Ann. Scient. Éc. Norm. Sup., 4e série, t. 38, 2005, p. 93 à 115. COUNTING RATIONAL POINTS ON HYPERSURFACES OF LOW DIMENSION BY PER SALBERGER ABSTRACT.–LetN(X,B) be the number of rational points of height at most B on a variety X ⊂ Pn defined over Q. We establish new upper bounds for N(X,B) for hypersurfaces of dimension at most four. We also study N(X,B) for the open complements X of all lines or all planes on such hypersurfaces. One 7/2−δ 5 of the goals is to show that N(X,B)=Od(B ),δ>0 for smooth hypersurfaces in P defined by aformF (x0,...,x5) of degree d 9. This improves upon a classical upper estimate of Hua for the form d d d d d d x0 + x1 + x2 − x3 − x4 − x5. 2005 Elsevier SAS RÉSUMÉ.–SoitN(X,B) le nombre de points rationnels de hauteur inférieure ou égale à B sur une variété X ⊂ Pn définie sur Q. Nous établissons des nouvelles majorations pour N(X,B) pour les hyper- surfaces de dimension 4. Nous étudions aussi N(X,B) pour les compléments X de toutes les droites 7/2−δ ou tous les plans sur ces hypersurfaces. Un des buts est de démontrer que N(X,B)=Od(B ),δ>0 5 pour les hypersurfaces lisses dans P definies par une forme F (x0,...,x5) de degré d 9. C´est une d d d d d d amélioration d´une majoration classique de Hua pour la forme x0 + x1 + x2 − x3 − x4 − x5. 2005 Elsevier SAS 0. Introduction Let Z be a locally closed subset of Pn defined over Q. To count rational points on Z,we n shall use the height H(x)=max(|x0|,...,|xn|) for a rational point x on P represented by a primitive integral (n +1)-tuple (x0,...,xn). We denote by N(Z,B) the number of rational points of height at most B on Z. This is clearly an abuse of notation since N(Z,B) depends both on the choice of the embedding Z ⊂ Pn and the choice of coordinates of Pn. We want to investigate the asymptotic behaviour of N(Z,B) when B →∞. The following result is due to Pila (see [23] and [24]). 0.1. THEOREM.– Let X ⊂ Pn be a geometrically irreducible projective variety of degree d defined over Q. Then, dim X+1/d+ε N(X,B)=Od,n,ε(B ). It is remarkable that the implied constant does not depend on X apart from its dependence on the degree and the dimension of X. The starting point and the most subtle ingredient of Pila’s proof is a uniform bound for curves due to Bombieri and Pila [3] (cf. also [24]). It is also possible to reduce to the case of hypersurfaces by means of finite birational projections. We shall in this paper give estimates for hypersurfaces of dimension at most four which are stronger than Theorem 0.1 and other known results. To obtain these, we apply the methods of ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 0012-9593/01/ 2005 Elsevier SAS. All rights reserved 94 P. SALBERGER Heath-Brown’s fundamental paper on rational points on curves and surfaces [19]. The most important new tool in [19] is Theorem 14, where determinants are used to construct auxiliary hypersurfaces containing the rational points of bounded height. We shall use the technique with auxiliary hypersurfaces when we count points on curves and surfaces. In higher dimensions we use a number of new arguments from algebraic geometry. We thus make essential use of the Kodaira dimension in our estimates for smooth hypersurfaces. We obtain in particular improvements of a well-known bound of Hua from 1938 concerning the number of non-trivial d d d − d − d − d positive solutions to the diophantine equation x0 + x1 + x2 x3 x4 x5 =0. This was also the aim of the paper of Browning and Heath-Brown [8], but our estimates are sharper and valid for general non-singular forms. This paper is organised as follows. In Section i, 1 i 4, we give uniform estimates for hyper-surfaces of dimension i. The approach is inductive and the proofs based on the estimates in lower dimensions. We shall apart from the determinant method also apply Siegel’s lemma in the inductive process. We use thereby an idea from [19, p. 581] to get non-trivial savings when summing over the “non-degenerate” hyperplane sections. To avoid unnecessary repetition, we give in Section 5 a more abstract and general formulation of this argument. In Section 6, we collect some lemmas from algebraic geometry which we use to prove the estimates for threefolds and fourfolds. 