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Diophantine Inequalities and Arakelov Theory

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Diophantine Inequalities and Arakelov Theory APPENDIX by Paul Vojta Diophantine Inequalities and Arakelov Theory In this appendix we consider a question of Parshin based on the Van de Ven-Bogomolov-Miyaoka-Yau inequality between the Chern classes for a compact complex surface. An arithmetic analogue of this inequality would imply a bound on the heights of rational points on a curve of genus g > 1, as well as the Fermat conjecture for sufficiently large expon­ ent. The idea dates back to Parshin's 1968 paper [Pa 1], which is the first to note that the Shafarevich conjecture implies the Mordell conjecture; at the same time Parshin notes that it would also suffice to bound Ki in the function field case. Commenting on Parshin's paper, Faltings wrote in [Fa 1], p. 411: "As in [P] we could derive the Mordell conjecture if we could bound <Wx , W x ) from above in terms of K, {} and the set of places v where X has bad reduction." One would then apply this bound to the complex or arithmetic surfaces produced by the Kodaira-Parshin construction relative to rational points on a fixed curve C of genus greater than one. This then gives a bound on the height of the rational point. Also, in an appendix to the Russian edition of Fundamentals of Diophantine Geometry (appearing presently in AMS translation) Parshin and Zarhin point out that the inequality ci ~ 3c 2 would suffice by the same argument. In October of 1986, Parshin gave a talk in Paris, in which he reformulated the Chern class inequality into a conjecture for arithmetic surfaces, and showed that the resulting conjecture implied a weak form of a conjecture of Szpiro between the conductors and discri­ minants of stable elliptic curves defined over Q; this in tum implies the asymptotic Fermat conjecture by an argument due to Frey. Here we give a more direct approach, using the effective bound for the heights of 156 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [App. §l] rational points on the curve X 4 + y4 = Z4 as a function of the normal­ ized discriminant of the field of definition of the point. The first section sets some notation. In §2, we discuss the function field case, which motivates the conjectures in the number field case. In §3, we show how a certain upper bound on (which may be called a canonical class inequality) implies a height in­ equality. We give another approach for such an implication in §4, sacri­ ficing concreteness in order to obtain uniformity in [F(P): F]. Conversely, we also show how conjectured height inequalities yield an in­ equality related to a canonical class inequality. Finally in §5 we give one application to the curve X 4 + y4 = Z4, showing that the height inequali­ ties for this curve imply the asymptotic Fermat conjecture and a weak form of the Masser-Oesterle abc conjecture. §1. GENERAL INTRODUCTORY NOTIONS Heights Let X be a regular arithmetic surface over Y = Spec(R) where R is the ring of integers of a number field F. We assume throughout that the generic fiber is geometrically irreducible. If D is a divisor on X F, then we have a height function defined on the set of algebraic points of X F , and well defined by D mod 0(1), i.e. modulo bounded functions. We say that hD is associated with D. Furthermore, hD depends only on the rational equivalence class of D. An example that will come up later is the case where the generic fiber C of XjY is embedded in p2 and D is the restriction of a hyper­ plane section to C. Then the height may be defined as where the sum is over all places of F(P) and [x: y: z] are homogeneous coordinates for the image of P in p2. On the other hand, heights can be defined via the methods of Arake­ lov intersection theory, as follows. If P E XF(Qa), then let Ep be the Zariski closure of P in X, so that Ep is an irreducible horizontal divisor said to correspond to P. Let D be an Arakelov divisor on X. [ApP. §1] GENERAL INTRODUCTORY NOTIONS 157 Proposition 1.1. The function 1 PI-+[F(P):Q] (D.Ep) is a height function in the class of heights mod 0(1) associated with the restriction of D to the generic fiber X F • Proof The height is a sum of local Weil-Neron functions, and for each absolute value w on F(P), the function PI-+(Dhor.Ep)w is a (normal­ ized) Weil function associated with the horizontal part Dhor' For the