Applications of Arithmetic Algebraic Geometry to ...vojta/ftp/cime.pdf · Applications of Arithmetic Algebraic Geometry ... theory to this proof, and prove Mordell’s conjecture

164

Applications of Arithmetic Algebraic Geometry

to Diophantine Approximations

Paul Vojta∗

Department of MathematicsUniversity of CaliforniaBerkeley, CA 94720 USA

Contents

1 History; integral and rational points2 Siegel’s lemma3 The index4 Sketch of the proof of Roth’s theorem5 Notation6 Derivatives7 Proof of Mordell, with some simplifications by Bombieri8 Proof using Gillet-Soule Riemann-Roch9 The Faltings complex

10 Overall plan11 Lower bound on the space of sections12 More geometry of numbers13 Arithmetic of the Faltings complex14 Construction of a global section15 Some analysis16 More derivatives17 Lower bound for the index18 The product theorem

Let us start by recalling the statement of Mordell’s conjecture, first proved byFaltings in 1983.

Theorem 0.1. Let C be a curve of genus > 1 defined over a number field k . ThenC(k) is finite.

In this series of lectures I will describe an application of arithmetic algebraic geom-etry to obtain a proof of this result using the methods of diophantine approximations(instead of moduli spaces of abelian varieties). I obtained this proof in 1989 [V 4]; itwas followed in that same year by an adaptation due to Faltings, giving the followingmore general theorem, originally conjectured by Lang [L 1]:

∗Partially supported by NSF grant DMS-9001372

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Theorem 0.2 ([F 1]). Let X be a subvariety of an abelian variety A , and let k bea number field over which both of them are defined. Suppose that there is nonontrivial translated abelian subvariety of A×k k contained in X ×k k . Then theset X(k) of k-rational points on X is finite.

In 1990, Bombieri [Bo] also found a simplification of the proof [V 4]. While it doesnot prove any more general finiteness statements, it does provide for a very elementaryexposition, and can be more readily used to obtain explicit bounds on the number ofrational points.

Early in 1991, Faltings succeeded in dropping the assumption in Theorem 0.2 thatX×k k not contain any translated abelian subvarieties of A , obtaining another conjec-ture of Lang ([L 2], p. 29).

Theorem 0.3 ([F 2]). Let X be a subvariety of an abelian variety A , both assumed tobe defined over a number field k . Then the set X(k) is contained in a finite union⋃

i Bi(k) , where each Bi is a translated abelian subvariety of A contained in X .

The problem of extending this to the case of integral points on subvarieties ofsemiabelian varieties is still open. One may also rephrase this problem as showingfiniteness for the intersection of X with a finitely generated subgroup Γ of A(Q) . Thesame sort of finiteness question can then be posed for the division group

g ∈ A(Q) | mg ∈ Γ for some m ∈ N ;

this has recently been solved by M. McQuillan (unpublished); see also [Ra].

Despite the fact that arithmetic algebraic geometry is a very new set of techniques,the history of this subject goes back to a 1909 paper of A. Thue. Recall that a Thueequation is an equation

f(x, y) = c, x, y ∈ Zwhere c ∈ Z and f ∈ Z[X, Y ] is irreducible and homogeneous, of degree at least three.Thue proved that such equations have only finitely many solutions.

The lectures start, therefore, by recalling some very classical results. These includea lemma of Siegel which constructs small solutions of systems of linear equations and,later, Minkowski’s theorem on successive minima.

Next follows a brief sketch of the proof of Roth’s theorem. It is this proof (or,more precisely, a slightly earlier proof due to Dyson) which motivated the new proof ofMordell’s conjecture.

After that, we will consider how to apply the language of arithmetic intersectiontheory to this proof, and prove Mordell’s conjecture using some of the methods ofBombieri. This will be followed by the original (1989) proof using the Gillet-SouleRiemann-Roch theorem. These proofs will only be sketched, as they are written indetail elsewhere, and newer methods are available.

Finally, we give in detail Faltings’ proof of Theorem 0.3, with a few minor simpli-fications.

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In this paper, places v of a number field k will be taken in the classical sense, sothat places corresponding to complex conjugate embeddings into C will be identified.Also, absolute values ‖·‖v will be normalized so that ‖x‖v = |σ(x)| if v corresponds toa real embedding σ : k → R ; ‖x‖v = |σ(x)|2 if v corresponds to a complex embedding,and ‖p‖v = p−ef if v is p-adic, where p is ramified to order e over a rational primep and f is the degree of the residue field extension. With these normalizations, theproduct formula reads

(0.4)∏v

‖x‖v = 1, x ∈ k, x 6= 0.

A line sheaf on a scheme X means a sheaf which is locally isomorphic to OX ; i.e.,an invertible sheaf. Similarly a vector sheaf is a locally free sheaf.

More notations appear in Definition 2.3 and in Section 5.

§1. History; integral and rational points

In its earliest form, the study of diophantine approximations concerns trying to provethat, given an algebraic number α , there are only finitely many p/q ∈ Q (written inlowest terms) satisfying an inequality of the form∣∣∣∣pq − α

∣∣∣∣ < c

|q|κ

for some value of κ and some constant c > 0 . It took many decades to obtain the bestvalue of κ : letting d = [Q(α) : Q] , the progress is as follows:

κ = d, c computable Liouville, 1844κ = d+1

2 + ε Thue, 1909κ = mind

s + s− 1 | s = 2, . . . , d+ ε Siegel, 1921κ =

√2d + ε Dyson, Gel’fond (independently), 1947

κ = 2 + ε Roth, 1955

Of course, stronger approximations may be conjectured; e.g.,∣∣∣∣pq − α

∣∣∣∣ < c|q|−2(log q)−1−ε.

See ([L 3], p. 71).Beginning with Thue’s work, these approximation results can be used to prove

finiteness results for certain diophantine equations, as the following example illustrates.

Example 1.1. The (Thue) equation

(1.2) x3 − 2y3 = 1, x, y ∈ Z

has only finitely many solutions.

Indeed, this equation may be rewritten

x

y− 3√

2 =1

y(x2 + 3√

2xy + 3√

4y2).

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But for |y| large the absolute value of the right-hand side is dominated by some multipleof 1/|y|3 ; if (1.2) had infinitely many solutions, then the inequalities of Thue, et al.would be contradicted.

For a second example, consider a particular case of Mordell’s conjecture (Theorem0.1).

Example 1.3. The equation

(1.4) x4 + y4 = z4, x, y, z ∈ Q

in projective coordinates (or x4 + y4 = 1 in affine coordinates) has only finitely manysolutions.

The intent of these lectures is to show that Theorem 0.1 can be proved by themethods of diophantine approximations. At first glance this does not seem likely, sinceit is no longer true that solutions must go off toward infinity. But let us start byconsidering how, in the language of schemes, these two problems are very similar.

In the first example, let W = Spec Z[X, Y ]/(X3 − 2Y 3 − 1) and B = Spec Z beschemes, and let π : W → B be the morphism corresponding to the injection

Z → Z[X, Y ]/(X3 − 2Y 3 − 1).

Then solutions (x, y) to the equation (1.2) correspond bijectively to sections s : B → Wof π since they correspond to homomorphisms

Z[X, Y ]/(X3 − 2Y 3 − 1) → Z, X 7→ x, Y 7→ y

and the composition of these two ring maps gives the identity map on Z .In the second example, let W = Proj Z[X, Y, Z]/(X4 +Y 4−Z4) and B = Spec Z .

Then sections s : B → W of π correspond bijectively to closed points on the genericfiber of π with residue field Q . In one direction this is the valuative criterion ofproperness, and in the other direction the bijection is given by taking the closure in W .These closed points correspond bijectively to rational solutions of (1.4).

Thus, in both cases, solutions correspond bijectively to sections of π : W → B .The difference between integral and rational points is accounted for by the fact that inthe first case π is an affine map, and in the second it is projective.

Note that, in the second example, any ring with fraction field Q can be used inplace of Z as the affine ring of B (by the valuative criterion of properness). But, in thecase of integral points, localizations of Z make a difference: using B = Spec Z[ 12 ] , forexample, allows solutions in which x and y may have powers of 2 in the denominator.

§2. Siegel’s lemma

Siegel’s lemma is a corollary of the “pigeonhole principle.” Actually, the idea dates backto Thue, but he did not state it explicitly as a separate lemma.

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Lemma 2.1 (Siegel’s lemma). Let A be an M × N matrix with M < N and havingentries in Z of absolute value at most Q . Then there exists a nonzero vectorx = (x1, . . . , xn) ∈ ZN with Ax = 0 , such that

|xi| ≤[(NQ)M/(N−M)

]=: Z, i = 1, . . . , N.

Proof. The number of integer points in the box

(2.2) 0 ≤ xi ≤ Z, i = 1, . . . , N

is (Z + 1)N . On the other hand, for all j = 1, . . . , N and for each such x , the jth

coordinate yj of the vector y := Ax lies in the interval [−njQZ, (N − nj)QZ] , wherenj is the number of negative entries in the jth row of A . Therefore there are at most

(NQZ + 1)M < (Z + 1)N

possible values of Ax . Hence there must exist vectors x1 6= x2 satisfying (2.2) andsuch that Ax1 = Ax2 . Then x = x1 − x2 satisfies the conditions of the lemma.

In order to further emphasize the arithmetic-geometric nature of the subject, allresults will be done in the context of a ring RS , obtained by localizing the ring R ofintegers of a number field k away from primes in a finite set S of places of k . We alsowill always assume that S contains the set of archimedean places of k .

In the case of Siegel’s lemma, the generalization to number fields is sufficient; thiswas proved at least as early as LeVeque ([LeV], proof of Thm. 4.14). The form that wewill use is due to Bombieri and Vaaler. First, however, we need to define some heights(cf. also Section 5). Recall that, classically, the height of an element x ∈ k \ 0 isdefined as

h(x) =1

[k : Q]

∑v

log max(‖x‖v, 1).

Definition 2.3.

(a). For vectors x ∈ kN , x 6= 0 ,

h(x) =1

[k : Q]

∑v

log max1≤i≤N

‖xi‖v.

(b). For P ∈ Pn(k) with homogeneous coordinates [x0 : · · · : xn] , h(P ) is definedas the height of the vector (x0, . . . , xn) ∈ kn+1 . By the product formula (0.4),it is independent of the choice of homogeneous coordinates.

(c). For x ∈ k , h(x) = h([1 : x]) , the height of the corresponding point in P1 .(d). For an M × N matrix A of rank M , h(A) is defined as the height of the

vector consisting of all M ×M minors of A .

For a number field k , let Dk/Q denote the discriminant and s the number ofcomplex places. Then the generalization of Siegel’s lemma to number fields is thefollowing.

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Theorem 2.4 (Siegel-Bombieri-Vaaler, ([B-V], Theorem 9)). Let A be an M×N matrixof rank M with entries in k . Then there exists a basis x1, . . . ,xN−M of thekernel of A (regarded as a linear transformation from kN to kM ) such that

N−M∑i=1

h(xi) ≤ h(A) +N −M

[k : Q]log((

2π

)s√|Dk|

).

The exact value of the constant will not be needed here; it has been included onlyfor reference.

Note that, in addition to allowing arbitrary number fields, this result gives infor-mation on all generators of the kernel of A ; this will be used briefly when discussingthe proof of Theorem 0.2 (Section 18).

§3. The index

Let Q(X, Y ) be a nonzero polynomial in two variables. Then recall that the multiplicityof Q at 0 is the smallest integer t such that aijX

iY j is a nonzero monomial in Qwith i + j = t . This definition treats the two variables symmetrically, whereas here itwill be necessary to treat them with weights which may vary. Therefore we will definea multiplicity using weighted variables, which is called the index.

Definition 3.1. Let

Q(X1, . . . , Xn) =∑

`1,...,`n≥0

a`1,...,`nX`1

1 · · ·X`nn =:

∑(`)≥0

a(`)X(`)

be a nonzero polynomial in n variables, and let d1, . . . , dn be positive real numbers.Then the index of Q at 0 with weights d1, . . . , dn is

t(Q, (0, . . . , 0), d1, . . . , dn) = min

n∑

i=1

`i

di

∣∣∣∣∣ a(`) 6= 0

Often the notation will be shortened to t(Q, (0, . . . , 0)) when d1, . . . , dn are clearfrom the context.

Note that, although stated for polynomials, the above definition applies equally wellto power series. Moreover, replacing some Xi with a power series b1X

′i + b2X

′2i + . . .

( b1 6= 0 ) does not change the value of the index. Likewise, the index is preserved ifQ is multiplied by some power series with nonzero constant term (i.e., a unit). Andfinally, there is no reason why one cannot allow several variables in place of each Xi .Thus the index can be defined more generally for sections of line sheaves on productsof varieties.

Definition 3.2. Let γ be a rational section of a line sheaf L , on a product X1×· · ·×Xn

of varieties. Let P = (P1, . . . , Pn) be a regular point on∏

Xi , and suppose thatγ is regular at P . Let γ0 be a section which generates L in a neighborhoodof P , and for each i = 1, . . . , n let zij , j = 1, . . . ,dim Xi , be a system of local

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parameters at Pi , with zij(Pi) = 0 for all j . Then γ/γ0 is a regular function ina neighborhood of P , so it can be written as a power series

γ/γ0 =∑

(`)≥0

a(`)X(`).

Here (`) = (`ij) , i = 1, . . . , n , j = 1, . . . ,dim Xi is a (dim X1 + · · · + dim Xn)-tuple. Also let d1, . . . , dn be positive real numbers. Then the index of γ at Pwith weights d1, . . . , dn , denoted t(γ, P, d1, . . . , dn) or just t(γ, P ) , is

min

n∑

i=1

dim Xi∑j=1

`ij

di

∣∣∣∣∣ a(`) 6= 0

.

