

combining (14) and (15) we recover $f_{\beta }$ from ${\mathcal {F}}_\alpha$ as in the statement the theorem. $\square$

5 Mock modular forms as overconvergent modular forms

We now let $\alpha$ range over the larger set of values (2), and interpret the exceptional value of $\alpha$ in Theorem 4.3 as the only value of $\alpha$ for which the ‘regularized’ mock modular form

$$\begin{aligned} {\mathcal {F}}_\alpha =F^+-\alpha E_f \end{aligned}$$

gives rise to an overconvergent modular form (see Definition 3.1) upon 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

For any $\alpha \in c^+(1)+{\mathbb {C}}_p$, define

$$\begin{aligned} {\mathcal {F}}_\alpha ^*:={\mathcal {F}}_\alpha -p^{1-k}\beta '{\mathcal {F}}_\alpha \vert V \end{aligned}$$

and write $D^{k-1}({\mathcal {F}}_\alpha ^*)=\sum _{n\gg -\infty }c_\alpha ^*(n)q^n$.

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

Assume that $v_p(\beta )< v_p(\beta ')$ and that $f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})$. Then, for all but at most one choice of $\alpha \in c^+(1)+{\mathbb {C}}_p$, we have

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{U^wD^{k-1}({\mathcal {F}}_\alpha ^*)}{c_\alpha ^*(p^w)}=f_{\beta }. \end{aligned}$$

Proof

Writing $\alpha =c^+(1)+\gamma$ with $\gamma \in {\mathbb {C}}_p$, an immediate calculation reveals that

$$\begin{aligned} D^{k-1}({\mathcal {F}}_\alpha ^*)=D^{k-1}({\mathcal {F}}_{c^+(1)})\vert (1-\beta 'V)-\gamma f_{\beta }. \end{aligned}$$

(17)

As in the proof of Theorem 4.2, we write

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{c^+(1)})]=t_1[f_\beta ]+t_2[f_{\beta '}] \end{aligned}$$

in $M_{\mathrm{dR},p}(f)$ with $t_1=\lim _{w\rightarrow +\infty }\beta ^{-w}c_{c^+(1)}(p^w)$. Applying the operator $1-\beta 'V$ to this last equality, and noting that $V=U^{-1}$ on cohomology, we obtain

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{c^+(1)})\vert (1-\beta 'V)]=t_1\frac{(\beta -\beta ')}{\beta }[f_{\beta }], \end{aligned}$$

and hence by (17):

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_\alpha ^*)]=\left( \frac{t_1(\beta -\beta ')}{\beta }-\gamma \right) [f_{\beta }]. \end{aligned}$$

(18)

Arguing again as in the proof of Theorem 4.2, we obtain the equalities

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{U^w(D^{k-1}({\mathcal {F}}_\alpha ^*))}{\beta ^w} =\left( \frac{t_1(\beta -\beta ')}{\beta }-\gamma \right) f_{\beta } \end{aligned}$$

(19)

and

$$\begin{aligned} \frac{t_1(\beta -\beta ')}{\beta }-\gamma =\lim _{w\rightarrow +\infty }\frac{c_{\alpha }^*(p^w)}{\beta ^w}. \end{aligned}$$

(20)

Therefore, except in the case where

$$\begin{aligned} \gamma =\frac{t_1(\beta -\beta ')}{\beta }=(\beta -\beta ')\lim _{w\rightarrow +\infty }\frac{c_{c^+(1)}(p^w)}{\beta ^{w+1}}, \end{aligned}$$

(21)

the combination of (14) and (15) recovers $f_{\beta }$ from ${\mathcal {F}}_\alpha ^*$ as in the statement. $\square$

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

Assume that $v_p(\beta )< v_p(\beta ')$ and that $f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})$. Then among all values of $\alpha \in c^+(1)+{\mathbb {C}}_p$, the value

$$\begin{aligned} \alpha =c^+(1)+(\beta -\beta ')\lim _{w\rightarrow +\infty }\frac{c_{c^+(1)}(p^w)}{\beta ^{w+1}} \end{aligned}$$

is the unique one such that ${\mathcal {F}}_\alpha ^*$ is an overconvergent modular form of weight $2-k$.

