diophantine approximation and diophantine equations

8/12/2019 Diophantine Approximation and Diophantine Equations

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Applications of Arithmetic Algebraic Geometryto Diophantine Approximations

Paul Vojta

Department of MathematicsUniversity of CaliforniaBerkeley, CA 94720 USA

Contents

1 History; integral and rational points

2 Siegels lemma3 The index4 Sketch of the proof of Roths 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 section

15 Some analysis16 More derivatives17 Lower bound for the index18 The product theorem

Let us start by recalling the statement of Mordells 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]; it

was 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 kk contained in Xkk . 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 thatXkk not contain any translated abelian subvarieties of A , obtaining another conjec-ture of Lang ([L 2], p. 29).

Theorem 0.3([F 2]). Let Xbe 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 unioni 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 showing

finiteness for the intersection of Xwith a finitely generated subgroup of A(Q). Thesame sort of finiteness question can then be posed for the division group

{gA(Q)| mg for some mN };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 Thueequationis 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, Minkowskis theorem on successive minima.

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

After that, we will consider how to apply the language of arithmetic intersectiontheory to this proof, and prove Mordells 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 thatxv =|(x)| if v corresponds toa real embedding : k R ;xv =|(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

xv = 1, xk, x= 0.

Aline sheafon a scheme Xmeans a sheaf which is locally isomorphic to OX; i.e.,an invertible sheaf. Similarly a vector sheafis 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 , there are only finitely many p/q Q (written inlowest terms) satisfying an inequality of the formpq

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

= d, c computable Liouville, 1844= d+12 + Thue, 1909

= min{ds + s 1|s = 2, . . . , d} + Siegel, 1921=

2d + Dyson, Gelfond (independently), 1947

= 2 + Roth, 1955

Of course, stronger approximations may be conjectured; e.g.,pq < c|q|2(log q)1.

See ([L 3], p. 71).Beginning with Thues 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 Zhas only finitely many solutions.

Indeed, this equation may be rewritten

x

y 32 = 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 Mordells conjecture (Theorem0.1).

Example 1.3. The equation

(1.4) x4 + y4 =z4, x, y, z Qin 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 = SpecZ[X, Y]/(X3 2Y3 1) and B = SpecZ beschemes, and let : W B be the morphism corresponding to the injection

Z Z[X, Y]/(X3 2Y3 1).Then solutions (x, y) to the equation (1.2) correspond bijectively to sections s : BWof since they correspond to homomorphisms

Z[X, Y]/(X3 2Y3 1) Z, Xx, Y yand the composition of these two ring maps gives the identity map on Z .

In the second example, let W= ProjZ[X, Y, Z ]/(X4 + Y4 Z4) and B = SpecZ .Then sections s : B W of correspond bijectively to closed points on the genericfiber of 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 : W B .The difference between integral and rational points is accounted for by the fact that inthe first case 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. Siegels lemma

Siegels 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 (Siegels 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

| (N Q)M/(NM) =:Z, i= 1, . . . , N .Proof. The number of integer points in the box

(2.2) 0xiZ, 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 j

th row of A . Therefore there are at most

(N QZ+ 1)M


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

NM

i=1

h(xi)h(A) +N

M

[k: Q] log 2s

|Dk| .The exact value of the constant will not be needed here; it has been included only

for 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 aijXiYj is a nonzero monomial in Qwith i+j =t . This definition treats the two variables symmetrically, whereas here it

will 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, . . . , X n) =

1,...,n0

a1,...,nX11 Xnn =:

()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

ni=1

idi

a()= 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 b1Xi+ b2X

2i +. . .

( b1= 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 be a rational section of a line sheafL, on a product X1 Xnof varieties. Let P = (P1, . . . , P n) be a regular point on

Xi, and suppose that

is regular at P. Let 0 be a section which generates L in a neighborhood

of 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 /0 is a regular function ina neighborhood of P, so it can be written as a power series

/0 =

()0a()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 at Pwith weights d1, . . . , dn, denoted t( , P , d1, . . . , dn) or just t(, P), is

min

ni=1

dimXij=1

ijdi

a()= 0

.