As noted already, this definition does not depend on the choices of γ0 or of localparameters zij .

As a special case of this definition, if n = 2 , if all Xi are taken to be P1 , and ifno Pi is ∞ , then γ is just a polynomial in two variables and this definition specializesto the preceding definition after P is translated to the origin.

§4. Sketch of the proof of Roth’s theorem

In a nutshell, the proof of Roth’s theorem amounts to a complicated system of inequal-ities involving the index. First we state the theorem, in general.

Theorem 4.1 (Roth [Ro]). Fix ε > 0 , a finite set of places S of k , and αv ∈ Q foreach v ∈ S . Then for almost all x ∈ k ,

(4.2)1

[k : Q]

∑v∈S

− log min(‖x− αv‖v, 1) ≤ (2 + ε)h(x).

If k = Q and S = ∞ , then this statement reduces to that of Section 1.We will only sketch the proof here; complete expositions can be found in [L 5] and

[Sch], as well as [Ro].First, we may first assume that all αv lie in k . Otherwise, let k′ be some finite

extension field of k containing all αv , let S′ be the set of places w of k′ lying overv ∈ S , and for each w | v let αw be a certain conjugate of αv . (In order to write‖x − αv‖v when αv /∈ k , some extension of ‖ · ‖v to k(αv) must be chosen; then theαw should be chosen correspondingly.) With proper choices of αw , the left-hand sidewill remain unchanged when k is replaced by k′ , as will the right-hand side.

The basic idea of the proof is to assume that there are infinitely many counterex-amples to (4.2), and derive a contradiction. In particular, we choose n good approxi-mations which satisfy certain additional constraints outlined below.

The proof will be split into five steps, although the often the first two steps aremerged, or the last two steps.

The first two steps construct an auxiliary polynomial Q with certain properties.Let α1, . . . , αm be the distinct values taken on by all αv , v ∈ S . Also let d1, . . . , dn

be positive integers. These will be taken large; independently of everything else in theproof, they may be taken arbitrarily large if their ratios are fixed.

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The polynomial should be a nonzero polynomial in n variables X1, . . . , Xn , andits degree in each Xi should be at most di , for each i . The first requirement is thatthe polynomial should have index ≥ n( 1

2−ε1) at each point (αi, . . . , αi) , i = 1, . . . ,m .Such polynomials can be constructed by solving a linear algebra problem in which thevariables are the coefficients of Q and the linear equations are given by the vanishing ofvarious derivatives of Q at the chosen points (αi, . . . , αi) . A nonzero solution exists ifthe number of linear equations is less than the number of variables (coefficients of Q ).Step 1 consists of showing that this is the case; the exact inequalities on ε1 and m willbe omitted, however, since they will not be needed for this exposition.

Thus, step 1 is geometrical in nature.Step 2 involves applying Siegel’s lemma to show that such a polynomial can be

constructed with coefficients in RS and with bounded height. (Here we let the heightof a polynomial be the height of its vector of coefficients.) From step 1, we know thevalues of M and N for Siegel’s lemma; the height will then be bounded by

(4.3) h(Q) ≤ c1

n∑i=1

di.

Here the constant c1 (not a Chern class!) depends on k , S , n , and α1, . . . , αm .This bound holds because M/(N − M) will be bounded from above, and coefficientsof constraints will be powers of α , multiplied by certain binomial coefficients.

Step 3 is independent of the first two steps; for this step, we choose elementsx1, . . . , xn ∈ k not satisfying (4.2). Further, the vectors(

− log min(‖xi − αv‖v, 1))v∈S

∈ R#S

all need to point in approximately the same direction, for i = 1, . . . , n . This is easy toaccomplish by a pigeonhole argument, since the vectors lie in a finite dimensional space.To be precise, there must exist real numbers κv , v ∈ S such that

− log min(‖xi − αv‖v, 1) ≥ κvh(xi), v ∈ S, i = 1, . . . , n

and such that

(4.4)1

[k : Q]

∑v∈S

κv = 2 +ε

2.

The points must also satisfy the conditions

h(x1) ≥ c2

andh(xi+1)/h(xi) ≥ r, i = 1, . . . , n− 1.

These conditions are easily satisfied, since by assumption there are infinitely many x ∈ knot satisfying (4.2), and the heights of these x go to infinity.

Having chosen x1, . . . , xn , let d be a large integer and let di be integers close tod/h(xi) , i = 1, . . . , n . For step 4, we want to obtain a lower bound for the index of

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Q at the point (x1, . . . , xn) . This is done by a Taylor series argument: if v ∈ S , thenwrite

Q(X1, . . . , Xn) =∑

(`)≥0

bv,(`)(X − αv)(`).

Bounds on the sizes ‖bv,(`)‖v can be obtained from (4.3) in Step 2; moreover bv,(`) = 0if

`1d1

+ · · ·+ `n

dn< n

(12− ε1

),

by the index condition in step 1. Then the only terms with nonzero coefficients arethose with high powers of some factors Xi − αv , so the falsehood of (4.2) implies aquite good bound on ‖Q(x1, . . . , xn)‖v for v ∈ S . Indeed, the nonzero terms in thisTaylor series are bounded by

n∏i=1

exp(−κv`ih(xi)) · other factors

≤ exp

(−κvd

n∑i=1

`i

di

)· other factors

≤ exp(−κvdn( 12 − ε1)) · other factors

At v /∈ S , we also have bounds on ‖Q(x1, . . . , xn)‖v , depending on the denominators inx1, . . . , xn . If these bounds are good enough, then the product formula is contradicted,implying that Q(x1, . . . , xn) = 0 . Indeed, taking the product over all v , the otherfactors come out to roughly exp([k : Q]dn) ; by (4.4) this gives∏

v

‖Q(x1, . . . , xn)‖v ≤ exp(−[k : Q](2 + ε

2 )dn( 12 − ε1)

)· exp([k : Q]dn) < 1.

Applying the same argument to certain partial derivatives of Q similarly gives vanish-ing, so we obtain a lower bound for the index of Q at (x1, . . . , xn) .

Note that the choice of the di counterbalances the varying heights of the xi , so infact each xi has roughly equal effect on the estimates in this step.

Finally, in step 5 we show that this lower bound contradicts certain other propertiesof Q . One possibility is to use the height h(Q) from step 2.

Lemma 4.5 (Roth, [Ro]; see also [Bo]). Let Q(X1, . . . , Xn) 6≡ 0 be a polynomial in nvariables, of degree at most di in Xi , with algebraic coefficients. Let x1, . . . , xn

be algebraic numbers, and let t = t(Q, (x1, . . . , xn), d1, . . . , dn) be the index of Qat (x1, . . . , xn) . Suppose that ε2 > 0 is such that

di+1

di≤ ε2

n−1

2 , i = 1, . . . , n− 1

and

(4.6) dih(xi) ≥ ε−2n−1

2 (h(Q) + 2nd1), i = 1, . . . , n.

Thent

n≤ 2ε2.

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Another approach is to use the index information from step 1. The following isa version which has been simplified, to cut down on extra notation. It is true moregenerally either in n variables, or on products of two curves.

Lemma 4.7 (Dyson, [D]). Let Vol(t) be the area of the set

(x1, x2) ∈ [0, 1]2 | x1 + x2 ≤ t,

so that Vol(t) = t2/2 if t ≤ 1 . Let ξ1, . . . , ξm be m points in C2 with distinct firstcoordinates and distinct second coordinates. Let Q be a polynomial in C[X1, X2]of degree at most d1 in X1 and d2 in X2 . Then

m∑i=1

Vol(t(Q, ξi, d1, d2)) ≤ 1 +d2

2d1max(m− 2, 0).

Historically, Dyson’s approach was the earlier of the two, but it has been revivedin recent years by Bombieri. I prefer it for aesthetic reasons, although the method ofusing Roth’s lemma is much quicker. In particular, it was the form of Dyson’s lemmawhich suggested the particular line sheaf to use in Step 1 of the Mordell proof.

Roth’s innovation in this area was the use of n good approximations instead oftwo; but for now we will just use two good approximations—it sufficed for Thue’s workon integral points.

§5. Notation

The rest of this paper will make heavy use of the language and results of Gillet andSoule extending Arakelov theory to higher dimensions. For general references on thistopic, see [So 2], [G-S 1], and [G-S 2].

In Arakelov theory it is traditional to regard distinct but complex conjugate em-beddings of k as giving rise to distinct local archimedean fibers. Here, however, wewill follow the much older convention of general algebraic number theory, that com-plex conjugate embeddings be identified, and therefore give rise to a single archimedeanfiber. This is possible to do in the Gillet-Soule theory, because all objects at complexconjugate places are assumed to be taken into each other by complex conjugation.

Then, if X is an arithmetic variety and v is an archimedean place, let Xv denotethe set X (kv) , identified with a complex manifold via some fixed embedding of kv intoC .

The notation also differs from Gillet’s and Soule’s in another respect. Namely,instead of using a pair D = (D, gD) to denote, say, an arithmetic divisor (i.e., anelement of Z1(X ) ), we will use the single letter D to refer to the tuple, and Dfin

to refer to its first component: D = (Dfin, gD) . Likewise, L will usually refer to ametrized line sheaf whose corresponding non-metrized line sheaf is denoted Lfin , etc.This notation is probably closer to that in Arakelov’s original work than the more recentwork of Gillet and Soule and others. Also, I feel that the objects with the additionalstructure at infinity are the more natural objects to be considering, and the notationshould reflect this fact.

174

In the theory of the Gillet-Soule Riemann theorem (cf. [G-S 3] and [G-S 4]), it isnatural to use the L2 norm to assign a metric to a global section γ of a metrized linesheaf; however, in this theory it is more convenient to use the supremum norm instead:

‖γ‖sup,v := supP∈X(kv)

‖γ(P )‖.

Here the norm on the right is the norm of L on Xv , since P ∈ X (kv) . If insteadwe had P ∈ X(k) , we would write ‖γ(P )‖v to specify the norm at the point (denotedPv ) on Xv corresponding to P ∈ X(k) . Also, if E ⊆ X is the image of the sectioncorresponding to P , then also let Ev equal Pv , as a special case of the notation Xv .

For future reference, we note here that often we will be considering Q-divisors orQ-divisor classes; these are divisors with rational coefficients (note that we do not tensorwith Q : this kills torsion, which leads to technical difficulties when converting to a linesheaf). Unless otherwise specified, divisors will always be assumed to be Cartier divisors.If L is such a Q-divisor class, then writing O(dL) will implicitly imply an assumptionthat d is sufficiently divisible so as to cancel all denominators in L . Also, the notationsΓ(X, L) and hi(X, L) will mean Γ(X, O(L)) and hi(X, O(L)) , respectively.

And finally, given any sort of product, let pri denote the projection morphism tothe ith factor.

§6. Derivatives

In adapting the proof of Roth’s theorem to prove Mordell’s conjecture, we replace P1

with an arbitrary curve C . Therefore instead of dealing with polynomials, we need toconsider something more intrinsic on Cn , namely, sections of certain line sheaves. Step4 of Roth’s proof therefore needs some notion of partial derivatives of sections of linesheaves at a point.

To begin, let π : X → B be an arithmetic surface corresponding to the curve C .By a theorem of Abhyankar [Ab] or [Ar], we may assume that X is a regular surface.Let W be some arithmetic variety which for now we will assume to be X ×B · · · ×B X(but actually it will be a slight modification of that variety); it is then a model for Cn .Let γ be a section of a metrized line sheaf L on W , let (P1, . . . , Pn) be a rationalpoint on Cn , and let E ⊆ W be the corresponding arithmetic curve, so that E ∼= Bvia the restriction of q : W → B .

Now the morphism π : X → B is not necessarily smooth, but that is not a problemhere, since we are dealing with rational points. Indeed, for i = 1, . . . , n let Ei denotethe arithmetic curve in X corresponding to Pi ; then the intersection number of Ei withany fiber is 1 (or rather log qv ), which means that Ei can only meet one local branchof the fiber and can meet that branch only at a smooth point. Thus the completed localring OX,Ei,v

is generated over OB,v by the local generator of the divisor Ei ; hence πis smooth in a neighborhood of Ei ; likewise q is smooth in a neighborhood of E .

Let x be a closed point in E , and let OB,q(x) be the completed local ring of Bat q(x) . For i = 1, . . . , n let zi be local equations representing the divisors Ei . Then,as noted above, the completed local ring OW,x of W at x can be written

OW,x∼= OB,q(x)[[z1, . . . , zn]].

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If γ0 is a local generator for L at x , then γ/γ0 is an element of this local ring, whichcan then be written

γ

γ0=∑

(`)≥0

b(`)z(`).

Then each term γ0b(`)z(`) lies in the subsheaf

L ⊗ O(−`1 pr−11 E1 − · · · − `n pr−1

n En) ⊆ L .

In general this element depends on the choices of γ0 and z1, . . . , zn . However, ifb(`′) = 0 for all tuples (`′) 6= (`) satisfying `′1 ≤ i1 , . . . , `′n ≤ in (i.e., if (`) isa leading term), then the restriction to E of this term is independent of the abovechoices. This is the definition of the partial derivative

D(`)γ(P1, . . . , Pn) ∈(L ⊗ O(−`1 pr−1

1 E1 − · · · − `n pr−1n En)

)∣∣∣E

.

In particular, this derivative is a regular section of the above sheaf. We will alsoneed the corresponding fact for the points on E at archimedean places. This translatesinto an upper bound on the metric of the above section at archimedean places, dependingon choices of metrics for O(Ei) .