Proof

Write $\alpha =c^+(1)+\gamma$ with $\gamma \in {\mathbb {C}}_p$. Since $[f_{\beta }]\ne 0\in M_{\mathrm{dR},p}(f)$ (see the proof of Proposition 3.4), we deduce from (18) and (21) that the class of $D^{k-1}({\mathcal {F}}_\alpha ^*)$ in $M_{\mathrm{dR},p}(f)$ vanishes only for the value of $\alpha$ in the statement. Since the restriction map

$$\begin{aligned} {\mathbb {H}}^1_{\mathrm{par}}(X_{K_p},{\mathcal {H}}_{k-2})\longrightarrow {\mathbb {H}}^1(W_2^\circ ,{\mathcal {H}}_{k-2}) \simeq \frac{H^0(W_2^\circ ,{\underline{\omega }}^k)}{\theta ^{k-1}H^0(W_2^\circ ,{\underline{\omega }}^{2-k})} \end{aligned}$$

is injective, the above value of $\alpha$ is also the unique one such that the class of $D^{k-1}({\mathcal {F}}_\alpha ^*)$ becomes trivial in ${\mathbb {H}}^1(W_2^\circ ,{\mathcal {H}}_{k-2})$, and hence such that ${\mathcal {F}}_\alpha ^*\in H^0(W_2^\circ ,{\underline{\omega }}^{2-k})$. $\square$

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

Definition 5.4

For any $\delta \in {\mathbb {C}}_p$, define

$$\begin{aligned} {\mathcal {F}}_{\alpha ,\delta }:={\mathcal {F}}_\alpha -\delta (E_f-\beta E_{f\vert V}). \end{aligned}$$

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

Assume that $v_p(\beta )< v_p(\beta ')$ and that $f_{\beta '}\not \in \mathrm{im}(\theta ^{k-1})$. Then ${\mathcal {F}}_{\alpha ,\delta }$ is an overconvergent modular form for a unique pair $(\alpha ,\delta )$. In fact, $\alpha$ is as in Theorem 5.3, and

$$\begin{aligned} \delta =\lim _{w\rightarrow +\infty }\frac{a_{{\mathcal {F}}_\alpha }(p^w)p^{w(k-1)}}{\beta '^w}. \end{aligned}$$

Proof

With the same notations as in the proof of Theorem 4.3, we can write the equality

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{\alpha ,\delta })]=\left( t_1-\gamma \frac{\beta '}{\beta -\beta '}\right) [f_\beta ] +\left( t_2+\gamma \frac{\beta }{\beta -\beta '}-\delta \right) [f_{\beta '}] \end{aligned}$$

(22)

in $M_{\mathrm{dR},p}(f)$. Since we may check the triviality of these classes upon restriction to $W_2^\circ$, it follows that ${\mathcal {F}}_{\alpha ,\delta }$ is an overconvergent modular form of weight $2-k$ if and only if the class $[D^{k-1}({\mathcal {F}}_{\alpha ,\delta })]$ vanishes. As in the proof of Proposition 3.4, the classes $[f_\beta ], [f_{\beta '}]$ form a basis for $M_{\mathrm{dR},p}(f)$, and hence ${\mathcal {F}}_{\alpha ,\delta }$ is an overconvergent modular form if and only if both coefficients in the right-hand side of (22) vanish. In particular (by the second coefficient), this shows that the value of $\gamma$ is given by (16), and therefore the necessary value of $\alpha =c^+(1)+\gamma$ is the same as in Theorem 5.3.

