Quadratic integral solutions to double Pell equations

Quadratic Integral Solutions to Double Pell Equations

FRANCESCOVENEZIANO(*)

ABSTRACT - We study the quadratic integral pointsÐthat is, (S-)integral points defined over any extension of degree two of the base fieldÐon a curve defined in P3by a system of two Pell equations. Such points belong to three families ex- plicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field, the equations defining the curve and the setS.

We exploit the peculiar geometry of the curve to adapt the proof of a theorem of Vojta, which in thiscase doesnot apply.

1. Introduction

LetCbe an irreducible curve defined over a number fieldk. We know, depending on the genus, the general structure of C(Q); on the other hand, of courseC(Q) is always infinite, so it is natural to ask what hap- pensif we consider algebraic pointsup to some fixed degree overQ.

Abramovich and Harrisin [AH91], and also Silverman and Vojta in [HS91, Voj91, Voj92] were among the first to study the set of points P2 C(Q) s uch that [k(P) :k]d, in particular whether it isinfinite or not.

In thispaper we deal with the analogousproblem forintegral points, sticking to the special case of dˆ2. This case was already studied in generality by Corvaja and Zannier in [CZ04, Corollary 1] using Schmidt's Subspace Theorem.

We will further specialise the problem to curves defined by a double Pell equation, such as, for example,

y²ˆ2x²‡1 z²ˆ3x²‡1:

( …1†

(*) Indirizzo dell'A.: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa.

E-mail: f.veneziano@sns.it

Double equationsof thiskind are historically relevant, being among the first curves of genus 1 ever studied.¹

Of course, by Siegel's theorem on integral points on curves, there are only finitely many solutions inZto (1), but it may be easily seen that there are infinitely many solutions in algebraic integers of degree 2 overQ: in fact it is classically known that each Pell equation x² dy²ˆ1 hasin- finitely many integral solutions whendis a positive integer not a square, so one can solve the first equation to find infinitely many (xn;yn) such that y²_n ˆ2x²_n‡1, and then setz_nˆ 

3x²_n‡1

p ; in thisway we can actually find three infinite families of solutions. We will fully describe the set of quadratic integral pointson these curvesand give a geometrical meaning to the familiesjust mentioned.

Going back to the general context, Abramovich and Harrisconjectured that the set ofP2 C(Q) such that [k(P) :k]disinfinite (up to a finite extension of the basefieldk) if and only if there exists some non constant morphismW:C !Xof degree at mostd, whereXiseitherP₁or an elliptic curveEwithjE(k)j ˆ 1.

Note that thiscondition isobviously sufficient; ifWisdefined overkthe preimage throughWof a rational point inXisa point of degree at mostdinC.

While the general conjecture wasproved false by Dabarre and Fahlaoui in [DF93], Abramovich and Harris managed to prove it in some cases as, for example, whendˆ2;3.

The first step in their proofs was to consider the d-fold symmetric product of the curve, C^(d); this is a variety whose set of points may be identified with the set of all unordered d-tuplesof pointsofC. Pointsof degree at mostdonCnaturally correspond tok-rational pointsonC^(d), s o the said authors could work on the varietyC^(d) and apply results by Falt- ingsafter mappingC^(d) in an abelian variety.²

If we consider only points of degree 2 over the base field, the existence of infinitely many quadratic rational points is surely necessary to have in- finitely many quadraticintegralpoints, but it is not sufficient, as we shall see.

In [CZ04] Corvaja and Zannier prove a theorem on integral pointson surfaces and apply it toC⁽²⁾to get the following theorem:

THEOREM1. LetC~be a projective non-singular curve defined over a number field k, and C ˆC n fQ~ ₁;. . .;Q_rg be an open affine subset, for

(¹) See for instance [Wei07] or [Mor69] for a discussion of some classical cases.

(²) For an account on thisproblem see [EE93].

distinct Qi2C(k). Then~

1. If r5 then C contains only finitely-many quadratic-integral points over k;

2. If rˆ4there exist finitely many rational mapsc: ~C !P1of degree 2 such that all but a finite number of the quadratic-integral points onC over k are sent toP1(k)by at least one of these maps.

Asit happensfor the structure of thek-integral points, what matters in thisproblem isthe number of pointsat infinity.

