AN ARITHMETIC RIEMANN-ROCH THEOREM FOR POINTED STABLE CURVES

Let (O, Sigma, F-infinity) be an arithmetic ring of Krull dimension at most 1, S = SpecO and (pi : chi -> S; sigma(1), ..., sigma(n)) an n-pointed stable curve of genus g. Write U = chi\U-j sigma(j)(S). The invertible sheaf omega(chi/s)(sigma(1) + ... + sigma(n)) inherits a hermitian structure parallel to.parallel to(hyp) from the dual of the hyperbolic metric on the Riemann surface U-infinity. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of omega(chi/s)(sigma(1) + ... + sigma(n))(hyp). The theorem is applied to modular curves X(Gamma), Gamma = Gamma(0)(p) or Gamma(1)(p), p >= 11 prime, with sections given by the cusps. We show Z'(Y(Gamma), 1) similar to e(a)pi(b)Gamma(2)(1/2)L-c(0, M-Gamma), with p equivalent to 11 mod 12 when Gamma = Gamma(0)(p). Here Z(Y(Gamma), s) is the Selberg zeta function of the open modular curve Y(Gamma), a, b, c are rational numbers, M-Gamma is a Suitable Chow motive and similar to means equality up to algebraic unit.