International Congress of Mathematicians

174 S. S. Kudlaseries with representation numbers of quadratic forms. The most complete exampleis that of anisotropic ternary quadratic forms (n = 1), so that the cycles are curvesand 0-cycles on the arithmetic surfaces associated to Shimura curves. Other surveysof the material discussed here can be found in [11] and [12].2. Shimura curvesLet F be an indefinite quaternion algebra over Q, and let D(B) be the productof the primes p for which B p = B ®Q Q P is a division algebra. The rational vectorspaceV = { x £ B | tr(x) = 0 }with quadratic form given by Q(x) = —x 2 = v(x), where tr(x) (resp. v(x)) is thereduced trace (resp. norm) of x, has signature (1,2). The action of F x on V byconjugationgives an isomorphism G = GSpin(V) ~ F x . LetD = { w £ V(C) | (w,w) = 0, (w,w) < 0 }/C x~ P^C) \P 1 (R)be the associated symmetric space. Let 0 B be a maximal order in F and let F = Oßbe its unit group. The quotient M(C) = Y\D is the set of complex points of theShimura curve M (resp. modular curve, if D(B) = 1) determined by F. This spaceshould be viewed as an orbifold [F\F]. For a more careful discussion of this andof the stack aspect, which we handle loosely here, see [18]. The curve M has acanonical model over Q. From now on, we assume that D(B) > 1, so that M isprojective. Drinfeld's model M for M over Spec (Z) is obtained as the moduli stackfor abelian schemes (.4, i) with an action i : 0 B < L -¥ End(.4) satisfying the 'special'condition, [3]. It is proper of relative dimension 1 over Spec(Z), with semi-stablereduction at all primes and is smooth at all primes p at which F splits, i.e., forp\D(B). We view M. as an arithmetic surface in the sense of Arakelov theory andconsider its arithmetic Chow groups with real coefficients CH (A4) = CH R (A4),as defined in [2]. Recall that these groups are generated by pairs (Z,g), where Zis an R-linear combination of divisors on A4 and g is a Green function for Z, withrelations given by R-linear combinations of elements div (/) = (div(/), ^log|/| 2 )where / £ Q(M) X is a nonzero rational function on A4. These real vector spacescome equipped with a geometric degree map deg Q : CH (A4) —^ CH 1 (MQ) —^ R,where MQ is the generic fiber of A4, an arithmetic degree map deg : CH (A4) —¥ R,and the Gillet-Soulé height pairing, [2],( , ) : CF^M) x CF^M) —• R.Let A be the universal abelian scheme over M. Then the Hodge line bundle u =e*(0^,_ M ) determined by A has a natural metric, normalized as in [18], section3, and defines an element û £ Pic(^M), the group of metrized line bundles on