manuscripta math. 98, 453 – 475 (1999)

© Springer-Verlag 1999

Carlo Gasbarri

On the number of points of bounded height on arithmetic projective spaces Received: 20 February 1998 / Revised version: 9 November 1998 Abstract. Let K be a number field and OK its ring of integers. Let E be a Hermitian vector bundle over Spec(OK ). In the first part of this paper we estimate the number of points of bounded height in P(E)(K) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d > 1 in P(E) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of PN K blown up at a linear subspace of codimension two.

1. Introduction Let K be a number field and OK be its ring of integers. Let E be a Hermitian vector bundle over Spec(OK ). Using the well known analogy between function fields and number fields (cf. [We]) we see that the OK -scheme P(E) can be seen as a projective bundle over the arithmetic curve Spec(OK ). Arakelov theory equips P(E) with an arithmetic intersection theory (cf. [GS]) and in particular with a height theory for cycles in P(E) (cf. [BoGS]). In this paper we deal with some arithmetic properties of P(E); with this we mean properties which depend on the field K and, change, in general, if we make a base extension (for instance the number of points of bounded height). After a very quick review of Arakelov theory of P(E), (which is there just to fix notations), we will prove an arakelovian analogous of the Schanuel estimate (cf. [Sc]) of the number of points of bounded height in a projective space. More precisely we prove Theorem 1.1. There exists a completely explicit constant α(N ; K) depending only on N and K such that, if E is a Hermitian vector bundle of rank N over Spec(OK ) and T is a positive real number and   λ(P(E); T ) = Card P ∈ P(E)(K) hN (P ) ≤ T ) C. Gasbarri: Universitá di Roma Tre, Dipartimento di Matematica, Piazza San L. Murialdo 1, I-00146 Rome, Italy. e-mail: [email protected] Mathematics Subject Classification (1991): 14G40, 14G05, 11G35

454

C. Gasbarri

then λ(P(E); T ) = α(N ; K) exp(dd eg(E)) exp(NT ) 1 )T ) + OE (exp((N − [K : Q] when T → ∞. The (very interesting) precise value of α(N ; K) is given in Sect. 3 (Theorem 3.6). The proof of this Theorem is similar to Schanuel original proof, but it is more intrinsic (for instance we never fix a basis of EK ) and systematically uses Arakelov language (cf. [Sz]). In Sect. 4 we give some applications of Theorem 1.1; we can estimate the number of hyperplanes in P(E) having normalized height less than T and containing a fixed linear subspace of P(E). This estimate is very precise (cf. Theorem 4.1). As a second application we give a less precise (but quite effective) estimate of the number of hypersurfaces of P(E) of degree d, normalized height less than T and containing a fixed linear subspace (cf. Theorem 4.2). In the papers [FMT] and [BM], the authors propose some interesting conjectures on the asymptotic estimation of the number of points of bounded height on Fano varieties. It would be interesting to see if methods similar to the ones developed in this paper allow to introduce the Arakelov theory in that circle of problems. The third application of Theorem 1.1 is in this direction; we give an estimation of the number of points of height, with respect to the anticanonical line bundle, less then T of PN K blown up at a linear subvariety of codimension two. Namely we can prove the following: Theorem 1.2. Let X˜ be the blown up of PN K at a linear subspace F of ˜ ˜ Then, codimension two and let T be the anticanonical line bundle of X. when T → ∞, n o  ˜ Card p ∈ X(K) hT˜ (x) ≤ T ∼ aT exp(T ) where a is a positive constant. In [T] there is a generalization of Schanuel’s computation to grassmannians. We would like to thank A. Abbes, J. B. Bost, M. McQuillan and C. Soulé for interesting conversations on the argument.

On the number of points of bounded height on arithmetic projective spaces

455

1.1. List of notations – – – – – – – – – – – – –

K: Number field. OK : Ring of integers of K. S∞ : set of archimedean absolute values of K. r1 : number of real embeddings of K. r2 : number of complex embeddings of K. [K : Q] = r1 + 2r2 . Cl(OK ): Class group of OK . h: class number of OK . w: number of roots of unity in K. U : group of units in OK . R: absolute regulator of K. DK : absolute discriminant of OK over Z. VN : volume of the N -dimensional sphere.

If σ ∈ S∞ , then we define Nσ = 1 if σ is real and Nσ = 2 if σ is complex. 2. Quick review of Arakelov intersection theory A Hermitian module E = (E; h·; ·iσ )σ ∈S∞ over Spec(OK ) is an OK -module of finite type E and, for each σ ∈ S∞ , if Kσ is the completion of K with respect to σ , we suppose that the Kσ -vector space Eσ = E⊗σ Kσ is equipped with a metric h·; ·iσ (which is a scalar product if σ is real and a Hermitian metric if σ is complex). If E is a projective module of rank 1, then the Hermitian module E is called a Hermitian line bundle and we define its arithmetic degree by the formula X dd eg(E) = log(Card(E/s OK )) − log kskσ σ ∈S∞

where s ∈ E \ {0} and, for s ∈ Eσ we define kskσ = hs; siNσ /2 . d d = deg(det(E)) If E is an arbitrary Hermitian module, we define deg(E) (cf. [Sz]; [MB]). Let M α : E −→ Eσ = ER σ

be the natural map; α(E) is a lattice in ER (and this last one is a Hermitian space by the direct sum Hermitian metric), we define then 
χ (E) = − log V ol ER α(E) + log Card(ET ors ) where ET ors is the torsion submodule of E. If E is locally free (projective) of rank N, we will say that E is a Hermitian vector bundle of rank N over Spec(OK ) (from now until the end of this