To metrize O(Ei) in a uniform way, for each archimedean place v choose a metricfor O(∆) on C(kv) × C(kv) . For example, one could use the Green’s functions ofArakelov theory, but any smooth metric will suffice. Then O(Ei) will be taken as therestriction of this metric to Ei,v × C(kv) .

The next lemma takes place entirely on the local fiber of an achimedean place v ;i.e., on a complex manifold. Therefore for this lemma let C be a compact Riemannsurface.

By a compactness argument there exists a constant ρ = ρ(C) > 0 such that for allP ∈ C there exists a neighborhood UP ⊆ C of P with coordinate zP : UP

∼→ Dρ suchthat zP (P ) = 0 and such that the norm at P of zP , regarded as a section of O(−P ) ,satisfies the inequality

(6.1) |zP (P )| ≤ 1.

Lemma 6.2. Fix a point P0 = (P1, . . . , Pn) ∈ Cn . Let L be a metrized line sheaf onW , and let γ ∈ Γ(W,L ) . Let γ0 be a section of L on U := UP1×· · ·×UPn whichis nonzero at P0 . Suppose (`) = (`1, . . . , `n) is a tuple for which the derivativeD(`)f

∗γ(P1, . . . , Pn) is defined. Then

− log ‖D(`)γ(P0)‖ ≥ − log ‖γ‖sup +n∑

i=1

`i log ρ + log infP∈U

‖γ0(P )‖‖γ0(P0)‖

.

Proof. Writing zP0 for the tuple of functions (pr∗1 zP1 , . . . ,pr∗n zPn) and writing

γ = γ0

∑(i)≥0

a(i)z(i)P0

176

gives

‖γ‖sup ≥ supP∈U

(‖γ0(P )‖

∣∣∣∣∣∑(i)≥0

a(i)z(i)P0

(P )

∣∣∣∣∣)

≥ infP∈U

‖γ0(P )‖ · supz∈Dn

ρ

∣∣∣∣∣∑(i)≥0

a(i)z(i)

∣∣∣∣∣≥ ‖γ0(P0)‖ inf

P∈U

‖γ0(P )‖‖γ0(P0)‖

|a(`)|ρ`1+···+`n

by Cauchy’s inequalities in several variables,

|a(`)|ρ`1+···+`n ≤ supz∈Dn

ρ

∣∣∣∣∣∑(i)≥0

a(i)z(i)

∣∣∣∣∣Thus by (6.1),

‖(γ0a(`)z(`)P0

)(P0)‖ρ`1+···+`n infP∈U

‖γ0(P )‖‖γ0(P0)‖

≤ ‖γ‖sup.

Of course, one can now transport this result into the arithmetic setting by usingthe formula ‖ · ‖v = | · |[kv :R] . Also, if P ∈ C(k) and v | ∞ , we shall write UP,v andzP,v in place of UPv and zPv , respectively.

Corollary 6.3. Let γ be a global section of a metrized line sheaf L on W . LetE ⊆ W correspond to some P0 = (P1, . . . , Pn) ∈ Cn . Let d1, . . . , dn be positivereal numbers. For each v | ∞ let γ0,v be a local generator of L

∣∣Uv

, where

Uv = UP1,v × · · · × UPn,v . Then the index t = t(γ, P0, d1, . . . , dn) is at least

t ≥−deg L

∣∣E−∑

v|∞ log ‖γ‖sup,v +∑

v|∞ log infP∈Uv

‖γ0,v(P )‖‖γ0,v(P0,v)‖

max1≤i≤n deg O(−diEi)∣∣Ei− [k : Q] log ρ max1≤i≤n di

.

Proof. Let (`) be a multi-index for which D(`)γ(P0) is defined and nonzero. ThenD(`)γ(P0) is a nonzero section of

L ⊗ O(−`1 pr∗1 E1 − · · · − `n pr∗n En)∣∣E

.

We obtain the inequality by computing the degree δ of this line sheaf in two ways.First, its degree is

δ = deg L∣∣E

+n∑

i=1

`i deg O(−Ei)∣∣Ei

≤ deg L∣∣E

+n∑

i=1

`i

di· max1≤i≤n

deg O(−diEi)∣∣Ei

.

177

On the other hand, δ can be computed from the degree of the arithmetic divisorD(`)γ(P0) on E , via Lemma 6.2:

δ ≥∑v|∞

− log ‖D(`)γ(P0)‖v

≥∑v|∞

− log ‖γ‖sup,v +∑v|∞

log infP∈Uv

‖γ0,v(P )‖‖γ0,v(P0,v)‖

+ [k : Q] log ρn∑

i=1

`i

≥∑v|∞

− log ‖γ‖sup,v +∑v|∞

log infP∈Uv

‖γ0,v(P )‖‖γ0,v(P0,v)‖

+n∑

i=1

`i

di· [k : Q] log ρ max

1≤i≤ndi

(here we assume ρ < 1 ). Combining these two inequalities gives the corollary, since wemay choose (`) such that

t =n∑

i=1

`i

di.

§7. Proof of Mordell, with some simplifications by BombieriTo introduce the various proofs of Mordell’s conjecture, etc., we start with a sketch ofa proof using ideas from Bombieri’s proof, since the methods do not involve as muchmachinery. More details on this proof can be found in [Bo].

First, we need some notation. Let C be the curve, and let g be its genus; theng > 1 . Let KC be a canonical divisor on C , and let F = KC/(2g− 2) , so that F hasdegree 1. Here F is a Q-divisor—a divisor with rational coefficients. On C × C , letpr1 and pr2 be the projections to the factors, let Fi = pr∗i F for i = 1, 2 , and let ∆be the diagonal on C × C . Let

∆′ = ∆− F1 − F2.

Let δ > 0 and r > 1 be rational; we require that a1(r) :=√

(g + δ)r also be rational.Then a2(r) :=

√(g + δ)/r is also rational. The goal of the first two steps of the proof,

then, will be to construct a certain global section of the line sheaf O(dY ) , where d isa large sufficiently divisible positive integer and

Y = Yr = ∆′ + a1F1 + a2F2.

In Bombieri’s proof, Step 1 is rather easy: by duality, h2(C × C, dY ) = 0 for dsufficiently large; therefore by the Hirzebruch Riemann-Roch theorem,

h0(C × C, dY ) ≥ d2Y 2

2≥ δd2,

since ∆′2 = −2g , ∆′F1 = ∆′F2 = F 21 = F 2

2 = 0 , and F1 . F2 = 1 .Step 2 requires a bit more cleverness. First, fix N > 0 such that NF is very

ample, and fix global sections x0, . . . , xn of O(NF ) giving a corresponding embeddinginto Pn . We also regard these as sections of O(NF1) on C × C via pr1 and letx′0, . . . , x

′n be the corresponding sections of O(NF2) defined via pr2 .

178

Also take s to be a sufficiently large integer such that

B := sF1 + sF2 −∆′

is very ample, and let y0, . . . , ym be global sections of O(B) on C × C giving acorresponding embedding into Pm .

Then one of Bombieri’s key ideas is to write

Y = δ1NF1 + δ2NF2 −B,

where δi = (ai + s)/N . Then, for d sufficiently large, a section γ of O(dY ) willhave the property that γyd

i will be represented by polynomials Φi of degree dδ1 inx0, . . . , xn and of degree dδ2 in x′0, . . . , x

′n . Indeed, let I be the ideal sheaf of the

image of C × C in Pn × Pn . Since O(δ1, δ2) (defined as pr∗1 O(δ1) ⊗ pr∗2 O(δ2) ) isample, for d sufficiently large we will have H1(Pn × Pn,I ⊗O(dδ1, dδ2)) = 0 , and bythe long exact sequence in cohomology, the map

H0(Pn × Pn,O(dδ1, dδ2)) → H0(C × C,O(dδ1, dδ2)∣∣C×C

)

= H0(C × C,O(d∆1NF1 + d∆2NF2))

will be surjective. Of course, it is not injective: instead, one chooses a subspace ofH0(Pn × Pn,O(dδ1, dδ2)) for which the map is injective, and for which the cokernel issufficiently small. If the coordinates x0, . . . , xn are chosen suitably generically, thenthese conditions hold for the subspace spanned by

xdδ10 (x′0)

dδ2 ·(

x1

x0

)a(x2

x0

)b(x′1x′0

)a′ (x′2x′0

)b′

,

0 ≤ a + b ≤ dδ1, 0 ≤ b ≤ N, 0 ≤ a′ + b′ ≤ dδ2, 0 ≤ b′ ≤ N.

Then, apply Siegel’s lemma to tuples (Φ1, . . . ,Φm) in the above subspace, subject tothe linear constraints

Φiydj = Φjy

di , i, j ∈ 0, . . . ,m.

A solution to this system then yields local sections Φi/ydi which patch together to give

a global sectionγ ∈ Γ(C × C,O(dY ))

with

(7.1) h(γ) ≤ cda1 + o(d).

Here the height h(γ) is defined as the height of the vector of coefficients of the polyno-mials Φ1, . . . ,Φm .

From this point Bombieri continues to proceed very classically, using Weil’s theoryof heights instead of arithmetic intersection theory. But instead we will continue theproof in more modern language.

Note, first of all, that one can fix metrics on F and ∆ , giving a metric on Y .Then we obtain a bound for the sup norm on Y , in terms of the sizes of the coefficientsof the polynomials Φi .

179

Step 3 now uses an idea from a 1965 paper of Mumford [M]. As in Roth’s theorem,it uses a pigeonhole argument, but this time the argument takes place in J(k) ⊗Z R ;here J denotes the Jacobian of C . By the Mordell-Weil theorem ([L 5], Ch. 6), thisis again a finite dimensional vector space.

Fixing a point P0 ∈ C , let φ0 : C → J be the map given by P 7→ O(P − P0) . By([M], pp. 1011–1012),

O(∆− P0 × C − C × P0) ' pr∗1 φ∗0Θ0 + pr∗2 φ∗0Θ0 − (φ0 × φ0)∗(pr1 +pr2)∗Θ0

where Θ0 is the divisor class of the theta divisor on J . But we need Θ0 to be asymmetric divisor class. Therefore fix a ∈ J such that (2g−2)a = O((2g−2)P0−KC) ,let φ(P ) = φ0(P ) + a , and let Θ be the class of the theta divisor defined relative tothe map φ . Then Θ is a symmetric divisor class, and

(7.2) ∆′ = −(φ× φ)∗P

on C × C , up to (2g − 2)-torsion, where

P := (pr1 +pr2)∗Θ− pr∗1 Θ− pr∗2 Θ

is the Poincare divisor class. See also ([L 5], Ch. 5, § 5).Let hΘ denote the Neron-Tate canonical height on J relative to Θ (cf. ([Sil],

Thm. 4.3) or ([L 5], Ch. 5, §§3, 6, 7)); it is quadratic in the group law and thereforedefines a bilinear pairing

(P1, P2)Θ = hΘ(P1 + P2)− hΘ(P1)− hΘ(P2)

which gives the vector space J(k)⊗Z R a dot product structure. The canonical heightalso satisfies

(7.3) hΘ(P ) = hΘ(P ) + O(1).

Let |P |2 = (P, P )Θ , so that |P |2 = 2hΘ(P ) . Then, assuming that C(k) is infinite,we may find an infinite subsequence such that all points in the subsequence point inapproximately the same direction: all P1 and P2 in this subsequence satisfy

(7.4) (P1, P2)Θ ≥ (cos θ)√|P1|2|P2|2

for a given θ ∈ (0, π) . Also choose P1 and P2 so that h(P1) is large, and so thath(P2)/h(P1) is large.

Mumford shows that φ∗Θ = gF ; thus by (7.2), (7.3), and (7.4),

h∆′(P1, P2) ≤ −2g(cos θ)√

hF (P1)hF (P2) + O(1).

Expressing this in terms of Y , and rewriting it in terms of metrized line sheaves onW := X ×B X gives

1[k : Q]

deg O(dY )∣∣E≤ d(a1hF (P1) + a2hF (P2)− 2g(cos θ)

√hF (P1)hF (P2) + O(1)

).

180

Taking r close to hF (P2)/hF (P1) and recalling the definitions of a1 and a2 then gives

1[k : Q]

deg O(dY )∣∣E≤ 2d

((√

g + δ − g cos θ)√

hF (P1)hF (P2) + O(1)).

For θ small enough, this is negative. If it is sufficiently small, then by the productformula (0.4), the restriction to E of any sufficiently small global section of O(dY )must vanish, so γ must vanish along E .

As was the case in Roth’s proof, Step 4 consists of applying a variant of the aboveargument to certain partial derivatives, giving a lower bound for the index of γ at(P1, P2) . But the work has already been done: take di = dai for i = 1, 2 and notethat the inequality of Corollary 6.3 gives:

t ≥2d(

√g + δ − g cos θ)

√hF (P1)hF (P2)− cda1 + o(d)

(2g − 2)da1hF (P1) + cda1.

Indeed, by (7.1) the term∑

v|∞ log ‖γ‖sup,v in the numerator of the expression inCorollary 6.3 is bounded by cda1+o(d), and the generators γ0,v for various O(dYr) maybe taken uniformly in d and r , so the term

∑v|∞ log infP∈Uv ‖γ0,v(P )‖

/‖γ0,v(P0,v)‖

also is bounded by cda1 .But now note that a1hF (P1) is approximately

√g + δ

√hF (P1)hF (P2) . Thus the

first terms in the numerator and denominator are dominant as the h(Pi) become large.This gives a lower bound for the index.

One can then project C × C down to P1 × P1 , take the norm of γ to get apolynomial, and apply Roth’s lemma to obtain a contradiction. We omit the detailsbecause they will appear in more generality in Section 18.

