Faltings Modular Height and Self-intersection of Dualizing Sheaf

is finite under the following equivalence (cf. Theorem 3.1). For stable curves X and Y over OK , X is equivalent to Y if X ⊗OK OK′ ≃ Y ⊗OK OK′ for some finite extension field K ′ of K. In §1, we will consider semistability of the kernel of H(C,L) ⊗ OC → L, which gives a generalization of [PR]. In §2, an inequality of self-intersection and height will be treated. Finally, §3 is devoted to finiteness of stable arithmetic surfaces with bounded self-intersections of dualizing sheaves We would like to thank Professor J.-B. Bost, S. Lang and L. Szpiro for their helpful comments.