Explicit uniform estimation of rational points II. Hypersurface coverings

DOI 10.1515/CRELLE.2011.139 angewandte Mathematik

(Walter de Gruyter BerlinNew York 2011

Explicit uniform estimation of rational points II. Hypersurface coverings

ByHuayi Chenat Paris

Abstract. We obtain an explicit uniform estimate for the number of rational points in a projective plane curve whose heights do not exceed the degree of the curve.

1. Introduction

This article is a continuation of [12]. LetK be a number field andX be a sub-variety ofP_Kⁿ of dimensiond and of degreed. The purpose of this article is to establish the follow- ing explicit estimate (see Theorem 4.2):

Theorem A. Lete>0and D be an integer such that D>max

ðe¹þ1Þ

2d^1dðdþ1Þ þd2

;2ðndÞðd1Þ þdþ2

:

There is an explicitly computable constant C ¼Cðe;d;n;d;KÞsuch that,for any Bfe^e,the set S1ðX;BÞ of regular rational points of X with exponential heighteB is covered by not more than CB^ð1þeÞd

1

dðdþ1Þ hypersurfaces of degreeeD not containing X .

This theorem generalizes some results of Heath-Brown [16], Theorem 14, and Bro- berg [6], Theorem 1, in the sense that we estimate explicitly the degree and the number of the auxiliary hypersurfaces needed to cover the set of rational points with bounded height.

The strategy of Heath-Brown in the proof of [16], Theorem 14, consists of establish- ing that a family of rational points having the same reduction modulo a ‘‘large’’ prime number are contained in one hypersurface (not containing X) with ‘‘low’’ degree. This idea is inspired by results of Bombieri–Pila [1] and Pila [22], and has been developed later in [6], [7], [8], [9], [10], [14], [17], [18], [23], [24].

Suggested by Bost, we adapt the above idea into the framework of his slope method [2], [3], [4]. Note that Bogomolov has asked a similar question on the possibility of replac- ing the method of Heath-Brown by arguments in Arakelov geometry (see [13], Question 34). We consider the evaluation map from the space of homogeneous polynomials to the space of values of these polynomials on a family of rational points. If the rational points

in the family have the same reduction modulo some finite placepofK such that the norm ofpis big, then the (logarithmic) height of this evaluation map is very negative. Hence by the slope inequality, the evaluation map cannot be injective and thus we obtain a non-zero homogeneous polynomial whose image by the evaluation map vanishes. The desired hyper- surface is obtained as the zero locus of the homogeneous polynomial.

The flexibility of the geometric framework (see Theorem 3.1) permits us to develop several interesting variants. For example, instead of considering the reduction modulo a fi- nite placep, we treat the case where the family of rational points has the same reduction modulo some power ofp. In other words, we can take a finite placepwith relatively lower norm and consider a family of rational points whosep-adic distances are very small. Such a family is contained in a hypersurface of lower degree. This argument permits us to prove that the constantC figuring in Theorem A depends on the degree ofK overQbut not on the discriminant. Another variant consists of taking into account the local Hilbert–Samuel functions of the variety, which generalizes a result of Salberger [23], Theorem 3.2. This per- mits us to sharpen the constantC in Theorem A in the case whereX is a plane curve andB is small. As a consequence, we obtain the following result.

Theorem B. Assume that X is an integral plane curve of degreed. Then,for anye>0, one has

KSðX;dÞf_K d^2þe:

This gives an answer to a question of Heath-Brown [13], Question 27.

To obtain an explicit upper bound for the number and the degree of the auxiliary hy- persurfaces, we need several e¤ective estimates in algebraic geometry and in Arakelov ge- ometry, which shall be recalled in the second section. In the third section, we explain the conditions which ensure that a family of rational points lies in the same hypersurface of low degree. Finally, in the fourth section, we estimate the number of hypersurface needed to cover rational points; in the fifth section, we discuss the plane curve case.

We keep Notation 1–8 introduced in [12], §2. Remind thatK denotes a number field andOK denotes its integer ring. We shall also use the following notation.

Notation. 9. Denote by nANnf0gan integer and by E thetrivialHermitian vector bundle of ranknþ1. In other words,E ¼O^lK^ðnþ1Þ, and for any embeddings:K !C, the canonical basis ofE is an orthonormal basis ofk k_s. See Notation 4 for the notion of Her- mitian vector bundles.

10. Denote by L the universal quotient sheaf on P_Oⁿ_K ¼PðEÞ, equipped with the Fubini–Study metrics.

11. Any point P¼ ðx0: . . . :xnÞAPⁿðKÞ gives rise to a uniqueOK-pointP APðEÞ.

The height of P (with respect to L) is by definition the slope (see Notation 6) ofPðLÞ, denoted byhðPÞ. Note that one has

hðPÞ ¼ 1

½K :Q

P

pASpmOK

log max

1eienjxij_pþ1 2

P

s:K!C

logPⁿ

j¼0

jxjj_s²

:

See Notation 2 for the definition of the absolute values j j_p and j j_s. Define HðPÞ:¼exp

½K :QhðPÞ

. Remind that here the logarithmic height functionhis absolute (i.e., invariant under finite field extensions ofK), while the exponential oneH is relative.

12. For any integerDf1, letEDbe theOK-moduleH⁰

PðEÞ;L^nD

, equipped with the John metricsk k_s;J associated to the sup-normk k_s;sup. We remind that the sup-norm is defined as follows:

EsAEDn

OK;sC; ksk_s;sup:¼ sup

xAP_sⁿðCÞ

ksðxÞk_s: The John normk k_s;J is a Hermitian norm onEDn

OK;sCsuch that ksk_s;supeksk_s;Je ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rkðEDÞ

p ksk_s;sup:

Denote byrðDÞthe rank ofED. One hasrðDÞ ¼ nþD D

.