§8. Proof using Gillet-Soule Riemann-Roch

In this case, we still use the same notations δ , r , a1 , a2 , F1 , F2 , ∆ , ∆′ , and Yas before. However, Step 1 is a little more complicated, in that we prove that if r issufficiently large, then Y is ample. For details on this and other parts of the proof, see[V 3] and [V 4].

Step 2 is the part which I wish to emphasize—this is where the Gillet-SouleRiemann-Roch theorem is used. First, we assume C has semistable reduction overk , and let X be the regular semistable model for C over B(= Spec R) ([L 7], Ch. V,§5). Then X ×B X is regular except at points above nodes on the fibers of each fac-tor. At such points, though, the singularity is known explicitly and can be resolved byreplacing it with a projective line. Let q : W → B be the resulting model for C × C .

The divisors F and ∆ on the generic fiber need to be extended to X and W ,respectively. To extend F , we take ωX/B at finite places, and fix a choice of metricswith positive curvature. The (Arakelov) canonical metric is one possible choice, but itis not required. To extend ∆ , we take its closure on W , and choose a metric for it.Again, the Arakelov Green’s function is one possible choice. Then F1 , F2 , ∆′ , and Ybecome arithmetic divisors on W as well. By the Gillet-Soule Riemann-Roch theorem,then,

2∑i=0

(−1)i deg Riq∗O(dY ) =d3Y 3

6+ O(d2).

181

We want a lower bound for deg q∗O(dY ) ; this is obtained as follows. First, since Yis ample, the free parts of Riq∗O(dY ) vanish for i > 0 if d is sufficiently large. Thetorsion part of the R1q∗ term is nonnegative, and the torsion part in the R2q∗ term iszero by a duality argument. Much longer arguments in a similar vein (but with their ownanalytic character) show that the same is true for analytic torsion, up to O(d2 log d) .Thus we find that

deg q∗O(dY ) ≥ d3Y 3

6−O(d2 log d).

Here Y 3 grows like −O(√

r) . Since the rank of O(dY ) is approximately δd2 , theratio deg q∗O(dY )

/rank q∗O(dY ) is approximately O(−d

√r) . Then it follows by a

geometry of numbers argument that there exists a global section γ of O(dY ) with∏v|∞

‖γ‖L2,v ≤ exp(cd√

r).

But we really need an inequality involving the sup norm. Of course, the trivialinequality

‖γ‖L2,v ≤ ‖1‖L2,v‖γ‖sup,v

holds, but this goes in the wrong direction.It is slightly more difficult to prove an inequality in the opposite direction.

Lemma 8.1. Fix a measure ν on a complex manifold X of dimension n . Then foreach metrized line sheaf L on X there exists a constant cL > 0 such that forall γ ∈ Γ(X, L ) ,

‖γ‖L2 ≥ cL ‖γ‖sup.

Moreover, if L ∼= L ⊗i11 ⊗ · · · ⊗L ⊗im

m , then we may take cL = ci1L1· · · cim

Lm.

Proof. By a compactness argument, there exists a constant ρ > 0 and for each P ∈ Xa local coordinate system on a neighborhood UP of P , zP : UP

∼→ Dnρ , such that

zP (P ) = 0 andddc|z1|2 ∧ · · · ∧ ddc|zn|2 ≤ ν.

Also, for each P and each L there exist local holomorphic sections γ0,P of L∣∣UP

such that ‖γ0,P (P )‖ = 1 and

c′L := infP∈XQ∈UP

‖γ0,P (Q)‖2

is strictly positive. This may require shrinking ρ , depending on L . Then, letting Pbe the point where γ attains its maximum,

‖γ‖2L2 ≥ c′L

∫Dn

ρ

∣∣∣∣( γ

γ0,P

)(Q)∣∣∣∣2 ddc|z1|2 ∧ · · · ∧ ddc|zn|2

≥ c2L ‖γ(P )‖2

for some suitable cL > 0 , by Parseval’s inequality (or harmonicity).The last statement follows by choosing the sections γ0,P for L compatibly with

those chosen for the Li .

182

Sharper bounds are possible (cf. ([V 4], 3.9)), but the above bound is sufficient forour purposes.

This proof of Mordell can then conclude with Steps 3–5 as before. Or, in eithercase, instead of Roth’s lemma, we can use Dyson’s lemma on a product of two curves.

Lemma 8.2 ([V 2]). Let ξ1, . . . , ξm be m points on C2 with distinct first coordinatesand distinct second coordinates. Let γ be a global section of a line sheaf L onC × C , and assume that (L . F1) ≥ d2 and (L . F2) ≥ d1 . Then, recalling thenotation Vol() from Lemma 4.7,

m∑i=1

Vol(t(γ, ξi, d1, d2)) ≤(L 2)2d1d2

+(L . F1)

2d1max(2g − 2 + m, 0).

In this case, di = dai as before, and L = O(dY ) . Then it follows that the firstterm on the right is δ/(g + δ) and the second term is (2g − 1)/r . Both can be madesmaller than t2/2 on the left, obtaining a contradiction.

It was this part of the argument that first led to some insight on the problem:instead of making certain terms on the left large, one could make Y 2 on the rightsmall. This is how one can prove finiteness for diophantine equations without usingdiophantine approximation per se.

§9. The Faltings complexThe remainder of these lectures will be devoted to proving Faltings’ generalization ofMordell’s conjecture, Theorem 0.3. This will be done in detail. See also [F 1] and [F 2].

As a first step towards generalizing the technique to more general subvarieties ofabelian varieties, recall the result of Mumford (7.2):

∆′ = (j × j)∗(pr∗1 Θ + pr∗2 Θ− (pr1 +pr2)∗Θ),

where pr1 +pr2 in the last term refers to the sum under the group law on the Jacobian.Then one can replace Θ with any symmetric ample divisor class L on a general abelianvariety A , and let the Poincare divisor class

P := (pr1 +pr2)∗L− pr∗1 L− pr∗2 L

play the role of (minus) ∆′ . But now the theorem of the cube implies that for a, b ∈ Z ,

(9.1) (a · pr1 +b · pr2)∗L = a2 pr∗1 L + b2 pr∗2 L + abP.

Then it follows that dY can be written (approximately) in the form

dY = (s1 · pr1−s2 · pr2)∗L− εs2

1 pr∗1 L− εs22 pr∗2 L.

In this case, however, it will be necessary to work on a product of n copies of A , solet us define

(9.2) Lδ,s =∑i<j

(si · pri−sj · prj)∗L + δ

n∑i=1

s2i pr∗i L

for rational δ and s ∈ Nn . By the theorem of the cube, this expression is homogeneousof degree two in s1, . . . , sn , so we also extend this definition to s ∈ Qn

>0 by homogeneity.

183

One aspect of the expression (9.2) is that it clearly points out a key idea in thewhole theory. Namely, on

∏Xi the first term is large (ample, in the case of Theorem

0.2), and the second term is small ( δ is taken negative but close to zero); however, onthe arithmetic curve corresponding to our point (P1, . . . , Pn) , the first term is smalland the second term then dominates.

Another benefit of this expression is that, by the theorem of the cube,

(a · pri−b · prj)∗L + (a · pri +b · prj)

∗L = 2a2 pr∗i L + 2b2 pr∗j L.

Thus, choosing global sections γ1, . . . , γm ∈ Γ(A,O(L)) which generate O(L) over thegeneric fiber A , for any X1, . . . , Xn we can form an injection

0 → Γ(∏

Xi, dLδ,s

)→ Γ

(∏Xi, d(2n− 2 + δ)

n∑i=1

s2i pr∗i L

)a

by tensoring with products of terms of the form (asi ·pri +asj ·prj)∗γb`ij

, where ba2 = d

and a is sufficiently divisible. Here the tuples (`)ij vary over 1, . . . ,mn(n−1)/2 .Likewise, one can extend this sequence to an exact sequence

0 → Γ(∏

Xi, dLδ,s

)→ Γ

(∏Xi, d(2n− 2 + δ)

n∑i=1

s2i pr∗i L

)a

→ Γ

(∏Xi, d

∑i<j

(si · pri−sj · prj)∗L + d(2n− 2 + δ)

n∑i=1

s2i pr∗i L

)b′

(9.3)

and embed this last term to obtain the Faltings complex

0 → Γ(∏

Xi, dLδ,s

)→ Γ

(∏Xi, d(2n− 2 + δ)

n∑i=1

s2i pr∗i L

)a

→ Γ

(∏Xi, d(4n− 4 + δ)

n∑i=1

s2i pr∗i L

)b

(9.4)

for some integer b . This integer will be quite large, but depends only on m and n .

§10. Overall plan

Now let us discuss the overall plan of the proof of Theorem 0.3. First of all, we mayassume that X is closed in A .

Next, recall some standard facts about subvarieties of abelian varieties. For now,let X ⊆ A be defined over C .

Definition 10.1. Let B(X) be the identity component of the algebraic group

a ∈ A | a + X = X.

184

Then the restriction to X of the quotient map A → A/B(X) gives a fibrationX → X/B(X) whose fibers are all isomorphic to B(X) . This fibration is calledthe Ueno fibration of X . It is trivial when B(X) is a point.

Theorem 10.2 (Ueno ([U], Thm. 10.9)). If B(X) is trivial, then X is of general type(and conversely).

Thus, Theorem 0.3 gives an affirmative answer to a special case of a question posedby Bombieri [N]: if X is a variety of general type defined over a number field k , is ittrue that X(k) is not dense in the Zariski topology? Furthermore, Lang has conjecturedin ([L 6], Conj. 5.8) that the higher dimensional part of the closure of X(k) should begeometric; i.e., independent of k . That conjecture is also answered in this special case,by the following theorem.

Theorem 10.3 (Kawamata structure theorem ([Ka], Theorem 4)). Let Z(X) denotethe union of all nontrivial translated abelian subvarieties of A contained in X .Then Z(X) is a Zariski-closed subset, and each irreducible component of it hasnontrivial Ueno fibration.

In particular, if X has trivial Ueno fibration, then Z(X) 6= X (and conversely).Returning to the situation over k , we note that B(X) and Z(X) are defined over

k , by simple Galois-theoretic arguments. (In general, Z(X) is a scheme: it may bereducible.)

For the proof of Theorem 0.3, we note first that it will suffice to assume that B(X)is trivial; otherwise the theorem on X follows from the theorem on X/B(X) .

Next, it will suffice to prove that X(k) \Z(k) is finite. Indeed, points in Z(k) canbe handled by Noetherian induction.

Now, for the main part of the proof, fix a very ample symmetric divisor class L onA , and an associated projective embedding. Degrees and heights of subspaces of X willrefer to this embedding. In particular, for heights, we use the definition from ([B-G-S],Section 1): the height of a subvariety of projective space is the intersection number witha “standard” linear subspace of complementary dimension in Pn ; a “standard” linearsubspace is one which is obtained by setting some of the coordinate functions equal tozero.

In the proof, we will fixn = dim X + 1

and work with points P1, . . . , Pn ∈ X(k) \ Z(k) . Let hL(Pi) denote the heights of thePi . These points will be chosen (later) so as to satisfy the usual Step 3 conditions, forcertain constants c1 , c2 ≥ 1 , and ε1 :(10.4.1). hL(P1) ≥ c1 ;(10.4.2). hL(Pi+1)/hL(Pi) ≥ c2 , i = 1, . . . , n− 1 ;(10.4.3). P1, . . . , Pn all point in roughly the same direction in A(k)⊗Z R : let hL(P )

denote the Neron-Tate canonical height associated to hL and let

(P,Q)L = hL(P + Q)− hL(P )− hL(Q)

185

be the associated bilinear form; then the assumption is that

(Pi, Pj)L ≥ (1− ε1)√

(Pi, Pi)(Pj , Pj) for all i, j .

We will also call these conditions CP (c1, c2, ε1) .The proof also uses subvarieties X1, . . . , Xn of X satisfying the following condi-

tions, denoted CX(c3, c4, P1, . . . , Pn) :(10.5.1). Each Xi contains Pi .(10.5.2). The Xi are geometrically irreducible and defined over k .(10.5.3). The degrees deg Xi satisfy deg Xi ≤ c3 .(10.5.4). The heights h(Xi) are bounded by the formula

n∑i=1

h(Xi)hL(Pi)

< c4

n∑i=1

1hL(Pi)

.

Here and from now on, constants c and ci will depend on A , X , k , the projectiveembedding associated to L , and sometimes the tuple (dim X1, . . . ,dim Xn) . They willnot depend on Xi , Pi , or (s) . Also, they may vary from line to line.

The overall plan of the proof, then, is to construct subvarieties X1, . . . , Xn of Xsatisfying the conditions (10.5). We start with X1 = · · · = Xn = X and successivelycreate smaller tuples of subvarieties, until reaching the point where dim Xj = 0 forsome j . In that case Xj = Pj , and h(Xj) = hL(Pj) . Then, by (10.5.4),

(10.6) 1 =h(Xj)hL(Pj)

≤n∑

i=1

h(Xi)hL(Pi)

≤ c4

n∑i=1

1hL(Pi)

≤ c4n

hL(P1)

and thus

(10.7) hL(P1) ≤ c4n.

This contradicts (10.4.1) if c1 is taken large enough.The inductive step of the proof takes place if all Xi are positive dimensional. For

i = 1, . . . , n let si be rational numbers close to 1/√

hL(Pi) . Let d be a positiveinteger. This is usually taken large and highly divisible, and may depend on practicallyeverything else. We shall construct a small section of O(dL−ε,s) for some ε > 0(depending on the dim Xi ). If the points P1, . . . , Pn were chosen suitably, then it ispossible to construct subvarieties X ′

i of each Xi such that the X ′i also satisfy (10.5)

(possibly with different constants), and such that some X ′i is strictly smaller.

186

To help clarify this step, the dependence of the constants, etc. can be writtensymbolically as follows.