ver­ tical part, we take into account that for any v on F, Nv[F(P): F] for v infinite, { (Xv, Ep)v = 10glk(v)I[F(P):F] for v finite. Therefore the terms corresponding to fibral components (finite or infin­ ite) do not matter, because their contribution (Dver.Ep)j[F(P):Q] re­ mains bounded. Discriminants For every number field F we define 1 d(F) = [F: Q] log DF/Q' where DF/Q is the absolute value of the discriminant of the number field F. We call d(F) the normalized logarithmic discriminant of F, or simply the logarithmic discriminant if the context is clear. If F c F' then d(F) ~ d(F'). If P E X ~Qa) we define d(P) = d(F(P». Stability In §5 of Chapter V the definition of a semistable arithmetic surface was given, and regularity was assumed as a part of the definition. There is also a notion of a stable arithmetic surface. This is an arithmetic surface (not necessarily regular) satisfying conditions SS l-SS 4, except that in SS 3, a non-singular rational component of a geometric fiber must meet the other components of the geometric fiber in at least three points. The concepts of stability and semistability are closely related, in the sense that a singularity on a stable model can be resolved by a sequence of 158 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §1] blowings-up, glVlng a chain of rational (-2)-curves as in the proof of Theorem 5.1 of Chapter V. If all singularities are resolved in this way, then the resulting surface is semistable. Conversely, all such chains on a semis table model can be blown down, resulting in a stable model. If X is a semis table model over Y, we denote by X# the correspond­ ing stable model. If X# /Y is a stable family of curves of genus g> 1, and y is a closed point of Y, then let 0: = number of double points on the (stable) geometric fiber over y. If X/Y is only semistable, we define 0: by first reducing to the stable model. It is known ([Ara 1], p. 1293) that Arakelov proves it in the complex case but it holds in arbitrary charac­ teristic. If X/Y is a semis table model, then for each closed point y of Y, we write Oy = number of double points on the (semistable) geometric fiber over y. Also let If v is the valuation, corresponding to the closed point y of Y, then we also write, and The direct image of the dualizing sheaf In this appendix we will be using a number of properties of the direct image 7r*Wx/y, and we summarize some of its properties here. First of all, since W XIY is torsion free and coherent, so is its direct image (see [Ha 1], II, 5.8.1 for the latter). Hence 7r*W x / y is a vector sheaf over Y; its rank is g. Also, it can be related to the structure of the relative Jaco­ bian-see [Ara 1]. Recall that the degree of a locally free sheaf is de­ fined as the degree of its highest exterior power so that The number deg 7r",.W x/y is an invariant of the family X/Y measuring its complexity in much the same way as the height measures the complexity of a point on a variety. [ApP. §1] GENERAL INTRODUCTORY NOTIONS 159 In fact, there exists a theory of moduli spaces of smooth curves; i.e., spaces whose closed points correspond bijectively to smooth curves. More precisely, there exists a moduli space .A9 and a (:ompactification .Ag with the following properties. MOD 1. Stable families X# /Y correspond in a "natural" way with morphisms mx: Y ~ .Ag • MOD 2. The correspondence in MOD 1 IS compatible with base change. In order to put the above conditions into a more rigorous framework, one should either define .A9 as an algebraic stack, or attach additional "level structure" (division points on the Jacobian) to the curves in ques­ tion. This is a quite deep subject area, and it would tak,~ another book to deal with the question in sufficient detail. For the purposes of discus­ sion we will behave as if .Ag was a scheme rather than a stack, and leave it up to the reader to keep in mind that often we will be using actually a morphism from a cover of Y to a cover of .Ag instead of directly from Y to .Ag • See also this book's Foreword. With the above caution in mind, we will often also write m(X y) to de­ note the point mx(Y) for a point y on Y. This is well defined on .Ag if Xy is smooth. If Y is a smooth projective complex curve and X/Y is a stable or semis table family of curves of genus g, we then have the Falt­ ings height function on .Ag: where '1Y denotes the generic point of Y.
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