APPENDIX by Paul Vojta

Diophantine Inequalities and

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

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 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

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 , 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 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 of Y. There is a parallel theory of a moduli space d 9 of (principally polar• ized) abelian varieties, and a morphism from .A9 to d 9 corresponding to the construction of the Jacobian. If J /Y is a family of a.belian varieties over a smooth projective complex curve Y, then we also have the Falt• ings height of J /Y:

hFalmj'1Y» = hFaP/Y) = deg 7T..OJ J / y.

If J is the Jacobian of a stable curve, then this definition is compatible with the preceding definition since

The same definitions of the Faltings heights carry directly over to the number field case once we have defined a metric on det 7T..OJ x / y ; for this we use the canonical metric as in Chapter V, §3.

Remark. Deligne has found an example when deg 7T..OJx/y can be neg• ative, because of the Green's functions at infinity. This i8 of course un• like the function field case, but this is of no consequence for §4. 160 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §2]

§2. THEOREMS OVER FUNCTION FIELDS

Throughout this section let n: X -> Y be a semistable family of curves of genus 9 > lover a base Y, which is a smooth projective complex curve.

The purpose of this section is to prove the following result which devel• oped from the work of Van de Ven, Bogomolov, Miyaoka, Yau, and Parshin.

Theorem 2.1 (Canonical class inequality). If X is a semistable model of genus 9 over a base Y of genus q, then

(WI/ y ) ~ 3 I b: + (2g - 2) max(2q - 2,0). YEY

Proof Let X # denote the associated stable model. For each singular point P E X # let Z p be the inverse image in X; then Z p will be a chain of ( - 2)-curves of length np. If q = 0 then let D be a divisor consisting of two distinct smooth fibers of n. Otherwise let D = O. Let Kx denote the canonical divisor of X. Then Kx + D is numerically effective; i.e., for all curves C on X, the intersection number (K x + D. C) ~ O. Theorem 1.1 of [Miy 2] (with N' = E = Ip Z p and N = 0) then gives

(1) (Kx + D)2 ~ 3(cz(X) - e(D» - 3I (n p + 1 __ 1 -). p np + 1

Here e(D) is the Euler number of the curve D; it equals - 2(2g - 2) if q = 0 and 0 otherwise. Thus, when D = 0, the left-hand side is c1(X)Z, so this inequality strengthens the inequality d ~ 3cz as the last term is non• positive. When D i= 0, the inequality can be regarded as the analogue of ct ~ 3('2 for the non-compact surface X\D. In any case, letting K x/y be a divisor such that W x/y ~ (l)x(K x/y), we have

c2(X) = I by + (2g - 2)(2q - 2) y

= I b: + I np + (2g - 2)(2q - 2); y p

<1. IIp + 1 [ApP. §2] THEOREMS OVER FUNCTION FIELDS 161

The value for C2 above follows from [Ara 1], p. 1288. The theorem then follows from the above equations and (1).

Corollary 2.2. Let s be the number of places of bad reduction. Then

(Wi/y) ~ (2g - 2)(2q - 2 + s).

Proof Note that the maximum has disappeared: one cannot have a curve over A I with good reduction everywhere, because adjoining divi• sion points to the Jacobian would produce a non-trivial etale cover of AI. Now, let Y' be a cover of Y of degree de, where d and e are natural numbers and Y' is ramified to order exactly e at all points lying over points of Y of bad reduction. Such a curve can be constructed by the Kodaira - Parshin construction if q ~ 1 and by more elementary means if q = O. (In the latter case we may require e to be odd.) Then, applying Theorem 2.1 to the minimal desingularization of X x y Y' gives

de(Wi/y) ~ 3d L 1J: + (2g - 2)(de(2q - 2) + dee - l)s). y

Taking e large then gives the result.

We note that the same method gives

(wi/y) ~ L min(31J:, 2g - 2) + (2g - 2)(2q _. 2) y if q ~ 1.

Lemma 2.3.

Proof By Noether's formula, we have

x(@x) = l2(Ki + c2} = l2«Wi/y) + 1J) + (g - l)(q - 1).

By [Ara 1], pp. 1287-1288,

(Note that E of [Ara 1] equals the dual of 1t* W x / y ; see p. 1293.) This gives the lemma. 162 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §2]

Note that Lemma 2.3 is close to the form in which Noether's formula carries over into the case of Arakelov theory-for the left-hand side use instead deg det Rn*(Wx/y), with notations and metrics as in Chapters V and VI. See [Fa 1]. Therefore Corollary 2.2 becomes g-l b deg n*Wx/y :;:;; -6- (2q - 2 + s) + 12·

Also, by a result of Xiao Gang [Xi], Corollary 1 of Theorem 2, p. 461, we have

b :;:;; (8 + Ddeg n*Wx/y·

See also Cornalba and Harris [C-H]. Therefore:

Theorem 2.4.

This agrees well with a result due to Arakelov (implicit in [Ara 1]), that

where go is the dimension of the trace of the Jacobian of X over Y. We shall now show how Conjecture 5.7.5 of [V], the (1, I)-form con• jecture, implies a weak form of Theorem 2.4. For the convenience of the reader, we shall restate the conjecture later in the context of number fields, in §4. Here we merely refer to it.