As noted already, this definition does not depend on the choices of 0 or of localparameters zij.

As a special case of this definition, if n= 2 , if all Xi are taken to be P1 , and if

no Pi is , then is just a polynomial in two variables and this definition specializesto the preceding definition after Pis translated to the origin.

4. Sketch of the proof of Roths theorem

In a nutshell, the proof of Roths 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 > 0 , a finite set of places S of k , and v Q foreach vS. Then for almost all xk ,

(4.2) 1

[k: Q]

vS

log min(x vv, 1)(2 + )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 v lie in k . Otherwise, let k

be some finiteextension field of k containing all v, let S

be the set of places w of k lying overv S, and for each w| v let w be a certain conjugate of v. (In order to writex vv when v /k , some extension of v to k(v) must be chosen; then thew should be chosen correspondingly.) With proper choices of w, 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 1, . . . , m be the distinct values taken on by all v, v S . Also let d1, . . . , dnbe 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, . . . , X n, andits degree in each Xi should be at most di, for each i . The first requirement is thatthe polynomial should have index n( 12 1) at each point (i, . . . , i) , 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 (i, . . . , i) . 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 1 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 Siegels 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 Siegels lemma; the height will then be bounded by

(4.3) h(Q)c1ni=1

di.

Here the constant c1 (not a Chern class!) depends on k , S, n , and 1, . . . , m.This bound holds because M/(N M) will be bounded from above, and coefficientsof constraints will be powers of , multiplied by certain binomial coefficients.

Step 3 is independent of the first two steps; for this step, we choose elementsx1, . . . , xnk not satisfying (4.2). Further, the vectors log min(xi vv, 1)vS R#Sall 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 v, vS such that

log min(xi vv, 1)vh(xi), vS, i= 1, . . . , n

and such that

(4.4) 1

[k: Q]

vS

v = 2 +

2.

The points must also satisfy the conditions

h(x1)c2and

h(xi+1)/h(xi)r, i= 1, . . . , n 1.These conditions are easily satisfied, since by assumption there are infinitely many xknot 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 to

d/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 vS, thenwrite

Q(X1, . . . , X n) =()0

bv,()(X v)().

Bounds on the sizesbv,()v can be obtained from (4.3) in Step 2; moreover bv,() = 0if1d1

+ + ndn

< n

1

2 1

,

by the index condition in step 1. Then the only terms with nonzero coefficients arethose with high powers of some factors Xiv , 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

ni=1

exp(vih(xi)) other factors

expvd

n

i=1i

di other factorsexp(vdn( 12 1)) other factors

At v /S, we also have bounds onQ(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)vexp[k : Q](2 + 2 )dn( 12 1) 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 in

fact 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 properties

of Q . One possibility is to use the height h(Q) from step 2.

Lemma 4.5 (Roth, [Ro]; see also [Bo]). Let Q(X1, . . . , X n)0 be a polynomial in nvariables, of degree at most di in Xi, with algebraic coefficients. Let x1, . . . , xnbe algebraic numbers, and let t= t(Q, (x1, . . . , xn), d1, . . . , dn) be the index of Qat (x1, . . . , xn) . Suppose that 2 > 0 is such that

di+1di

2n12 , i= 1, . . . , n 1

and

(4.6) dih(x

i)

2n1

2 (h(Q) + 2nd

1), i= 1, . . . , n .

Thent

n 22.


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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+ x2t},so that Vol(t) = t2/2 if t1 . Let 1, . . . , m bem 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

mi=1

Vol(t(Q, i, d1, d2))1 + d22d1

max(m 2, 0).

Historically, Dysons 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 Roths lemma is much quicker. In particular, it was the form of Dysons lemmawhich suggested the particular line sheaf to use in Step 1 of the Mordell proof.

Roths innovation in this area was the use of n good approximations instead oftwo; but for now we will just use two good approximationsit sufficed for Thues 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 archimedean

fiber. 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, ifX 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 Gillets and Soules in another respect. Namely,instead of using a pair D = (D, gD) to denote, say, an arithmetic divisor (i.e., an

element of Z1(X) ), we will use the single letter D to refer to the tuple, and Dfinto 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 Arakelovs 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 notation

should reflect this fact.


