An arithmetic hilbert-samuel theorem for pointed stable curves

Let .O,∑ ∞/ be an arithmetic ring of Krull dimension at most 1, S D SpecO and .X → S; _o1;o _n;)n/ a pointed stable curve. Write U D =u oi (S) For every integer k > 0, the invertible sheaf ω ^k+1 X=S.ko1 + +kσn inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface U∞. In this article we define a Quillen type metric || || Q on the determinant line λk+1=λ & omega;D k+1 Xi=S.kσ ₁+ +kσ))and compute the arithmetic degree of .λk+1; || ||kQ/ by means of an analogue of the Riemann-Roch theorem in Arakelov geometry. As a byproduct, we obtain an arithmetic Hilbert-Samuel formula: the arithmetic degree of.λk+1; || || _L2|| || kL2 / admits an asymptotic expansion in k, whose leading coefficient is given by the arithmetic self-intersection of omega;XS.( σ1 +σn), hyp|| ||).Here || || kL ² and || ||hyp denote the L2 metric and the dual of the hyperbolic metric, respectively. Examples of application are given for pointed stable curves of genus 0.