Modular Curves and Semistable Reduction

Let $X_0(N)$ be the modular curve associated to the subgroup $\Gamma_0(N)$ of $\SL_2(\mathbf{Z})$ that consists of those matrices which are upper triangular modulo $N$. The algebraic curve $X_0(N)_{\mathbf{C}}$ can be constructed as a Riemann surface as the quotient

$$\Gamma_0(N)\backslash \left(\left\{z : z \in \mathbf{C}, \text{Im}(z)>0\right\}\cup \mathbf{P}^1(\mathbf{Q})\right),$$

and $X_0(N)$ has a canonical structure of algebraic curve over  $\mathbf{Q}$.

It is well known that the $p$-new part of the Jacobian $J_0(N)$ of $X_0(N)$ has purely toric reduction at $p$ when $p\mid\mid N$. Let us briefly recall the reason, writing $N = Mp$. Using the description of closed fibers of modular curves [10, Ch. 13] and Raynaud's result relating Néron models and Picard functors (as summarized in [2, Ch. 9]), the standard finite flat degeneracy maps $X_0(Mp) \rightarrow X_0(M)$ over $\mathbf{Z}_{(p)}$ induce a ``pushfoward'' map on Néron model connected components

$$\Pic^0_{X_0(Mp)/\mathbf{Z}_{(p)}} \longrightarrow \Pic^0_{X_0(M)/\mathbf{Z}_{(p)}} \times \Pic^0_{X_0(M)/\mathbf{Z}_{(p)}}$$

which on the closed fiber is the map induced by pullback to the two components $X_0(M)_{/\mathbf{F}_p}$ in $X_0(Mp)_{/\mathbf{F}_p}$. The kernel of this latter map is a torus [2, Ex. 9.2.8], yet this kernel is visibly isogenous to the semistable mod $p$ fiber of the dual of $J_0(Mp)^{\new}$, whence the purely toric conclusion.