2. Setting and statement

Asmentioned above, thispaper will study quadratic integral pointson some special curves inP3of genus1 and with 4 pointsat infinity, defined by a double Pell-like equation; let thena;b;c;dbe algebraic integerssuch that abcd6ˆ0 andad bc6ˆ0, and letCbe the affine curve inA³defined by

y²ˆax²‡c z²ˆbx²‡d:

( …2†

Let~Cbe itsprojective completion, defined by homogeneousequations Y²ˆaX²‡cW²

Z²ˆbX²‡dW²: (

Let usindicate with P1;P2;P3;P4 the four pointsat infinity of ~C n C, which, in the coordinates(X:Y:Z:W) are the points

P₁:(1: 

pa : 

pb :0) P₂:(1: 

pa

: 

pb :0) P₃:(1: 

pa : 

pb :0) P₄:(1: 

pa

: 

pb :0):

Letkbe a number field containing 

pa

; 

pb

;candd, and letS Mka finite set of absolute values ofkcontaining all the archimedean onesand all the primesincd(bc ad); letsbe the cardinality ofS.

THEOREM2. The set of quadratic S-integral points onCis the union of:

Three families consisting of the preimages through the three

mapsC !P1

(x;y;z)7!(x;y) (x;y;z)7!(x;z) (x;y;z)7!(y;z) of the S-integral points ofP1;

A finite set of cardinality at most2^2835s‡3;

A finite and effectively computable set whose cardinality is at most32^{1121(s‡H 1)‡1}, where H is the class number of OS.

REMARK3. The three familiesof quadratic pointsare indeed easy to spot. If for example the first of the two Pell equations definingC hasin- tegral solutions (xn;yn), then (xn;yn; 

bx²_n‡d

p ) are quadratic integral pointsonC. Similarly if (xn;zn) are integral solutions to the second equa- tion, then we have a second family (x_n; 

ax²_n‡c

p ;z_n), and if (y_n;z_n) are integral solutions toby²_n az²_n ˆbc ad, we get



y²_n c a r

;yn;zn

! for a subsequence of (y_n;z_n).

REMARK4. We also note that this theorem allows for a bound on the number of exceptional solutions outside the three infinite families, while Theorem 1 doesnot.

While the Subspace theorem isnot effective, there isa semi-effective version due to Evertse which provides an explicit bound for thenumberof exceptional hyperplanes; it is not possible, however, to use it in the proof of Theorem 1 to get such a bound for the quadratic integral points. This is because, in the proof of Theorem 1, the Subspace theorem is applied to a surface, and each exceptional hyperplane for the Subspace Theorem gives an exceptional curve on thissurface which may contain pointscorre- sponding to quadratic integral points; but even if we can bound the number of such curves, they are not effectively computable and so it is not possible to bound the number of pointson them.

REMARK 5. We finally remark that we can not expect in general a bound for the finite sets of points which is uniform in the coefficients a;b;c;d; for example it isknown³that there isa constantC>0 such that for infinitely many positive integers A, the number of positive integer solutions to the equationX³‡Y³ ˆAexceedsC 

logA p3

.

(³) This result is essentially due to Mahler in [Mah35] and improved by Silverman in [Sil83].

3. Sketch of the proof

To study quadratic integral points we proceed as [AH91] and [CZ04]

and consider the symmetric square of the affine curveC, which isdefined asthe quotient of the productC Cby the involution which exchangesthe couple (P;Q) with the couple (Q;P). Thisquotient isan irreducible surface which can be identified with the set of unordered couples of points, and the quotient map is

C C !^W C⁽²⁾ (P;Q)7! fP;Qg:

Consider a pointPdefined on a field of degree 2 overk, and letP⁰be its conjugate. Then the point fP;P⁰g on C⁽²⁾ isfixed by every Galoisauto- morphism ofQ=k, and hence isdefined over k. If furthermorePisan in- tegral point so is the pointfP;P⁰g.⁴

The special geometry of curves defined by double Pell equations carries to their symmetric square and allows one to study quadratic integral points directly and to give an explicit description of the maps mentioned in Theorem 1; for the proof we will mimic the proof of a theorem by Vojta ([Voj87]):

THEOREM 6 (Vojta). Let V a projective, nonsingular variety over a number field k. Let r be the rank of the group of k-rational points ofPic⁰(V), rthe rank of the NuÃÁron-Severi group of V and D a divisor with at least dimV‡r‡r‡1distinct irreducible components, all defined over k. Then all sets of quasi-S-integral points on V n jDjare degenerate.

