Let $p\neq{ 2,3}$ be a prime number and let $\Gamma \subset \mathrm{ SL}_{ 2}(\mathbb{ Z})$ be a congruence subgroup with modular curve $X_{ \Gamma}/K$ and Jacobian $J(X_{ \Gamma})$. In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of $J(X_{ \Gamma})$ at the finite places of $K$ lying over $p$. We first produce a suitable deformation retract of the minimal Berkovich skeleton of $X_{ \Gamma}$ in terms of Hecke-Iwahori double coset spaces. We call this deformation retract the pruned skeleton of the curve. Our description of this skeleton includes a group-theoretic formula for the edge lengths, allowing us to give the component group of the modular curve as the quotient of a lattice using the monodromy pairing. For $X_{ 0}(N)$, $X_{ 1}(N)$, $X_{ sp}(N)$ and $X_{ sp}^{ +}(N)$, we explicitly determine the pruned skeleta using a set of coset schemes over $\mathbb{ Z}$. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of $X_{ 0}(p^{ n})$ for $n\leq{ 4}$. Finally, we determine the geometric Tamagawa number and the prime-to-$2$ structure of the component group of $X_{ 0}(N)$ over the extension given by Krir's theorem.
Categories: math.NT math.AG