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Line Bundles on Arithmetic Surfaces and Intersection Theory

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Line Bundles on Arithmetic Surfaces and Intersection Theory Line bundles on arithmetic surfaces and intersection theory J¨org Jahnel∗ Abstract For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by Θ. We recover the classical formula due to Faltings and Hriljac for the N´eron-Tate height on the Jacobian in terms of the intersection pairing on the arithmetic surface. 1 Introduction In this paper we will suggest a construction for height functions for line bundles on arithmetic varieties. Following the philosophy of [BoGS] heights should be objects in arithmetic geometry analogous to degrees in algebraic geometry. So let K be a number field, OK its ring of integers and X /OK an arithmetic variety by which we will mean a scheme, projective and flat over OK . In order to have a good intersection product available we assume X to be regular. Its generic fiber will be denoted by X/K. Then we have to fix a metrized line bundle (T , k.k) or, equivalently, its first Chern class ∧ ∧ 1 c1 (T , k.k) = (T, gT ) ∈ CH (X ). The height of a line bundle L on X should be the arithmetic degree of the ∧ dim X intersection of c1 (L) with (T, gT ) . For this a natural hermitian metric has to be chosen on L. We fix a K¨ahlermetric ω0 on X (C), invariant under complex conjugation F∞, as in [Ar]. Then it is well known that the condition on the Chern form to be harmonic defines k.k up to a locally constant factor. In order to determine this factor we require ∧ ∧ deg c1 det Rπ∗L, k.kQ = 0. Here π : X −→ Spec OK is the structural morphism and k.kQ is Quillen’s metric ([Qu], [BGS]) at the infinite places of K. ∗Mathematisches Institut, Universit¨atG¨ottingen, Bunsenstraße 3-5, 37073 G¨ottingen, Ger- many; email: [email protected] Jahnel 2 1.1 Fact. a) If the Euler characteristic χ(LK ) does not vanish, such a metric exists. ∧ b) c1 (L, k.k) is uniquely determined up to a summand (0,C), where C = (Cσ)σ:K,→C is a system of constants on X ×SpecK,σ Spec C with X Cσ = 0 (and Cσ = Cσ¯). σ:K,→C ∧ 1.2 Fact. Such (0,C) ∈ CH1 (X ) are numerically trivial. 1.3 We note that in order to prove the Fact 1.1 one mainly uses the following Lemma. Let f : X −→ Y be a smooth proper map of complex manifolds, where X is equipped with a K¨ahlerform ω and Y is connected, and E be a holomorphic vector bundle on X. For a hermitian metric k.k on E and a constant factor D > 0 we have χ(EK ) k.kQ,(E,D·k.k) = k.kQ,(E,k.k) · D on the line bundle det Rπ∗E on Y . 1.4 Now we can state our fundamental Definition. The height of the line bundle L is given by ∧ h ∧ dim X i hT ,ω0 (L) := deg π∗ c1 (L, k.k) · (T, gT ) , where k.k is one of the distinguished metrics specified above. 1.5 An arithmetic surface is an arithmetic variety of absolute dimension 2. In this paper we will analyze Definition 1.4 in that case. Our main result is Theorem (Equivalence) . Let π : C −→ OK be a regular arithmetic surface. Denote by C := C ×SpecOK Spec K its generic fiber. Assume C to be connected and to admit a K-valued point x ∈ C(K) and let Θ be the Theta divisor on the Jacobian J = Picg(C) (transferred from Picg−1(C) by x), where g denotes the genus of C. On a CC := C ×SpecK,σ Spec C (C) σ:K,→C let ω be a K¨ahlerform invariant under F∞. Then, for line bundles L ∈ Pic(C), fiber-by-fiber of degree g and of degree of absolute value less than H on every irreducible component of the special fibers of C (with some constant H ∈ N) hx,ω(L) = hΘ(LK ) + O(1), where hΘ is the height on J defined using the ample divisor Θ. 1.6 Remark. Another connection between heights on the Jacobian of a curve and arithmetic intersection theory was obtained by Faltings [Fa84] and Hriljac [Hr]. Recently it has been generalized to higher dimensions and higher codimen- sion Chow groups by K¨unnemann [K¨u].They write down an explicit formula for Jahnel 3 the N´eron-Tate height pairing on the Jacobian (higher Picard variety) in terms of arithmetic intersection theory. The main point is that they