456

C. Gasbarri

paper we will suppose that N ≥ 2). In this case we can associate to E an Arakelov variety P(E) = Proj(Sym(E ∗ )) (E ∗ is the dual of E). For each σ ∈ S∞ , the projective space P(E)σ = P(E) ⊗σ C is endowed with the Fubini-Study metric induced by the metric on Eσ . The arithmetic variety X = P(E) is equipped with an arithmetic intersection theory (cf. [GS]), more precisely we have: – – – –

\ i (X); for each integer i ∈ [0; N], an arithmetic Chow group CH \ \ \ i (X) ⊗ CH j (X) → CH i+j (X); an intersection product: (·; ·) : CH \ N (X) → R; d : CH an arithmetic degree map: deg for each cycle of dimension p, Z ∈ Zp (X), an arithmetic fundamental N−p (X) (this class depends on the chosen metric on E). class [Z] ∈ CH\

Moreover, the arithmetic variety X is equipped with the universal ample Hermitian line bundle OX (1) = L. We can then define a normalized height function for subvarieties of arbitrary dimension of X: let Z ⊂ X be a (closed) subscheme of dimension p, we define the normalized height hN (Z) of Z by the formula: d p · [Z]) deg(L hN (Z) = p deg(ZK ) where ZK is the generic fibre of Z and deg(ZK ) is the usual degree of a closed subscheme of a projective space. For an extended account on height theory cf. [BoGS], [F]. We recall the following formula from [BoGS] (cf. [BoGS] Prop. 4.1.2, p. 964): If F ⊂ E is a saturated submodule of rank p, and P(F ) ⊂ P(E) is the corresponding linear subspace, then hN (P(F )) = − where σp =

p+1 2

Pp+1

1 m=2 m

d ) [K : Q]σp−1 deg(F + p p

if p > 0 and σ0 = 0.

Remark. A Hermitian vector bundle over Spec(OK ) is the generalisation of a lattice. A Hermitian vector bundle of rank N over Spec(Z) is just a lattice 3 in a R-vector space V of dimension N (a discrete subgroup generated by a basis of V ). If E = 3 = ZN ⊂ RN (the standard lattice in RN with the standard scalar product), then we are just considering the standard Weil N−1 height over PN−1 Z : if p ∈ PZ (Q), then we can write p in homogeneus coordinates p = [x0 ; . . . ; xN ] with pPxi ∈ Z coprime numbers (and this in a |xi |2 . In general E may not be trivial (as unique way); then hN (p) = log an OK -module) and the metrics may be arbitrary (in particular it is possible that any orthonormal basis of Eσ is contained in the image of E).

On the number of points of bounded height on arithmetic projective spaces

457

3. Points of bounded height on arithmetic projective spaces In this chapter we will estimate the number of points of bounded height in a projective space P(E), where E is a Hermitian vector bundle of rank N over Spec(OK ). This will generalize a theorem by Schanuel [Sc] where the trivial (naïve) height was considered. In our proof we systematically used the language of Arakelov theory and we made a special effort to keep arguments as intrinsic as possible (for instance we never fix a basis of EK ). Due to the intrinsic nature of our arguments, this estimate can be applied to very general situations. We will see some applications in the next chapter. By completely different methods, a similar problem for flag varieties is studied in [FMT] and [Pe]; our approach is simpler in this situation. 3.1. Compactified divisors over Spec(OK ) Cf. [Sz]. A compactified divisor over Spec(OK ) is a formal sum X X D= mp [p] + λσ [σ ] p∈MK

σ ∈S∞

where MK is the set of finite places of K, mp ∈ Z are almost all zero and λσ ∈ R. The degree of a compactified divisor is given by X X d deg(D) = mp log N (p) + λσ σ

p∈MK

where N(p) is the absolute norm of the maximal ideal p over Z. We will denote by Divc (OK ) the group of compactified divisors over OK . As we can see, P every compactified divisor D is the difference P of its "finite part" Df in = p∈MK mp [p] and its "infinite part" D∞ = σ ∈S∞ λσ [σ ]. If a is a fractional ideal of OK we can associate to a a "finite" compactified divisor X vp (a)[p]. D(a) = p∈MK

If t ∈ K, we associate to t the compactified divisor X X div(t) = vp (t)[p] − log ktkσ [σ ]. p∈MK