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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 of a metrized linesheaf; however, in this theory it is more convenient to use the supremum norm instead:

sup,v := sup

PX(kv) (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(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 nottensorwith 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 notations(X, L) and hi(X, L) will mean (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 Roths theorem to prove Mordells 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 Roths proof therefore needs some notion of partial derivatives of sections of linesheaves at a point.

To begin, let : X B be an arithmetic surface corresponding to the curve C.By a theorem of Abhyankar [Ab] or [Ar], we may assume that Xis a regular surface.Let W be some arithmetic variety which for now we will assume to be XB BX(but actually it will be a slight modification of that variety); it is then a model for Cn .

Let be a section of a metrized line sheaf L on W , let (P1, . . . , P n) 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 : XB 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 Xcorresponding to Pi; then the intersection number ofEi 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 local

ringOX,Ei,v is generated overOB,v by the local generator of the divisor Ei; hence is smooth in a neighborhood of Ei; likewise qis smooth in a neighborhood of E.

Let x be a closed point in E, and letOB,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 ringOW,x of W at x can be writtenOW,x=OB,q(x)[[z1, . . . , zn]].


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If 0 is a local generator for L at x , then /0 is an element of this local ring, whichcan then be written

0=()0

b()z().

Then each term 0b()z()

lies in the subsheaf

L O(1pr11 E1 npr1n En)L.In general this element depends on the choices of 0 and z1, . . . , zn. However, ifb() = 0 for all tuples (

)= () satisfying 1 i1, . . . , n in (i.e., i f () isa leading term), then the restriction to Eof this term is independent of the abovechoices. This is the definition of the partial derivative

D()(P1, . . . , P n)L O(1pr11 E1 npr1n En)

E

.

In particular, this derivative is a regular section of the above sheaf. We will alsoneed the corresponding fact for the points on Eat archimedean places. This translatesinto an upper bound on the metric of the above section at archimedean places, depending

on choices of metrics for O(Ei) .To metrize O(Ei) in a uniform way, for each archimedean place v choose a metric

for O() on C(kv)C(kv) . For example, one could use the Greens 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 = (C)> 0 such that for all

P C there exists a neighborhood UP C of P with coordinate zP: UP D 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, . . . , P n)Cn . Let L be a metrized line sheaf on

W, and let (W,L) . Let 0 be a section ofL on U :=UP1 UPn whichis nonzero at P0. Suppose () = (1, . . . , n) is a tuple for which the derivativeD()f

(P1, . . . , P n) is defined. Then

log D()(P0) log sup+ni=1

ilog + log infPU

0(P)0(P0) .

Proof. Writing zP0 for the tuple of functions (pr1 zP1 , . . . , pr

n zPn) and writing

= 0 (i)0a(i)z

(i)P0


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gives

sup supPU

0(P)

(i)0

a(i)z(i)P0

(P)

infPU

0(P) supzDn

(i)0

a(i)z(i)

0(P0) infPU

0(P)0(P0)|a()|

1++n

by Cauchys inequalities in several variables,

|a()|1++n supzDn

(i)0

a(i)z(i)

Thus by (6.1),

(0a()z()

P0 )(P0)1++n

infPU 0(P)

0(P0) sup. Of course, one can now transport this result into the arithmetic setting by using

the 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 be a global section of a metrized line sheaf L on W. LetE W correspond to some P0 = (P1, . . . , P n)Cn . Let d1, . . . , dn be positivereal numbers. For each v | let 0,v be a local generator of L

Uv

, where

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

t degL

Ev|log sup,v+ v|log infPUv 0,v(P)0,v(P0,v)

max1indegO(

diEi)Ei [k: Q]log max1in di

.

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

L O(1pr1 E1 nprnEn)E

.

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

= degLE

+ni=1

idegO(Ei)Ei

degLE

+n

i=1idi

max1in

degO(diEi)Ei.