Theorem 2.5. Assume Conjecture 5.7.5 of [V] holds. Fix g and e > O. Fix a smooth projective complex curve Yo of genus qo, and a finite sub• set S of closed points of Yo. Let Y denote a smooth projective curve of genus q mapping onto Yo and let [Y: Yo] denote the degree of that morphism. Then, as X varies over semistable families of curves of genus g over Y with good reduction at all points of Y not lying over S,

(2)

and

(3) (Wi/y) :;:;; (6g + e)(2q - 2) + O([Y: Yo]). [ApP. §3] CONJECTURES OVER NUMBER FIELDS 163

Proof For (2), use [V], (5.7.5)-(5.7.7), together with the definition

For (3), use (2) and Lemma 2.3, noting that {) ~ O. All three of these bounds on deg 7t.OJx/y then give the Shafarevich conjecture (for curves over function fields with fixed degeneracies), pro• vided one also shows that there are no infinite families with bounded heights. In the number field case, if any analogue of the:~ bounds could be proved, then one would have the Shafarevich conjecture without need• ing any additional results. Because of the connections with the Shafarevich conjecture, it appears that an arithmetic analogue of Theorem 2.1 would still be quite deep. Finally, we note that in all of the above discussion, the quantity (2q - 2)/[Y: Yo] is the function field analogue of d(F); also an analogue of s in the number field case would be

s = L 10glk(v)l. veS

With these identifications, much of the above discussion ,~arries over into the number field case (conjecturally, at least). The next two sections will cover this in more detail.

§3. CONJECTURES OVER NUMBER FIELDS

Notation. Let F, F', F", etc. denote number fields with rings of integers R, R', R" and let

Y = Spec(R), Y' = Spec(R'), Y" = Spec(R"),

respectively. We also continue to let X denote a semistable family of curves of genus g over a base Y.

One basic height conjecture is stated in [V], 5.5.0.1, as follows.

Conjecture 3.1. Let C be a smooth projective curve defined over a number field F. Let K denote the canonical divisor c.f C. Let e > O. Then for P varying over all algebraic points of C, we have

hK(P) ~ (1 + e)d(P) + 0(1),

where the constant implicit in O( 1) depends on C and B, but is indepen• dent of P. 164 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §3]

As shown in [V], Chapter 5, Appendix ABC, this conjecture implies a large number of other diophantine conjectures, e.g., the Masser-Oesterle abc conjecture, the Hall and Szpiro conjectures, etc. Conjecturally, the factor 1 + E: is best possible. One can weaken the conjecture by replacing 1 + E: with larger factors. For instance, in the function field case I can prove it with a factor of 2 + E:. Also, Theorems 4.5 and 3.5 hold also in the function field case, giving theorems in the function field case corre• sponding to Conjectures 4.4 and 3.4. In the last section, we shall remind the reader how weaker forms of Conjecture 3.1 imply the asymptotic Fermat conjecture and weaker forms of the abc conjecture. We wish to see how Conjecture 3.1 is related to Arakelov theory. We shall see that an upper bound on (Wi/y) proposed by Parshin, and ana• logous to the Canonical Class Inequality 2.1, implies a weaker form of Conjecture 3.1, where instead of 1 + E: we have a factor linear in g. Thus an inequality in Arakelov theory implies a height inequality. Conversely, in the next section, we shall show how a suitable conjectured height in• equality implies a bound on deg 1r*Wx/y. The relationship between

deg det 1[*Wx / y and deg det R1r*W x /y is not clear, and corresponds to the more subtle formulation of the Noether formula in the number field case, as distinguished from the func• tion field case in Lemma 2.3. Presumably a bound for (Wi/y) would also emerge from the clarification of the situation. We start by stating Parshin's analogue of Theorem 2.1. He prefers it to be stated as a question rather than as a conjecture.

Question 3.2 (Parshin). Do there exist effectively computable positive numbers ao, aI' a 2 , with aI' a 2 absolute constants, and ao depending on g, such that for all number fields F and all semistable families X/Y of genus g the inequality

(Wi/y) ~ a 2 L <5: 10glk(y)1 + a I (2g - 2)[F: Q]d(F) + ao[F: Q]

holds? (The sum is taken over all closed points of Y.) There is no clear understanding today of what the constants ao, al' az would be like. I would suggest that a l could be taken as 1 + E:, in which case ao would also depend on 6. One could also formulate a conjectural analogue of Corollary 2.2; it would hold if the above inequality holds. We leave the details to the reader. We note that the canonical class inequality in Question 3.2 is uniform for all semistable families of a given genus g. In many applications, and in particular in the rest of this section, we use uniformity over a smaller family: we fix X/Y and consider the family {Xp} of ramified coverings of X F obtained by the Parshin-Kodaira construction, as P varies over alge• braic points of X. [App. §3J CONJECTURES OVER NUMBER FIELDS 165