σ

458

C. Gasbarri

Remark. dd eg(div(t)) = 0 (product formula). P P P 0 0 P Let D0 = p∈MK mp [p] + σ ∈S∞ λσ [σ ] and D = p∈MK mp [p]0 + σ ∈S∞ λσ [σ ] be two compactified P divisors. P We will say that D ≥ D if mp ≥ m0p for all p ∈ MK and σ λσ ≥ σ λ0σ . Remark. If D ≥ D 0 then dd eg(D) ≥ dd eg(D 0 ). Let E be a Hermitian vector bundle over Spec(OK ). Let s ∈ EK = E ⊗OK K and let Ls be the Hermitian line bundle K · s ∩ E (with induced metrics). We define the compactified divisor div(s) in the following way: Let t ∈ OK , such that ts ∈ Ls ; we then have a map of line bundles OK −→ Ls 1 −→ ts if we dualize this map, giving 0 → L∗s −→ OK ϕ −→ ϕ(ts); we can identify L∗s with an ideal a of OK ; we then define X div(s) = D(a) − log ktskσ − div(t). σ

Remark. This definition do not depend on the chosen t ∈ OK : to prove this it suffices to see that if ts ∈ Ls and t 0 ts ∈ Ls , with t ∈ OK and t 0 ∈ K, then div(t 0 ts) = div(ts) + div(t 0 ), and this is straightforward. As before we will write div(s) = div(s)f in − div(s)∞ . 3.2. Number of compactified divisors in EK bigger than D We fix a compactified divisor D and a Hermitian vector bundle E of rank N over Spec(OK ). Let U be the group of units of OK . Let    L(D; E) = s ∈ EK div(s) ≥ D U and let λ(D; E) = Card(L(D; E)). We will estimate λ(D; E) in terms of the degree of D.

On the number of points of bounded height on arithmetic projective spaces

459

Remark. If D is linearly equivalent to D 0 , which means that there exists a t ∈ K such that D = D 0 + div(t), then it is easy to see that λ(D; E) = λ(D 0 ; E); indeed the map

L(D; E) −→ L(D 0 ; E) s −→ ts

is a bijection. Theorem 3.1. Let D be a compactified divisor and E be a Hermitian vector bundle of rank N over Spec(OK ); then L(D; E) is a finite set and r2 Nr2 N r1 +r2 −1 VNr1 V2N 2 R q exp(dd eg(E)) exp(−N dd eg(D)) λ(D; E) = N w DK

+ OE (exp(−(N −

1 )dd eg(D))) [K : Q]

d when deg(D) → +∞. The constant in OE depends only on E. Proof. Let a1 ; . . . ; ah be ideals of OK representing the classes in Cl(OK ); by the remark it suffices to prove that for all compactified divisor D such that Df in = D(ai ) we have that λ(D; E) =

r2 Nr2 2 R N r1 +r2 −1 VNr1 V2N d d q exp(deg(E)) exp(−N deg(D)) N w DK

1 d ∞ ))). )deg(D [K : Q] P So, let a be one of the ai ’s and D = D(a) + σ λσ [σ ] = D(a) + D∞ . Looking at the exact sequence  0 → E ⊗ a −→ E −→ E ⊗ OK a → 0 + OE (exp((N −

we see that, if s ∈ EK is such that div(s) ≥ D, then s ∈ E ⊗ a; so we must compute the L number of elements s in E ⊗ a with P div(s) ≥ D modulo units. Let V = σ R and H ∈ V the subspace xσ = 0. Let d : U −→ V u −→ (log kukσ ) then (Dirichlet Units Theorem) d(U ) is a lattice in H with discriminant R (the absolute regulator of K).

460

C. Gasbarri

We fix the two following elements in V : – v = (Nσ )σ ; – n = (1; 1; . . . ; 1). We see that H = {x ∈ V / (x; n) = 0} (here we consider the usual scalar product on V ' Rr1 +r2 ); and that U acts (via d) on H . Let’s consider the maps M (Eσ \ {0}) −→ V η: σ

(xσ ) −→ (log kxσ kσ ) and

P r : V −→ H  y −→ y −

 1 (y; n)v . (v; n) L The composite map γ = (P r ◦ η) : σ (Eσ \ {0}) → H is a U -map. So if F is a fundamentalLdomain of H modulo U , then 1 = γ −1 (F ) is a {0}) fundamental domain σ (Eσ \   of L Pmodulo U . d Let R(D∞ ) = (zσ ) ∈ σ Eσ σ log kzσ kσ ≤ deg(D∞ ) . The set R(D∞ ) is U -stable
and we must compute the number of U -orbits in R(D∞ ) ∩ (E ⊗ a \ {0}) . Let 1(D∞ ) = R(D∞ ) ∩ 1. By what we just said we see that wλ(D; E) = Card(E ⊗ a ∩ 1(D∞ )). Remark. The set R(D∞ ) depend only on the degree of D∞ Lemma 3.2. a) For all t ∈ R∗ , we have that t1 = 1 d ∞ ))R(0); b) R(D∞ ) = exp( [K:1Q] deg(D d ∞ ))1(0) where 0 ∈ Divc (OK ) is the “zero” c) 1(D∞ ) = exp( [K:1Q] deg(D compactified divisor. Proof. c) is a consequence of a) and b). a) η(t (xσ )) = log |t|.v + η((xσ )), but P r(v) = 0 and P r is linear, so a) follows. P b) If ρ(z) = σ log kzkσ , then ρ(tz) = [K : Q] log |t| + ρ(z); then b) easily follows. u t We recall now the following classical statement on the number of points of a lattice in an expanding domain. For a proof see [La] Chap. VI §2 (where you can also find the definition of a Lipschitz–parametrizable set).