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

v| log D()(P0)v

v|

log sup,v+v|

log infPUv

0,v(P)0,v(P0,v) + [k : Q]log

ni=1

i

v|

log sup,v+v|

log infPUv

0,v(P)0,v(P0,v) +

ni=1

idi

[k: Q]log max1in

di

(here we assume 1. Let KCbe a canonical divisor on C, and let F =KC/(2g 2), so that F hasdegree 1. Here F is a Q-divisora 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 >0 and r >1 be rational; we require that a1(r) := (g+ )r also be rational.

Then a2(r) := (g+ )/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 Bombieris 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)d2Y2

2

d2,since 2 =2g , F1 = F2 = F21 =F22 = 0, and F1. F2 = 1 .

Step 2 requires a bit more cleverness. First, fix N > 0 such that N F is veryample, and fix global sections x0, . . . , xn of O(N F) giving a corresponding embeddinginto Pn . We also regard these as sections of O(N F1) on C C via pr1 and letx0, . . . , x

n be the corresponding sections of O(N F2) defined via pr2.


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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 P

m

.Then one of Bombieris key ideas is to write

Y =1N F1+ 2N F2 B,where i = (ai + s)/N. Then, for d sufficiently large, a section of O(dY) willhave the property that ydi will be represented by polynomials i of degree d1 inx0, . . . , xn and of degree d2 in x

0, . . . , x

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

image of CC in Pn Pn . Since O(1, 2) (defined as pr1 O(1)pr2 O(2)) isample, for d sufficiently large we will have H1(Pn Pn,I O(d1, d2)) = 0, and bythe long exact sequence in cohomology, the map

H0(Pn Pn,O(d1, d2))H0(C C,O(d1, d2)CC

)

=H0(C

C,O(d1N F1+ d2N F2))

will be surjective. Of course, it is not injective: instead, one chooses a subspace ofH0(Pn Pn,O(d1, d2)) 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

xd10 (x0)d2

x1x0

ax2x0

bx1x0

a x2x0

b,

0a + bd1, 0bN, 0a + b d2, 0b N.Then, apply Siegels lemma to tuples (1, . . . , m) in the above subspace, subject tothe linear constraints

iydj = jy

di , i, j {0, . . . , m}.

A solution to this system then yields local sections i/ydi which patch together to givea global section

(C C,O(dY))with

(7.1) h()cda1+ o(d).Here the height h() is defined as the height of the vector of coefficients of the polyno-mials 1, . . . , m.

From this point Bombieri continues to proceed very classically, using Weils 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 i.


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Step 3 now uses an idea from a 1965 paper of Mumford [ M]. As in Roths theorem,it uses a pigeonhole argument, but this time the argument takes place in J(k) Z R ;here Jdenotes 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 0 : C

Jbe the map given by P

O(P

P0). By([M], pp. 10111012),

O( {P0} C C {P0})pr1 00+ pr2 00 (0 0)(pr1+ pr2)0where 0 is the divisor class of the theta divisor on J. But we need 0 to be asymmetric divisor class. Therefore fix aJsuch that (2g2)a= O((2g2)P0KC) ,let (P) = 0(P) + a , and let be the class of the theta divisor defined relative tothe map . Then is a symmetric divisor class, and

(7.2) =( )Pon C C, up to (2g 2)-torsion, where

P:= (pr1+ pr2) pr1 pr2

is the Poincare divisor class. See also ([L 5], Ch. 5,5).Let h denote the Neron-Tate canonical height on J relative to (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) = h(P1+ P2) h(P1) h(P2)which gives the vector space J(k) Z R a dot product structure. The canonical heightalso satisfies

(7.3) h(P) = h(P) + O(1).

Let|P|2 = (P, P), so that|P|2 = 2h(P) . Then, assuming that C(k) is infinite,we may find an infinite subsequence such that all points in the subsequence point in

approximately the same direction: all P1 and P2 in this subsequence satisfy

(7.4) (P1, P2)(cos )

|P1|2|P2|2

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

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

h(P1, P2) 2g(cos )

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

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

1

[k: Q]degO(dY)

Ed

a1hF(P1) + a2hF(P2) 2g(cos )

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


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

1

[k: Q]degO(dY)

E2d(g+ g cos )hF(P1)hF(P2) + O(1).