∀ c3, c4 and ∀ δ1, . . . , δn ∈ N∃ c1, c2, ε1, c

′3, c

′4 such that

∀ P1, . . . , Pn ∈(X \ Z(X)

)(k) satisfying CP (c1, c2, ε1) and

∀ X1, . . . , Xn ⊆ X satisfying CX(c3, c4, P1, . . . , Pn) and dim Xi = δi ∀ i

∃ X ′1, . . . , X

′n with X ′

i ⊆ Xi ∀ i and X ′i 6= Xi for some i,

and satisfying CX(c′3, c′4, P1, . . . , Pn).

(10.8)

To conclude the proof, we now assume that X(k) \ Z(k) is infinite. Then it ispossible to choose P1, . . . , Pn in X(k) \ Z(k) to satisfy Conditions (10.4) for all c1 ,c2 , and ε1 which occur in the finitely many times that the above main step can takeplace. This leads to a contradiction, so X(k) \Z(k) must be finite, and the theorem isproved.

For future reference, let

(10.9) di = ds2i .

§11. Lower bound on the space of sections

Recall that X1, . . . , Xn are positive dimensional subvarieties of X , not lying in theKawamata locus of X .

Lemma 11.1. If n ≥ dim X + 1 , then the morphism f :∏

Xi → An(n−1)/2 given by(x1, . . . , xn) 7→ (xi − xj)i<j is generically finite over its image.

Proof. If P ∈ Xi,reg , then the tangent space TXi,P may be identified with a linearsubspace of the tangent space TA,0 at the origin of A via translation. Then we maychoose points Pi ∈ Xi,reg such that f is smooth at P := (P1, . . . , Pn) , and such that

n⋂i=1

TXi,Pi = (0).

Then any tangent to the fiber of f at P must be zero, so f is a finite map there.

Corollary 11.2. If n ≥ dim X + 1 , then the intersection number((L0,1

∣∣QXi

)Pdim Xi

)> 0.

Proof. The Q-divisor class L0,1 is the pull-back to∏

Xi of an ample divisor class onAn(n−1)/2 via a generically finite morphism.

For the remainder of the proof of Theorem 0.3, fix n = dim X + 1 .

Next we prove a homogeneity result in s .

187

Lemma 11.3. Fix an embedding of k into C . Then the cohomology class in H1,1

∂(An)

corresponding to the divisor class

Pij := (pri +prj)∗L− pr∗i L− pr∗j L

is represented over A(C)n by a form in

pr∗i E 1,0(A)⊗ pr∗j E 0,1(A) + pr∗i E 0,1(A)⊗ pr∗j E 1,0(A) ⊆ E 1,1(An).

Proof. Let d = dim A , and let A be given local coordinates z1, . . . , zd obtained fromthe representation of A as Cd modulo a lattice. Let O(L) be given the metric withtranslation invariant curvature, which can then be written

d∑α,β=1

aαβdzα ∧ dzβ ,

where aαβ are constants. For α = 1, . . . , d let uα = pr∗i zα and vα = pr∗j zα . Usingthe above choice of metric on L , the curvature of Pij is

d∑α,β=1

aα,β(duα ∧ dvβ + dvα ∧ duβ).

By counting degrees we immediately obtain the main homogeneity lemma:

Corollary 11.4. Any intersection product∏i<j

Peij

ij .n∏

i=1

pr∗i Lei

of maximal codimension on∏

Xi vanishes unless

2ei +∑j<i

eji +∑j>i

eij = 2dim Xi, i = 1, . . . , n.

Consequently, since

(si · pri−sj · prj)∗L = s2

i pr∗i L + s2j pr∗j L + sisjPij ,

it follows that the highest self-intersection number of Lδ,s on∏

Xi is homogeneousof degree 2 dim Xi in each si .

The next proposition is the main result of this section. The proof is due to M.Nakamaye and G. Faltings, independently.

Proposition 11.5. There exist constants c > 0 and ε > 0 , depending only on X ,A , L , dim X1, . . . ,dim Xn , and the bounds on deg Xi , such that for all tupless = (s1, . . . , sn) of positive rational numbers,

h0(∏

Xi, dL−ε,s

)> cd

Pdim Xi

n∏i=1

s2 dim Xii

188

for all sufficiently large d (depending on s ).

Proof. By Seshadri’s criterion ([H 1], Ch. I, §7), Lδ,s is ample for δ > 0 . By Riemann-Roch, it follows that

h0(∏

Xi, dLδ,s

)= d

Pdim Xi

(LP

dim Xi

δ,s )(∑

dim Xi)!(1 + o(1)).

To shorten notation, let N =∑

dim Xi for the remainder of the proof. For each indexj , let Hj be the subscheme of

∏Xi cut out by some section of pr∗j O(L) . Then, as

above,

h0(Hj , dLδ,s

∣∣Hj

)= dN−1

(pr∗j L . LN−1δ,s )

(N − 1)!(1 + o(1)).

For each i (including j ), pr∗i O(L) is represented by an effective divisor on Hj , so

h0

(Hj ,

(dLδ,s −

∑mi pr∗i L

)∣∣∣Hj

)≤ dN−1

(pr∗j L . LN−1δ,s )

(N − 1)!(1 + o(1))

for each tuple m1, . . . ,mn of nonnegative integers. Note that the o(1) term does notdepend on m1, . . . ,mn .

The exact sequence in cohomology attached to the short exact sequence

0 → O(dLδ,s −

∑mi pr∗i L− pr∗j L

)→ O

(dLδ,s −

∑mi pr∗i L

)→ O

(dLδ,s −

∑mi pr∗i L

) ∣∣∣Hj

→ 0

gives the inequality

h0(∏

Xi, dLδ,s −∑

mi pr∗i L− pr∗j L)

≥ h0(∏

Xi, dLδ,s −∑

mi pr∗i L)− h0

(Hj , dLδ,s −

∑mi pr∗i L

)≥ h0

(∏Xi, dLδ,s −

∑mi pr∗i L

)− dN−1

(pr∗j L . LN−1δ,s )

(N − 1)!(1 + o(1)).

Therefore, since dL−ε,s = dLδ,s − d(δ + ε)∑

i s2i Hi , this estimate gives

h0(∏

Xi, dL−ε,s

)≥ h0

(∏Xi, dLδ,s

)− d(δ + ε)

n∑i=1

s2i h

0(Hi, dLδ,s

∣∣Hi

)≥ dN

((LN

δ,s)N !

− (δ + ε)n∑

i=1

s2i

(pr∗i L . LN−1δ,s )

(N − 1)!

)(1 + o(1)).

By Corollary 11.4, this lower bound equals

dNn∏

i=1

s2 dim Xii

((LN

δ,1)N !

− (δ + ε)n∑

i=1

(pr∗i L . LN−1δ,1 )

(N − 1)!

)(1 + o(1)).

189

The quantity inside the parentheses is a polynomial in δ and ε whose constant termis positive, by Corollary 11.2. Therefore we may take sufficiently small δ > 0 , ε > 0 ,and c > 0 such that

0 < c <(LN

δ,1)N !

− (δ + ε)n∑

i=1

(pr∗i L . LN−1δ,1 )

(N − 1)!

The fact that these intersection numbers are taken on varying Xi is not a seriousproblem: since the degrees of the Xi are bounded, they can lie in only finitely manynumerical equivalence classes.

§12. More geometry of numbers

Let Γ be a metrized, finitely generated lattice over the ring of integers R of k . For allarchimedean places v of k , let the completion Γv of Γ at v be given a measure suchthat, for each set γ1, . . . , γδ of R-linearly independent vectors in Γ which generate Γmodulo torsion, the product over all v of the covolumes of the lattices generated overZ (if kv

∼= R ) or Z[i] (if kv∼= C ) by γ1, . . . , γδ equals

(Γ :

∑δi=1 Rγi

). Also let

V (Γ) be the product, over all v , of the volumes (relative to the above measures) of theunit balls (relative to ‖ · ‖v ) in Γv . Define a length function `(γ) =

∏v ‖γ‖v , and for

i = 1, . . . , δ define successive minima λi to be the minimum λ such that there existR-linearly independent elements γ1, . . . , γi ∈ Γ such that `(γj) ≤ λ for all j = 1, . . . , i .

Lemma 12.1. In this situation,

(a). There exist constants c1 and c2 depending only on k such that

(c2δ)−δ ≤ λ1 · · ·λδV (Γ) ≤ cδ3.

(b). Let β : Γ1 → Γ2 be a homomorphism of metrized R-modules. Let δ0 andδ2 be the ranks (over R ) of the kernel and image of β , respectively. Alsoassume that C is a constant such that

(12.2)∏v|∞

‖β(γ)‖v ≤ C∏v|∞

‖γ‖v for all γ ∈ Γ1 .

Then

(12.3) V (Γ1) ≤ 2[k:Q]δ0V (Kerβ)Cδ2V (Im β).

Proof. Part (a) is a generalization of Minkowski’s theory of convex bodies. It followsfrom ([V 1], 6.1.11). There, it is proved only for parallelepipeds, but the proof holdsmore generally for convex length functions. The proof of part (b) is straightforward.

We note that this lemma can be used to prove a Siegel lemma over number fields,although it is not as sharp as Theorem 2.4. Indeed, metrize Γ1 := RN by a metric‖x‖2 = |x1|2 + · · · + |xN |2 at real places, and ‖x‖ = |x1|2 + · · · + |xN |2 at complexplaces. Then h(xi) = (1/[k : Q]) log λi , in the language of Theorem 2.4. Also, in(12.2), let C =

∏v|∞ N [kv :R] maxi,j ‖aij‖ ; this will play the role of Q in Lemma 2.1

190

or exp(h(A)) in Theorem 2.4. Then V (Γ1) and V (Im β) are functions only of therespective ranks, so (12.3) gives a lower bound for V (Kerβ) , depending only on C andthe various ranks. Then part (a) of the lemma gives an upper bound on

∑N−Mi=1 h(xi) .

We will use Lemma 12.1 to determine a lower bound for V (Kerβ) . It will thenfollow from part (a) that a global section γ exists, whose metric in Γ1 is small (boundedby exp

(cd∑

s2i

)), provided a suitable lower bound on δ0 can be shown. This will be

done in Section 14.

§13. Arithmetic of the Faltings complex

To obtain a small section of O(dL−ε,s) , one would like to extend the Faltings complexto a model A for A over the integers, and apply Siegel’s lemma. However, whendiscussing the arithmetic of sections of (a · pri +b · prj)∗O(L) , some care is needed:usually the morphism a · pri +b · prj does not extend to a morphism of arithmeticschemes. Therefore, we use the theorem of the cube in order to convert such sectionsinto sections of O(a2 · pr∗i L + b2 · pr∗j L + abPij) ; the isomorphism is to be extendedover Spec R so as to give an isomorphism over the zero section. Here, as in Section 9,Pij denotes the Poincare class

Pij := (pri +prj)∗L− pr∗i L− pr∗j L.

Then integrality of a section of (a · pri +b · prj)∗O(L) is to be understood via thecorresponding isomorphism

(13.1) (a · pri +b · prj)∗O(L) ∼= O(a2 · pr∗i L + b2 · pr∗j L + abPij).

Here and in what follows, let W0 be some model for An such that Pij is defined as aCartier divisor class, for all i and j .

Lemma 13.2 ([F 1], Lemma 5.1). Fix a model A for A over B , an arithmetic divisorclass L on A , and sections γ1, . . . , γm ∈ Γ(A ,O(L)) which generate O(L) overthe generic fiber A . Let A 2 be some model for A×A (as above). Then there existconstants c5,v , with c5,v = 0 for almost all v , satisfying the following property.Let a, b ∈ Z , not both zero. Then for all archimedean places v the sections

(a · pr1 +b · pr2)∗γi ∈ O(a2 pr∗1 L + b2 pr∗2 L + abP12)

satisfy the following conditions.

(a). For all i = 1, . . . ,m and for all P ∈ A2(kv) ,

log ‖((a · pr1 +b · pr2)∗γi)(P )‖v ≤ c5,v(a2 + b2).

(b). For all P ∈ A2(kv) ,

max1≤i≤m

log ‖((a · pr1 +b · pr2)∗γi)(P )‖v ≥ −c5,v(a2 + b2).

191

Furthermore, let X be a closed integral subscheme of A2 and let X be its closurein A 2 . Let v be a non-archimedean place, let F be a component of the fiber ofX at v , and let e be the multiplicity with which it occurs in the fiber at v . Thenfor all such X , v , and F ,

(c). For all i = 1, . . . ,m the multiplicity at F of the divisor ((a ·pr1 +b ·pr2)∗γi)is bounded from below by −c5,ve(a2 + b2) .

(d). The minimum over i = 1, . . . ,m of the multiplicities at F of the divisors((a · pr1 +b · pr2)∗γi) is at most c5,ve(a2 + b2) .

Proof. First, note that if v is a place of good reduction, then (c) and (d) hold withc5,v = 0 , by the theorem of the cube on the fiber. Also note that the constants areindependent of X .

We prove this result by stating two assertions about Weil functions, which simul-taneously imply (a) and (c), and (b) and (d), respectively. For references on Weilfunctions, see ([L 5], Ch. 10). Indeed, let Di = ((a · pr1 +b · pr2)∗γi) for i = 1, . . . ,m .Then the assertions of (a) and (c) follow from the inequality

(13.3) λDi(P ) ≥ −c5,v(a2 + b2) for all i = 1, . . . ,m and all P ∈ A2(kv)

and the assertions of (b) and (d) follow from

(13.4) min1≤i≤m

λDi(P ) ≤ c5,v(a2 + b2) for all P ∈ A2(kv).