REMARK 7. Thistheorem hasbeen improved later by Vojta himself, removing the assumptions on the Pic⁰, in [Voj96] and also by Noguchi and Winkelmann in [NW02].

While thistheorem doesnot apply to our case, the proof adaptswell because on the curveCthe difference of any two pointsat infinity istorsion in the Picard group.

The first step of the proof, following the strategy already illustrated, will be to study instead the structure of the set of integral points on the surfaceC⁽²⁾obtained by taking the symmetric square of the original curve.

(⁴) See, for example, [Ser88] for more on the symmetric product.

We will then follow the proof of Theorem 6 and build three functions a;b;g without zeroesand poleson C⁽²⁾. These functions, up to constant factors, take integral points of the symmetric square toS-units.

We will then find a relation amonga;b;g, so that taking them as co- ordinatesgivesa map fromC⁽²⁾ to the subvariety of G³_m defined by this relation; it will turn out that thisrelation islinear.

Thismeansthat the functionsa;b;g take integral pointsofC⁽²⁾ to so- lutionsof theS-unit equation, which isthe object of the following theorem.

THEOREM8 (Evertse, [Eve95]). Let k be a number field, let S be a finite set of places of k containing all the archimedean ones, let sˆ jSjand let a₁;. . .;a_n2Q. Let A⁰(a₁;. . .;a_n;O_S) be the number of non degenerate solutions x₁;. . .;x_n2 O_Sto the equation

a₁x₁‡. . .‡a_nx_nˆ1;

…3†

where a solution(x1;. . .;xn)is calleddegenerateif there is a proper van- ishing subsum in the left hand side of(3).Then

A⁰(a₁;. . .;a_n;O_S)2^35n4s:

Using this theorem we will then proceed to bound the number of non degenerate solutions and to examine degenerate solutions, which come from special subvarieties.

We will show that three of these special subvarieties are curves of genus 1, hence each givesonly a finite number of quadratic integral points, while the other three have genus0 and give three familiesof quadratic integral points.

The strategy outlined here would also work for any variety such that the difference of any two components of the divisor at infinity is torsion in the Picard group.

4. Proof Theorem 2 Three functions onC

Let usconsider the functions f ˆY‡ 

pa X

W ˆ cW

Y 

pa X gˆZ‡ 

pb X

W ˆ dW

Z 

pb X hˆ

b p Y 

pa Z

W ˆ (bc ad)W

pb

Y‡ 

pa Z:

They are functionson the curveCdefined overk.

By explicit computation one has xˆf² c

2 

pa

f ˆg² d 2 

pb g yˆf²‡c

2f ˆbc ad‡h² 2 

pb h zˆg²‡d

2g ˆbc ad h² 2 

pa

h ;

so we have that

k(f)ˆk(x;y); k(g)ˆk(x;z); k(h)ˆk(y;z);

andk(C) hasdegree 2 over each of them.

The divisors of the three functions are (f)ˆP₃‡P₄ P₁ P₂ (g)ˆP2‡P4 P1 P3

(h)ˆP₁‡P₄ P₂ P₃:

These three functions are ink[C], as we can see by their explicit expres- sions and the expressions for their inverses, or by observing that their di- visors are supported on the points at infinity.

By direct computation using the first and then the second definition of f;g;hone immediately checks that they satisfy the linear relations

b p f 

pa gˆh (4a)

c 

pb f

d 

pa g ˆh:

(4b)

Three functions onC⁽²⁾

Let us now consider the Cartesian productC Cgiven by equations y²ˆax²‡c

z²ˆbx²‡d y⁰²ˆax⁰²‡c z⁰²ˆbx⁰²‡d;

8>

>>

>>

<

>>

>>

>:

and let usindicate byf⁰;g⁰;h⁰ the functionscorresponding tof;g;hin the primed variables. Let C⁽²⁾ be the symmetric product of C obtained from C Casa quotient by the action ofZ=2Zthat actsswapping the two co- ordinates.

The ring of regular functionsoverC⁽²⁾isAˆk[C C]^Z=2Z, the subring of k[C C] consisting of the functions invariant for this action, and the pointsofC⁽²⁾can be thought asunordered couplesof pointsonC.

C C !^p C⁽²⁾

k[x;y;z;x⁰;y⁰;z⁰]A:

Let usdenote bypthe quotient map fromC CtoC⁽²⁾.