13. Let X be an integral closed subscheme of P_Kⁿ ¼PðEKÞ. Letd be the dimension ofX and d be the degree ofX. Recall that one has d¼deg

c1ðLKÞ^d ½X

. Denote by X the Zariski closure ofX inPðEÞ. The (relative) Arakelov height ofX is denoted byh_LðXÞ.

Recall that

h_LðXÞ:¼ 1

½K :Qdegdegd

cc^1ðLÞ^dþ1 ½X :

14. For any integerDf1, letFDbe the saturation (inH⁰ðX;Ljⁿ_X^DÞ) of the image of the restriction map

h_X;D:ED;K ¼H⁰

PðEK;Lⁿ_K^DÞ

!H⁰ðX;Ljⁿ_X^DÞ;

namely FD is the largest sub-OK-module of H⁰ðX;Ljⁿ_X^DÞ containing Imðh_X;DÞ and such that FD;K ¼Imðh_X;DÞ_K. We equip FD with the quotient metrics (from the metrics of ED) so that FD becomes a Hermitian vector bundle on SpecOK. Denote by r1ðDÞ the rank ofFD.

15. Let pbe a maximal ideal ofOK with residue field F_p. For any point in XðFpÞ, denote byOx the local ring ofX atxand by mxthe maximal ideal ofOx. Note thatOx is a local algebra overOK;p. Denote byHx:N!Nthe Hilbert–Samuel function ofOx=pOx

(which is the local ring ofXF_p atx), namely, HxðkÞ ¼rkF_p

ðmx=pOxÞ^k=ðmx=pOxÞ^kþ1 : Let

qxðmÞ

mf1 be the increasing sequence of non-negative integers such that the integer kAN appears exactly HxðkÞ times. Let QxðmÞ ¼qxð1Þ þ þqxðmÞ. Denote by m_x the multiplicity of the local ringOx=pOx. Recall that one has

HxðkÞ ¼ m_x

ðd1Þ!k^d1þoðk^d1Þ:

16. For any real number B>0, let SðX;BÞ be the subset of XðKÞ consisting of points P such that HðPÞeB (see Notation 11 for the definition of HðÞ). Denote by S1ðX;BÞ the subset of SðX;BÞ of regular points. Define NðX;BÞ and N1ðX;BÞto be the cardinality ofSðX;BÞandS1ðX;BÞrespectively.

17. For any maximal ideal pofOK, xAXðFpÞ and B>0, denote by SðX;B;xÞthe set of pointsPASðX;BÞwhose reduction modulo pisx. Define

S1ðX;B;pÞ ¼ S

xAXðF_pÞ xregular

SðX;B;xÞ;

wherexregular means thatxis a regular point ofXF_p, or equivalently,Ox=pOxis a regular local ring.

18. More generally, for any maximal ideal pand anyaANnf0g, denote byA_p^ðaÞ the Artinian local ringOK;p=p^aOK;p. For any pointhAXðA_p^ðaÞÞ, denote bySðX;B;hÞthe set of points inSðX;BÞwhose reduction modulop^acoincides withh. We shall use the fact that

EaANnf0g; ExAXðFpÞ; SðX;B;xÞ ¼ S

hAXðA_p^ðaÞÞ x¼ðhmodpÞ

SðX;B;hÞ:

19. We introduce several constants as follows:

C1¼ ðdþ2Þmm^_max

S^dðE⁴Þ þ1

2ðdþ2Þlog rkðS^dEÞ þd

2 log

ðdþ2ÞðndÞ þd

2ðdþ1Þlogðnþ1Þ;

C2¼r

2log rkðS^dEÞ þ1

2log rkðL^ndEÞ þlog ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðndÞ!

p þ ðndÞlogd;

C3¼ ðndÞC1þC2:

Recall that the constantC1has been defined in [12], (21).1)With the notation of [12], The- orem 3.8, the constantC2is justC2ðE;nd;dÞ(see also Remark 3.9 loc. cit.). Finally, the constantC3appears in [12], Theorem 3.10. Recall that one hasC3f_n;dd(see Theorem 3.10 loc. cit.).

20. By the e¤ective version of Chebotarev’s theorem (cf. [20], see also [25], Theorem 2) there exists an explicitly computable constantaðKÞsuch that, for any real numberxf1, there exists a finite place pASf such that N_pA

x;aðKÞx

. This is an analogue of Ber- trand’s postulate for number fields.

2. Reminders

We recall in this section several results that we shall use in the sequel. They are either well known or described in [12].

1) SinceS^dEis a direct sum of Hermitian line bundles, the quantity%^ðdþ2Þ G^dðEÞ

vanishes (see [12], §2.2).

Furthermore, whenEis trivial, one hasmm^_maxðL^ndEÞ ¼0.

2.1. LetðPiÞ_iAI be a collection of distinct rational points ofX (see Notation 13) and Df1 be an integer. Assume that the evaluation map f :FD;K !L

iAI

P_iL^nDis an isomor- phism (see Notation 14). Then the equality

m^

mðFDÞ ¼ 1 r1ðDÞ

P

iAI

DhðPiÞ þhðL^r1ðDÞfÞ

holds. In particular, one has m^ mðFDÞ

D esup

iAI

hðPiÞ þ 1

Dr1ðDÞhðL^r1ðDÞfÞ;

ð1Þ

wherehðL^r1ðDÞfÞis defined as hðL^r1ðDÞfÞ ¼ 1

½K :Q P

p

logkL^r1ðDÞfk_pþ P

s:K!Q

logkL^r1ðDÞfk_s

:

A slight variant of this argument shows that, ifðPiÞ_iAJ is a family of rational points ofX such that

sup

iAJ

hðPiÞ<mm^_maxðFDÞ

D 1

2logðnþ1Þ;

ð2Þ

then there exists a hypersurface of degreeDinP_Kⁿ not containingX which contains all ra- tional pointsPi. See [12], Proposition 2.12, for details.

