An arithmetic riemann-roch theorem for pointed stable curves

Let (O, σ, F_∞) be an arithmetic ring of Krull dimension at most 1,S = SpecO and (π: X → S; σ₁, ⋯, σ_n) an n-pointed stable curve of genus g. Write U = X \ U_jσ_j (S). The invertible sheaf w_x/s (σ₁ + · · · + σ_n) inherits a hermitian structure ∥ · ∥_hyp from the dual of the hyperbolic metric on the Riemann surface U_∞. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of w_x/s(σ₁ + ·+ σ_n)_hyp. The theorem is applied to modular curves X(γ), γ = γ₀(p) or γ₁(p), p ≤ 11 prime, with sections given by the cusps. We show Z′(Y(γ),1) ∼ e^aπ^b γ2(1/2)^cL(0, M _γ), with p ≡ 11 mod 12 when σ = σ₀(p). Here Z(Y(σ),s) is the Selberg zeta function of the open modular curve Y(σ), a, b, c are rational numbers, M _σ is a suitable Chow motive and ∼ means equality up to algebraic unit.