On the number of points of bounded height on arithmetic projective spaces

461

Proposition 3.3. Let 0 be a bounded set of Rk with Lipschitz parametrizable boundary. Let 3 be a lattice in Rk ; then Card(3 ∩ t0) =

m(0) k t + O(t k−1 ) det(3)

for t → ∞. Where m(0) is the Lebesgue measure of the closure of 0. The constant in 0 depends only on 3 and on the Lipschitz constants of the boundary of 0. L We can prove that 1(0) ⊂ σ Eσ is a bounded set with Lipschitz parametrizable boundary (it is even C 1 parametrizable) in the same way as in [Sc]. In particular the fact that 1(0) is bounded can be seen easily from the definitions: there exists open neighborhoods Bσ of 0 in Eσ (∀σ ) such L L that σ Bσ ∩ 1 = ∅ and R(0) ∩ ( Eσ \ Bσ ) is a bounded set. We apply Proposition 3.2 in our situation and we get Card(E ⊗ a ∩ 1(D∞ )) = Card(E ⊗ a ∩ exp( =

m(1(0)) det(E ⊗ a)

1 dd eg(D∞ ))1(0)) [K : Q]

d ∞ )) exp(N deg(D

1 d ∞ )) )deg(D [K : Q] d ∞ )) = m(1(0)) exp(χ(E ⊗ a)) exp(N deg(D 1 d ∞ )). )deg(D + OE (exp(N − [K : Q] + OE (exp(N −

But we have the exact sequence  0 → E ⊗ a −→ E −→ E ⊗ OK a → 0 so we get

 χ (E ⊗ a) = χ (E) − N log(Card(OK a) = χ (E) − N log N (a).

d Now, using Arakelov–Riemann–Roch (χ   (E) = deg(E)+N χ(OK ) cf. [Sz]) and recalling that χ (OK ) = log Card(E ⊗ a ∩ 1(D∞ )) =

2 r2 1/2 DK

, we finally find

m(1(0))2Nr2 q exp(dd eg(E)) exp(−N dd eg(D)) N DK

+ OE (exp(N −

1 )dd eg(D∞ )). [K : Q]

462

C. Gasbarri

Finally we must compute m(1(0)). We recall that 1(0) = 1 ∩ R(0), where ) ( M  Y kzσ kσ ≤ 1 Eσ and 1 = γ −1 (F ). R(0) = (zσ ) ∈ σ

We consider polar coordinates in Eσ , ρσ and θi;σ with ρσ ≥ 0 and 0 < θσ ≤ 2π (i = 1; . . . ; N − 1 if σ is real, and i = 1; . . . ; 2N − 1 if σ is complex). Since the definition of 1(0) does not depend on the θi;σ ’s, we can integrate with respect to them and we obtain Z Y Y Y r1 r2 m(1(0)) = (N VN ) (2N V2N ) ρσN −1 ρσ2N−1 dρσ σ real

σ complex

σ ∈S∞

where the integration domain is n o Y  (ρσ ) 0 < ρσNσ ≤ 1; P r(log(ρσNσ ) ∈ F . We then put tσ = ρσNσ and we get m(1(0)) = N

r1 +r2

r2 VNr1 V2N

Z Y

tσN −1 dtσ ;

σ ∈S∞

where we integrate over the domain n o Y  (tσ ) 0 < tσ ≤ 1; P r((log(tσ ))) ∈ F . It is possible  to find r1 + r2 − 1 linear functions τj : H → R such that F = x ∈ H 0 ≤ τ (x) < 0 . We consider the change of variables Y u= tσ ξj = τj (P r(log(tσ ))); The jacobian of this change of variables is ±R. So we get Z 1 Z 1 Z r1 +r2 r1 r2 N−1 m(1(0)) = N VN V2N R u du dξ1 · · · 0

0

r2 = N r1 +r2 −1 VNr1 V2N R.