For 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 must vanish along E.

As was the case in Roths proof, Step 4 consists of applying a variant of the aboveargument to certain partial derivatives, giving a lower bound for the index of 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:

t2d(

g+ g cos )hF(P1)hF(P2) cda1+ o(d)(2g 2)da1hF(P1) + cda1 .

Indeed, by (7.1) the term

v|log sup,v in the numerator of the expression inCorollary 6.3 is bounded by cda1+o(d), and the generators 0,v for various O(dYr) maybe taken uniformly in d and r , so the termv|log infPUv0,v(P)0,v(P0,v)also is bounded by cda1.

But now note that a1hF(P1) is approximately

g+

hF(P1)hF(P2) . Thus thefirst 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 to get apolynomial, and apply Roths 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 , 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 emphasizethis 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 XBX 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 X/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 Greens function is one possible choice. Then F1, F2, , and Ybecome arithmetic divisors on W as well. By the Gillet-Soule Riemann-Roch theorem,

then, 2i=0

(1)i deg RiqO(dY) = d3Y3

6 + O(d2).


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We want a lower bound for deg qO(dY) ; this is obtained as follows. First, since Yis ample, the free parts of RiqO(dY) vanish for i > 0 if d is sufficiently large. Thetorsion part of the R1q term is nonnegative, and the torsion part in the R

2q 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 qO(dY)d3Y3

6 O(d2 log d).

Here Y3 grows like O(r) . Since the rank of O(dY) is approximately d2 , theratio deg qO(dY)

rank qO(dY) is approximately O(d

r) . Then it follows by a

geometry of numbers argument that there exists a global section of O(dY) withv|

L2,vexp

cd

r

.

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

L2,v

1

L2,v

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 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 (X,L) ,

L2 cLsup.Moreover, if L= Li11 Limm , then we may take cL =ci1L1 cimLm .

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

Dn , such thatzP(P) = 0 and

ddc

|z1

|2

ddc

|zn

|2

.

Also, for each P and each L there exist local holomorphic sections 0,P of LUPsuch that0,P(P)= 1 and

cL := inf

PXQUP

0,P(Q)2

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

2L2 cLDn

0,P

(Q)

2 ddc|z1|2 ddc|zn|2c2L(P)2

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

those chosen for the Li.


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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 35 as before. Or, in eithercase, instead of Roths lemma, we can use Dysons lemma on a product of two curves.

Lemma 8.2([V 2]). Let 1, . . . , m be m points on C2

with distinct first coordinatesand distinct second coordinates. Let 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,

mi=1

Vol(t(, i, d1, d2)) (L2)

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 /(g+) 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 Y2 on the right

small. This is how one can prove finiteness for diophantine equations without usingdiophantine approximation per se.

9. The Faltings complex

The remainder of these lectures will be devoted to proving Faltings generalization ofMordells 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):

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

P:= (pr1+ pr2)L pr1 L pr2 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 pr1 L + b2 pr2 L + abP.Then it follows that dY can be written (approximately) in the form

dY = (s1 pr1 s2 pr2)L s21pr1 L s22pr2 L.In this case, however, it will be necessary to work on a product of n copies of A , solet us define

(9.2) L,s= i0 by homogeneity.


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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 ( is taken negative but close to zero); however, onthe arithmetic curve corresponding to our point (P1, . . . , P n) , the first term is smalland the second term then dominates.

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

(a prib prj)L + (a pri+b prj)L= 2a2 pri L + 2b2 prjL.Thus, choosing global sections 1, . . . , m(A,O(L)) which generate O(L) over thegeneric fiber A , for any X1, . . . , X n we can form an injection

0

Xi, dL,s

Xi, d(2n 2 + )ni=1

s2ipri L

aby tensoring with products of terms of the form (asi pri+asj prj)bij , where ba2 =dand a is sufficiently divisible. Here the tuples ()ij vary over {1, . . . , m}n(n1)/2 .Likewise, one can extend this sequence to an exact sequence

0

Xi, dL,s

Xi, d(2n 2 + )ni=1

s2ipri L

a

Xi, di


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185

be the associated bilinear form; then the assumption is that

(Pi, Pj)L(1 1)

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

We will also call these conditions CP(c1, c2, 1) .