As in (13.1), these Weil functions are to be derived from fixed Weil functions on certainrepresentations of L and P by the theorem of the cube. The non-archimedean cases(c) and (d) follow from the above statements by choosing a point P ∈ X(kv) whosesection crosses F transversally.

To prove these assertions, we make canonical choices of Weil functions, called Neronfunctions, using the following result:

Theorem 13.5. Let A be an abelian variety defined over kv . To each divisor D on A

which is not supported at 0 , there exists a Weil function λD , uniquely determinedby the following properties.

(1). λD(0) = 0 .(2). If D and D′ are divisors on A not supported at 0 , then

λD+D′ = λD + λD′ .

(3). If D = (f) is principal, then λD(P ) = v(f(P )/f(0)) .

(4). We have λ[2]∗D = λD [2].

Such functions λD also satisfy the property that if φ : B → A is a homomorphismof abelian varieties defined over kv , then

(13.6) λφ∗D = λD φ.

Proof. See ([L 5], Ch. 11, Thm. 1.1). The normalization (1) eliminates the constantsused there.

192

Since the difference between λD and λD is bounded in absolute value for any fixeddivisor D , it will suffice to prove (13.3) and (13.4) for λ . This requires that the Di

not pass through 0 , which we may assume after translating by a fixed point in A(kv) ,if necessary.

Then, by (13.6), λDiis the same whether it is computed using (13.1) or directly

from the definition of (a · pri +b · prj)∗ . Since the divisors (γi) are all effective, we canchoose c5,v such that

λ(γi)(P ) ≥ −c5,v

for all P and all i , which implies (13.3). Also, since these divisors have no geometricpoint in common on the generic fiber, we can choose c5,v such that

min1≤i≤m

λ(γi)(P ) ≤ c5,v

for all P , implying (13.4).

This result is also proved in a weaker form in ([V 5], Lemma 6.5). In this case astronger statement is needed, however, since the subscheme X may vary in parts (c)and (d).

Applying this lemma to the Faltings complex (9.4), parts (c) and (d) of Lemma13.2 give the following corollary.

Corollary 13.7. The Faltings complex can be extended to a sequence of R-modules bymultiplying by integers of size at most exp

(cd∑

s2i

), and the cohomology of the

complex will then be torsion, annihilated by similarly bounded integers.

We also include here two lemmas which will also be used in the next section.

Lemma 13.8 ([F 1]). Let X1, . . . , Xn be subvarieties of X , and let X1, . . . ,Xn betheir closures in A . Then for all sufficiently large natural numbers d1, . . . , dn ,

V(Γ(∏

Xi,∑

di pr∗i L))

≥ exp(−c · h0

(∏Xi,

∑di pr∗i L

)∑di

)for some constant c independent of X1, . . . , Xn and d1, . . . , dn .

Proof. Embed A into projective space by a very ample multiple of L . Then it followsthat the direct sum

⊕d≥0 Γ(X , dL) is finitely generated over the ring of homogeneous

polynomials. Therefore

V (Γ(A , dL)) ≥ exp(−c · h0(X , dL) · d).

By ampleness of L (on the generic fiber), restricting to Xi will not increase this c .The lemma then follows, since Γ

(∏Xi,

∑di pr∗i L

)is a direct sum of tensor products

of such modules.

Lemma 13.9. Let X1, . . . , Xn , X1, . . . ,Xn , and d1, . . . , dn be as above, and assumealso that the degrees of the Xi are bounded. Then for all sufficiently larged1, . . . , dn and all nonzero γ ∈ Γ

(∏Xi,O

(∑di pr∗i L

)),∏

v|∞

‖γ‖sup,v ≥ exp(−c∑

dih(Xi)− c′∑

di

)

193

for some constants c and c′ independent of γ , X1, . . . , Xn , and d1, . . . , dn .

Proof. As in (18.5) (below), there exist projections πi : Xi → Pmi

Spec R ( mi = dim Xi )of degree Ni such that the norm of γ is an integral section γ′ of O(e1, . . . , en) with

(13.10) ei = di

n∏j=1j 6=i

Nj ,

with norms at infinity derived from the Fubini-Study metric, and such that for allarchimedean places v of k and all P ∈ (Pm1 ×· · ·×Pmn)(kv) over which π1×· · ·×πn

is finite,

(13.11)∏

Q∈(π1×···×πn)−1(P )

‖γ(Q)‖ ≥ ‖γ′(P )‖ exp(c∑

dih(Xi) + c′′∑

di

).

The set of points in Pm whose homogeneous coordinates are all roots of unity is densein the Zariski topology. Therefore in (13.11) we take P = (P1, . . . , Pn) such that eachPi is of this form, and let

E ⊆ Pm1Spec R ×Spec R · · · ×Spec R Pmn

Spec R

denote the arithmetic curve corresponding to P . Then

1[k : Q]

∑v|∞

− log ‖γ‖sup,v ≤1

[k(P ) : Q]

∑w|∞

− log ‖γ′(P )‖v + c∑

dih(Xi) + c′′∑

di

≤ 1[k(P ) : Q]

∑w|∞

deg O(e1, . . . , en)∣∣E

+ c∑

dih(Xi) + c′′∑

di

≤n∑

i=1

ei(h(Pi) + c′′′) + c∑

dih(Xi) + c′′∑

di

≤ c∑

dih(Xi) + c′∑

di

This last step follows from (13.10), together with the fact that (naıve) heights of all thePi vanish. Also, the places w range over archimedean places of k(P ) .

We also note that in the application of this lemma, the full version of (18.5) can beused (with the Poincare divisors), and then (9.3) can be used in place of (9.4) for theFaltings complex.

§14. Construction of a global section

These past three sections now provide all the tools needed to construct a small globalsection.

194

Proposition 14.1. Let (s) and X1, . . . , Xn satisfy the conditions (10.5) and let ε be asin Proposition 11.5. Let W denote the closure of X1 × · · · ×Xn in W0 . Then forall sufficiently large d ∈ N (depending on s ), there exists an integral section

γ ∈ Γ(W , dL−ε,s)

such that

(14.2)∏v|∞

‖γ‖sup,v ≤ exp(cd∑

s2i

).

Proof. Let β : Γ1 → Γ2 be the last arrow in the Faltings complex (9.4). We have nowextended the picture to schemes over Spec R , so that for example

Γ1 = Γ(W , d(2n− 2 + ε)

∑s2

i pr∗i L)a

.

This is metrized by taking the largest of the sup norms of its components. Also let

δ0 = rank(Kerβ);

δ1 = rank(Γ1);

δ2 = rank(Im β).

The proof will follow by applying Lemma 12.1 to β .First, we will need to replace

∏Xi with W in Lemmas 13.8 and 13.9. This

is easy to do, since for any fixed divisor class F on∏

Xi , there is an injection ofΓ(∏

Xi, dF)

into Γ(W , dF ) whose cokernel is annihilated by an integer independentof d . Then Lemma 13.8 implies that

(14.3) V (Γ1) ≥ exp(−δ1cd

∑s2

i

).

By Lemma 13.9 and (10.5.4) (and 10.9), for all γ /∈ Ker β ,∏v|∞

‖β(γ)‖ ≥ exp(−cd

∑s2

i

)and therefore

V (Im β) ≤ exp(cdδ2

∑s2

i

).

Combining this with (14.3) and Lemma 12.1b gives the bound

V (Kerβ) ≥ exp(−cdδ1

∑s2

i

).

Clearly δ1 ≤ cdP

dim Xi∏

s2 dim Xii ; thus, by Proposition 11.5, δ1/δ0 is bounded; hence

Lemma 12.1a gives a section γ ∈ Ker β with

(14.4)∏v|∞

‖γ‖v ≤ exp(cd∑

s2i

).

Here we used Lemma 13.2a to give (12.2). By Lemma 13.2b, then, the bound (14.4)holds also for the (sup) norms of γ as an element of Γ(W , dL−ε,s) .

195

§15. Some analysis

This section contains some analytic results which will be needed for working with partialderivatives in the next section. Because of technical difficulties, the analytic approachof Lemma 6.2 is necessary at all places of k .

For archimedean places v , let Cv = kv ; after taking the algebraic closure the fieldis still complete. At non-archimedean places, this is no longer the case: only after takingthe completion Cv of kv is the resulting field both complete and algebraically closed.On such fields, functions defined by power series often behave similarly to functions onCm . There are also differences, though: for example Cv is totally disconnected, so it isoften necessary to specify that a given function is defined by the same power series onall of the domain. This set of ideas is often referred to as rigid analysis. For more detailson p-adic analysis, see [Ko]; for rigid analysis, see [T] or, for another perspective, [Be].

In the non-archimedean case, the absolute value ‖ ·‖v extends to an absolute valueon Cv ; when discussing analysis on Cv , we often write the absolute value as | · | ,omitting the subscript v . In the archimedean case, let | · | be the usual absolute value.For vectors x = (x1, . . . , xm) ∈ Cm

v let |x|2 = |x1|2 + · · · + |xm|2 if v is archimedeanand |x| = max(|x1|, . . . , |xm|) if it is non-archimedean. Let the open and closed discsDρ and Dρ be the sets z ∈ Cm

v | |z| < ρ and z ∈ Cmv | |z| ≤ ρ , respectively. This

notation is inherited from the archimedean case, and is slightly misleading because inthe non-archimedean case both sets are simultaneously open and closed. Also note thatin the non-archimedean case, there is no difference between polydiscs and balls.

For non-archimedean v , Cauchy’s inequalities still hold:

Lemma 15.1. Let v be a non-archimedean place and assume that the multivariablepower series

f(z) =∑(i)≥0

a(i)z(i)

converges (absolutely) in an open disc Dρ of radius ρ > 0 . Then for all tuples(`) ≥ 0 ,

|a(`)|ρ`1+···+`m ≤ supz∈Dρ

|f(z)|.

Proof. As in the archimedean case, it will suffice to prove the inequalities in the one-variable case. Also, it will suffice to show that |a`|(ρ′)` ≤ supz∈Dρ′

|f(z)| for all ρ′ < ρ ;the lemma then follows by taking the limit. By dilation, we may assume ρ′ = 1 ; thenthe power series converges at z = 1 , so |ai| → 0 . Multiplying by a constant, we assumemaxi |ai| = 1 ; it will then suffice to show that |f(z)| = 1 for some z in the closed unitdisc. But by the theory of the Newton polygon ([Ko], Ch. IV, §3), together with theWeierstrass Preparation Theorem, this holds for any z in the boundary of the discwhich is not close to any of the roots of the Weierstrass polynomial corresponding tof .

We will also use Newton’s method and its p-adic corollary, Hensel’s lemma. Thiswill follow Lang ([L 4], Ch. 2, §2, Prop. 2), but will be done in the context of severalvariables. To begin, let K be a field with non-archimedean valuation | · | , and letA := x ∈ K | |x| ≤ 1 be its valuation ring. Note that the valuation does not need to

196

be discrete. For vectors x ∈ Km let |x| be defined as before, and for m×m matricesJ let

|J | = minx6=0

|Jx||x|

.

Note that if the entries of J lie in A , then |J | ≥ |det J | , so that the Hensel’s lemmagiven here strengthens the one given in ([G], Prop. 5.20). Also, if |J | 6= 0 then J isnonsingular and

(15.2) |J−1x| ≤ |x||J |

for all x ∈ Km .

Then the following is a multivariable Hensel’s lemma.

Lemma 15.3. Let f = (f1, . . . , fm) be a vector of polynomials in A[X1, . . . , Xm]m , andlet

Jf :=(

∂fi

∂Xj

)i,j

denote its jacobian matrix. Suppose α0 ∈ Am is such that

(15.4) |f(α0)| < |Jf (α0)|2.

Then there is a unique root α of f(X) in Am with

(15.5) |α− α0| ≤|f(α0)||Jf (α0)|

< 1,

and the sequence

(15.6) αi+1 = αi − Jf (αi)−1 · f(αi)

converges to it.

Proof. First, consider α satisfying (15.5). Clearly α ∈ Am . Also, we show that|Jf (α)| = |Jf (α0)| . Indeed, consider the Taylor expansion of Jf : for x ∈ Km with|x| = 1 ,

Jf (α) · x = Jf (α0) · x + β · (α− α0)

for some matrix β (depending on x ) with entries in A . Then by (15.5) and (15.4)

|β · (α− α0)| < |Jf (α0)|.

In any case this implies that |Jf (α)| ≤ |Jf (α0)| ; taking x so that |Jf (α0)·x| is minimal,we find the opposite inequality.

To prove uniqueness, assume α and α′ are two roots satisfying (15.5). Then theTaylor expansion for f about α gives

f(α′) = f(α)− Jf (α) · (α′ − α) + t(α′ − α) · β · (α′ − α)

for some m-tuple β of matrices with entries in A . But f(α′) = f(α) = 0 , the secondterm on the right has absolute value at least |Jf (α0)||α−α′| by the result just proved,and the last term has absolute value at most |α − α′|2 . This contradicts (15.5) and(15.4) unless α = α′ .

197

Finally, let c = |f(α0)|/|Jf (α0)|2 < 1 . We show inductively that(i). αi satisfies (15.5) and

(ii).|f(αi)||Jf (αi)|2

≤ c2i

.

These two conditions obviously imply the lemma: by (ii) the αi converge to a root,and by (i) this root satisfies (15.5).

The conditions with i = 0 are obvious; then assume them for i . To show (i),

|αi+1 − αi| ≤ c2i

|Jf (αi)| ≤ c|Jf (α0)| =|f(α0)||Jf (α0)|

.