Notice that in Question 3.2 we are summing over finite places. This differs from Parshin's formulation, in which he uses Faltings' definition of (\ for archimedean v. (See the definition after Theorem 3.3 of Chapter VI; we will not discuss it further here. See also [Fa IJ, Section 6, as well as [B-BJ and [B-G-S].) The problem here is that we are dealing with

two concepts, bv and b:. Before discussing this in detail, we shall digress briefly to discuss the general setup. Recall from Section I the discussion of the moduli space vi! 9 of stable curves of genus g. Let v be an archi• medean place of F; it corresponds to an injection (J: F c.. C. Applying (J to the generic fiber of X/Y then produces a complex curve which we will write as X()". This corresponds to a point m(X()") on vl!iC). The desired functions b()" = b()",x/y (resp. b: = b:' x/y) should then be obtained by eva• luating some function on vl!iC) at the point m(X()"). Now, returning to the matter at hand, we point out that Faltings' b" is an analogue of our bv because he uses it for Noether's formula on a semistable model. This makes sense also because bv at non-archimedean places depends on the fiber over the local ring, and it makes sense to translate a p-adic analytic definition into a complex analytic definition. On the other hand, b: depends only on the special fiber" so on the mo• duli space, it is a question of which Zariski closed subsets of the bound• ary contain the point m(X v), At archimedean places, the generic fiber of X /Y is assumed to be smooth, therefore m(X ()") never lies on the bound• ary of the moduli space. Hence the archimedean analogue of b: should be zero. If the reader is not comfortable with the above argument, however, the results of this section still remain valid for ba coming from any con• tinuous function on vi! iC). Indeed, we will be applying the inequality of Question 3.2 to surfaces X;/Y; obtained from (varying) algebraic points P on a fixed X/Y via the Kodaira-Parshin construction. For any corre• sponding embedding (J: F; ~ C, it is also true that Xp,o' coincides with the curve obtained by applying the Kodaira-Parshin construction to the point (J(P) on X a' In particular, m(X p,a) varies continuously with (J(P) on vi! y(C). Since continuous functions on compact sets are bounded, b()" would then be bounded for all archimedean places (J. Th.e bound, how• ever, would necessarily depend on the generic fiber C of X/Y, and it is not clear how effective this bound is. The Parshin construction of ramified coverings of a given curve there• fore leads us to formalize some properties of the families of curves ob• tained in this way. Let:!l' = {X/Y} be a family of semistable arithmetic surfaces. We shall say that :!l' is a limited family if it satisfies the follow• ing conditions.

LIM 1. There exists Yo and a finite set of closed points So of Yo such that Y ranges over finite covers of Yo and X/Y is smooth over all points not lying above So. 166 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §3]

LIM 2. There exists a fixed compact subset of the moduli space .Ag(C), containing all points m(Xa), as X varies over the members of ~ and a varies over all embeddings a: F ..... C.

We may now reformulate a weaker form of the inequality proposed by Parshin.

Conjecture 3.3. Let ~ be a limited family of semistable arithmetic sur• faces. Then for all elements XjY in ~ we have

(WilY) [F: Q] ~ a1(2g - 2)d(F) + a;t',

where a l is the constant of Question 3.2 and a;t' depends on !!l". Conjecture 3.3 is implied by 3.2. Indeed, the term involving the numbers <5 # can be absorbed into the term a;t' because each <5: is bounded by 3g - 3. Also, if one uses a different definition of <5: at archimedean places, then this contribution can also be absorbed into a;t' by LIM 2. The purpose of this section is to show that following conjecture fol- lows from Conjecture 3.3.

Conjecture 3.4. Let C be a smooth projective curve defined over a number field F, and let K denote the canonical divisor. Then for P vary• ing over all algebraic points of C of bounded degree over F, we have

where a 1 is the same as in Question 3.2 and a4 depends vnly on C, F, and [F(P): F].

Theorem 3.5. Conjecture 3.3 implies Conjecture 3.4.

It is still an open problem to show that a4 can be effectively com• puted in terms of the constants in Question 3.2, but that is likely to be the case. The difficulty lies with proving bounds on the analytical quan• tities used in the metrics at infinity. We shall now prove Theorem 3.5. The inequality of Conjecture 3.4 is unaffected by base change, so we may assume C corresponds to a semi• stable family X/Yo For the rest of this section we fix n: X ..... Y, a semistable family of curves of genus g> 1, and we let C be the generic fiber of n. Let P denote an algebraic point on C, let F' = k(P), and let X' be a minimal desingularization of X x Y Y'. Let Ep denote the corresponding horizon• tal divisor on X and let E~ denote an irreducible horizontal divisor on X', of degree lover Y' and lying over Ep. [App. §3] CONJECTURES OVER NUMBER FIELDS 167

Lemma 3.6. For each point P as above there exists a semistable arith• metic surface

and a morphism fp: X; -+ X, such that the family

forms a limited family, and satisfies the following conditions (where we omit the subscript P for simplicity): (a) The generic fiber C" of X" has genus 4g - 2, and C" is ramified to order 2 above P and unramified elsewhere; (b) The degree [F": F] is bounded (depending only on X and [F(P) : F]), and

deg f = 4[F": F];

(c) d(F");::;; id(P) + as, where as depends only on X and [F(P): F]; (d) We have

where E~ and E; are sections of n" lying over E;, and D" is supported only on fibers of nil; and (e) We have

(E;'. D") ;::;; a7 deg f, i = 1,2;

(E'{ .E;) ~ a8 deg f;

where a6 , a7 , as depend only on X and [F(P): F].

Proof The proof draws many of its ideas from Parshin and Szpiro [Sz 3]. We break it down into six steps.

