Maass–Shimura operator

In number theory, specifically the study of modular forms, a Maass–Shimura operator is an operator which maps modular forms to almost holomorphic modular forms.

Definition

The Maass–Shimura operator on (almost holomorphic) modular forms of weight $k$ is defined by $$\delta _{k}f(z):={\frac {1}{2\pi i}}\left({\frac {k}{2iy}}+{\frac {\partial }{\partial z}}\right)f(z)$$ where $y$ is the imaginary part of $z$.

One may similarly define Maass–Shimura operators of higher orders, where $$\delta _{k}^{(n)}:=\delta _{k+2n-2}\delta _{k+2n-4}\cdots \delta _{k+2}\delta _{k}={\frac {1}{(2\pi i)^{n}}}\left({\frac {k+2n-2}{2iy}}+{\frac {\partial }{\partial z}}\right)\left({\frac {k+2n-4}{2iy}}+{\frac {\partial }{\partial z}}\right)\cdots \left({\frac {k+2}{2iy}}+{\frac {\partial }{\partial z}}\right)\left({\frac {k}{2iy}}+{\frac {\partial }{\partial z}}\right),$$ and $\delta _{k}^{(0)}$ is taken to be identity.

Properties

Maass–Shimura operators raise the weight of a function's modularity by 2. If $f$ is modular of weight $k$ with respect to a congruence subgroup $\varGamma \subseteq \mathrm {SL} _{2}(\mathbb {Z} )$, then $\delta _{k}f$ is modular with weight $k+2$: $$(\delta _{k}f)(\gamma z)=(\delta _{k}f(z))(cz+d)^{k+2}\quad {\text{for any }}\gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \varGamma .$$ However, $\delta _{k}f$ is not a modular form due to the introduction of a non-holomorphic part.

Maass–Shimura operators follow a product rule: for almost holomorphic modular forms $f$ and $g$ with respective weights $k$ and $\ell$ (from which it is seen that $fg$ is modular with weight $k+\ell$), one has $$\delta _{k+\ell }(fg)=(\delta _{k}f)g+f(\delta _{\ell }g).$$

Using induction, it is seen that the iterated Maass–Shimura operator satisfies the following identity: $$\delta _{k}^{(n)}=\sum _{r=0}^{n}(-1)^{n-r}{\binom {n}{r}}{\frac {(k+r)_{n-r}}{(4\pi y)^{n-r}}}{\frac {1}{(2\pi i)^{r}}}{\frac {\partial ^{r}}{\partial z^{r}}}$$ where $(a)_{m}=\Gamma (a+m)/\Gamma (a)$ is a Pochhammer symbol.

Lanphier showed a relation between the Maass–Shimura and Rankin–Cohen bracket operators: $$(\delta _{k}^{(n)}f(z))g(z)=\sum _{j=0}^{n}{\frac {(-1)^{j}{\binom {n}{j}}{\binom {k+n-1}{n-j}}}{{\binom {k+\ell +2j-2}{j}}{\binom {k+\ell +n+j-1}{n-j}}}}\delta _{k+\ell +2j}^{(n-j)}([f,g]_{j}(z))$$ where $f$ is a modular form of weight $k$ and $g$ is a modular form of weight $\ell$.