To determine the value of $\delta$, we rewrite Eq. (13) for the above value of $\alpha$ (so that the first summand in the right-hand side of that equation vanishes):

$$\begin{aligned} D^{k-1}({\mathcal {F}}_\alpha )=\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) f_{\beta '}+\theta ^{k-1}h. \end{aligned}$$

Equating the $p^w$-th coefficients in this equality we obtain

$$\begin{aligned} c_\alpha (p^w)=\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) \beta '^w+O(p^{w(k-1)}), \end{aligned}$$

and hence dividing by $\beta '^w$ and letting $w\rightarrow +\infty$ we deduce

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{c_\alpha (p^w)}{\beta '^w}=\left( t_2+\gamma \frac{\beta '}{\beta -\beta '}\right) . \end{aligned}$$

(23)

(Note that the assumption $v_p(\beta ')<k-1$ is being used here.) Finally, substituting (23) into (22) we see that the necessary value for $\delta$ is given by

$$\begin{aligned} \delta =\lim _{w\rightarrow \infty }\frac{c_\alpha (p^w)}{\beta '^w} =\lim _{w\rightarrow \infty }\frac{a_{{\mathcal {F}}_\alpha }(p^w)p^{w(k-1)}}{\beta '^w}, \end{aligned}$$

as was to be shown. $\square$

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

Assume that 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

$$\begin{aligned} \lim _{w\rightarrow +\infty }\frac{a_{D^{k-1}(F^+)}(p^{2w+1})}{\beta ^{2w+1}}=t_1-t_2. \end{aligned}$$

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 $\beta '=-\beta$.

Upon restriction to $W_2^\circ$, we can write

$$\begin{aligned} D^{k-1}(F^+)=t_1f_\beta +t_2f_{\beta '}+\theta ^{k-1}h \end{aligned}$$

(24)

for some $h\in H^0(W_2^\circ ,{\underline{\omega }}^{2-k})$. Taking $p^{2w+1}$-st coefficients in this identity, we obtain

$$\begin{aligned} a_{D^{k-1}(F^+)}(p^{2w+1})&=t_1\beta ^{2w+1}+t_2\beta '^{2w+1}+O(p^{(2w+1)(k-1)})\\&=(t_1-t_2)\beta ^{2w+1}+O(p^{(2w+1)(k-1)}), \end{aligned}$$

and hence dividing by $\beta ^{2w+1}$ and letting $w\rightarrow +\infty$ the result follows. $\square$

Definition 6.2

For any $\alpha \in {\mathbb {C}}_p$, define

$$\begin{aligned} \widetilde{{\mathcal {F}}}_\alpha :=F^+-\alpha E_{f\vert V}. \end{aligned}$$

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

Assume that $p\not \mid N$ is inert in M, and for any ${\widetilde{\alpha }}\in {\mathbb {C}}_p$ define

$$\begin{aligned} G_{{\widetilde{\alpha }}}:=F^+-{\widetilde{\alpha }}(E_f-\beta E_{f\vert V}). \end{aligned}$$

Then, there exists a unique value of ${\widetilde{\alpha }}$ such that $G_{{\widetilde{\alpha }}}$ is an overconvergent modular form of weight $2-k$, and it is given by

$$\begin{aligned} {\widetilde{\alpha }}=\lim _{w\rightarrow +\infty }\frac{a_{D^{k-1}(F^+)}(p^{2w+1})}{\beta ^{2w+1}}. \end{aligned}$$

Proof

We will deduce this result by first determining the values of $\alpha$ and $\delta$ for which the form ${\mathcal {F}}_{\alpha ,\delta }$ of Definition 5.4 is an overconvergent modular form. Note that this case is not covered by Theorem 5.5, since its proof exploits the assumption that $v_p(\beta )<v_p(\beta ')$. However, $[f_\beta ]$ and $[f_{\beta '}]$ still form a basis for $M_{\mathrm{dR}}(f)$, and so Eq. (22) for $[D^{k-1}({\mathcal {F}}_{\alpha ,\delta })]$ applies, yielding (setting $\gamma =\alpha$ by the algebraicity of $c^+(1)$)

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{\alpha ,\delta })]=\left( t_1-\frac{\alpha }{2}\right) [f_\beta ] +\left( t_2-\frac{\alpha }{2}-\delta \right) [f_{\beta '}]. \end{aligned}$$