The proof also uses subvarieties X1, . . . , X n of X satisfying the following condi-tions, denoted CX(c3, c4, P1, . . . , P n) :

(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 Xic3.(10.5.4). The heights h(Xi) are bounded by the formula

ni=1

h(Xi)

hL(Pi)< c4

ni=1

1

hL(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 will

not 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, . . . , X n 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)

ni=1

h(Xi)

hL(Pi) c4

ni=1

1

hL(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. Fori = 1, . . . , n let si be rational numbers close to 1hL(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,s) for some > 0(depending on the dim Xi). If the points P1, . . . , P n were chosen suitably, then it ispossible to construct subvarieties Xi of each Xi such that the X

i also satisfy (10.5)

(possibly with different constants), and such that some Xi is strictly smaller.


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

c3, c4 and1, . . . , n N c1, c2, 1, c3, c4 such that

P1, . . . , P n

X\ Z(X)(k) satisfying CP(c1, c2, 1) andX1, . . . , X nXsatisfyingCX(c3, c4, P1, . . . , P n) and dim Xi = i i

X1, . . . , X n with XiXi i and Xi=Xi for somei,and satisfying CX(c

3, c

4, P1, . . . , P n).

(10.8)

To conclude the proof, we now assume that X(k)\Z(k) is infinite. Then it ispossible to choose P1, . . . , P n in X(k) \ Z(k) to satisfy Conditions (10.4) for all c1,c2, and 1 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, . . . , X n 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(n1)/2 given by(x1, . . . , xn)(xi xj)i0.Proof. The Q-divisor class L0,1 is the pull-back to

Xi of an ample divisor class on

An(n1)/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 .


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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 pri L prjL

is represented over A(C)

n

by a form inpri E

1,0(A) prj E0,1(A) + pri E0,1(A) prj E1,0(A) E1,1(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,=1

adz dz ,

where a are constants. For = 1, . . . , d let u = pri z and v = pr

jz. Using

the above choice of metric on L , the curvature ofPij is

d,=1

a,(du dv+ dv du).

By counting degrees we immediately obtain the main homogeneity lemma:

Corollary 11.4. Any intersection producti 0 and > 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,

h0Xi, dL,s > cdP dimXi ni=1

s2 dimXii


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for all sufficiently large d (depending on s ).

Proof. By Seshadris criterion ([H 1], Ch. I,7), L,s is ample for >0 . By Riemann-Roch, it follows that

h0Xi, dL,s =dP dimXi (LP

dimXi

,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 , dL,sHj

=dN1

(prjL . LN1,s )

(N 1)! (1 + o(1)).

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

h0

Hj ,

dL,s

mipri LHj

dN1 (pr

jL . L

N1,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

0O

dL,s

mipri L prjL

O

dL,s

mipr

i L

O

dL,s

mipri L Hj

0

gives the inequality

h0

Xi, dL,s

mipri L prjL

h0

Xi, dL,s

mipr

i L

h0

Hj , dL,s

mipr

i L

h0Xi, dL,smipri L dN1 (prjL . LN1,s )(N 1)! (1 + o(1)).Therefore, since dL,s= dL,s d(+ )

i s

2iHi, this estimate gives

h0

Xi, dL,s

h0

Xi, dL,s

d(+ )ni=1

s2ih0

Hi, dL,sHi

dN

(LN,s)

N! (+ )

ni=1

s2i(pri L . L

N1,s )

(N 1)!

(1 + o(1)).

By Corollary 11.4, this lower bound equals

d

Nn

i=1 s2 dimXii (LN,1)

N! (+ )

n

i=1(pri L . L

N1,1 )

(N 1)! (1 + o(1)).


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

0< c 0. Thenext step consists of controlling this radius.

