Toric ranks and component groups of Jacobians of modular curves

Let p≠2,3 be a prime number and let Γ ⊂ SL₂(Z) be a congruence subgroup with modular curve XΓ/K and Jacobian J(XΓ). In this paper we give an explicit group-theoretic description of the semistable toric rank and component group of J(XΓ) at the finite places of K lying over p. We first produce a suitable deformation retract of the minimal Berkovich skeleton of XH 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), Xₛₚ(N) and Xₛₚ⁺(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(pⁿ) for n≤4. 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.