2.2. For any integerDf1, one has the following estimates:

Dþdþ1 dþ1

Ddþdþ1 dþ1

er1ðDÞ:¼rkðFDÞed Dþd d

ð3Þ :

See [11] for the upper bound and [26] for the lower bound.

2.3. For any integerDf2ðndÞðd1Þ þdþ2, one has m^

mðFDÞ

D f d!

dð2dþ2Þ^dþ1h_LðXÞ logðnþ1Þ 2^d; ð4Þ

whereh_LðXÞis the Arakelov height ofX. See [12], Theorem 4.8 and Remark 4.9, for details.

2.4. SinceFDis a quotient ofED, one has (see Notation 12) m^

mðFDÞfmm^_minðEDÞf1

2Dlogðnþ1Þ:

ð5Þ

We refer to [12], Corollary 2.9, for the proof. Note that this bound is much less precise than (4). However, it works for any integerDf1.

2.5. For any integerDf1, one has m^

mðFDÞ D e 1

dh_LðXÞ þ1

2logðnþ1Þ:

ð6Þ

See [12], Remark 4.11.

2.6. There exists a Hermitian vector subbundleM ofS^ðd1ÞðndÞE such that (i) mm^_minðMÞfðndÞh_LðXÞ C3,

(ii) the subscheme of PðEÞ defined by vanishing of M contains the singular loci of fibres ofXbut not the generic point ofX,

where the constantC3is defined in Notation 19. This result has been proved in [12], Theo- rem 3.10. In particular, the singular locus of X is contained in a hypersurface of degree ðd1ÞðndÞnot containingX.

2.7. Suppose thatPAXðKÞis a regular point andP is theOK-point ofPðEÞextend- ingP. For any maximal idealpofOK, if the reduction ofP modulopis a singular point of XFp, we writea_pðPÞ ¼1, else we writea_pðPÞ ¼0. We have shown in [12], Proposition 2.11, that, for any real numberN0>0, the following inequality holds:

P

N_pfN0

a_pðPÞeðndÞðd1ÞhðPÞ þ ðndÞh_LðXÞ þC3

ðlogN0Þ=½K :Q : ð7Þ

In fact, it su‰ces to apply [12], Proposition 2.11, to the special caseI ¼M, whereM is as in §2.6.

2.8. The following estimates of binomial coe‰cients will be used:

ðNkþ1Þ^k

k! e N

k e

N ðk1Þ=2k

k! ; Nfkf1:

The second inequality comes from the comparison of the arithmetic and the geometric means:

NðN1Þ ðNkþ1Þe Nþ ðN1Þ þ þ ðNkþ1Þ k

k

:

3. Existence of the auxiliary hypersurface The purpose of this section is to establish the following theorem.

Theorem 3.1. Let S¼ ðp_jÞ_jAJ be a finite family of maximal ideals of OK and ðajÞ_jAJ AðNnf0gÞ^J. For eachp_j,let h_j be a point inXðAp^ðaÞj Þ(see Notation18)whose reduc- tion modulopis denoted byxj. Assume thatðxjÞ_jAJ are distinct. Consider a familyðPiÞ_iAI of

rational points ofXK such that, for any iAI and any jAJ, the reduction of Pi modulop_j^aj coincides withh_j. Assume that(see Notation11, 14and15)

sup

iAI

hðPiÞ<mmðF^ DÞ

D logr1ðDÞ

2D þ 1

½K :Q P

jAJ

Qx_j r1ðDÞ

Dr1ðDÞ logN_p^a_j^j: ð8Þ

Then there exists a section sAED;K which does not vanish identically on XK and such that PiAdivðsÞfor any iAI .

This theorem generalizes a result of Salberger [23], Theorem 3.2, in two aspects. On one hand, we treat projective varieties over a number field; on the other hand, we consider a family of thickenings of points over finite places.

The proof of Theorem 3.1 consists of adapting the idea of Bombieri–Pila and Heath- Brown in the framework of the slope method. Note that Broberg has generalized [16], The- orem 14, to the number field case, which corresponds to the case wherejJj ¼1 andaj¼1 here. However, his method is di¤erent from ours. In fact, the slope method permits us to avoid using Siegel’s lemma. Moreover, in (8), there appears only the degree of the number fieldK but not the discriminant.

The following subsections are devoted to the proof of Theorem 3.1 and to discuss several applications. We first estimate the heights of the determinants of some evaluation maps. This stage is quite similar to the determinant argument of Bombieri–Pila and Heath-Brown. Then we use the slope inequality to obtain the desired result. To apply the theorem, we need explicit estimates of the functionsQx_j andr1ðDÞ, which we discuss in the end of this section.

3.1. Estimation of norms.

Lemma 3.2. Let A be a ring and M be an A-module.

(i) If N is a sub-A-module of M such that M=N is generated by q elements, then for any integer mfq,we haveL^mM ¼ ðL^mqNÞ5ðL^qMÞ.