1

dξr1 +r2 −1

0

And finally we find λ(D; E) =

r2 Nr2 N r1 +r2 −1 VNr1 V2N 2 R d d q exp(deg(E)) exp(−N deg(D)) N w DK

+ OE (exp(−(N −

1 )dd eg(D))). [K : Q]

which is the statement of the Theorem. u t

On the number of points of bounded height on arithmetic projective spaces

463

3.3. The Möbius inversion formula and the number of points of bounded height in P(E) Let I be the multiplicative semigroup of the non zero integral ideals of K, and let R(I ) be the group of functions f : I → R; it is actually a ring by the multiplication rule X χ0 • χ1 (a) = χ0 (a1 )χ1 (a1 ). 



a0 ·a1 =a

Let A(E) = sup dd eg(L) L is a Hermitian subline bundle of E and let   d >A . RE (Divc (OK )) = f : Divc (OK ) → R f (D) = 0 if deg(D) The group RE (Divc (OK )) is an R(I )-module by the following product: if χ ∈ R(I ) and f ∈ RE (Divc (OK )), then X χ • f (D) = χ (b)f (D + D(b)) b∈I

The sum on the R.H.S. is actually a finite sum. Let µ be the Möbius function of K: – – – –

µ(p) = −1 for all non zero prime ideal p of OK ; µ(pn ) = 0 if n > 1; µ(1) = 0; µ(a · b) = µ(a)µ(b).

And let χ0 be the constant function χ0 (a) = 1; the classical Möbius inversion formula can be written µ • χ0 = 1I ; where 1I (1) = 1 and 1I (a) = 0 if a 6= 1. So we have the classical lemma Lemma 3.4.

where ζK (s) =

P

X µ(a) 1 = s N (a) ζK (s) a∈I

1 a∈I N(a)s

is the classical zeta function of K.

Lemma 3.5. Let f ; g ∈ RE (Divc (OK )) then d a) if f (D) = α exp(−s deg(D)) where s > 1 is a real number, then µ • f (D) =

α d exp(−s deg(D)) + O(exp(− deg(D))); ζK (s)

b) if g(D) = O(exp(−t dd eg(D)) then d µ • g(D) = O(exp(−t deg(D))).

464

C. Gasbarri

Proof. a) Let D ∈ Divc (OK ) then X αµ(a) exp(−s dd eg(D + D(a))) µ • f (D) = d N(a)≤exp(deg(D)+A(E))

=

X

αµ(a) exp(−s dd eg(D − s log N (a)))

N(a)≤...

=

X

α exp(−s dd eg(D))

N(a)≤...



= α exp(−s dd eg(D)) 

µ(a) N (a)s

1 − ζK (s)

X d N (a)≥exp(deg(D)+A(E))

 µ(a)  N (a)s

while X X µ(a) 1 ≤ N (a)s N(a)≥... N (a)s N(a)≥exp(deg(D)+A(E)) d = O(exp(−(1 − s)(dd eg(D) + A(E)))). b) g(D) = O(exp(−t dd eg(D))) so   X d µ(a) exp(−t deg(D) − t log N (a) µ • g(D) = O  d N(a)≤exp(−deg(D)+A(E))



 µ( a) d = O(exp −t deg(D)) ) N(a)s N(a)≤... X

P while N(a)≤exp(−deg(D)+A(E)) d



µ(a) N(a)s

= 0(1). u t

We can now prove the main Theorem of this section. Theorem 3.6. Let E be a Hermitian vector bundle of rank N over Spec(OK ) and T a real number; let   P (P(E); T ) = p ∈ P(E)(K) hN (p) ≤ T ; then Card(P (P(E); T )) =

r2 N r2 2 R hN r1 +r2 −1 VNr1 V2N q exp(dd eg(E)) exp(NT ) N wζK (N ) DK

+ OE (exp((N − when T → +∞.

1 )T )) [K : Q]

On the number of points of bounded height on arithmetic projective spaces

465

Proof. Let D = Df in + D∞ be a compactified divisor over OK ; Df in determines a well defined fractional ideal a(Df in ) of K. Let cl(Df in ) ∈ Cl(OK ) be its class in the class group of OK . More generally, if L is a projective OK -module of rank one, we will denote by cl(L) its class in Pic(OK ) ' Cl(OK ). Let P(E) be the projective space associated to E; for all p ∈ P(E)(K) let Lp be the saturated subline bundle of E determined by p. We know that eg(Lp ) (cf. Sect. 2). hN (p) = −dd We first estimate the cardinality of the set ˜ L(D; E) =   = p ∈ P(E)(K) cl(Lp;f in ) = cl(Df in ) and hN (p) ≤ −dd eg(D)   eg(Lp ) ≥ dd eg(D) . = p ∈ P(E)(K) cl(Lp;f in ) = cl(Df in ) and dd ˜ ˜ Let λ(D; E) = Card(L(D; E)). ˜ It is easy to see that λ(D; E) and λ(D; E) are elements of RE (Divc (OK )). Let  L(D; E) = {s ∈ EK \ {0} div(s)f in = Df in and  d d deg(div(s) ∞ ) ≤ deg(D∞ )} U ; there exists a bijection ˜ L(D; E) ←→ L(D; E) U · s ←→ Ls , and L(D; E) =

[

L(D + D(a); E);

a∈I

this means that

λ(D; E) =

X

˜ λ(D + D(a); E)

a∈I

˜ E). = χ0 • λ(D; So

˜ µ • λ(D; E) = µχ0 λ(D; E) = λ˜ (D; E);

now; by Theorem 3.1, λ(D; E) = f (D) + g(D) with r2 Nr2 2 R N r1 +r2 −1 VNr1 V2N d d q exp(deg(E)) exp(−N deg(D)) f (D) = N w DK

and g(D) = O(exp(−(N −

1 )dd eg(D))). [K : Q]

