Harmonic $1-$ form on the upper half-plane $\mathbb{H}$

complex-analysisdifferential-formshyperbolic-geometrymodular-formsmodular-group

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.

A function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ of dimension $-2$ if the following conditions are satisfied:

  1. $f(g(z)) = (cz+d)^2 f(z)$, for all $z \in \mathbb{H}, g\in \Gamma$.
  2. $f$ is holomorphic in $\mathbb{H}$.
  3. $f$ is holomorphic at the cusps of $\Gamma$.

Let $C$ be the set of all cusps of $\Gamma$.

Let $f$ be an entire modular form of dimension $-2$ taking real values at the cusps. It is then mentioned that the real part of $f(z)dz$ is a harmonic $1-$ form on the upper half-plane $\mathbb{H}$.

The questions are:

  1. How is $f(z)dz$ defined?
    Is $f(z)dz=f(z)dx+if(z)dy$? Or what?
  2. what does harmonic $1-$ form mean?

Best Answer

Yes, now write $f=u+iv$ and read off the real part.

A harmonic $1$-form $\omega$ is closed and co-closed. That is, $d\omega=d{\star}\omega=0$. The metric is the usual hyperbolic metric on $\Bbb H$, we assume.

Related Question