(ii) If M ¼M1IM2I IMiIM_iþ1I is a decreasing sequence of sub-A- modules of M such that, for any if1, Mi=M_iþ1 is isomorphic to a principal ideal of A, then for any integer rf1,we have

L^rM¼M15M25 5Mr:

Proof. (ii) is a consequence of (i). To prove (i), by induction it su‰ces to establish the case wherem¼rþ1. SinceM=N is generated byqelements, we haveL^rþ1ðM=NÞ ¼0 (see [5], Chapter III, §7, n3, Proposition 3). Furthermore, since the kernel of the canonical homomorphism of exterior algebrasLM!LðM=NÞis the ideal generated byN (loc. cit.), we obtain thatL^rþ1MHN5ðL^rMÞ. r

Lemma 3.3. Let k be a field equipped with a non-archimedean absolute value j j,U and V be two k-linear ultranormed spaces of finite rank and j:U !V be a k-linear

homomorphism. Let m be the rank of U. For any integer1eiem,let li ¼ inf

WHU codimW¼i1

kjj_Wk:

If i>m,letli ¼0. Then for any integer r>0,we have kL^rjkeQ^r

i¼1

li: ð9Þ

Proof. Lete>0 be an arbitrary positive real number. We shall construct a decreas- ing filtration ofU,

U ¼U1lU2l lUm; ð10Þ

such that kjj_Uikeliþe. By definition, there exists a vector xmAU of norm 1 such that kjðxmÞkelmþe. Suppose that we have chosenU_iþ1I IUmsuch thatkjj_Ujkeljþe for anyiþ1ejem. SinceUiþ1has codimensioniinU, the set of vectorsxAU of norm 1 withkjðxÞkeliþecan not be contained inUiþ1. Pick an elementxi AUnUiþ1of norm 1 withkjðxiÞkeliþe. Let Ui be the linear subspace generated byxi andUiþ1. Since the norm of U is ultrametric, one obtains kjj_Uikeliþe. By induction we can construct the filtration as announced. By Lemma 3.2, one obtains

kL^rjkeQ^r

i¼1

ðliþeÞ:

Sincee>0 is arbitrary, the proposition is proved. r

3.2. A preliminary result on local homomorphisms. Letpbe a maximal ideal ofOK

and xbe anF_p-point of X. Suppose given a familyðfiÞ_1eiem of local homomorphisms of OK;p-algebras fromOx(see Notation 15) toOK;p. LetEbe a free sub-OK;p-module of finite type ofOxand let f be theOK;p-linear homomorphismðfij_EÞ_1eiem:E!O_K;p^m . As f1 is a homomorphism of OK;p-algebras, it is surjective. Let a be the kernel of f1. One has Ox=aGOK;p. Furthermore, sinceOxis a local ring of maximal ideal mx, one has mxIa.

Moreover, since f1 is a local homomorphism, the equality aþpOx¼mx holds. For any integer jf0,a^j=a^jþ1is anOx=aGOK;p-module of finite type, and

Fpn

OK;pða^j=a^jþ1ÞGða=pOxÞ^j=ða=pOxÞ^jþ1Gðmx=pOxÞ^j=ðmx=pOxÞ^jþ1:

By Nakayama’s lemma, the rank of a^j=a^jþ1 over OK;p is equal to the rank of ðmx=pOxÞ^j=ðmx=pOxÞ^jþ1overF_p, that is,HxðjÞaccording to Notation 15. The filtration

Ox¼a⁰Ia¹I Ia^jIa^jþ1I ofOxinduces a filtration

F :E¼EXa⁰IEXa¹I IEXa^jIEXa^jþ1I ð11Þ

ofEwhose j-th subquotientEXa^j=EXa^jþ1is a freeOK;p-module of rankeHxðjÞ.

Assume that aANnf0gis such that the reductions of fi modulo p^a are the same (in other words, the composed homomorphisms Ox!OK;p !OK;p=p^aOK;p

fi

are the same), then the restriction of f on EXa^j has normeN_p^ja. In fact, for any 1eiem, one has

fiðaÞHp^aOK;p and hence fiða^jÞHp^ajOK;p.

By Lemma 3.3, we obtain the following result.

Proposition 3.4. Let p be a maximal ideal of OK and xAXðF_pÞ. Suppose that ðfiÞ_1eiemis a family of localOK;p-linear homomorphisms fromOxtoOK;pwhose reductions modulo p^a are the same, where aANnf0g. Let E be a free sub-OK;p-module of finite type ofOxand f ¼ ðfij_EÞ_1eiem. Then, for any integer rf1,one has

kL^rfKkeN_p^QxðrÞa; ð12Þ

where N_p is the degree ofF_p over its characteristic field. See Notation 15 for the definition of Qx.

Proof. Consider the filtration (11) above. The restriction of f on EXa^j has normeN_p^ja, which implies that (see Notation 15 for the definition ofqx)

WHEinfK

codimW¼j1

kfKj_WkeN_p^qxðjÞa;

where we have used the fact that rkðEXa^jÞ rkðEXa^jþ1ÞeHxðjÞ. The inequality (12) then follows from Lemma 3.3. r

3.3. Proof of Theorem 3.1. LetDf1 be an integer. LetFDandr1ðDÞ ¼rkFDbe as in Notation 14. Assume that the section predicted by the theorem does not exist. Then the evaluation map f :FD;K !L

iAI

P_iLK is injective. By possibly replacingI by a subset, we may suppose that f is an isomorphism. For any embeddings:K !C, one has

1

r1ðDÞ logkL^r1ðDÞfk_selogkfk_selog ffiffiffiffiffiffiffiffiffiffiffiffi r1ðDÞ

p ;

where the second inequality comes from the definition of metrics of John (see Notation 12).

Furthermore, f is induced by a homomorphism ofOK-modulesFD!L

iAI

PiL^nD, where Pi denotes the OK-point of X extending Pi. Hence for any finite place p of K, one has logkL^r1ðDÞfk_pe0.