466

C. Gasbarri

We apply Lemma 3.5 and get ˜ E) = λ(D;

r2 Nr2 N r1 +r2 −1 VNr1 V2N 2 R q exp(dd eg(E)) exp(−N deg(D)) N wζK (N ) DK

+ O(exp(−(N −

1 )dd eg(D))). [K : Q]

Let a1 ; . . . ; ah be ideals of OK which represent the classes of Cl(OK ) and let Di be compactified divisors such that Di;f in = D(ai ) and dd eg(Di ) = −T . We find that Card(P (P(E); T )) =

h X

λ˜ (Di ; E)

i=1

r2 N r2 2 R hN r1 +r2 −1 VNr1 V2N q = exp(dd eg(E)) exp(NT ) N wζK (N ) DK

+ O(exp((N −

1 )T )). [K : Q]

t u

Remark. From now until the end of this paper we will denote α(K; N ) the r r2 N r 2 hN r1 +r2 −1 VN1 V2N √ N2 R . number wζK (N )

DK

Remark (on the constant in O(. . . )). The constant which is in O((exp((N − 1 )T ) evidently depends on E; but if F ⊂ E is a subbundle, looking [K:Q] carefully to the proof, we can see that we have a commutative diagram L L σ Fσ σ Eσ ηF

so 1E (0) ∩

L σ

η

E L? s σ R

Fσ = 1F (0). This implies that (M = rk(F )):

Card(P (P(F ); T )) = α(K; M) exp(dd eg(F )) exp(MT ) 1 )T )) + OE (exp((M − [K : Q] with the constant in OE (exp((M − [K:1Q] )T )) depending only on E! To prove this you must read carefully the last part of Theorem 2 page 128 in [La] and the proof of Lemma 3.5.

On the number of points of bounded height on arithmetic projective spaces

467

4. Applications In this chapter we give some applications of Theorem 3.6, firstly we show how Theorem 3.6 can be used to estimate the number of hyperplanes and hypersurfaces of bounded height containing a fixed linear subspace. Secondly we estimate the number of points of anticanonical height less then T on PN blown up at a linear subspace of codimension two. Both applications strongly use the fact that Theorem 3.6 (and its proof) is very intrinsic. 4.1. Hyperplanes of bounded height containing a fixed linear subvariety Let E be a Hermitian vector bundle of rank N over Spec(OK ) and let A ⊂ E be a saturated submodule of rank M. lf F ⊂ E is a saturated sub bundle of rank N − 1, subsequently we will call “hyperplane” the closed subvariety P(F ) ⊂ P(E). Let T be a real number. We can then consider the set n  L(A; T ) = P(F ) ⊂ P(E) P(F ) o hyperplane s. t. P(F ) ⊃ P(A) ; hN (P(F )) ≤ T . Theorem 4.1. Let E and A ⊂ E be as before, and let p = N − M; then Card(L(A; T )) = α(K; p) exp(−p[K : Q]σN −2 ) d d × exp(deg(A) + (p − 1)deg(E)) exp(p(N − 1)T ) 1 )(N − 1)T )) + OE (exp((p − [K : Q] when T → +∞. Proof. We have the exact sequence 0 → A −→ E −→ (E/A) → 0 and its dual

∗

0 → (E/A)∗ −→ E −→ A∗ → 0. If P(F ) is an hyperplane in P(E), it corresponds to a sub line bundle LF ⊂ E ∗ and dd eg(LF ) [K; Q]σN −1 − dd eg(E) + . hN (P(F )) = − N −2 N −1 Moreover we have that P(F ) ⊃ P(A) if and only if LF ⊂ (E/A)∗ .

468

C. Gasbarri

Then there is a bijective correspondence between L(A; T ) and the set n  M(A; T ) = p ∈ P((E/A)∗ )(K) hN (p) eg(E)) ≤ T (N − 1) − ([K; Q]σN −2 − dd

o

By using Theorem 3.6, we get Card(L(A; T )) = α(K; p) exp(dd eg(A) − dd eg(E)) × exp(p(N − 1)T − p([K; Q]σN −2 − dd eg(E))) 1 )(N − 1)T )) + OE (exp((p − [K : Q] eg(A) = α(K; p) exp(−p[K : Q]σN −2 ) exp(dd d + (p − 1)deg(E)) × × exp(p(N − 1)T ) + OE (exp((p −

1 )(N − 1)T )) [K : Q]

which is the desired result. u t 4.2. Some remarks on the fundamental class of hypersurfaces in P(E) Theorem 3.6 can be also used to estimate the number of hypersurfaces of degree d in P(E) with bounded height and containing a fixed linear subspace P(A) ⊂ P(E). An hypersurface of degree d in P(E) is the Zariski closure of a hypersurface of degree d in P(EK ). Let P(A) ⊂ P (E) be a linear subspace of dimension r; we want to estimate the number hd (E; A; T )   = Card S ⊂ P(E) S hyp. of deg. d s. t. S ⊃ P(A) and hN (S) ≤ T we will prove the following Theorem 4.2. There exist two absolute constants Ai = Ai (N ; d; r; [K : Q]) (i = 1; 2) depending only on N ; d; r and [K : Q], such that if P(A) ⊂ P(E) is as above, then, when T → +∞, I1 (E; A; T ) ≤ hd (E; A; T ) ≤ I2 (E; A; T )