(25)

By Theorem 3.4, the classes [f] and [V(f)] form a basis for $M_{\mathrm{dR}}(f)$, and rewriting (25) in terms of them we arrive at

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_{\alpha ,\delta })]=(t_1+t_2-\alpha -\delta )[f]+\beta (t_1-t_2-\alpha -\delta )[V(f)]. \end{aligned}$$

(26)

Now, ${\mathcal {F}}_{\alpha ,\delta }$ is an overconvergent modular form if and only if both coefficients in Eq. (26) vanish; in particular, we need to have

$$\begin{aligned} \alpha +\delta =t_1-t_2=\lim _{w\rightarrow +\infty }\frac{a_{D^{k-1}(F^+)}(p^{2w+1})}{\beta ^{2w+1}}, \end{aligned}$$

(27)

where we used Lemma 6.1 for the second equality. The necessary vanishing of (26) also forces the vanishing of $t_2$ and hence from (25) we deduce that $\delta =-\frac{\alpha }{2}$, or equivalently, $\alpha +\delta =\frac{\alpha }{2}$. Finally, noting that

$$\begin{aligned} {\mathcal {F}}_{\alpha ,\delta }&=F^{+}-\frac{\alpha }{2}\left( E_f-\beta E_{f\vert V}\right) =G_{\frac{\alpha }{2}}, \end{aligned}$$

we conclude from (27) that $G_{{\widetilde{\alpha }}}$ is an overconvergent modular form of weight $2-k$ if and only if ${\widetilde{\alpha }}$ is given by the p-adic limit in the statement. $\square$

Corollary 6.4

Assume that $p\not \mid N$ is inert in 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

$$\begin{aligned} \alpha =\lim _{w\rightarrow +\infty }\frac{a_{D^{k-1}(F^+)}(p^{2w+1})}{\beta ^{2w}}. \end{aligned}$$

Proof

Comparing the definitions of $\widetilde{{\mathcal {F}}}_\alpha$ and $G_{{\widetilde{\alpha }}}$, we see that

$$\begin{aligned} G_{{\widetilde{\alpha }}}=\widetilde{{\mathcal {F}}}_\alpha -{\widetilde{\alpha }}E_f, \end{aligned}$$

with $\alpha ={\widetilde{\alpha }}\beta$. Since $E_f$ is easily seen to be an overconvergent modular form of weight $2-k$ under our running hypotheses (see [1, Prop. 4.2], which remains true in our case $p\not \mid N$), the result follows from Theorem 6.3. $\square$

Finally, we deal with the case in which 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 $\alpha =\gamma$)

$$\begin{aligned} {\mathcal {F}}_\alpha ^*=(F^+-\alpha E_f)\vert (1-p^{1-k}\beta 'V)={\mathcal {F}}_0^*-\alpha E_{f_{\beta }}, \end{aligned}$$

(28)

we can easily prove the following result (cf. [1, Thm. 1.2]).

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

As we have already argued in preceding proofs, ${\mathcal {F}}_{\alpha }^*$ is an overconvergent modular form of weight $2-k$ if and only if the class $[D^{k-1}({\mathcal {F}}_\alpha ^*)]$ vanishes, and from (28) we see that

$$\begin{aligned} {[}D^{k-1}({\mathcal {F}}_\alpha ^*)]=0\quad \Longleftrightarrow \quad \alpha [f_{\beta }]=[D^{k-1}({\mathcal {F}}_0^*)]. \end{aligned}$$

In particular, this shows that ${\mathcal {F}}_\alpha ^*$ is an overconvergent modular form of weight $2-k$ for $\alpha =0$, and so $[D^{k-1}({\mathcal {F}}_0^*)]=0$. On the other hand, since $[f_{\beta }]\ne 0$ (see the proof of Proposition 3.4), the above equivalence shows that $[D^{k-1}({\mathcal {F}}_\alpha ^*)]\ne 0$ for $\alpha \ne 0$, yielding the result. $\square$

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.