Let jAJ. For eachi AI, theOK-pointPi defines a local homomorphism fromOx_j to OK;pj which isOK;pj-linear. By taking a local trivialization of Latxj, we identifyFDwith a

sub-OK;pj-module ofOxj. Proposition 3.4 then implies that logkL^r1ðDÞfk_pjeQx_j

r1ðDÞ

logNp^aj^j: We then obtain (see §2.1)

m^ mðFDÞ

D e sup

iAI

hðPiÞ þ 1

2D logr1ðDÞ 1

½K :Q P

jAJ

Qxj

r1ðDÞ

Dr1ðDÞ logNp^aj^j;

which leads to a contradiction. Thus the evaluation homomorphismFD;K !L

iAI

P_iL^nD is not injective. In other words, there exists a homogeneous polynomial of degree D which is not identically zero onX but vanishes on eachPi.

3.4. Applications. Letpbe a maximal ideal ofOK and xbe a rational point of XF_p. Recall (see Notation 15) thatOx denotes the local ring of Xat x,mx denotes its maximal ideal, and the local Hilbert–Samuel function ofxis defined as

HxðkÞ:¼rkF_p

ðmx=pOxÞ^k=ðmx=pOxÞ^kþ1 :

In some particular cases, the local Hilbert–Samuel function of x can be explicitly estimated.

(i) If x is regular (i.e., the local ring Ox=pOx is regular), then one has HxðkÞ ¼ kþd1

d1

for anykf0.

(ii) Assume that the local ring Ox=pOx is one-dimensional and Cohen–Macaulay (that is, mx=pOx contains a non zero-divisor ofOx=pOx), then by [21], Theorem 1.9, one has HxðkÞem_x for any integer kf0, where m_x denotes the multiplicity of the local ring Ox=pOx. Moreover, ifkfm_x1, then one hasHxðkÞ ¼m_x (see [19], Theorem 2).

Proposition 3.5. Letpbe a maximal ideal ofOK,xbe a rational point ofXF_p,and r be an integer,rf1.

(i) If theF_p-pointxis regular,then(see Notation15for the definition of Qx)

QxðrÞ>ðd!Þ^1d d

dþ1r^1þ1d dþ3 2dþ2dr:

ð13Þ

(ii) If d ¼1andxis Cohen–Macaulay,then

QxðrÞf r² 2m_x r

2m_x: ð14Þ

Proof. LetUxbe the partial sum function ofHx. Namely, UxðkÞ:¼Hxð0Þ þ þHxðkÞ:

One has

Qx

UxðkÞ

¼P^k

j¼0

jHxðjÞ:

Moreover, ifrA

Uxðk1Þ;UxðkÞ

, then one hasQx

Uxðk1Þ

eQxðrÞeQx

UxðkÞ .

(i) In the case where xis regular, one has UxðkÞ ¼P^k

j¼0

jþd1 d1

¼ kþd d

ð15Þ : Therefore Qx

UxðkÞ

¼P^k

j¼0

jHxðjÞ ¼P^k

j¼0

j jþd1 d1

¼P^k

j¼0

d jþd1 d

¼d kþd dþ1

:

Letrbe an integer in

Uxðk1Þ;UxðkÞ

. One has QxðrÞ ¼Qx

Uxðk1Þ þk

rUxðk1Þ ð16Þ

¼krþd kþd1 dþ1

k kþd1 d

¼kr kþd dþ1

¼krkþd

dþ1Uxðk1Þ> d

dþ1ðk1Þr;

where in the last inequality, we have used the estimateUxðk1Þ<r. Note that (see §2.8) reUxðkÞ ¼ kþd

d

e

kþ ðdþ1Þ=2d

d!

implies

kfðrd!Þ^d1 ðdþ1Þ=2:

ð17Þ

Combining with (16), we obtain that (13) holds.

(ii) Assume that d ¼1 and Ox=pOx contains a non zero-divisor, then one has 1eHxðkÞem_x for any integer kf1. Let ðakÞ_kf1 be the increasing sequence of non- negative integers such that the integer 0 appears exactly one time, and other integers appear exactlym_xtimes. Note that one hasqxðkÞfak for anykANnf0g. Hence

QxðrÞ ¼ P^r

k¼1

qxðkÞfP^r

k¼1

ak ¼m_x

2 AðAþ1Þ þ ðAþ1Þðr1m_xAÞ

¼ ðAþ1Þðr1Þ m_x

2 AðAþ1Þ ¼ ðAþ1Þðr1m_xA=2Þ;

whereA¼ r1 m_x

. Using the fact that r1

m_x m_x1

m_x eAer1 m_x ;

we obtain

QxðrÞf r

m_x r1r1 2

f r² 2m_x r

2m_x: r

Remark 3.6. When d¼1, the estimate (13) is less precise than (14). The reason is that in the last inequality of (16), we have used the estimate Uxðk1Þ<r but not the more precise oneUxðk1Þer1.

Corollary 3.7. Let ðp_jÞ_jAJ be a finite family of maximal ideals ofOK and e>0. For any jAJ, let ajANnf0g, xj AXðFpjÞ be a regular rational point ofXF_pj and h_j AXðAp^ðaj^jÞÞ whose reduction modulop_j is xj. If

P

jAJ

logN_p^a_j^jfð1þeÞ

logBþ ½K :Qlogðnþ1Þ

d^d1dþ1 d ; ð18Þ

then,for any integer D such that

D>ðe¹þ1Þ

d^d1ðdþ3Þ=2þd2 ð19Þ ;

there exists a hypersurface of degree D ofT P_Kⁿ not containing X which contains

jAJ

SðX;B;h_jÞ.

