Maass–Shimura operator

The Maass–Shimura operator on (almost holomorphic) modular forms of weight $k$ is defined by

$$\backslash delta\_kf(z):=\backslash frac\{1\}\{2\backslash pi\; i\}\backslash left(\backslash frac\{k\}\{2iy\}+\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\backslash right)f(z)$$

where $y$ is the imaginary part of $z$.

One may similarly define Maass–Shimura operators of higher orders, where

$$\backslash delta\_k^\{(n)\}:=\backslash delta\_\{k+2n-2\}\backslash delta\_\{k+2n-4\}\backslash cdots\backslash delta\_\{k+2\}\backslash delta\_k=\backslash frac\{1\}\{(2\backslash pi\; i)^n\}\backslash left(\backslash frac\{k+2n-2\}\{2iy\}+\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\backslash right)\backslash left(\backslash frac\{k+2n-4\}\{2iy\}+\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\backslash right)\backslash cdots\backslash left(\backslash frac\{k+2\}\{2iy\}+\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\backslash right)\backslash left(\backslash frac\{k\}\{2iy\}+\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\backslash right),$$

and $\backslash delta\_k^\{(0)\}$ is taken to be identity.

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 $\backslash varGamma\backslash subseteq\backslash mathrm\{SL\}\_2(\backslash Z)$, then $\backslash delta\_kf$ is modular with weight $k+2$:{{Cite journal |last=Shimura |first=Goro |authorlink=Goro Shimura |year=1975 |title=On some arithmetic properties of modular forms of one and several variables |journal=Annals of Mathematics |volume=102 |pages=491–515 |doi=10.2307/1971041}}

However, $\backslash delta\_kf$ 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 $\backslash ell$ (from which it is seen that $fg$ is modular with weight $k+\backslash ell$), one has

$$\backslash delta\_\{k+\backslash ell\}(fg)=(\backslash delta\_kf)g+f(\backslash delta\_\backslash ell\; g).$$

Using induction, it is seen that the iterated Maass–Shimura operator satisfies the following identity:

$$\backslash delta\_k^\{(n)\}=\backslash sum\_\{r=0\}^n(-1)^\{n-r\}\backslash binom\{n\}\{r\}\backslash frac\{(k+r)\_\{n-r\}\}\{(4\backslash pi\; y)^\{n-r\}\}\backslash frac\{1\}\{(2\backslash pi\; i)^r\}\backslash frac\{\backslash partial^r\}\{\backslash partial\; z^r\}$$

where $(a)\_m=\backslash Gamma(a+m)/\backslash Gamma(a)$ is a Pochhammer symbol.{{cite book |last=Zagier |first=Don |authorlink=Don Zagier |chapter=Elliptic Modular Forms and Their Applications |title=The 1-2-3 of Modular Forms |publisher=Springer |year=2008}}

Lanphier showed a relation between the Maass–Shimura and Rankin–Cohen bracket operators:{{Cite journal |last=Lanphier |first=Dominic |year=2008 |title=Combinatorics of Maass–Shimura operators |journal=Journal of Number Theory |volume=128 |issue=8 |pages=2467–2487 |doi=10.1016/j.jnt.2007.10.010 |doi-access=free}}

$$(\backslash delta\_k^\{(n)\}f(z))g(z)=\backslash sum\_\{j=0\}^n\backslash frac\{(-1)^j\backslash binom\{n\}\{j\}\backslash binom\{k+n-1\}\{n-j\}\}\{\backslash binom\{k+\backslash ell+2j-2\}\{j\}\backslash binom\{k+\backslash ell+n+j-1\}\{n-j\}\}\backslash delta\_\{k+\backslash ell+2j\}^\{(n-j)\}([f,g]\_j(z))$$

where $f$ is a modular form of weight $k$ and $g$ is a modular form of weight $\backslash ell$.

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Category:Modular forms