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 ${\displaystyle k}$ is defined by $${\displaystyle \delta _{k}f(z):={\frac {1}{2\pi i}}\left({\frac {k}{2iy}}+{\frac {\partial }{\partial z}}\right)f(z)}$$ where ${\displaystyle y}$ is the imaginary part of ${\displaystyle z}$.

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

Properties

Maass–Shimura operators raise the weight of a function's modularity by 2. If ${\displaystyle f}$ is modular of weight ${\displaystyle k}$ with respect to a congruence subgroup ${\displaystyle \varGamma \subseteq \mathrm {SL} _{2}(\mathbb {Z} )}$, then ${\displaystyle \delta _{k}f}$ is modular with weight ${\displaystyle k+2}$:^[1] $${\displaystyle (\delta _{k}f)(\gamma z)=(\delta _{k}f(z))(cz+d)^{k+2}\quad {\text{for any }}\gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \varGamma .}$$ However, ${\displaystyle \delta _{k}f}$ 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 ${\displaystyle f}$ and ${\displaystyle g}$ with respective weights ${\displaystyle k}$ and ${\displaystyle \ell }$ (from which it is seen that ${\displaystyle fg}$ is modular with weight ${\displaystyle k+\ell }$), one has $${\displaystyle \delta _{k+\ell }(fg)=(\delta _{k}f)g+f(\delta _{\ell }g).}$$

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

Lanphier showed a relation between the Maass–Shimura and Rankin–Cohen bracket operators:^[3] $${\displaystyle (\delta _{k}^{(n)}f(z))g(z)=\sum _{j=0}^{n}{\frac {(-1)^{j}{\binom {n}{j}}{\binom {k+n-1}{n-j}}}{{\binom {k+\ell +2j-2}{j}}{\binom {k+\ell +n+j-1}{n-j}}}}\delta _{k+\ell +2j}^{(n-j)}([f,g]_{j}(z))}$$ where ${\displaystyle f}$ is a modular form of weight ${\displaystyle k}$ and ${\displaystyle g}$ is a modular form of weight ${\displaystyle \ell }$.