Proof. Assume that such hypersurface does not exist. By Theorem 3.1, one has logB

½K :Q fmmðF^ DÞ

D logr1ðDÞ

2D þP

jAJ

Qx_j

r1ðDÞ Dr1ðDÞ

logN_p^a_j^j

½K :Q: ð20Þ

Moreover, sincex_j is regular, Proposition 3.5 shows that Qxj

r1ðDÞ

fðd!Þ^1d d

dþ1r1ðDÞ^1þ1d dþ3

2dþ2dr1ðDÞ:

Hence

Qx_j r1ðDÞ

Dr1ðDÞ fðd!Þ^1d d dþ1

r1ðDÞ^1d

D ðdþ3Þd ð2dþ2ÞD: By a result of Sombra recalled in §2.2, one has (forDfd2)

r1ðDÞf Dþdþ1 dþ1

Ddþdþ1 dþ1

¼P^d

j¼1

Ddþdþ j d

fdðDdþ2Þ^d

d! :

Combining with (5) and the trivial estimater1ðDÞeðnþ1Þ^D, (20) implies logB

½K :Q f1

2logðnþ1Þ 1

2logðnþ1Þ þ d^1d d dþ1

Ddþ2

D ðdþ3Þd ð2dþ2ÞD

P

jAJ

logN_p^a_j^j

½K :Q: Or equivalently

d^d1 d

dþ1 P

jAJ

logN_p^a_j^j

½K :Q logB

½K :Qlogðnþ1Þ

DeP

jAJ

logN_p^a_j^j

½K :Q d^d1 d

dþ1ðd2Þ þ dþ3 2dþ2d

:

By the hypothesis (18), the left side is not less than e

1þed^1d d dþ1

P

jAJ

logN_p^a_j^j

½K :QD;

which implies that

Deðe¹þ1Þ

d^1dðdþ3Þ=2þd2 :

This contradicts (19). r

Corollary 3.8. Assume thatX is Cohen–Macaulay and d¼1. Letðp_jÞ_jAJ be a finite family of maximal ideals ofOK and e>0. For any j AJ, let ajANnf0g, x_j AXðFpjÞ and h_j AXðAp^ðaj^jÞÞwhose reduction modulop_j isx_j. If

P

jAJ

logN_p^a_j^j

m_xj fð1þeÞ2 d

logBþ ½K :Qlogðnþ1Þ ð21Þ ;

then for any integer D such that

D>ð1þe¹Þðd2þd¹Þ;

ð22Þ

there exists a hypersurface of degree D ofPⁿnot containing X which contains T

jAJ

SðX;B;h_jÞ.

Proof. The proof is quite similar to that of Corollary 3.7. By Proposition 3.5, one has the estimate

Qx_j

r1ðDÞ

Dr1ðDÞ f r1ðDÞ 2m_xjD 1

2m_xjD:

Assume that the hypersurface does not exist. By Theorem 3.1, one has logB

½K :Qþlogðnþ1ÞfP

jAJ

logN_p^a_j^j

½K :Q d

2m_xjDdþ2

D 1

2m_xjD

!

;

or equivalently D

P

jAJ

logN_p^a_j^j

½K :Q d

2m_xj logB

½K :Qlogðnþ1Þ

eP

jAJ

logN_p^a_j^j

½K :Qmx_j

dðd2Þ

2 þ1

2

:

By the assumption (21), one obtains D e

1þe P

jAJ

logN_p^a_j^j

½K :Q d

2m_xj eP

jAJ

logN_p^a_j^j

½K :Qm_xjd²2dþ1

2 ;

Deð1þe¹Þðd2þd¹Þ:

The last formula leads to a contradiction. r

4. Covering rational points by hypersurfaces

In this section, we explain how to suitably cover S1ðX;BÞand SðX;BÞby hypersur- faces of low degree. If p is a maximal ideal ofOK and x is a singular rational point of XðFpÞ, there seems to be no general explicit estimate of the local Hilbert–Samuel func- tionQx.2) The idea of Heath-Brown is to consider only regular points. The di‰culty then comes from the fact that the reduction modulopof a regular pointPin XðKÞis not nec- essarily regular. Hence we need to estimate the ‘‘smallest’’ maximal idealpsuch thatPspe- cializes to a regular point modulop. This has been obtained in [16] and in [6] by using the Jacobian criterion. Here we prove that the singular loci of fibres of X are actually con- tained in a divisor whose degree and height are controlled.

Lemma 4.1. Let N0>0be a real number and r the integral part of the number ðndÞðd1ÞlogBþ

ðndÞh_LðXÞ þC3

½K :Q logN0

þ1;

ð23Þ

where the constant C3 is defined in Notation19.Ifp₁;. . .;p_r are r distinct finite places of K such that N_pifN0for any i,then

S1ðX;BÞ ¼ S^r

i¼1

S1ðX;B;p_iÞ:

Proof. With the notation of §2.7, ifPis a rational point inS1ðX;BÞwhich does not lie in anyS1ðX;B;p_iÞ, then one hasa_piðPÞf1 for anyi¼1;. . .;r. Hence, by §2.7, one has

re P

N_pfN0

a_pðPÞe ðndÞðd1ÞhðPÞ þ ðndÞh_LðXÞ þC3

ðlogN0Þ=½K :Q ; which leads to a contradiction. r

Theorem 4.2. Lete>0be an arbitrary positive real number. Let D be an integer such that

D>max

ðe¹þ1Þ

d^1dðdþ3Þ=2þd2

;2ðndÞðd1Þ þdþ4

:

There exists an explicitly computable constant Cðe;d;n;d;KÞ such that, for any Bfe^e, the set S1ðX;BÞ is covered by not more than Cðe;d;n;d;KÞB^ð1þeÞd

1

dðdþ1Þ hypersurfaces of de- gree D not containing X .

Proof. In the first stage, we assume that h_LðXÞeð2dþ2Þ^dþ1

d! d logB

½K :Qþ3

2logðnþ1Þ þ2^d

:

2) In the case whereXis Cohen–Macaulay, there are explicit estimates (see for example [27]). However, they are far from optimal.