Thus (15.5) holds.The proof of (ii) again uses the Taylor expansion for f :

f(αi+1) = f(αi)− Jf (αi) · Jf (αi)−1 · f(αi) + t(Jf (αi)−1 · f(αi)

)· β ·

(Jf (αi)−1 · f(αi)

)for some m-tuple β of matrices with entries in A . The first two terms on the rightcancel; by (15.2) this gives

(15.7) |f(αi+1)| ≤(|f(αi)||Jf (αi)|

)2

.

This gives (ii).

The archimedean analogue of this lemma is slightly more complicated.

Lemma 15.8. Let f = (f1, . . . , fm) be a vector of C2 functions from Cm to C andlet Jf be the Jacobian determinant, as in Lemma 15.3. Suppose α0 ∈ Cm andB ∈ R>0 satisfy

(15.9) |f(α0)| <|Jf (α0)|2

2B;

and

(15.10) B > sup

∣∣∣∣∣m∑

i=1

m∑j=1

∂2f(α)∂zi∂zj

viwj

∣∣∣∣∣as α ranges over the ball Bρ(α0) of radius ρ centered at α0 , and v and w varyover the unit ball in Cm ; here

ρ =|f(α0)||Jf (α0)|

∞∑i=0

(1 + c)i

(c

2(1− c)2

)2i−1

,

and

c =B|f(α0)||Jf (α0)|2

.

Then the sequence (15.6) converges to a root α of f(X) with

(15.11) |α− α0| ≤|f(α0)||Jf (α0)|

∞∑i=0

(1 + c)i

(c

2(1− c)2

)2i−1

198

Proof. First we note that if α and α′ both lie in Bρ(α0) , then

(15.12) |Jf (α)| −B|α′ − α| ≤ |Jf (α′)| ≤ |Jf (α)|+ B|α′ − α|.

To prove this, find some x with |x| = 1 and |Jf (α)x| = |Jf (α)| ; then by Taylor’sformula applied to Jf · x and by (15.10),

|Jf (α′)| ≤ |Jf (α′)x|≤ |Jf (α)x|+ B|α′ − α|≤ |Jf (α)|+ B|α′ − α|.

This proves the second half of the inequality; the first half follows by symmetry.Next, as in the non-archimedean case, we prove inductively that

(i). |αi − α0| ≤|f(α0)||Jf (α0)|

i−1∑j=1

(1 + c)j

(c

2(1− c)2

)2j−1

;

(ii). |Jf (αi)| ≤ (1 + c)i|Jf (α0)| ; and

(iii).B|f(αi)||Jf (αi)|2

≤ c

(c

2(1− c)2

)2i−1

.

By (15.9), c < 1/2 ; thus c2(1−c)2 < 1 . Then (ii) and (iii) imply that the αi converge

to a root, and (i) implies (15.11).To carry out the induction, we first observe that (ii) and (iii) for αi imply (i) for

αi+1 :

|αi+1 − αi| ≤|f(αi)||Jf (αi)|

≤ c|Jf (α0)|B

(1 + c)i

(c

2(1− c)2

)2i−1

≤ |f(α0)||Jf (α0)|

(1 + c)i

(c

2(1− c)2

)2i−1

.

Next, by the second half of (15.12) and by (iii) for αi ,

|Jf (αi+1)| ≤ |Jf (αi)|+B|f(αi)||Jf (αi)|

≤ |Jf (αi)|(1 + c).

This gives (ii).To prove (iii), we first note that, as before,

|Jf (αi+1)| ≥ |Jf (αi)|(1− c).

As in the non-archimedean case, we obtain in place of (15.7) the formula

|f(αi+1)| ≤B

2

(|f(αi)||Jf (αi)|

)2

.

199

Combining these two formulas gives

B|f(αi+1)||Jf (αi+1)|2

≤ B2|f(αi)|2

2(1− c)2|Jf (αi)|4

≤ c2

2(1− c)2

(c

2(1− c)2

)2i+1−2

.

This implies (iii).

Remark. In the archimedean case the uniqueness statement is not as clean as before:let ρ′ = |Jf (α0)|/2B ; then if (15.10) holds for α ∈ Bρ′(α0) , then this ball contains atmost one zero of f . Note also that if c is sufficiently small, then ρ′ ≥ ρ . However,neither uniqueness statement will be needed here.

For this paper, Hensel’s lemma will be applied to a finite morphism q : X → Cmv ,

where X is an analytic subvariety of CMv . Let P be a point where q is etale; then

there exists a regular sequence

f1, . . . , fM−m ∈ A[X1, . . . , XM ].

for some neighborhood of P in X , as a subset of CMv . Then let gi = fi for

i = 1, . . . ,M −m , and let gM−m+1, . . . , gM be the coordinates of q . Hensel’s lemmagives a (pointwise) inverse function of q ; moreover, the inverse function theorem impliesthat this function is locally given by power series in the variables. But what we needis a lower bound on the radius of a polydisc on which this power series both convergesand gives a local inverse for q . This can be easily shown by applying Hensel’s lemmato a different field, as follows.

Corollary 15.13. Let Jg be the matrix (∂gi/∂xj)i,j , and assume that Jg is nonsingularat a point P ∈ Cm

v .

(a). Assume P ∈ Am . Then the inverse of q is given by a single power series inthe polydisc of radius |Jg(P )|2 about q(P ) .

(b). Let ρ and B be as in Lemma 15.8; then the inverse of q is given by a singlepower series in the ball of radius ρ about q(P ) .

Proof. The power series is obtained by applying Hensel’s lemma to the field

Cv((X1, . . . , XM )),

using the index as a valuation (the weights of the variables may be chosen arbitrarily).It still must be shown that the power series converges in the indicated polydisc, and thatevaluating it gives the function obtained earlier. This follows by comparing the αi inthis application of Hensel’s lemma with the αi obtained when applying Hensel’s lemmapointwise. The two sets of convergents coincide, in the sense that evaluating the powerseries αi at a given point gives the same value as the corresponding convergent in themethod given above. Moreover, on any polydisc of strictly smaller radius, the boundson |αi+1−αi| are uniform; thus the power series converges on the given polydisc.

200

§16. More derivatives

The higher dimensional generalization due to Faltings also uses partial derivatives onX1×· · ·×Xn , but now each Xi may be of higher dimension and may have singularities.The p-adic analysis of the preceding section is necessary in order to handle the singular-ities. But also, the varieties Xi may vary, so the theory of Chow coordinates is needed.Therefore let C be the Chow variety of subvarieties of Pm of a given degree, and let Γbe the universal variety sitting over C . For this purpose we use the chosen embeddingof X (over k ) into Pm given by the very ample divisor L . At finitely many placesthis fails to extend to an embedding over Spec R , but this will be addressed later.

We assume that the varieties C are projective. Points in C not corresponding toan integral subvariety of Pm form an algebraic subset; we may assume that none of theXi corresponds to such a point. If not, C can be replaced by a Chow variety of strictlysmaller degree. Also, note that since the degrees of the Xi are bounded, finitely manysuch C will suffice. On each such C , the height of the point corresponding to Xi isrelated to the height h(Xi) defined in Section 10; for details, see ([So 1], Theoreme 3).

Proposition 16.1. Let L = O(dL−ε,s) , and let γ be a global section of L on the modelW for X1 × · · · × Xn obtained by taking the closure in the model W0 for An .Let P0 = (P1, . . . , Pn) , E = E1 ×B · · · ×B En , and d1, . . . , dn be as in Corollary6.3. Then there exist constants c1 , c2 , c3 , c4 , c5 , and c6 , depending only onW0 , X , and deg X1, . . . ,deg Xn , and subsets Zi ( Xi satisfying deg Zi ≤ c1 andh(Zi) ≤ c2h(Xi) for all i , and such that either

(a). Pi ∈ Zi for some i , or(b). the index t = t(γ, P0, d1, . . . , dn) satisfies the inequality

t ≥−deg L

∣∣E−∑

v|∞ log ‖γ‖sup,v − c3

∑di − c4

∑dih(Xi)

c5 max1≤i≤n dihL(Pi) + c6

∑di

.

Proof. For each i = 1, . . . , n , recall that Xi is given with an embedding into Pm .Let mi = dim Xi and choose one of the standard projections from Pm to Pmi suchthat the projection is well defined at Pi . Blow up Pm so that the projection becomesa morphism everywhere on the blowing-up. Let Zi be the ramification locus of thisprojection. Clearly the degree and height of Zi satisfy the required conditions. Fromnow on, therefore, we assume that Pi /∈ Zi , and concentrate on part (b) of the lemma.Also, by permuting coordinates, we may assume that Pi does not lie over the hyperplaneat infinity on Pmi . Thus we have projections qi : Xi → Ami .

We now extend this picture, both over the Chow variety and over Spec R . Let Ci

denote the Chow variety appropriate for Xi , and restrict to the subset correspondingto subvarieties of X . Let Ci be defined similarly, as a Chow scheme over Spec R . LetΓi ⊆ Ci×Spec R X be the universal family; we blow up as before to make Γi project toPmi , and extend the model so that L extends to a line sheaf on Γi ; this can be doneindependently of s , etc. Now extend C and Ci to make them proper over Spec k andSpec R , respectively. Also, we assume Γi is projective over Ci : Γi ⊆ PM

Cifor some

M . And finally let the bad subset of Ci be the set whose corresponding subvariety iseither not integral or does not map surjectively to Pmi .

201

Note that, so far, the choices do not depend on the place v . For each place v , qi

extends to a v-adic analytic morphism from a subset of Xi(Cv) to Cmiv . This is locally

(by assumption on Pi ) biholomorphic on some open polydisc of a radius ρi,v > 0 . Thenext step consists of controlling this radius.

By a slight change of model, we regard Xi as a subset of Γi , and let Ei be thearithmetic curve in Xi corresponding to Pi . This does not affect the index, which isdefined on the generic fiber.

Lemma 16.2. For each i there exists an arithmetic divisor Di on Γi , depending only onCi , the choice of projections to Pmi , and the permutation of coordinates on Pmi ,such that for all places v of k , the map qi gives a biholomorphic map from a neigh-borhood Ui,v of Pi(Cv) (in Xi(Cv) ) to the polydisc of radius exp(−(Di . Ei)v)and center qi(Pi) in Cmi

v . Moreover, Ei is not contained in the support of Di .And finally, at archimedean places we can place an upper bound on the radius inAM of the sets Ui,v .

Proof. First consider non-archimedean places v . Since Pi /∈ Zi , qi is etale at Pi ,and therefore Γi is smooth over Ci at Pi (on the generic fiber over Spec k ). Recallthat Ei,v denotes the closed point in Ei lying over v ∈ Spec R . By smoothness ([F-L], Ch. IV, Prop. 3.11), there exists a regular sequence f1, . . . , fM−mi for the idealof some neighborhood of Ei,v in Γi , as a subset of some affine space AM

Ci. These

are functions in local coordinates x1, . . . , xM in PM , and in local coordinates on Ci .Let y1, . . . , ymi denote the coordinates of qi ; then also y1 − y1(Pi), . . . , ymi − ymi(Pi)form a regular sequence for Pi on Xi . Now specialize to Xi(Cv) , and note thatthese functions now describe Xi(Cv) in the open unit disc about Pi(Cv) . Sincef1, . . . , fM−mi , y1 − y1(Pi), . . . , ymi − ymi(Pi) form a regular sequence, their Jacobiandeterminant is nonzero. Let the radius equal the square of the absolute value of thisdeterminant. By Hensel’s lemma (Corollary 15.13), the desired biholomorphic mapexists.

As the points Ei,v vary over all of a certain Zariski-open subset of Γi , only finitelymany systems f1, . . . , fmi are needed (quasi-compactness of the Zariski topology), sothere exists a divisor Di on Γi (on the algebraic part; i.e., not an arithmetic divisoryet) which dominates the squares of these determinants; moreover, it can be chosen sothat its vertical (over Ci ) components on the generic fiber correspond only to the badset discussed earlier.

At archimedean places one can similarly construct a Green’s function gDi for Di

such that the desired biholomorphic map exists, with a polydisc of radius exp(−gDi) .

This is done using compactness of Γi(Cv) .

A second estimate is needed, but it is much easier. First, choose a hermitian metricon ΩΓi/Ci

.

Lemma 16.3. For each i there exists an arithmetic divisor Fi on Γi , depending onlyon Ci , the choice of projections to Pmi , and the permutation of coordinates onPmi , such that for all places v of k , the metric of the element q∗i yj − yj(qi(Pi))of the ideal sheaf of Pi in Xi satisfies

− log ‖q∗i yj − yj(qi(Pi))‖v ≥ −(Fi . Ei)v.

202

(At non-archimedean places, the above metric is given by the scheme structure.Also, we inject the ideal sheaf of Pi into ΩΓi/Ci

via ([H 2], II 8.12).)

Proof. It will suffice to take a divisor Fi which dominates the torsion sheaf given bythe relative differentials of qi . Adding q∗i H∞ (where H∞ denotes the hyperplane atinfinity in Pmi ) corrects for the growth at infinity.

For each place v , let Uv be the product of the neighborhoods Ui,v from Lemma16.2. Also let γ0,v be a local generator for L on Uv . This should be chosen uniformlyin terms of generators on pr∗i O(L) and the Poincare divisors Pij ; then it will bepossible to ensure that

(16.4) log infP∈Uv

‖γ0,v(P )‖‖γ0,v(P0)‖v

≥ −c3,v

n∑i=1

di.

At archimedean places this is possible since L is defined on X , and we can limit theradius of Uv , as noted following the proof of Lemma 16.2. At places where A has goodreduction, this infimum is just 1 . Other places should be treated as in the archimedeancase; this is due to the change in model needed to define the Poincare divisors. Also,let ‖γ‖sup,v = 1 for non-archimedean v .

