Recall the notion of a "nearly holomorphic modular form" introduced by Shimura:
A function $f : \mathfrak h \to \mathbb C$ is said to be nearly holomorphic of level $\Gamma_1(N)$, weight $k$ and order $\leq r$ if it satisfies the following conditions:
- $f$ is $C^\infty$;
- $f$ is invariant under the weight $k$ action of $\Gamma_1(N)$;
- $f$ can be written as \begin{equation*} f(\tau) = f_0(\tau) + \frac{1}{y} f_1(\tau) + \dots + \frac{1}{y^r}f_r(\tau) \end{equation*} where the $f_j(\tau)$ are holomorphic, $y = \frac{\tau - \overline{\tau}}{2}$;
- $f$ has finite limits at the cusps.
Alternatively, a nearly holomorphic modular form can be viewed as a global section of the vector bundle $\mathcal H^r_k = \underline{\omega}^{\otimes(k-r)} \otimes \mathcal H^{\otimes k}$ on the modular curve $X_1(N)$, where $\mathcal H =\mathcal H^1_{dR}(\mathcal E/X_1(N))$ is the relative de Rham cohomology of the universal semi-abelian scheme $\mathcal E$ over $X_1(N)$ (with suitable care taken to define $\mathcal H$ properly at the cusps).
Let $\mathcal N_k^r$ denotes the $\mathbb C$-vector space of such forms. It is a finite dimensional complex vector space, we have $\mathcal N_k^r \subseteq {\mathcal N_k^{r+1}}$ and we define $\mathcal N_k^\infty = \cup_{r\geq 0}\mathcal N_k^r$.
By invariance of $f$ under $\Gamma_1(N)$, and from the invariance of the function $y$ under $\tau \mapsto \tau+1$, we deduce from the expansion (3) (which is unique) that the individual functions $f_j(\tau)$ are invariant under $\tau \mapsto \tau+1$, so they admit $q$-expansions. Furthermore, one can show that the map $\mathcal N_k^\infty \to \mathbb C[[q]]$ given by $f \mapsto f_0(q)$ is injective. Therefore, it is reasonable to call $f_0(q)$ "the" $q$-expansion of the nearly holomorphic modular form $f$. Although a priori this $q$-expansion would have been an element of $\mathbb C[[q]][1/y]$, the injectivity of $f \mapsto f_0$, motivates the following definition (a special case of a definition of Urban in the setting of nearly overconvergent modular forms)
A power series $f_0 \in \mathbb C[[q]]$ is said to be the $q$-expansion of a nearly holomorphic modular form of level $\Gamma_1(N)$, weight $k$ and order $r$ if there exists a nearly holomorphic form $f$ of weight $k$ and order $r$ such that $f_0(q)$ is the "constant term" of $f$ in its expansion as a polynomial in $1/y$ with coefficients in $\mathbb C[[q]]$.
The Maass-Shimura derivative $\delta$ is a degree $2$ derivation of the graded ring $\bigoplus_{k=0}^\infty \mathcal N_k^\infty$, given on a form of weight $k$ by
$$\delta_k f = \frac{1}{2\pi i}\left(\frac{\partial f}{\partial \tau} + \frac{k}{2iy}f\right).$$
Since $d/d \tau = 2 \pi i q \times d/d q$, we see that $\delta_k$ acts on $q$-expansions by
$$(\delta_k f)_0(q) = q\frac{d}{dq}f_0(q).$$
Let us write $\theta = q\frac{d}{dq}$. This is Serre's differential operator on $p$-adic modular forms. Remark that its effect on $q$-expansions is independent of the weight $k$; and so for an integer $t$, the operator $\theta^t$ on $\mathbb C[[q]]$ corresponds to $\delta^t$.
Remark that $\theta^t$ acts on $q$-expansions by
$$\sum a_n q^n \mapsto \sum n^ta_n q^n.$$
This expression has been used (I believe first by Darmon and Rotger) to define $p$-adic powers of the Serre operator. If we $p$-deplete a holomorphic modular form by deleting the Fourier coefficients with index divisible by $p$, in other words
$$f^{[p]}(q) = f(q)-VUf(q) = \sum_{n \not\mid p} a_n q^n$$
then we can use the fact that the function $t \mapsto n^t$ interpolates $p$-adically when $(n,p)=1$. Thus we can view $\theta^t f^{[p]}$ as a family of $p$-adic modular forms passing through $f^{[p]}$, and depending on the $p$-adic parameter $t$. This embodies the idea that if $f$ is an eigenform, then we can twist the Galois representation $V(f)$ attached to $f$ by a power of the cyclotomic character which depends on the same $p$-adic variable $t$, yielding a family of Galois representations which specializes to $V(f)$ at $t=0$.
My questions: What about complex powers of the Serre operator? If $f \in \mathbb C[[q]]$, and $s$ is a complex number, then we have an operator
$$\theta^s : \sum a_n q^n \mapsto \sum n^s a_n q^n$$
which preserves the ring of holomorphic functions on the open unit disc $\mathcal D$, and satisfies $\theta^s \theta^{s'} = \theta^{s+s'}$.
If $f_0(q)$ is the $q$-expansion of a nearly holomorphic modular form $f$, we may view $\theta^sf_0(q) = \sum n^s a_n q^n$ as a holomorphic function on $\mathbb C \times \mathcal D$. Its specialization at an integer $s=t$ is the $q$-expansion of the nearly holomorphic modular form $\delta^t f$. In other words the Maass-Shimura derivatives of $f$ lie naturally in a complex-analytic family, just as the Maass-Shimura derivatives of $f^{[p]}$ lie naturally in a $p$-adic family.
If $s$ is not an integer, and $f$ is a holomorphic modular form, what kind of object is the holomorphic function $\theta^s f$? In the $p$-adic world, if $s$ were a $p$-adic variable, it would be a $p$-adic modular form (of non-integral weight). But in the complex world, I have no idea what it is. Can it be viewed as a section of a sheaf on $X_1(N)$? This seems like it would require doing something akin to taking complex powers of the Hodge bundle $\underline{\omega}$ on $X_1(N)$, and I'm doubtful that such a thing can be done.
In particular the operator $\theta$ has $n$-th roots for any $n$. Is there a meaningful sense in which the object $\theta^{1/2} f$ has weight $k+1$ if $f$ has weight $k$ (apart from the fact that its coefficients have the growth rate expected of a modular form of weight $k+1$)?
Any thoughts are most welcome!