LetPbe a quadratic integral point onC, and letP⁰be itsconjugate.

The pair (P;P⁰) isa quadratic integral point onC C, and the unordered couplefP;P⁰g isan integral point onC⁽²⁾ which isfixed by any Galoisau- tomorphism overkbecause any such morphism either fixes bothP;P⁰or swaps them.

The pointfP;P⁰gistherefore defined overk.

Let usnow define three functions aˆ cd

ff⁰gg⁰ˆ(y 

pa

x)(y⁰ 

pa

x⁰)(z 

pb

x)(z⁰ 

pb x⁰)=cd bˆc(bc ad)

ff⁰hh⁰ ˆ(y 

pa

x)(y⁰ 

pa x⁰)( 

pb

y‡ 

pa z)( 

pb

y⁰‡ 

pa

z⁰)=c(bc ad) gˆd(ad bc)

gg⁰hh⁰ ˆ(z 

pb

x)(z⁰ 

pb x⁰)( 

pb

y‡ 

pa z)( 

pb

y⁰‡ 

pa

z⁰)=d(ad bc) 1

aˆff⁰gg⁰

cd ˆ(y‡ 

pa

x)(y⁰‡ 

pa

x⁰)(z‡ 

pb

x)(z⁰‡ 

pb x⁰)=cd 1

bˆ ff⁰hh⁰

c(bc ad)ˆ(y‡ 

pa

x)(y⁰‡ 

pa x⁰)( 

pb

y 

pa z)( 

pb

y⁰ 

pa

z⁰)=c(bc ad) 1

gˆ gg⁰hh⁰

d(ad bc)ˆ(z‡ 

pb

x)(z⁰‡ 

pb x⁰)( 

pb

y 

pa z)( 

pb

y⁰ 

pa

z⁰)=d(ad bc):

We clearly see that they belong to k[C⁽²⁾], and so do their inverses; the functionsa;b;gare the three regular and nonvanishing functions that we need to follow Vojta'sstrategy and viewC⁽²⁾asa subvariety ofG³_m.

The functionsa;b andgare defined on a surface, so they must be al- gebraically dependent. Our next step is to find a relation between them.

Multiplying together equation (4a) for the primed and unprimed vari- ablesone gets

hh⁰ˆbff⁰‡agg⁰ 

pab

(f⁰g‡fg⁰);

…5†

doing the same for equation (4b) gives hh⁰ˆc²b

ff⁰ ‡d²a

gg⁰ cd 

pab 1 f⁰g‡ 1

fg⁰

: …6†

If we now multiply (6) by ff⁰gg⁰

cd and subtract it from (5), after some tidying up and using the definitions fora;b;gwe obtain

a‡b‡gˆ1:

Computing degrees

From what we have said until now we have the mappings C C !^p C^(2)(a;b;g)! HG³_m

whereHisthe subvariety ofG³_mdefined byX‡Y‡Zˆ1.

The corresponding homomorphisms between the rings of regular functionsare

k X;Y;Z; 1 XYZ

[Zƒƒƒƒƒƒƒ!7!1 X Y]k X;Y; 1

XY(1 X Y)

X7!a Y7!b

ƒ!Ak[C C]:

To find the degree of the mapC^(2)(a;b;g)! Hwe must find the degree [k(C⁽²⁾):k(a;b;g)]:

One findsdirectly that (ff⁰)²ˆ c²g

ab, so that [k(ff⁰;gg⁰):k(a;b;g)]ˆ2, and from

f² c pa

f ˆg²  d pb

g (ff⁰)² cf²

pa

(ff⁰)f ˆ(gg⁰)² dg² pb

(gg⁰)g

followsthat [k(f;f⁰;g;g⁰):k(ff⁰;gg⁰)]4, so by our previous remarks we have

[k(C C):k(a;b;g)]ˆ[k(f;f⁰;g;g⁰):k(a;b;g)]ˆ

ˆ[k(f;f⁰;g;g⁰):k(ff⁰;gg⁰)][k(ff⁰;gg⁰):k(a;b;g)]8:

Obviously [k(C C):k(C⁽²⁾)]ˆ2, asthere are two ordered pairsfor a generic unordered couple; therefore we can conclude that the degree of the map given bya;b;gbetweenC⁽²⁾andHisat most four.

