INTERSECTION THEORY OF DIVISORS ON AN ARITHMETIC SURFACE

In this article it is explained how to construct for a nonsingular model of a curve defined over a number field a theory analogous to the theory of divisors, and the intersection numbers of divisors, on a compact algebraic surface. Introduction In this article we define the notion of a divisor class on the nonsingular model of an algebraic curve defined over an algebraic number field K. This notion replaces the usual one of the class of divisors on an ordinary algebraic variety with respect to linear equivalence. By a method analogous to that used by Neron we can construct an intersection theory for such classes which is fairly geometric in spirit, and which is reminiscent of intersection theory on an algebraic surface. §0. The definition Let Κ be a number field, ACfi the ring of integers. The symbol oo will denote any embedding of Κ in the field of complex numbers; there are thus η infinities, where η = [Κ: Q], and Σ Μ will denote summation over all of these infinities. Consider a smooth complete curve X over K. Χχ will denote the Riemann surface of the curve X^® C. Let V be any smooth and complete model of X over Λ. By a finite divisor on V we will mean a divisor in the usual sense 'where k{ are integers, and C.C V ace irreducible closed subsets of codimension 1. The curves XM are the fibers of the morphism /: V -» Spec Λ over the places of Κ at infinity. We will call these the components at infinity; by definition these components can occur in a divisor with real coefficients: AMS (MOS) subject classifications (1970). Primary 14G25, 141125; Secondary 53C55. Copyright £ 1976. American Mathematical Society 1167 1168 S. Ju. ARAKELOV D = 2 kid -v 2 λ,χ» = Dlia with έ. e Ζ, λ^ e R. Divisors of this type form a group, which we denote by Div'(V). We can also consider the group Div"(K)— this will be the group of divisors of the form Ρ where the £ are the prime ideals of the field K, mf el and \ x e R. In this group the principal divisors are, by definition, divisors of the form S ι>ρ(α). P + 2(-log|a|J, ρ oo with a e K, and la^l = |oo(a)|. The degree of the divisor d is the real number degd=2P °S^ + 2 λ. ; the degree of the principal divisor is ρ °° by the product formula. W« now define the principal divisor of an element / e K(X). For every », / is a meromorphic function on the Riemann surface X^, and we want to define ι/χ^/) as the mean values of (log |/|) on XM, For this, consider and fix on each Χχ a Hermitian metric ds^; we will denote the corresponding volume element by Ίμ^· We will suppose that άμχ is normalized in such a way that /χ άμχ = 1. Define v^f) OO by the formula f By definition, a principal divisor is where (/),. is the usual divisor of a rational function on the two-dimensional scheme V. Let P"(V) be the group of principal divisors. We define ΟΓ(Κ) =Div~(V)/P'(VA. We can fix up the same sort of group for Spec K: CT{Ki = Oiv~(K)/P~(K). We have homomorphisms Cl" V -» Cl V and Ci~ Κ -» Cl Κ of the groups just defined into the usual divisor class groups. Both of these homomorphisms are surjective, and DIVISORS ON AN ARITHMETIC SURFACE 1169 have the same kernel, which we denote by G. G is the quotient group of real «-dimensional space by the image of the group of units of Κ under the logarithm map. Of course, it might be natural to identify complex conjugate embeddings of Κ and consider conjugate metr ics άμ^, but it seems to us that this i s necessary only in more detailed study of the situation, and for the time being there is no part icular need for i t . This means that the group G turns out a bit bigger than one might have expected. Finally, note that we have the inverse image homomorphism /*: Cl Κ -» C1"(V). Thus the group G comes from below. § 1 . Intersection numbers Firs t ly we recal l the definition of the usual intersection numbers of two finite divisors D χ and Dγ Let f χ and / 2 be local equations for t h e s e , and let Ρ e V be a O-dimensional point of V; let f(P) = \i e Spec Λ. Then the local intersect ion number (D j · D2)p i s defined by (Dt-Dt),=l{O,,rl{fu h))-\ogN (P), where N(P) is the number of elements of the field k(P) = 0 p v/mp v , and l(M) denotes the length of the 0p y-module M. If one of the divisors, say D2, is irreducible and horizontal (that is, f(D2) = Spec A), then it can be written in the form £>2 = f(Spec Γ), where Γ is the ring of integers of a field L finite over K, and i: Spec Γ -» V is a morphism over SpecA. If ) 0 j , . , b, are the prime ideals of Γ lying over the point P, then