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

One may similarly define Maass–Shimura operators of higher orders, where $$\begin{array}{cc}{\delta }_k^{(n)}& :={\delta }_{k+2n-2}{\delta }_{k+2n-4}\cdots {\delta }_{k+2}{\delta }_k\\ & =\frac{1}{(2\pi i{)}^n}(\frac{k+2n-2}{2iy}+\frac{\partial }{\partial z})(\frac{k+2n-4}{2iy}+\frac{\partial }{\partial z})\cdots (\frac{k+2}{2iy}+\frac{\partial }{\partial z})(\frac{k}{2iy}+\frac{\partial }{\partial z}),\end{array}$$ 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 $𝛤\subseteq {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, then ${\delta }_{k}f$ is modular with weight $k+2$: $$({\delta }_{k}f)(\gamma z)=({\delta }_{k}f(z))(cz+d{)}^{k+2}\text{for any }\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in 𝛤.$$ 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 identity $${\delta }_k^{(n)}=\sum _{r=0}^n(-1{)}^{n-r}(\genfrac{}{}{0ex}{}{n}{r})\frac{(k+r{)}_{n-r}}{(4\pi y{)}^{n-r}}\frac{1}{(2\pi i{)}^r}\frac{{\partial }^r}{\partial z^r}$$ where $(a{)}_m=\mathrm{\Gamma }(a+m)/\mathrm{\Gamma }(a)$ is a Pochhammer symbol for a rising factorial.

Lanphier showed a relation between the Maass–Shimura and Rankin–Cohen bracket operators: $$({\delta }_k^{(n)}f(z))g(z)=\sum _{j=0}^n\frac{(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(\genfrac{}{}{0ex}{}{k+n-1}{n-j})}{(\genfrac{}{}{0ex}{}{k+\ell +2j-2}{j})(\genfrac{}{}{0ex}{}{k+\ell +n+j-1}{n-j})}{\delta }_{k+\ell +2j}^{(n-j)}([f,g{]}_j(z))$$ where $f$ is a modular form of weight $k$ and $g$ is a modular form of weight $\ell$. Additionally, Lanphier showed that the Rankin–Cohen bracket definition can be written in terms of Maass–Shimura operators as $$[f,g{]}_n=\sum _{r+s=n}(-1{)}^r(\genfrac{}{}{0ex}{}{k+n-1}{s})(\genfrac{}{}{0ex}{}{\ell +n-1}{r})({\delta }_k^{(r)}f)({\delta }_{\ell }^{(s)}g).$$