Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
I think that it is well-known that a function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ of dimension $-2$ if:
- $f(g(z)) = (cz+d)^2 f(z)$, for all $z \in \mathbb{H}, g\in \Gamma$.
- $f$ is holomorphic in $\mathbb{H}$.
- $f$ is holomorphic at the cusps of $\Gamma$.
Let $C$ be the set of all cusps of $\Gamma$.
From the Riemann-Roch theorem, it is known that the dimension of the space of entire modular forms is $|C|-1+g$ (where $g$ is the genus of the compact Riemann surface $R= (\mathbb{H}\cup C)/\Gamma$).
It is then mentioned that The $\mathbb{R}-$dimension of the space $M_{\mathbb{R}}$ of entire forms taking real values at the cusps is $2g+|C|–1$.
How could this be done?
I mean how the space $M_{\mathbb{R}}$ of entire forms taking real values at the cusps, that is a subspace of the space of entire modular forms of dimension $-2$, has dimension bigger than the original space?
Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. Page 26.
Best Answer
The easy answer to your question as stated, not telling much because the important clues/theorems are left hidden in your statement, is that $g$ is the (complex vector space) dimension of the holomorphic 1-forms on $(\mathbb{H}\cup C)/\Gamma$, which is the same as $S_2(\Gamma)$, which is the kernel of the map $$f\mapsto (\Gamma c(i\infty)\in C\mapsto f|_2 \Gamma c(i\infty))$$ sending $f$ to its values at the cusps.
$\sum_{c\in C} f|_2 c(i\infty)=0 $ because otherwise $\sum_{\gamma\in \Gamma\backslash SL_2(\Bbb{Z})} f|_2 \gamma$ would be non-zero in $M_2(SL_2(\Bbb{Z}))$.
Whence for any set of values at the cusps summing to zero there is some $f\in M_2(\Gamma)$ taking those values (since otherwise $\dim_\Bbb{C}M_2(\Gamma)$ would be less than $g+|C|-1$).
And hence with $M_2(\Gamma)_\Bbb{R}$ the real vector subspace of $M_2(\Gamma)$ taking real values at the cusps we get that $M_2(\Gamma)_\Bbb{R}/ S_2(\Gamma) \cong \Bbb{R}^{|C|-1}$ ie. $\dim_\Bbb{R}(M_2(\Gamma)_\Bbb{R})=2g + |C|-1$