Toric ranks and component groups of Jacobians of modular curves

Let p2, 3 be a prime number and let SL₂ (Z) be a congruence subgroup with modular curve X_/K and Jacobian J (X_). In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of J (X_) at the finite places of K lying over p. We first produce a suitable deformation retract of the minimal Berkovich skeleton of X₇ 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 X₀ (N), X₁ (N), Xₒ₏ (N) and Xₒ₏^+ (N), we explicitly determine the pruned skeleta using a set of coset schemes over Z. This in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima on the semistable reduction type of X₀ (p^n) for n4. Finally, we determine the geometric Tamagawa number and the prime-to-2 structure of the component group of X₀ (N) over the extension given by Krir's theorem.