By a slight change of model, we regard Xi as a subset of i, 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 divisorDi on i, depending only onCi, the choice of projections to P

mi , and the permutation of coordinates on Pmi ,such that for all placesv 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 C

miv . 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 i is smooth over Ci at Pi (on the generic fiber over Spec k ). Recall

that 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, . . . , f Mmi for the idealof some neighborhood of Ei,v in i , as a subset of some affine space AMCi . 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, . . . , f Mmi, 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 Hensels lemma (Corollary 15.13), the desired biholomorphic mapexists.

As the points Ei,v vary over all of a certain Zariski-open subset of i, only finitelymany systems f1, . . . , f mi are needed (quasi-compactness of the Zariski topology), so

there exists a divisor Di on i (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 Greens function gDi for Disuch that the desired biholomorphic map exists, with a polydisc of radius exp(gDi) .This is done using compactness of i(Cv) .

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

Lemma 16.3. For each i there exists an arithmetic divisor Fi on i, depending onlyon Ci , the choice of projections to P

mi , and the permutation of coordinates onPmi , such that for all places v of k , the metric of the element qi yj

yj(qi(Pi))

of the ideal sheaf of Pi in Xi satisfies

log qi yj yj(qi(Pi))v (Fi. Ei)v.


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

infinity in Pm

i ) 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 0,v be a local generator for L on Uv. This should be chosen uniformlyin terms of generators on pri O(L) and the Poincare divisors Pij ; then it will bepossible to ensure that

(16.4) log inf PUv

0,v(P)0,v(P0)v c3,v

ni=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,

letsup,v = 1 for non-archimedean v .Now let () = (11, . . . , 1m1 , . . . , nmn) be a tuple such that

(16.5)ni=1

mij=1

ijdi

=t

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

D():= 1

11! nmn !

y11

11

ynmn

nmn= 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()(P0)v

log sup,v+ni=1

mij=1

ij

log i,v (Fi. Ei)v

+ log infPUv

0,v(P)0,v(P0)v

log sup,v+ni=1

mij=1

ij

log i,v (Fi. Ei)v c3,v n

i=1

di,

(16.6)

by (16.4). Here D()(P0) is regarded as a section of the vector sheafL S11++1m1 X1/SpecR Sn1++nmnXn/SpecR

E

,


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where Eas usual is the arithmetic curve on 1 SpecR SpecRn corresponding toP0 = (P1, . . . , P n) . See also ([Laf],4). Thus

v log D()(P0)vdegLE

+n

i=1mi

j=1ijdegMi

Ei,

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

(16.7)ni=1

mij=1

ij

degMi

Ei

v

log i,v+ (Fi. Ei)

degLEv|

log sup,v c3ni=1

di.

But the quantity inside the parentheses on the left is an intersection number on i.From the structure of i as a subset of a product, there exist divisors Gi on P

M andHi on Ci such that

MiO(Fi+ Di)O(Gi+ Hi)relative to the cone of very ample line sheaves on the generic fiber of i over Spec k .Thus

degMiEi

+ (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) becomes

n

i=1mi

j=1 ijc5hL(Pi) + c4h(Xi) + c6 degLEv| log sup,v c3n

i=1 di.But now, for all i ,

mij=1 ijdi; therefore modifying c4 gives (using also (16.5))

c4

ni=1

dih(Xi)+c5t max1in

dihL(Pi)+c6tni=1

di degLE

v|

log sup,vc3ni=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) ZR is a finite dimensional vector space. LettingL replace the theta divisor, it follows that

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


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

(Pi, Pj)L(1 1)

|Pi|2|Pj |2

2(1 1)hL(Pi)hL(Pj)(17.1)

for some given 1 > 0 and for all i < j . From fundamental properties of the canonicalheight,

deg

(s2i pris2j prj)LE

= s2i hL(Pi) + s2j hL(Pj) sisj(Pi, Pj)L+ O(s2i + s2j)

si

hL(Pi) sj

hL(Pj)

2+ 21sisj

hL(Pi)hL(Pj) + O(s

2i + s

2j).

Letting si be rational and close to 1

hL(Pi) , the square in the above expression

approaches zero and we obtain

(17.2) deg L,sEn(n 1)1 n + O

s2i

.