Let MANnf0g be the least common multiple3) of 1;2;. . .;½K :Q. Let N0Að0;þyÞ be such that

logN0¼ ð1þeÞd^d1dþ1 dM

logBþ ½K :Qlogðnþ1Þ :

Letrbe the natural number as in Lemma 4.1. Note that one has

reA1logBþA2

logN0

þ1;

where

A1¼ ðndÞðd1Þ þð2dþ2Þ^dþ1

d! ðndÞd;

A2¼ ½K :Q

C3þð2dþ2Þ^dþ1 d! d 3

2 logðnþ1Þ þ2^d

:

Recall that the constant C3 is defined in Notation 19. Since we have assumed that logBfe, the value of r is bounded from above by a constant A3 which depends only onM,e,n,d andd:

A3:¼M A1þe¹A2

ð1þeÞd^1dðdþ1Þ=dþ1:

By Bertrand’s postulate, there exist r distinct prime numbers p1;. . .;pr such that N0epie2ⁱN0for anyiAf1;. . .;rg. We choose, for eachi, a maximal idealp_iofOK lying over pi. By Lemma 4.1, one has S1ðX;BÞ ¼ S^r

i¼1

S1ðX;B;p_iÞ. Note that, for anyi, N_pi is a power of piwhose exponent fi dividesM (since fie½K :Q). Letai ¼M=fi.

Letx be an arbitrary regularF_pi-point ofXF_pi. By Corollary 3.7, we obtain that, for anyhAXðAp^ðai^iÞÞwhose reduction modulop_i is x, SðX;B;hÞis contained in a hypersurface of degreeDnot containingX. Note that there exists at mostN_p^ða_i^i1Þdpoints inXðA_p^ða_i^iÞÞ(see Notation 15) whose reduction modulop_i equalsxi and the cardinal ofXðF_piÞdoes not ex- ceedddN_p^d

i. HenceS1ðX;B;p_iÞis covered by at most

ddN_p^a_i^id ¼ddp_i^aifid¼ddp_i^Mde2^iMdddN₀^Md ð24Þ

hypersurfaces of degreeDnot containingX. Therefore, S1ðX;BÞis covered by at most

ddN₀^MdP^r

i¼1

2^iMdeddr2^rMd

ðnþ1Þ^½K:QBð1þeÞd

1 dðdþ1Þ

3) One has 2^½K:QeMe½K:Q^pð½K:QÞ. See [28], p. 30, for a proof.

such hypersurfaces. So the theorem is proved with the constant Cðe;d;n;d;KÞ ¼ddA32^A3Mdðnþ1Þ^ð1þeÞd

1

dðdþ1Þ½K:Q

ð25Þ :

Now we treat the case where logB

½K :Q< d!

dð2dþ2Þ^dþ1h_LðXÞ 3

2 logðnþ1Þ 2^d:

By §2.1, inequality (2) and §2.3, we obtain that the setSðX;BÞis contained in a hypersur- face of degreeDinPⁿwhich does not containX. The theorem is also true in this case. r Corollary 4.3. With the notation of Theorem4.2,assume that d ¼1. For any positive real number Bfe^e,one has

KSðX;BÞe

Cðe;d;n;d;KÞ þ1

dDB^ð1þeÞ2=d: ð26Þ

Proof. By Be´zout’s theorem, the intersection of each hypersurface in the conclusion of Theorem 4.2 and X contains at most dD rational points. Hence the corollary follows from Theorem 4.2 (see also §2.6). r

Remark 4.4. (i) Observe that one has A1f_n;d d and A2f_n;dd and hence A3f_n;d;ed^1þ1d. Therefore, one has

logCðe;d;n;d;KÞf_n;d;K;ed^1þd1:

Moreover, the constantCðe;d;n;d;KÞ does not depend on the discriminant ofK (but on the degree ofK overQ).

(ii) The original strategy of Heath-Brown corresponds essentially to the case where ai¼1 for any i. By taking a larger N0, his strategy also allows to obtain an explicit upper bound with the same exponent. However, the choice of maximal ideals forces us to use Bertrand’s postulate for the number fieldK where the discriminant ofK is inevitable, according to a counter-example of Heath-Brown that Browning has communicated to me.

5. The case of a plane curve

In this section, we assume thatX is an integral plane curve (that is,d ¼1 andn¼2).

Note that the model X of X is Cohen–Macaulay since it is a subscheme of P_O²_K defined by one homogeneous equation. We obtain, for ‘‘small’’ value ofB, an explicit estimate of KSðX;BÞ.

Theorem 5.1. Assume that d ¼1and n¼2. Let D¼ b2ðd2þd¹Þc þ1. Then,for any real number BAðe;e^d2Þ,one has

KSðX;BÞeC4ðK;BÞdD;

ð27Þ

where

C4ðK;BÞ ¼ ð ffiffiffiffiffiffiffiffiffiffiffi logB

p þ1ÞaðKÞ² ffiffiffiffiffiffiffiffi

logB

p þ2exp 8logBþ ½K :Qlog 2 ffiffiffiffiffiffiffiffiffiffiffi logB p

" #

þ ðlogBÞ ffiffiffiffiffiffiffiffi

logB

p þ1 d1

d ffiffiffiffiffiffiffiffiffiffiffi logB p

! ffiffiffiffiffiffiffiffiplogB þ1

;

aðKÞbeing the constant introduced in Notation20.