On the number of points of bounded height on arithmetic projective spaces

469

where

  d d +r Ii (E; A; T ) = Ai α(K; n(N ; r; d)) exp( dd eg(A) r r      N −1 d +N d +r +d − dd eg(E)) N N r × exp(d(N − 1)n(N ; r; d)T ) 1 )d(N − 1)T )) + OE (exp((n(N ; r; d) − [K : Q]

and n(N ; r; d) = N+d − r+d . d d

Before we start the proof, we will make some remarks on the fundamental classes of hypersurfaces in P(E). Let P(F ) ⊂ P(E) be a hyperplane. By the theory of Levine forms, we can compute the fundamental class OP(E) (P(F )) ∈ Picc (P(E)) (cf. [BoGS] p. 921 ff.). The fundamental class of a hypersurface S ⊂ P(E) is given by a cycle [1 (P(E)) where [S] = (S; gS;σ ) and the gS;σ ’s are functions on [S] ∈ CH P(E σ ) such that dd c gS;σ + δSσ = c1 (OP(E σ ) (deg(SK )) where δSσ is the current of integration along Sσ ; OP(E σ ) (1) is the line bundle OP(E σ ) (1) with the Fubini–Study metric induced by the Hermitian metric on Eσ ; and Z N −1

P(E σ )

gS;σ c1 (OP(E σ ) (1)

= 0.

We recall the construction of the Levine forms: let V be a Hermitian vector space of dimension M; let P(F ) ⊂ P(V ) be a hyperplane ( and F the corresponding linear subspace of codimension one in V ). The Levine function 3P(F ) is a function such that a) dd c 3P(F ) + δP(F ) = c1 (OP(E σ ) (1)) b) if H0;0 (P(V )) ' C is the space of space of the harmonic functions (with respect to the Fubini–Study metric) and H : A0 (P(V )) → C is the harmonic projection; then H (3P(F ) ) = 2(σM−1 − σM−2 ). Let’s recall the construction of 3P(F ) : We have a decomposition V = F ⊕ F⊥

470

C. Gasbarri

where F ⊥ = L∗ is a C-vector space of dimension one; let P r : V −→ L∗ be the orthogonal projection; for all x 6∈ F we then define 3P(F ) (x) = log

kxk2 . kP r(x)k2

It is not difficult to prove that this 3P(F ) (x) defines a function on P(V ) and that it has the properties we want. Let E be a Hermitian vector bundle of rank N over Spec(OK ) and let F ⊂ E be a saturated sub bundle of rank N − 1, subsequently we will call such a submodule a “hyperplane”; the fundamental class of P(F ) is then given by the cycle [1 (P(E)). [P(F )] = (P(F ); 3P(F ) − 2(σN−1 − σN −2 )σ )σ ∈S∞ ∈ CH

We have a natural isomorphism [1 (P(E)) −→ Picc (P(E)) α : CH

(cf. [GS2], [SABK]). Let’s compute the image of [P(F )] in Picc (P(E): The ∗ subbundle F ⊂ E corresponds to a Hermitian sub line bundle LF ⊂ E ; we see then that α([P(F )]) = OP (P(F )) = OP (1) ⊗ π ∗ (L∗F ) ⊗ π ∗ (OK (−σN −1 + σN −2 )), where π : P(E) → Spec(OK ) is the structural morphism and OK (−σN−1 + σN −2 ) is the line bundle OK with the metrics k1kσ = exp(σN −1 − σN −2 ). Indeed, by definition, α([P(F )]) = (O(P(F ); k1P(F ) k2σ (x) =

kP r(x)k2σ exp(2(σN −1 − σN −2 )); kxk2σ

and, if Zσ : Eσ → C is the linear function defining Fσ , we have that k1P(F ) k2σ (x) =

kZσ (x)k2σ 1 exp(2(σN −1 − σN −2 )). kxk2σ kZk2σ

Let’s look now at the case of hypersurfaces of arbitrary degree: let d > 1 ∗ ∗ be an integer and S d (E ) be the d-th symmetric product of E (with the symmetric product metrics (cf. [BoGS])): Let ∗

id : P(E) −→ P((S d (E ))∗ )