Also, let Sl denote the set of places of F where C has bad reduction, together with all places lying over 2. Step 1. Construct an (almost) etale cover of X. Let C' be an etale cover of C of degree two. Let F 1 => F be a field over whil~h C' is defined and attains semistable reduction. Let X 1 be a minimal desingularization of X x y Y1. Then the function field of C' is obtained from the function field of C by adjoining the square root of a function, say, k(C') = k( C)( Jh), where h is a rational function on Xl. 168 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §3]

Let F 2 be a field containing F 1 such that, for every place v of F 1 ly• ing over S l' all places w of F 2 have the property that the corresponding local fields F z. w contain all extensions of Fl." of degree at most [F':F]. Let X 2 be the semistable model of C obtained by desingularizing X x y Y2' By construction, F~ = F 2(P) is unramified over F 2 at all places ly• ing over S l' Therefore, X 2 x Y2 Y~ is also regular, and therefore the irre• ducible horizontal divisor on X 2 lying over E p does not pass through any of the double points on the fibers. Also let F 3 be a quadratic exten• sion of F2 ramified at all places v of F 2 for which v 12 or the divisor (h) is supported on a component of the fiber of X 2 over v. Note that none of the choices made so far depends on P, only on [F(P): F]. Now let F~ = F iP) and let X 3 = X 2 X Y2 Y~. Let E3 be the irreduc• ible horizontal divisor on X 3 lying over E~. We note that X 3 might not be regular, but will be regular at all points of E 3 • Let X 4 be the nor- malization of X 3 in the function field k(X 3)(Jh). Then we claim that, if v is a place of F'J not lying over 2 or 00, then the two inverse images of E3 in X 4 do not meet above v. Indeed, let x denote the point where E3 meets the fiber over v; the local ring will be isomorphic to

O.,[[TJ], where (!" is the local ring of F'3 at v. This holds because the local ring is regular and the fiber is irreducible at x. This ring is a UFD, so we can factor has,

h = uteh~l ... h~r, where u is a unit in (0 v[[T]], t is a uniformizer in (!) v' and hi"" A are distinct primes. By assumption, e l , ... ,er must be even, and we may therefore assume.

by absorbing squares into h. But also e must be even, as F 3 is ramified over F 2 and h comes from a function on X 2' Thus, a local ring on the inverse image of x in X 4 is etale over X 3 at x, so the inverse images of x are distinct. This analysis fails for places v 12, because the discriminant of X 2 - u is no longer a unit in (!).,[[T]]. However, we can blow up the point x, which corresponds to replacing T with tT'. After at most v(2) blowings• up, we will find that there exists an element a E (!)v[[T]] such that a2 == u mod 2 and therefore a+~ 2 is integral over (!j,,[[T]] with discriminant equal to a unit. [App. §3] CONJECTURES OVER NUMBER FIELDS 169

Finally, let F 5 be a quadratic extension of F 3(P) such that P lifts to two rational points on C(Fs). By the Chevalley-Weil theorem ([La 1], Chapter 2, Theorem 8.1), d(F 5) - d(F 3) is bounded as po varies. Let X 5 be a minimal desingularization of X 4 X Y 4 Ys, and let E s and E~ be the irreducible horizontal divisors on X 5 lying over E 3 • Thus, we have a regular family ?t: X 5 -+ Y5 , and the pull back of E' (as a Cartier divisor) is E5 + E'5 + D 5 , where D5 is supported only on fibers of ?t, and E 5 and E'5 do not meet. Also, note that when we desing• ularized X 4 X Y4 Y5 to produce X 5' the singular points didl not lie over E'. Therefore, D5 is supported only at fibers over 2 and 00. Also, the bad reduction of each fiber over v is independent of P if v,r 2 and varies over a finite set of possibilities if v 12. This family also has good reduction outside of places over S l'

Step 2. Divide (!)(E5 - E~) in two. This is an invertible sheaf of degree zero on the generic fiber; therefore there exists a finite extension F 6 of F 5' of bounded degree, and ramified only at places above Sl' such that the divisor E5 - E~ can be divided by two in PicO(C)(F 6)' Thus there exists a rational function h' on X 6 = X 5 X Y, Y6 such that,

(h') = Es - E~ + 2D + D', where D' is supported only on fibers of ?t6: X 6 -+ Y6 • Restricting D' to smooth fibers gives a fractional ideal of R 6 ; by Minkowski-type compu• tations, it is linearly equivalent to an ideal b of norm bounded by,

where N = [F 6 : Q] and DF6 is the absolute value of the discriminant of F 6' See [La, 5], Chapter V, Theorem 4. Therefore we may assume that D' is supported only on places over S 1> plus places occurring in the ideal b. Let S2 be the set of places on which D' is supported and not lying over Sl'

Step 3. Construct a ramified cover. Let F 7 be an extension of F 3 rami• fied to even order over places in S l' above and beyond any ramification indicated by the Chevalley-Weil argument in Step 1. Then F~ = F7F6 is ramified to even order over F 6 at all places lying over S l.' Moreover, F 7 does not depend on P. Then, by the same arguments as in Step 1, we can let X 7 be the normalization in F 7 k(X6 )(fi) of a blow-up of X6 x Y. Y~. Then inverse images of E5 and E~ will remain disjoint. Ex• cept for primes above S2, components of fibers meeting E5 and E~ will be reduced; fibers at primes above S2 will be smooth but may occur with multiplicity one or two. 170 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §3]

