Toric ranks and component groups of modular curves

Abstract

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.