1. Curves The following result is due to Heath-Brown [19] in the case where C is geometrically irreducible and n =2or 3. The general case is due to Broberg [4], who also proves the same uniform estimate for curves over arbitrary number fields. 1.1. THEOREM.– Let C ⊂ Pn be an irreducible projective curve over Q of degree d. Then, 2/d+ε N(C,B)=Od,n,ε(B ). Note that all rational points on C are singular if C is irreducible but not geometrically irreducible. We have thus at most Od(1) rational points on such curves. There are also results on the number of rational points in boxes with sides of different lengths in [19] and [4]. We now deduce some consequences of Theorem 3 in [19] for plane curves. 2 1.2. Notation.–LetC ⊂ P be a plane curve over Q and B0,B1,B2 three positive integers. (a) N(C,B0,B1,B2) denotes the number of Q-points on C with a primitive integral representative (x0,x1,x2) satisfying |xi| Bi for i =0, 1, 2. (b) V = B0B1B2. 1.3. THEOREM.– Let C be an irreducible plane curve over Q of degree d. Then, 1/(2d−m)+ε N(C,B0,B1,B2)=Od,ε(V ) for the maximum multiplicity m max(d −1, 1) of rational points on C. (If C(Q) is empty, then m is defined to be zero.) If there are no rational points on C of multiplicity >d/2, then 2/3d+ε N(C,B0,B1,B2)=Od,ε(V ). ◦ 4e SÉRIE – TOME 38 – 2005 – N 1 COUNTING RATIONAL POINTS ON HYPERSURFACES 95 e f g ⊂ 2 Proof. – Let F (x0,x1,x2)= ae,f,gx0x1 x2 be a form defining C P and let T be the e f g maximum of all B0B1 B2 with ae,f,g =0. Then Heath-Brown [19, Th. 3] proved that (cf. also [4, Cor. 2]): 1/d+ε 1/d2 (1.4) N(C,B0,B1,B2)=Od,ε(V /T ). It thus suffices to show the following inequalities: (1.5) (a) T V d/3 if m d/2, (b) T V (d−m)d/(2d−m) if m>d/2. We may also assume that B0 B1 B2. Suppose first that P0 =(1, 0, 0) ∈/ C. Then T = d d/3 ∈ B0 V . Suppose next that P0 C and let m0 be the multiplicity of C at P0. Then − e f g d−m m ae,f,g =0for some triple (e, f, g) with e d m0. Hence T B0B1 B2 B0 B2 .Also, since F is irreducible, we get that ae,f,g =0 for some triple (e, f, g) with g =0. Therefore, e f d T B0B1 B1 . We have thus: { d−m m d} d d−m d T max B0 B2 ,B1 = B2 max (B0/B2) , (B1/B2) d d(d−m)/(2d−m) B2 (B0/B2)(B1/B2) d−m d−m m d/(2d−m) =(B0 B1 B2 ) . Also, since B0 B1 B2 1 we conclude that: d−m d−m m (2d−m)/3 B0 B1 B2 V if m d/2, d−m d−m m d−m B0 B1 B2 V if m>d/2. This completes the proof of (1.5) and Theorem 1.3. 2 One can deduce a more precise result from (1.4). If P is a rational point on C,letmP be its multiplicity and nP be the maximum multiplicity of the rational lines in the tangent cone of C at P . 1.6. THEOREM.– Let C be a geometrically irreducible plane curve over Q of degree d>1. Then, 2/3d+ε N(C,B0,B1,B2)=Od,ε(V ), if mP + nP d for all rational points on C. Otherwise, there exists a rational point P on C with mP + nP >dsuch that (d+n−m)/d(2d+n−2m)+ε N(C,B0,B1,B2)=Od,ε(V ) for m = mP , n = nP . The proof is similar to the proof of Theorem 1.3. We leave the details to the reader, as we shall not use this result in the sequel. 1.7. DEFINITION.– LetΠ ⊂ Pn be a rational hyperplane given by the equation y0x0 + ···+ ynxn =0 n∨ for a rational point y =(y0,...,yn) in the dual projective space P . Then the height H(Π) of Π is defined as the height of the rational point y in Pn∨. ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 96 P. SALBERGER The following corollary of Theorem 1.3 will be important. 1.8. COROLLARY.– Let C ⊂ P3 be an irreducible projective curve over Q of degree d>1 contained in a plane Π ⊂ P3. Then, 3 1/(d+1)+ε N(C,B)=Od,ε (B /H(Π)) +1 .