Step 4. Pass to a semistable model. Let F s be an extension of F 6F 7 over which the 15-level structure of the Jacobian of X 7 is rational; this has bounded degree and is only ramified over F 6 F 7 above places in S l' Let F" be an extension field of F s over which X 7 attains stable reduc• tion. Let X" be the resulting semistable model. By a moduli space argu• ment, we see that F"/Fs is unramified at places not lying over S1 or S2' and is ramified to order at most two at places over S2' Note that, when passing to the stable model, the neighborhood above E5 and E~ will be unaffected except that some of the blowings-up of Step 3 will be undone (but the blowings-up of Step 1 will remain). Also note that, by construction, conditions (a), (b), (c), and (d) of the lemma hold. Step 5. Check condition (e). For the first equation, this amounts to controlling D" in part (d). The divisor D" contains contributions related to fibers over S l' as well as the archimedean fibers. As noted earlier, the types of fibers over S 1 range over a finite set, therefore their contribution to D" is bounded. There is also a contribution to D" at the infinite places which amounts to a Coo function on CIf, and therefore is bounded. Moreover, this function varies continuously as P varies, therefore the contribution to D" is again bounded. This is admittedly a weak point in the argu• ment, but hopefully an expert in differential equations would be able to produce an effective bound. For a few more explanations see the end of the section. The other two equations can be proved by similar arguments. Step 6. Show that the surfaces X; form a limited family. Condition LIM 1 is clear, since all surfaces X; have good reduction at places not lying over S1' and S1 does not depend on P. Now consider condition LIM 2. If we were to replace the base Y with a curve C and replace P with the point ~ on C x C, defined over the function field k(C), then the above construction would still be valid, except for the Minkowski argument in Step 2. For that part (noting that S 1 = 0), we see that the vertical fibers of (hi) define a divisor on C (the base) of even degree; therefore it can be divided in two since C (the base) is defined over C and Jac(C)(C) is a divisible group. Thus, we obtain a diagram as in (*) in the proof of Theorem 4.5, and we take the compact subset of uHiC) to be the image of the compact set C(C). Finally, it is necessary to check that the process of taking a point P on C to X; is compatible with the correspondence (*). We leave these details to the reader. This concludes the proof of Lemma 3.6.

Lemma 3.7. With the above notations, [ApP. §3] CONJECTURES OVER NUMBER FIELDS 171

Proof By part (e) of Lemma 3.6,

By the adjunction formula on X",

CE'?) = -(f*Wx/y ® (!J(E'{ + E'2 + D").ED, and likewise for (E'22). Therefore,

2(CE'{ + E'2)2) = - (f*Wx/y ® (!J(D"). E'{ + E'2) + 2CE'{. E'2)

~ -(f*Wx/y.E'{ + E'2) - 2a7 degf + 2as degf.

Combining this with the earlier equation gives

(Wi"w) > (W2) 3(f*Wx/y·E'{ + E'2) _ a - a + a . degf = XjY + 2 d eg f 6 7 8

But the difference between the pull back of E~ and 2(E'{ + E'2) is sup• ported on exceptional divisors for the morphism from X" to X'; there• fore,

2(f*Wx/y·E~ + E'2) _ (WX'/y,.E~) degf - [F(P): F] .

Moreover,

Therefore,

(Wi"w) > (W2) 3(Wx/y·Ep ) _ _ . degf = X/y + 4[F(P): F] a6 a7 + as·

This concludes the proof of Lemma 3.7.

Proof of Theorem 3.5. We combine Lemma 3.7 with the inequality of Conjecture 3.3 applied to the family {X;/Y;}Pex to give

(WX /Y • Ep) :$; a (8g - 6) [F: Q] d(F") [F(P):F] - 1 3

4( 2 ) [F: Q] + '3 -(Wx/y) + a6 + a7 - as + ~-3- a!Z. 172 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §3]

Recall that degf = 4[F" : F] and that the genus in Conjecture 3.3 equals 4g - 2 for the present family. Consolidating a few constants and taking note of condition (c) of Lemma 3.6 gives

The left-hand side is hK(P). and the proof is complete.

We shall now comment briefly on the argument at infinity at the end of Step 5. We deal more generally with a

f: C -+ C of complete smooth curves over the complex numbers. We let g' and g be the Green's functions on C and C respectively. viewed as functions on the product minus the diagonal. Let

11' = Closure in C x C ofthe set of points (Q. Q') such that f(Q) = f(Q') and Q -F Q'. Then there exists a smooth function

a: C x C - 11' -+ R such that g'=go(fxf)+a.