IfPisan S-integral point onC⁽²⁾the values a(P);b(P);g(P) will be S- integers, and so will be their inverses 1

a(P); 1 b(P); 1

g(P).

The pointPwill then provide a solution inO_S of the equation x1‡x2‡x3ˆ1:

Non degenerate solutions

Theorem 8 tellsusthat there are only finitely-many non degenerate solutions, and that we can bound their number. If we apply the theorem with nˆ3 we obtain that the number of non degenerate triples (a;b;g)2Hisat most 2^3534sˆ2^2835s.

We already bounded in the previousparagraph the degree of the map (a;b;g), which isat most four. For every integral point on C⁽²⁾ we have two quadratic integral pointsonC, hence the number of quadratic integral pointson C corresponding to non degenerate solutions is at most 2^2835s‡3.

Degenerate solutions

The degenerate solutions to a‡b‡gˆ1 are those with a subsum equal to 0, that isthose for which one of the three functionsa;b;gisequal to 1; let usthen define

Wa:aˆ1; W_b :bˆ1; Wg :gˆ1 the subsets ofC⁽²⁾thusobtained.

Using the definition ofa;b;gwe see that, for example,

aˆ1; b ˆ g

ff⁰gg⁰ ˆcd; c(bc ad)

ff⁰hh⁰ ˆ d(ad bc) gg⁰hh⁰ ff⁰cgg⁰

d ˆc²; ff⁰ˆcgg⁰ d (ff⁰)² ˆc²; gg⁰ˆff⁰ c d

and similarly for the other two cases, so we see thatWa;Wb;Wgare com- posed of two subvarieties each, and

WaˆW_x [W_x^‡ W_bˆW_y [W_y^‡ W_gˆW_z [W_z^‡; where

W_x :ff⁰ˆc;gg⁰ˆd W_x^‡:ff⁰ˆ c;gg⁰ˆ d W_y :ff⁰ˆ c;hh⁰ˆad bc W_y^‡:ff⁰ˆc;hh⁰ˆbc ad W_z :gg⁰ˆ d;hh⁰ˆbc ad W_z^‡:gg⁰ˆd;hh⁰ˆda bc:

These six subvarieties give all degenerate solutions; to understand them better we use the following simple lemma:

LEMMA9. Let Pˆ((x;y;z);(x⁰;y⁰;z⁰))2 C C. Then ff⁰(P)ˆ c)xˆ x⁰and yˆ y⁰;

gg⁰(P)ˆ d)xˆ x⁰and zˆ z⁰;

hh⁰(P)ˆ (bc ad))zˆ z⁰and yˆ y⁰. PROOF. For example, ifff⁰ˆcthen

x⁰ˆ(f⁰)² c 2 

pa

f⁰ ˆc²=f² c 2 

pa

c=f ˆc f² 2 

pa

f ˆ x;

and similarly for the other five cases. p

If we define six subvarieties ofC C

V_x:xˆ x⁰;yˆ y⁰;zˆ z⁰ V_y:xˆ x⁰;yˆ y⁰;zˆ z⁰ V_z:xˆ x⁰;yˆ y⁰;zˆ z⁰;

then the lemma tellsusthat each of them correspondsthrough pto the similarly named subvarietyW_x;y;z ofC⁽²⁾(we denote byV_x;y;z any of the six varietiesV_x;V_x^‡;V_y;V_y^‡;V_z;V_z^‡, and we do the same withW_x;y;z ). In short we have that

pj_V

x;y;z :V_x;y;z !W_x;y;z

Note also thatV_x;y;z ' Cthrough the projection on the first component of C C, so that these six curves all have genus one.

Points on W_x;y;z^‡

Consider for example

pj_Vx^‡ :V_x^‡!W_x^‡

((x;y;z);(x; y; z))7! f(x;y;z);(x; y; z)g:

A generic point in W_x^‡ hastwo preimages, obtained by exchanging the order of the pair; the points which have just one primage are those such thatyˆzˆ0, but we see from the defining equations (2) ofCthat thiscan never happen becauseby² az²ˆbc ad6ˆ0. The mappj_Vx^‡istherefore an unramified covering ofW_x^‡, and the Riemann-Hurwitz formula tellsus that

0ˆ2g(V_x^‡) 2ˆ2(2g(W_x^‡) 2) g(W_x^‡)ˆ1:

The same is true forW_y^‡andW_z^‡, because there is no point onCwhere two ofx;y;zboth vanish.