Proof. LetN0Að0;þyÞbe such that

logN0¼4logBþ ½K :Qlog 2 ffiffiffiffiffiffiffiffiffiffiffi logB

p :

Let r¼ d ffiffiffiffiffiffiffiffiffiffiffi logB

p e. Choose a family ðpiÞ_i¼1^r of distinct maximal ideals ofOK such that N0eN_pieaðKÞⁱN0, where aðKÞ is the constant of Bertrand’s postulate introduced in Notation 20. For anyðxiÞ_i¼1^r A Q^r

i¼1

XðF_piÞ, let

S

X;B;ðxiÞ_i¼1^r :¼ T^r

i¼1

SðX;B;x_iÞ:

Note that one has

SðX;BÞ ¼ S^r

i¼1

S

xAXðFpiÞ m_xed= ffiffiffiffiffiffiffiffi

plogB

SðX;B;xÞ

W S

ðxiÞ_i¼1^r AU

r i¼1

XðF_piÞ m_xi>d= ffiffiffiffiffiffiffiffi

logB

p S

X;B;ðxiÞ_i¼1^r ð28Þ :

LetpAfp₁;. . .;p_rg. Assume thatxis anF_p-point ofXF_p whose multiplicitym_xsatis- fiesm_xed= ffiffiffiffiffiffiffiffiffiffiffi

logB

p . One has

logN_p

m_x f logN0

d= ffiffiffiffiffiffiffiffiffiffiffi logB

p ¼4

dðlogBþ ½K :Qlog 2Þ:

By Corollary 3.8 (the case wheree¼1 andjJj ¼1), there exists a hypersurface of degreeD not containingX which containsSðX;B;xÞ. Note that the cardinal of the set

S^r

i¼1

fxAXðFpiÞ jm_xed= ffiffiffiffiffiffiffiffiffiffiffi logB

p g

does not exceed

P^r

i¼1

KP²ðFpiÞeP^r

i¼1

N_p²

ieraðKÞ^2rN₀²:

ð29Þ

LetiAf1;. . .;rg. By Be´zout’s theorem (see [15], 5–22, p. 115), one has P

xAXðFpiÞ

m_xðm_x1Þedðd1Þ:

ð30Þ

Hence

KfxAXðF_piÞ jm_x>d= ffiffiffiffiffiffiffiffiffiffiffi logB

p geðlogBÞ d1 d ffiffiffiffiffiffiffiffiffiffiffi

logB

p ;

which implies that the number ofr-tubesðxiÞ_i¼1^r A Q^r

i¼1

XðFpiÞwithm_xifd= ffiffiffiffiffiffiffiffiffiffiffi logB

p does not

exceed

ðlogBÞ^r d1 d ffiffiffiffiffiffiffiffiffiffiffi

logB p

!r

ð31Þ :

Note that the inequality (30) also implies that m_xed for any xAXðF_piÞ. Therefore, if ðx_iÞ_i¼1^r is an element in Q^r

i¼1

XðF_piÞ, then one has

P^r

i¼1

logNpi

m_xi frlogN0

d f4

dðlogBþ ½K :Qlog 2Þ;

where the second inequality comes from the estimate rf ffiffiffiffiffiffiffiffiffiffiffi logB

p . Still by Corollary 3.8, one obtains thatS

X;B;ðxiÞ_i¼1^r

is contained in a hypersurface of degree D not contain- ingX.

By (28), (29) and (31), the set SðX;BÞ is contained in a family of hypersurfaces of degree D not containing X, and the number of the hypersurfaces in the family does not exceed

raðKÞ^2rN₀²þ ðlogBÞ^r d1 d ffiffiffiffiffiffiffiffiffiffiffi

logB p

!r

eC4ðK;BÞ:

By Be´zout’s theorem the intersection of each hypersurface withX contains at most dDra- tional points. Therefore, we obtain

KSðX;BÞeC4ðK;BÞdD: r

Remark 5.2. The logarithmic of the first summand ofC4ðK;dÞis 8logdþ ½K :Qlog 2

ffiffiffiffiffiffiffiffiffiffi logd

p þ ð2 ffiffiffiffiffiffiffiffiffiffi logd

p þ2ÞlogaðKÞ þlogð ffiffiffiffiffiffiffiffiffiffi logd

p þ1Þf ffiffiffiffiffiffiffiffiffiffi logd

p ðd!yÞ;

while the logarithmic of the second summand is ð ffiffiffiffiffiffiffiffiffiffi

logd p þ1Þ

0

@log logdþlog d1 d ffiffiffiffiffiffiffiffiffiffi

logd p

!1

Af ffiffiffiffiffiffiffiffiffiffi logd

p log logd ðd!yÞ:

Hence there exists a constantMK which only depends onK such that C4ðK;dÞeM

ffiffiffiffiffiffiffi plogd

log logdþ ffiffiffiffiffiffiffi

logd

p

K f_K d^e

for anye>0.

Corollary 5.2. Assume that X is an integral plane curve of degree d. Then, for any e>0,one has

KSðX;dÞf_K d^2þe:

Acknowledgment. Inspired by a talk of P. Salberger at Go¨ttingen University, J.-B.

Bost suggested me to use the slope method to study the density of rational points and shared with me his personal notes. S. David and P. Philippon kindly explained to me their work on the lower bound of arithmetic Hilbert–Samuel function. During the preparation of this work, I benefited from discussions with D. Bertrand, J.-B. Bost, R. De la Brete`che, T. Browning, A. Chambert-Loir, M. Chardin, D. R. Heath-Brown, H. A. Helfgott, H.

Randriambololona, P. Salberger, C. Soule´ and B. Teissier and received many helpful sug- gestions. I would like to express my deep gratitude to them all. Part of this work has been written during my visit to theInstitut des Hautes E´ tudes Scientifiques. I would like to thank the institute for hospitalities. Finally I am grateful to the referees for their valuable re- marks.

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Institut de Mathe´matiques de Jussieu, Universite´ Paris Diderot—Paris 7, France e-mail: chenhuayi@math.jussieu.fr

Eingegangen 12. November 2009, in revidierter Fassung 17. Januar 2011