Let R be the ramification divisor of f. so

R = L (ea, - l)(Q'). Q' where the sum is taken over all complex points Q' of c. If Q is not a ramification point of f. then g' - g 0 (f x f) extends by continuity to (Q. Q) since g' and go (f x f) have the same singularity at (Q. Q). Ex• tending g' bilinearly to divisors. so

g'(Q. R) = L (ea, - l)g'(Q. Q'). Q' we see that the function fJ(Q) = g'(Q. R) + a(Q. Q) is defined and smooth on the set of points Q ¢ R. But again by compar• ing singularities. we see that fJ: C-+R [ApP. §4] ANOTHER HEIGHT INEQUALITY 173 is defined as a smooth function on all of C. Then the following ques• tions arise. (a) What are max f3 and min f3 on C? (b) Suppose C varies in an algebraic family, so we write {C;}tET with some parameter curve T. Then {C;} is the fibration of a surface S, and the family {f3t} may be regarded as a function f3 on S. The problem is to show that f3 is continuous, and to determine its maximum and minimum on S. The function f3 is precisely what is needed to compare the metrics coming from the Green's functions on the sheaves

and f*Wc ® l!J(R).

Such a comparison was needed in Step 5, where we dealt with the Parshin family {C;}, or rather {C;'} where T is an etale covering of C.

§4. ANOTHER HEIGHT INEQUALITY

Just as Conjecture 3.1 arose from Nevanlinna theory, there is another conjecture stemming from a different type of Nevanlinna theory in [V], 5.7.5, which we restate here briefly, assuming that the reader is ac• quainted with some terminology of differential geometry.

Conjecture 4.1 (The (1, I)-form conjecture). Let V be a quasi-projective variety defined over a number field contained in C and let D be a nor• mal crossings divisor on V. Let w be a hermitian (1, I)-form on V\D, whose holomorphic sectional curvatures are bounded from above by -c < O. Let 2:' be a line sheaf on V such that c i (2:') ~ w. Let E be an almost ample divisor on V. Then for all D-integralizable sets of closed points P of V,

1 h!AP) ~ ~ d(P) + shEep) + 0(1). c

All terms used in the statement of the conjecture are: defined in [V]. We only want to help the reader with a quick statement.

We want again to see how this conjecture relates to conjectures aris• ing from Arakelov theory. Instead of using the Parshin Inequality 3.2 to derive a height inequality as in Conjecture 3.4, we shall use the number field analogue of (2) from Section 2. We shall see that the (1, I)-form conjecture implies the analogue of (2), which in turn implies a streng• thening of Conjecture 3.4. The analogue of (2) is the following. 174 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §4]

Conjecture 4.2. Fix a base Yo, g> 1, a finite set of finite places S, and s > O. For all extensions Y of the base Yo and all semistable families n: X --> Y of genus g, with good reduction at points of Y not lying over S,

___ 1___ deg n (jJ :-s; (~ + s) d(F) + 0(1). [F: Q] * X/Y - 2

Theorem 4.3. The (1, I)-form conjecture implies Conjecture 4.2.

Proof We refer to the notation of [V], Example 5.7.6. Basically, Con• jecture 4.2 is (5.7.7) of op. cit. First, though, it is necessary to indicate how det n*(jJx/y is metrized. This is done via the Jacobian:

in turn the determinants are metrized by Faltings' canonical metric (Chapter V, §3). As noted in [Fa 2], however, this metric extends to a singular metric on any compactification of the moduli space d 9 of abe• lian varieties of dimension g. It is possible to compactify d 9' obtaining

.rdg , such that the boundary is a normal crossings divisor D. The singularities on the canonical metric are then known to be bounded by O(log( Y. D)J for any infinite place v; therefore replacing the singular ca• nonical metric with a smooth metric affects the left-hand side of inequali• ty 4.2 by at most o(hD)' But also, by [V], 1.2.9h, hD ~ O(hK+ D). Therefore, as noted in [V], the difference between the (smoothly mctrized) height occurring in the (1, I)-form conjecture and the Faltings height used in Conjecture 4.2 can be absorbed into the s.

Conjecture 4.4. Let C be a smooth projective curve defined over a number .field F, and let K denote the canonical divisor. Fix 6 > O. Then for P varying over all algebraic points of C, we have

hK(P) ~ (8g - 4 + 6) d(P) + OCt),

where the constant in 0(1) depends only on C, F, and f..

We note that the bounds do not depend on [F(P): FJ See also [V], 5.5.1.

Theorem 4.5. Conjecture 4.2 implies Conjecture 4.4. [ApP. §4] ANOTHER HEIGHT INEQUALITY 175

Proof By [Sz 4], X.1, there exists an etale cover C of C mapping into J{ 4g _ 2 (the moduli space of curves of genus 4g - 2):

c

Let K (resp. K') denote the canonical divisor of C (resp. C). If P' E C lies over PEe, then

hK(P) = hK,(P') + 0(1) by functoriality

~ (4 + B')hFa.{t(P'») + 0(1) see justification below

~ (8g - 4 + e) d(t(P'» + 0(1) by Conjecture 4.2

~ (8g - 4 + B) d(P') + 0(1) because F(t(P'») c F(P')

~ (8g - 4 + B) d(P) + 0(1) by Chevalley-1iVeil.

To justify the first inequality, we note that the diagram (*) is obtained by applying the Kodaira-Parshin construction as in L(:mma 3.6 to the diagonal ,1 on C x C, regarded as a point on C rational over k(C). This produces a surface n: W --+ C which also maps to C x C; the morphism f: W --+ C x C has degree 4 deg p and is ramified only over,1. Thus

W~le' = !*(wrg;c/c ® (f)cxcC,1»; (W~/C,) = (degf)«W2xc;d + (Wcxc/c ',1) + i(,1.,1») = (degf)(O + (2g - 2) - ¥2g - 2») = i( deg f)(2g - 2) = 3( deg p )(2g - 2).