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(, (P1, . . . , P n), d1, . . . , dn) satisfies

tn( (n 1)1) c

s2ic5+ c6

s2i

.

If the heights hL(Pi) are sufficiently large and 1 sufficiently small, which we nowassume, then the s2i will be small, and the above inequality becomes

(17.3) t

2

for some 2 > 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 = Pm1 Pmn be a product of projective spaces

over a field of characteristic zero, and let 3 > 0 be given. Then there existnumbers r , c1, c2, and c3 with the following property. Suppose is a nonzeroglobal section of the sheaf O(e1, . . . , en) on which has index 3 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


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

(18.2)

eih(Yi)c2v|

log sup,v+ c3i

ei.

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

We note that this theorem generalizes Roths 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, . . . , X nbe geometrically irreducible subvarieties of Xdefined over k whose degrees (rela-tive to the chosen projective embedding of X) are bounded. Let > 0 and 2 > 0be given. Then there exist numbers r , c1, c2, c3, and c4 with the following

property. Let be a nonzero global section of O(dL,s) which has index 2relative to (d1, . . . , dn) (di = ds

2

i ) at some point (P1, . . . , P n) with Pi Xi(k)for all i . If di/di+1 r for all i = 1, . . . , n1 , then there exist subvarietiesXiXi, not all of which are equal to Xi, such that

(i). each Xi contains Pi;(ii). each Xi is geometrically irreducible and defined over k ;

(iii). the degrees of Xi are bounded by c1; and(iv). the heights h(Xi) satisfy the inequality

(18.4)

dih(Xi)c2

v|

log sup,v+ c3i

dih(Xi) + c4i

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 is

a 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 . Therefore, for each

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

a model for Ki such that the rational maps Xi Xi corresponding to all injections

K(Xi) Ki over K(Pmi) are morphisms. The product of the pull-backs of O(L)via these morphisms is a line sheaf on Xi which is isomorphic to the pull-back of somemultiple of O(1) from Pmi . Expanding this to the Chow family i over Ci/ Spec Ras in Section 16, the isomorphism holds up to a divisor on Ci, fibral componentsover SpecR , 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 P

miCi

SpecR PmjCj , and over each point of CiCj


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it is trivial. But for each point Q PmiCi , the restriction of this norm to {Q} PmjCj

is algebraically equivalent to zero, because it is obtained from the restriction of Pijto sets of the form {Q} X, for Q i 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 denominators

are of the form expch(Xi)h(Xj) + c . This is indeed fortunate, because the divisorclasses Pij occur with multiplicities sisj in L,s, and

sisj

h(Xi)h(Xj)1

2

s2ih(Xi) + s

2jh(Xj)

,

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

(18.5)v|

sup,vv|

sup,v exp

c

dih(Xi) + c

di

.

We now calculate ei. First note that the map

Xi has degree N :=

Ni;

then ei= di

N/Ni. Since the Ni are bounded, the index of at the point (x1, . . . , xn)

lying below (P1, . . . , P n) 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+1r implies ei/ei+1r .

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

dih(Yi)c

eih(Yi)

cv|

log sup,v+ c

dih(Xi) + c

di+ c

ei

cv|

log 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 Xi equal Xi.

It remains only to ensure that (ii) holds. But we may intersect Xi with finitelymany (at most dimXi ) of its conjugates over k until the geometrically irreduciblecomponent containing Pi is defined over k . Replacing Xi 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 Xi 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 Xi has dimension strictlysmaller than dim Xi. This concludes the main part of the proof of Theorem 0.3.


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

P1, . . . , P nX(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, . . . , n1 , h(Pi+1)/h(Pi) > r , where r is the largest of ther 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 1, 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,s is ample. Shrinking a little, it ispossible to obtain an upper bound on the dimension of the space of sections ofO(dL,s)

which have index at the point (P1, . . . , P n) , for some suitable >0 . This boundis bounded away from h0(Xn, dY,s) , so the more precise form (2.4) of Siegels lemmaallows us to construct a global section with index at (P1, . . . , P n) . 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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