An Arakelov theoretic proof of the equality of conductor and discriminant

Let K be a number field, OK be the ring of integers of K, and S be Spec(OK). Let f : X → S be an arithmetic surface. By this we mean a regular scheme, proper and flat over S, of relative dimension one. We also assume that the generic fiber of X has genus≥ 1, and that X/S has geometrically connected fibers. Let ωX be the dualizing sheaf of X/S. The Mumford isomorphism ([Mumf], Theorem 5.10) det Rf∗(ω ⊗2 X )⊗K → (det Rf∗ωX) ⊗13 ⊗K, which is unique up to sign, gives a rational section ∆ of (det Rf∗ωX) ⊗13 ⊗ (det Rf∗(ω ⊗2 X )) . The discriminant ∆(X) of X/S is defined as the divisor of this rational section ([Saito]). If p is a closed point of S, we denote the coefficient of p in ∆(X) by δp. On the other hand X/S has an Artin conductor Art(X) (cf. [Bloch]), which is similarly a divisor on S. We denote the coefficient of p in Art(X) by Artp. Let S ′ be the strict henselization of complete local ring at p, with field of fractions K . Let s be its special point, η be its generic point, and η be a geometric generic point corresponding to an algebraic closure K ′ of K . Let l be a prime different from the residue characteristic at p. Then