Then, by Lemma 2.3,

deg n* WW/e' = !C deg p )(2g - 2).

Therefore, by functoriality of heights (the projection fommla, Chapter III, Theorem 4.1; see also [La 1], Chapter 4, Theorem 5.1),

hFa1(t(P'» = hDCP') + 0(1) for some divisor D of degree tcdeg p)(2g - 2) on C. Since deg K' = (deg p)(2g - 2), we get the desired inequality by a property of heights ([La I], Chapter 4, Proposition 3.3.). 176 DIOPHANTINE INEQUALITIES AND ARAKELOV THEORY [ApP. §5]

One should also note that on page 264 of [Sz 4], the equation

initially holds up to a divisor supported on fibers, but then it can be shown that this divisor has even degree and therefore it also is linearly equivalent to a 2-divisible divisor. This approach has the advantage of being elegant, and the constants do not depend on [F(P): F], but it would likely be harder to prove effective bounds in this case.

§5. APPLICATIONS

All the applications we can think of at this time in diophantine analysis come from a height inequality. In [V], 5.7 we saw how the (1, I)-form conjecture implies the Shafarevich conjecture that given a finite set of places S of Y = Spec(R), there is only a finite number of isomorphism classes of curves of given genus g> lover F with good reduction out• side S. A large number of other applications depend only on a weaker form of a height inequality, essentially the form given in 3.4, namely for a fixed curve Cover F,

hK(P) ~ ac d(P) + 0(1) with P varying over the algebraic points of C, and ac is some constant, depending on C. In [V], Chapter 5, §5 and Appendix, a number of con• sequences were already listed, e.g. the abc conjecture, Hall and Szpiro conjectures, etc. Here we repeat two of these applications just to show how the constants enter in these consequences. We apply the height inequality to the Fermat curve of degree 4. Let a, b, and e be pairwise relatively prime integers with an + b" = en and n odd. Let X be the curve X 4 + y4 = Z4 C p2, defined over Q. It has genus g = 3. Then we have a point P = [a"/4: b"/4: C"/4] on X defined over Q(yIa, ib, ifc)·

Lemma 5.1.

3 d(P) ~ 210g 2 + - L log p 4 plabc

~ 2 log 2 + ! log max( 1ai, 1b I, 1e I). [ApP. §5] APPLICATIONS 177

Lemma S.2.

Proof. The canonical sheaf of C is the restriction to C of CD( 1) on p2; this gives the lemma by Proposition 1.1 and the comments preceding that proposition.

Lemma 5.3. max(lal, Ibl, lei) ;?; 2n + 2.

Proof. We may assume that 0 < a < b < e. Letting x := e - 1, we have b :S; x; a :S; x-I; therefore,

(x - I)" + x" ;?; (x + 1)"; (,,-1)/2 (n) f(x) == x" - 2.L 2' Xli;?; O. • =0 l

By "Descartes' rule of signs," f has a unique positive n~al root (); since f(2n) < 0, () > 2n. Therefore e is an integer strictly greater than 2n + 1.

No doubt better elementary bounds exist. Combining Lemmas 5.1 and 5.2 with Conjecture 3.4 gives

n 15al 4 log max(lal, Ibl, lei) ~ a4 + a9 + -2- (2 log 2 + £log max(lal, Ibl, leI».

Therefore, by Lemma 5.3,

135a1 60a1 log 2 + 4a4 + 4a9 n:S;--+ . - 2 10g(2n + 2)

Likewise, we obtain the abe conjecture with a different exponent. Let a, b, e be pairwise relatively prime integers with a + b + e = 0 and

N = Radical(abe) == n p. plabc We obtain 45a 1 log max(lal, Ibl, leI) ~ -2-log N + 6Oa 1 10g 2 + 4a4 + 4a9'

The derivation is similar. 178 DIOPHANTINE INEQUALITIES AND ARAKL ('Y THEORY [ApP. §5]

We note that here we obtained a weak form of the abc conjecture from the weakening 3.4 of Conjecture 3.1. If we l.icd Conjecture 3.1 in• stead, we would get the strong form of the abc ce'll.1ecture; see [V], Ap• pendix 5.ABC.

Remark. As Lang points out, there might be applications of Arakelov theory other than to diophantine analysis. For instance, one may trans• late Theorems 1 and l' from [Xi] to give a conjectured condition under which, for a semistable family X/Y, the image of t11,l: fundamental group

is trivial. Thus if a family X/Y satisfied a suita:,b bound on (WilY)' then no coverings of X would come from a coverillg of the generic fiber. References

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CH: 52 CH(X, A): Arakelov Chow group 76 c1(p): Chern form of metric 11 y(E): volume discrepancy 106 yv(f): component in Arakelov divisor of a function 8, 75 b(X): Faltings delta 144 by, b: : number of double points on geometric fiber 158 d, de: differential operators 10 D: discriminant 93, 111 d: log D 93, 99, 100, 111 dEfY 93 d;.: log A-discriminant 99, 111 doo 99 1 d(P): [F(P): Q] log DF(p)fQ 157 degF : F-degree 73 degz(D): local degree of a divisor 80 deg.(2', p): degree of a metrized line sheaf 81 degz(2', p): global degree 81 Divo: divisors of degree zero 61, 77 Divy: fibral divisors 58 (D. E): intersection symbol 72

