Why is it so difficult to use geometric methods to construct weight one modular forms?

Question

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be helpful if someone could give me an overview?

Answer 1

There are no nonzero modular forms of odd weight for $SL(2,\mathbb{Z})$. Because of the invariance of a modular form $f$ under the action of $−id$, where $id$ is the identity matrix in $SL(2,\mathbb{Z})$, it follows that $f$ of odd weight $m$ is zero: $f(z) =(−1)^mf(z)$. So we may assume that $m=2k$ is even. It is easy to see that for $k=0$ there are only the constant modular forms. There is also no modular form of weight $2$ for $SL(2,\mathbb{Z})$, and no modular form of neagtive weight. For each $k>2$ however we have nontrivial modular forms of weight $2k$.

On the other hand, the situation is different in general for congruence subgroups $\Gamma(N)$ of $SL_2(\mathbb{Z})$. There is a large literature on this subject (see also Deligne-Serre construction of Galois representations, geometric constructions using $\ell$-adic cohomology of modular curves $X(N)$, and other topics).

Answer 2

If $\Gamma$ is subgroup of $ \mathrm{SL}_2(\Bbb{Z})$ which $-\mathrm{id}\notin \Gamma$, then there is weight $1$ modular form such as

$$ \mathbb{G}_1(z)=\frac{1}{2}L(0,\chi)+\sum^{\infty}_{n=1}\left(\sum_{d|n}\chi(d)d\right)e^{2\pi i n z}$$

over $ \Gamma(4)$. where $\chi:(\Bbb{Z}/n\Bbb{Z})^{\times}\to \Bbb{C}$ is Dirichlet character. but if $\Gamma=\mathrm{SL}(2,\Bbb{Z})$, there is no modular form of weight $1$.