Modular Forms

Fix a positive integer $N$. The group

$$\Gamma_0(N) = \left\{ \left(\begin{matrix}a&b c&d\end{matrix}\right) \in \SL_2(\mathbf{Z}) : N \mid c \right\}$$

acts by linear fractional transformations on the extended complex upper halfplane $\mathfrak{h}^*$. As a Riemann surface, $X_0(N)(\mathbf{C})$ is the quotient $\Gamma_0(N)\backslash{}\mathfrak{h}^*$. There is a standard model for $X_0(N)$ over  $\mathbf{Q}$ (see [Shi94, Ch. 6]), and the Jacobian $J_0(N)$ of $X_0(N)$ is an abelian variety over  $\mathbf{Q}$ of dimension equal to the genus $g$ of $X_0(N)$, which is equipped with an action of the Hecke algebra $\mathbf{T}= \mathbf{Z}[\ldots T_n \ldots]$. The space $S_2(\Gamma_0(N))$ of cuspforms of weight $2$ on $\Gamma_0(N)$ is a module over  $\mathbf{T}$ and $S_2(\Gamma_0(N))\cong H^0(X_0(N),\Omega_{X_0(N)})$ as $\mathbf{T}$-modules.