The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

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:

1. $f$ is $C^\infty$;
2. $f$ is invariant under the weight $k$ action of $\Gamma_1(N)$;
3. $f$ 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, $y = \frac{\tau - \overline{\tau}}{2}$;
4. $f$ 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!

• Regarding the very last bullet - similar half-derivatives appeared in a comment by @rlo to one of my questions. Partial information about its level and the "deviation from modularity" is in my answer there. On the whole the key concepts here must be (imo) related to the treatment of mock modular forms in Zwegers' thesis. – მამუკა ჯიბლაძე Sep 3 '15 at 8:03