We have thus shown that the three subvarietiesW_x;y;z^‡ are all curvesof genus 1, so they carry only a finite number of integral points, which in turn correspond to a finite number of quadratic integral points onC.

We might also argue the finiteness of quadratic integral points coming from the curvesW_x;y;z^‡ asfollows: letKˆk( 

pe

) be a quadratic extension of k, we may supposee2 O_S; letPˆ(x₀;y₀;z₀) be aK-integral point on C, andP⁰ˆ(x⁰₀;y⁰₀;z⁰₀) itsconjugate. Suppose that (P;P⁰) belongstoV_x^‡; if it is so, after enlarging S to anS⁰ so thatO_S⁰ hastrivial classgroup, we can writex₀ ˆt;y₀ˆu 

pe

;z₀ˆv 

pe

for somet;u;v2 O_S⁰.

Substituting back into the equations forCwe getbeu² aev²ˆbc ad, soedividesbc ad; differente differing only by a square factor give the same extension, and given that O_S0=(O_S0)² isfinite we see that K must

belong to a finite set of extensions ofk, whose cardinality may be bounded in termsof the cardinality ofS⁰.

We know, by Siegel'stheorem, that there are only finitely many in- tegral pointsonCdefined over a fixed number field; sinceChasfour points at infinity, thisnumber may be bounded effectively, asdone in [CZ03], in termsof the degree ofK and the cardinality of the extension ofS⁰ toK, which in turn are both bounded in termsofkandjS⁰j.

The computationsinvolved in thisbound are quite heavy, but again the special structure of the curve C providesuswith a simpler argument:

equation (4a) givesthe unit equation

b p f

h (P)

a p g

h (P)ˆ1;

…7†

so the number of solutions may be easily bounded using again Theorem 8.

For a fixedKand extensionS⁰⁰of the set of absolute valuesS⁰, there are at most 2^3524jS00jˆ2^1120jS0j solutions; the number of extensions K isat most jO_S0=(O_S0)²j ˆ2^jS0j; the functionsf=handg=hhave degree two, aswe can see computing their divisors, so each solution gives at most two points. Com- bining all, we have that the number of quadratic integral pointsPsuch that (P;P⁰) lieson any of theW_x;y;z^‡ isat most 32^1121jS0j‡1ˆ32^{1121(s‡H 1)‡1}, whereHisthe classnumber ofO_S.

I thank the anonymous referee for suggestions about this bound.

We should also remark that, since the relevant curve has genus 1, it is in fact possible, as is well known, to boundthe heightof the solutions them- selves; tha same can be done on equation (7), using effective results ontwo- termunit equations(derived from Baker'stheory of linear formsin loga- rithms). So the quadratic integral points arising from W_x;y;z^‡ are, in fact, effectively computable (in contrast with those arising from the ``non de- generate solutions'').

Points on W_x;y;z

Reasoning as we did before, we consider, for example pj_Vx :V_x !W_x

((x;y;z);( x;y;z))7! f(x;y;z);( x;y;z)g:

Thisisagain a map of degree 2.

In this case however, we see that it is ramified at points wherexˆ0, and there are 4 such points onC, namely (0; 

pc

; 

pd

), each of them of

course ramified of index 2. Therefore this time applying the Riemann- Hurwitz formula we obtain

0ˆ2g(V_x) 2ˆ2(2g(W_x 2)‡4 g(W_x)ˆ0;

asW_x hasgenus0, it may contain infinitely many integral points.

The composition of the maps

C ! V_x ! W_x ^pj!^Wxfby² az²ˆbc adg (x;y;z)7!((x;y;z);( x;y;z))7!f(x;y;z);( x;y;z)g 7! (y;z) givesa map of degree two fromCto a curve of genus0, that takesquadratic integral pointsarising fromW_x to integral points; this map together with the same compositions forW_y andW_z, that is,

C ! fz²ˆbx²‡dg (x;y;z) 7! (x;z);

and

C ! fy²ˆax²‡cg (x;y;z) 7! (x;y);

are the mapsin Theorem 1.

Acknowledgments The author thanks Professors Corvaja, Evertse and Zannier for the substantial advice received in the preparation of this article, and the anonymousreferee for the thorough report and for having suggested some improvements.

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Manoscritto pervenuto in redazione l'11 ottobre 2010.