consider line bundles (cycles) algebraically equivalent to zero. So there is no need to scale a metric (to specify the infinite part of the arithmetic cycles occuring). We recover the Faltings-Hriljac formula for the N´eron-Tate height by considering some kind of asymptotic limit over the heights constructed here. 1.7 Theorem (Asymptotic behaviour) . Let π : C −→ OK be a regular arith- metic surface. Equip CC with a K¨ahlerform ω invariant under complex conjuga- tion F∞. Let E and M be line bundles on C equipped with a distinguished hermitian ∧ metric und L ∈Pic (C) be a hermitian line bundle. Assume n ∈ N is chosen such ⊗n that χ(EK ⊗ MK ). Then ⊗n ⊗n χ(EK ⊗ MK )hL(E ⊗ M ) = χ(EK )hL(E) + n [(deg(MK )hL(E) +χ(EK )D − deg(LK )B] A + n2[deg(M )D − deg(L ) ]. K K 2 Here χ always denotes the Euler characteristic and A, B and D are abbreviations for the arithmetic intersection numbers ∧ ∧ 2 A := deg π∗(c1 (M) ), ∧ ∧ ∧ 1 ∧ B := deg π c (M) c (E) + c (T ) and ∗ 1 1 2 1 f ∧ ∧ ∧ D := deg π∗(c1 (M) c1 (L)). ∧ By c1 (Tf ) we mean the first arithmetic Chern class of the relative tangent bundle equipped with the metric associated with ω. 1.8 Remark. Consider the special case, where deg(EC) = g, deg(LC) = 1 and deg(M ) = 0. Then h (E ⊗ M⊗n) is a quadratic polynomial in n, whose C L ∧ 1 1 ∧ 2 leading coefficient is just − 2 A = − 2 deg π∗(c1 (M) ), one half of the term considered by Faltings and Hriljac. In fact we have reproven the coincidence of the N´eron-Tate height on the Jacobian with the arithmetic self-intersection ∧ ∧ 2 number − deg π∗(c1 (M) ) since the N´eron-Tate height is actually defined as a limit of the type considered. 1.9 Corollary. Let Θ be the Theta divisor on Pic0(C) (transferred from Picg−1(C) by x). Then 1 ∧ h (M ) = − deg π (c∧ (M)2). NT,Θ K 2 ∗ 1 1.10 Corollary (Faltings, Hriljac). ∧ ∧ 2 hNT,Θ+Θ− (MK ) = − deg π∗(c1 (M) ). Proof. The endomorphism [−1] does not change the right hand side. 2 Jahnel 4 1.11 Remark. The proof of the equivalence theorem is relatively long but mainly consists of using elementary algebraic geometry and analysis. There is another (even more involved?) proof for that theorem in [J1]. To the contrary the asymptotic behaviour is an almost direct application of the arithmatic Riemann- Roch theorem. We note once more that Definition 1.4 works in every dimension. We deal with the case of dimension greater than two in another paper [J2]. 2 Proof of the Equivalence Theorem 2.1 Divisors versus points of the Jacobian 2.1.1 The remainder of this paper is devoted to the proof of Theorem 1.5. So let C/K be a regular proper algebraic curve of genus g with C(K) 6= ∅. g We consider a regular projective model C/OK . Denote by J = PicC/K the Jacobian of C. When x ∈ C(K) is chosen we have a canonical isomorphism g−1 g PicC/K −→ PicC/K = J and thus the divisor Θ on J. Θ induces a closed 0 N embedding i : J,→ PK and a ”naive” height for K-valued points of J: Y hΘ(D) := log max {ki(D)0kν,..., ki(D)N kν} . ν∈MK 0 ∗ g j i N Accordingly j (Θ) induces a morphism i : C −→ J −→ PK and a height g function hj∗(Θ) for K-valued points of C . Here j denotes the natural map sending a divisor to its associated line bundle. A general construction for heights defined by a divisor, the ”height machine”, is given in [CS, Chapter VI, Theorem 3.3]. N The underlying height h for K-valued points of PK is a height in the sense of Arakelov theory [BoGS] as follows: We choose the regular projective model PN ⊇ PN . Every K-valued point y of PN can be extended uniquely to an OK K K O -valued point y of PN . Let O(1) be the hermitian line bundle on PN , K OK OK where the hermitian metrics at the infinite places are given by − 1 2 2! 2 x0 xN kxik := + ... + 1 + ... + . xi xi Then h = hO(1) is the height defined by O(1) in the sense of [BoGS, Definition 3.1.; formula (3.1.6)]. 2.1.2 Remark. We need a better understanding of O(j∗(Θ)). By 1 g−1 Riemann’s Theorem [GH, Chapter 2, §7] one has Θ = (g−1)! j∗((x) × C ), where j : Cg −→p C(g) −→c J factors into a morphism finite flat of degree g! and a birational morphism.
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