N v: local degree [Fv: Qv] 71 j1: canonical volume form 28 C(J*: direct image 53 <1>: canonical 2-form on X x X 29 Pic(X, 2): Arake10v Picard group 85 Rat(X, 2) 76 P: metric 5 P;",D: metric on (!J(D) defined by Weil function 7, 143 e: theta divisor 139 v(z): -log Izl" 1 vF(z) = -log Ilzllv 71 WEIY: Dedekind complementary module 93 WEIY: canonical sheaf or dualizing sheaf 89 Index

A Chern form 11, 33 Chow group 52, 76 Adjunction formula 88, 99, 115 Coleman forms 34 Admissible 85, 101, 140 Complementary module 93 Anti-holomorphic forms 34 Cornalba-Harris 162 Arakelov adjunction formula 99 Curve 50 Arakelov Chow group 76 Arakelov divisor 75, 83 Arakelov-Picard group 85 D Arithmetic curve 80, 102 Arithmetic genus 117 ddc lemma 17 Arithmetic surface 49 Dedekind complementary module 93 Artin's theorem 122 Degree, of a curve over another 57 Associated metric 85 of an Arakelov divisor 84 of a Cartier divisor 80 of a divisor over a field 73 B of a line sheaf 81 Base change 62, 65, 122-127 of a Weil divisor on a curve 59, 73 Bernoulli polynomials 43, 150 Determinant 104, 133 Different 93 Differential sheaf 90 c Dimension 50 Canonical Arakelov divisor 101 Direct image 53 Canonical class inequality 156, .160 Discriminant 93, 97, 99, 111, 113, 157 Canonical form 28, 33, 143 Distributions 26, 47 Canonical Green function 29 Divisor 50, 59, 75, 83 Canonical height pairing 77 Divisors of degree zero 61, 67, 77 Canonical sheaf 89, 91 Dualizing module 93 base change 126 positivity 128 E Cartier divisor 2, 50, 59 CH 52,76 Effective divisor 49, 118 Chern class 12 Eigenvalue expansion 150 186 INDEX

Elkies' theorem 150 Local complete intersection 86, 88 Elliptic curves 44, 130 Local degree 81 Euler characteristic 103, 104, 106, Logarithmic discriminant 93, 99, 100, 112, 115 157 base change 125 M F Metric on determinant (volume) 105 F AL conditions 110, 140 associated with Arakelov divisor 85 Faltings (j 144, 165 on line sheaf 5, 80 Faltings height 159 and Weil function 7, 85, 143 Faltings metric 110, 140, 142, 147 Metrized line sheaf 5 Faltings Riemann-Roch 113 on an arithmetic curve 81 F-degree 73 on an arithmetic surface 84 Fibral divisor 52 Minimal desingularization 122 Fibral intersections 60, 75, 79, 122, 130 N G Negative intersection form 61, 77 Negative Neron symbol 77 Geometric fiber 121 Neron divisor 2 Global degree 81 Neron family 3 Global Neron symbol 77 Neron function 3, 69 GR conditions 21 on 43 33 Green's function 21. 29, Neron symbol 67, 77 on elliptic curve 44 Normalized Haar measure 103 Green's operator 28 Normalized volume form 21, 103 NF conditions 3 H NS conditions 67 Haar measure 103, 105 Harmonic forms 34, 39 o Height 77, 156 Height inequalities 158 Order 49 Height pairing 77 Orthogonalization 62, 79 Hodge Index 77 Horizontal divisor 52 P Parshin 155, 164-167 I Parshin inequality 164 Intersection numbers 55, 64, 71, 72, Pic(X, A.) 85 74, 77, 86 Poincare sheaf 136 base change 65 Positive form or metric 12, 21 Inverse image 63, 64, 65, 77, 82, 125 Positivity of canonical sheaf 128 Irreducible divisor 49 Prime divisor 49 Principal divisor 50 Principal Neron divisor 3 K Projection formula 64 Kunz-Waldi 94 Proper transform 63

L R Laplacian 27 Rank 57 Limited family 165 Rat 52, 76 Line sheaf 1 Rational equivalence 52 INDEX 187

Rational singularity 125 T Regular imbedding 86 Regular morphism 90 Tensor product of metrics 6 Relative differential divisor 97 Theta divisor 139 Relative Euler characteristic 104 Theta functions 143 Relative logarithmic discriminant 93, Trivial metric 83 157 Reproduction formula 31 V Residue isomorphism 93 Residue theorem 94 Variety 49 Riemann-Roch on curves 104 115 Vector sheaf 1 Riemann-Roch Faltings 113 ' Vertical divisor 52 Volume, on a vector space 105 of fundamental domain 106 Volume discrepancy 106 s Volume exact 106, 111 Volume form 21 Self-intersection 58 Semipositive 12 Semistable 121 w SS conditions 121 Weil divisor 49, 59 Stable 157 Weil function 1 Standard metric 6, 13, 83 and metric 7 Subvariety 49 Support 52 X Symmetry 24 Szpiro 118, 128, 167 Xiao Gang 162, 178