Now let (`) = (`11, . . . , `1m1 , . . . , `nmn) be a tuple such that

(16.5)n∑

i=1

mi∑j=1

`ij

di= t

and such that (letting yij , j = 1, . . . ,mi , denote the coordinates of Cmiv )

D(`)γ :=1

`11! · · · `nmn!

(∂

∂y11

)`11

· · ·(

∂

∂ynmn

)`nmn

γ 6= 0,

where we define this partial derivative via the maps qi . Then, as in the proof of Lemma6.2, we find that the norm of the partial derivative satisfies

− log ‖D(`)γ(P0)‖v

≥ − log ‖γ‖sup,v +n∑

i=1

mi∑j=1

`ij

(log ρi,v − (Fi . Ei)v

)+ log inf

P∈Uv

‖γ0,v(P )‖‖γ0,v(P0)‖v

≥ − log ‖γ‖sup,v +n∑

i=1

mi∑j=1

`ij

(log ρi,v − (Fi . Ei)v

)− c3,v

n∑i=1

di,

(16.6)

by (16.4). Here D(`)γ(P0) is regarded as a section of the vector sheaf(L ⊗ S`11+···+`1m1 ΩX1/ Spec R ⊗ · · · ⊗ S`n1+···+`nmn ΩXn/ Spec R

)∣∣∣E

,

203

where E as usual is the arithmetic curve on Γ1×Spec R · · · ×Spec R Γn corresponding toP0 = (P1, . . . , Pn) . See also ([Laf], §4). Thus∑

v

− log ‖D(`)γ(P0)‖v ≤ deg L∣∣E

+n∑

i=1

mi∑j=1

`ij deg Mi

∣∣Ei

,

for some sufficiently large line sheaves Mi on Γi . Combining this with (16.6) thengives

(16.7)n∑

i=1

mi∑j=1

`ij

(deg Mi

∣∣Ei−∑

v

log ρi,v + (Fi . Ei)

)

≥ −deg L∣∣E−∑v|∞

log ‖γ‖sup,v − c3

n∑i=1

di.

But the quantity inside the parentheses on the left is an intersection number on Γi .From the structure of Γi as a subset of a product, there exist divisors Gi on PM andHi on Ci such that

Mi ⊗ O(Fi + Di) ≤ O(Gi + Hi)

relative to the cone of very ample line sheaves on the generic fiber of Γi over Spec k .Thus

deg Mi

∣∣Ei

+ (Fi . Ei) + (Di . Ei) ≤ (Gi . Ei) + (Hi . Ei) + O(1).

Also we have(Gi . Ei) ≤ c5hL(Pi) + O(1)

and(Hi . Ei) ≤ c4h(Xi) + O(1).

Then by Lemma 16.2, (16.7) becomesn∑

i=1

mi∑j=1

`ij

(c5hL(Pi) + c4h(Xi) + c6

)≥ −deg L

∣∣E−∑v|∞

log ‖γ‖sup,v − c3

n∑i=1

di.

But now, for all i ,∑mi

j=1 `ij di ; therefore modifying c4 gives (using also (16.5))

c4

n∑i=1

dih(Xi)+c5t max1≤i≤n

dihL(Pi)+c6tn∑

i=1

di ≥ −deg L∣∣E−∑v|∞

log ‖γ‖sup,v−c3

n∑i=1

di.

Solving for t then gives the proposition.

§17. Lower bound for the index

As was the case in Section 7, the Mordell-Weil theorem implies that A(k) is a finitelygenerated abelian group; hence A(k)⊗Z R is a finite dimensional vector space. LettingL replace the theta divisor, it follows that

(P1, P2)L := hL(P1 + P2)− hL(P1)− hL(P2)

204

defines a nondegenerate dot product structure on A(k)⊗Z R . Also let |P |2 = (P, P )L .We now assume that P1, . . . , Pn have been chosen such that

(Pi, Pj)L ≥ (1− ε1)√|Pi|2|Pj |2

≥ 2(1− ε1)√

hL(Pi)hL(Pj)

(17.1)

for some given ε1 > 0 and for all i < j . From fundamental properties of the canonicalheight,

deg((s2

i · pri−s2j · prj)

∗L)∣∣

E= s2

i hL(Pi) + s2j hL(Pj)− sisj(Pi, Pj)L + O(s2

i + s2j )

≤(

si

√hL(Pi)− sj

√hL(Pj)

)2

+ 2ε1sisj

√hL(Pi)hL(Pj) + O(s2

i + s2j ).

Letting si be rational and close to 1/√

hL(Pi) , the square in the above expressionapproaches zero and we obtain

(17.2) deg L−ε,s

∣∣E≤ n(n− 1)ε1 − nε + O

(∑s2

i

).

We now apply Proposition 16.1. By (14.2) and (10.5.4), the second and fourth termsin the numerator of the fraction in (16.1) are bounded by c

∑di ; hence by (17.2), the

index t = t(γ, (P1, . . . , Pn), d1, . . . , dn) satisfies

t ≥ n(ε− (n− 1)ε1)− c∑

s2i

c5 + c6

∑s2

i

.

If the heights hL(Pi) are sufficiently large and ε1 sufficiently small, which we nowassume, then the s2

i will be small, and the above inequality becomes

(17.3) t ≥ ε2

for some ε2 > 0 depending only on the usual list X , A , . . . , (dim X1, . . . ,dim Xn) .

§18. The product theorem

The last step of the proof consists of applying the product theorem, as was done in([F 1], §6) or [F 2].

Theorem 18.1 ([F 1], §3). Let Π = Pm1 × · · · × Pmn be a product of projective spacesover a field of characteristic zero, and let ε3 > 0 be given. Then there existnumbers r′ , c1 , c2 , and c3 with the following property. Suppose γ′ is a nonzeroglobal section of the sheaf O(e1, . . . , en) on Π which has index ≥ ε3 relative to(e1, . . . , en) at some point (x1, . . . , xn) . If ei/ei+1 ≥ r′ for all i = 1, . . . , n − 1 ,then there exist subvarieties Yi ⊆ Pmi , not all of which are equal to Pmi , suchthat

(i). each Yi contains xi ;(ii). the degrees of Yi are bounded by c1 ; and

205

(iii). the heights h(Yi) satisfy the inequality

(18.2)∑

eih(Yi) ≤ c2

∑v|∞

log ‖γ′‖sup,v + c3

∑i

ei.

Proof. See ([F 1], Thm. 3.1 and Thm. 3.3).

We note that this theorem generalizes Roth’s lemma (4.5), although without theexplicit constants. Indeed, let m1 = · · · = mn = 1 . Then at least one of the resultingYi must equal xi ; thus (18.2) contradicts (4.6).

This theorem implies a result which is more suitable for the problem at hand:

Corollary 18.3. Let X be a projective variety defined over a number field k . Fix aprojective embedding of X , and let L be an ample divisor on X . Let X1, . . . , Xn

be geometrically irreducible subvarieties of X defined over k whose degrees (rela-tive to the chosen projective embedding of X ) are bounded. Let ε > 0 and ε2 > 0be given. Then there exist numbers r , c1 , c2 , c3 , and c4 with the followingproperty. Let γ be a nonzero global section of O(dL−ε,s) which has index ≥ ε2relative to (d1, . . . , dn) ( di = ds2

i ) at some point (P1, . . . , Pn) with Pi ∈ Xi(k)for all i . If di/di+1 ≥ r for all i = 1, . . . , n − 1 , then there exist subvarietiesX ′

i ⊆ Xi , not all of which are equal to Xi , such that

(i). each X ′i contains Pi ;

(ii). each X ′i is geometrically irreducible and defined over k ;

(iii). the degrees of X ′i are bounded by c1 ; and

(iv). the heights h(X ′i) satisfy the inequality

(18.4)∑

dih(X ′i) ≤ c2

∑v|∞

log ‖γ‖sup,v + c3

∑i

dih(Xi) + c4

∑i

di.

Proof. Let mi = dim Xi . Let Pm be the projective space in which X is embedded,and for each i fix a standard projection from Pm to Pmi whose restriction to Xi isa generically finite rational map. Let Ni be its degree. In fact, Ni = deg Xi , so thatNi is bounded. We would like to take the norm from

∏Xi to Π . Therefore, for each

i let K∗i be a finite extension of K(Xi) which is normal over K(Pmi) . Let X∗

i bea model for K∗

i such that the rational maps X∗i → Xi corresponding to all injections

K(Xi) → K∗i over K(Pmi) are morphisms. The product of the pull-backs of O(L)

via these morphisms is a line sheaf on X∗i which is isomorphic to the pull-back of some

multiple of O(1) from Pmi . Expanding this to the Chow family Γi over Ci/ Spec Ras in Section 16, the isomorphism holds up to a divisor on Ci , fibral componentsover Spec R , and, correspondingly, a change of metric at archimedean places. Herewe metrize O(1) via the Fubini-Study metric. Thus, the isomorphism holds up todenominators bounded by exp(ch(Xi) + c′) , and the metrics correspond up to a factorbounded by a similar bound.

For the Poincare divisors Pij , similar bounds are not good enough. This is dueto the fact that they occur with coefficients of size sisj . However, a more refinedargument gives the required bounds. Let Ci denote the generic fiber of Ci . Then thenorm of Pij is a divisor class on Pmi

Ci×Spec R Pmj

Cj, and over each point of Ci × Cj

206

it is trivial. But for each point Q ∈ Pmi

Ci, the restriction of this norm to Q × Pmj

Cj

is algebraically equivalent to zero, because it is obtained from the restriction of Pij

to sets of the form Q′ × X , for Q′ ∈ Γi lying over Q ∈ X . The same argumentholds after interchanging the factors; thus the difference is a divisor of Poincare type onCi × Cj . Therefore the bounds on the ratio of norms and on the size of denominatorsare of the form exp

(c√

h(Xi)h(Xj) + c′). This is indeed fortunate, because the divisor

classes Pij occur with multiplicities sisj in L−ε,s , and

sisj

√h(Xi)h(Xj) ≤

12(s2

i h(Xi) + s2jh(Xj)

),

which is exactly the sort of bound that is needed!After cancelling denominators, then, the norm γ′ of γ has metrics satisfying

(18.5)∏v|∞

‖γ′‖sup,v ≤∏v|∞

‖γ‖sup,v · exp(c∑

dih(Xi) + c′∑

di

).

We now calculate ei . First note that the map∏

Xi → Π has degree N :=∏

Ni ;then ei = di ·N/Ni . Since the Ni are bounded, the index of γ′ at the point (x1, . . . , xn)lying below (P1, . . . , Pn) is bounded from below, even after switching from weights(d1, . . . , dn) to (e1, . . . , en) . The boundedness of the Ni also implies that there existssome r such that di/di+1 ≥ r implies ei/ei+1 ≥ r′ .

Then applying Theorem 18.1 to γ′ gives subvarieties Yi ⊆ Pmi . By (18.2) and(18.5), ∑

dih(Yi) ≤ c∑

eih(Yi)

≤ c∑v|∞

log ‖γ‖sup,v + c′∑

dih(Xi) + c′′∑

di + c′′′∑

ei

≤ c∑v|∞

log ‖γ‖sup,v + c′∑

dih(Xi) + c′′∑

di.

Now pull back these Yi to subsets X ′i of Xi . Then h(X ′

i) is bounded in terms ofh(Yi) (using the definition from Section 10); hence Condition (iv) holds. Condition (i)holds by construction, and (iii) is easy to check. Since not all of the Yi are equal toPmi , not all of the X ′

i equal Xi .It remains only to ensure that (ii) holds. But we may intersect X ′

i with finitelymany (at most dim Xi ) of its conjugates over k until the geometrically irreduciblecomponent containing Pi is defined over k . Replacing X ′

i with this irreducible com-ponent gives (ii), and since the number of intersections is bounded, (iii) and (iv) stillhold (after adjusting the constants).

This corollary can be applied directly to the situation of Theorem 0.3. Indeed,either (16.1a) holds, which gives the inductive step rather directly, or (16.1b) holds, sothat by (17.3) and Corollary 18.3, we obtain subvarieties X ′

i of Xi satisfying (10.5.1)–(10.5.3), possibly with a different set of constants. Also, (10.5.4) holds for X ′

i , by(14.2), (10.5.4) (for Xi ), and (18.4). Moreover, at least one X ′

i has dimension strictlysmaller than dim Xi . This concludes the main part of the proof of Theorem 0.3.

207

Returning to the overall plan of Section 10, then, we now choose

P1, . . . , Pn ∈ X(k) \ Z(X)(k)

such that(a). The height hL(P1) is sufficiently large to contradict (10.7) and to ensure that

(17.3) holds.(b). For i = 1, . . . , n − 1 , h(Pi+1)/h(Pi) > r′′ , where r′′ is the largest of the

r occurring in all applications of the product theorem; we also assume thatr′′ ≥ 1 .

(c). Condition (17.1) holds for all possible (dim X1, . . . ,dim Xn) .In particular, r and ε1 , as well as the various constants c , depend on the tuple(dim X1, . . . ,dim Xn) , but only finitely many such tuples occur. Then the inductionmay proceed as outlined in Section 10, leading to a contradiction.

This concludes the proof of Theorem 0.3.We conclude with a few remarks on how this proof differs from the proof of Theorem

0.2. In that case it is possible to show that L−ε,s is ample. Shrinking ε a little, it ispossible to obtain an upper bound on the dimension of the space of sections of O(dL−ε,s)which have index ≥ σ at the point (P1, . . . , Pn) , for some suitable σ > 0 . This boundis bounded away from h0(Xn, dY−ε,s) , so the more precise form (2.4) of Siegel’s lemmaallows us to construct a global section γ with index ≤ σ at (P1, . . . , Pn) . Thus Step 5is incorporated into Step 2. One then obtains a contradiction in Step 4, without needingthe induction on the subvarieties Xi .

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