On the number of points of bounded height on arithmetic projective spaces

471

be the Veronese embedding. We have that id∗ (OP((S d (E ∗ ))∗ ) (1)) ' OP(E) (d), and this is an isometry. ∗ Every hyperplane of P((S d (E ))∗ ) intersect id (P(E)) in the image of a hypersurface of degree d and every hypersurface of degree d in P(E) is obtained in this way. ∗ Let P(F ) ⊂ P((S d (E ))∗ ) be a hyperplane; it corresponds to a subline ∗ bundle LF ⊂ S d (E ) and OP((S d (E ∗ ))∗ ) (P(F )) = OP((S d (E ∗ ))∗ (1)⊗πd∗ (L∗F )⊗ ∗ πd∗ (OK (C)) where πd : P((S d (E ))∗ ) → Spec(OK ) is the structural morphism and C is some completely explicit constant (it is useless to write it now explicitely). So, we see from the construction of the fundamental class of a cycle in P(E) (cf. [BoGS] par. 2.3.2, p. 942) that, for all σ ∈ S∞ there exists a conti∗ ∗ nous function Cσ on P(S d (E )σ (C) such that, if P(F ) ⊂ P((S d (E ))∗ ) is a hyperplane and SF = id∗ (P(F )) is the corresponding hypersurface of P(E), then, if we denote by OP(E) (SF ) the fundamental class of SF in Picc (P(E)), we have O OP(E) (SF ) = OP(E) (d) ⊗ π ∗ (L∗F ) ⊗ π ∗ (OK (Cσ (xF ))) (1) σ

where OK (Cσ (xF )) is the line bundle (OK ; k1kσ = Cσ (xF ) and k1kτ = ∗ 1 if τ 6= σ ) and xF is the point in P(S d (E )σ (C) corresponding to P(F ). Since the functions Cσ are continous (and a projective space is compact), we can find universal constants A(N ; d) and B(N ; d), depending only on ∗ N an d such that, for all hyperplane P(F ) ⊂ P((S d (E ))∗ ), A(N ; d) ≤ Cσ (xF ) ≤ B(N ; d). Let S ⊂ P(E) be a hypersurface of degree d; from (1) we get d P (1)N−1 · [S]) deg(O d(N − 1) P d F ) + σ Cσ (xF ) dN hN (P(E)) − deg(L = d(N − 1)

hN (S) =

Finally, we can find two real positive numbers A1 (N ; d) and A2 (N ; d) depending only on N and d such that A1 (N; d)[K : Q] ≤ hN (S) −

d d F) deg(E) deg(L + ≤ A2 (N ; d)[K : Q]. N −1 d(N − 1)

This formula is interesting because it reduces the computation of the normalized height of a hypersurface to the computation of the degree of a Hermitian line bundle; in some way it linearize the problem.

472

C. Gasbarri

Proof of Theorem 4.2. Firstly we remark that, there exists an absolute constant C(d; M) ∈ R such that, if V is a Hermitian vector bundle of rank M over Spec(OK ), then d d (V )) = deg(S

  d ) M +d d deg(V + C(d; M)[K : Q] M d

(cf. [BoGS], p. 984 ff. and [Ga]). Consider the exact sequence ∗

0 → Kd (E; A) −→ S d (E ) −→ S d (A∗ ) → 0 we have that rk(Kd (E; A)) = n(N ; r; d) and there exists an absolute constant C1 = C1 (N ; r; d) such that dd eg(Kd (E; A)) =

    d d +N d d +r d deg(A) − dd eg(E) + C1 [K : Q]. r d N d

By what we proved before, we have that P2 (E; A; T ) ≤ hd (E; A; T ) ≤ P1 (E; A; T ) where Pi (E; A; T ) = Card{p ∈ P(Kd (E; A)(K)  d hN (p) ≤ d(N − 1)T − d deg(E) − d(N − 1)Ai [K : Q]}. It suffices now to apply Theorem 3.6 to conclude. u t Remark. In Theorem 4.1 and 4.2, the subbundle A can be the trivial bundle A = {0}. In this case the Theorems give an estimate for the number of hypersurfaces of bounded height and degree. If A is a subline bundle of E, then we are computing the number of hypersurfaces of bounded height passing through a fixed point. Remark. A qualitative interpretation of Theorems 4.1 and 4.2 can be the following: There are less hypersurfaces (hyperplanes) of bounded height through a linear subspace of large height then through a linear subspace of small height.

On the number of points of bounded height on arithmetic projective spaces

473

4.3. Number of points of bounded height on PN K blown up at a linear subspace of codimension two This application have been suggested by M. McQuillan. As a third application, we give a (qualitative) estimation of the number of points of bounded height, with respect to the anticanonical line bundle, on a projective space blown up at a linear subspace of codimension two. The result we find agree with the Batyrev–Manin conjectures in this context. This application is a generalization of the computations we can find in [Pe] (where the case P2 was studied). We want to remark again how simple and natural is the proof using this language (and Theorem 3.6). Theorem 4.3. Let X˜ be the blown up of PN K at a linear subspace F of ˜ Then, codimension two and let T˜ be the anticanonical line bundle of X. when T → ∞, n o  ˜ Card p ∈ X(K) hT˜ (x) ≤ T ∼ aT exp(T ) where a is a positive constant. Proof. In order to simplify notations we will suppose that T is a positive natural number. We can suppose that F is given by the equations x0 = x1 = 0 (where xi are homogeneus coordinates on PN K. 1 It is well known that X˜ can be realized as a closed subset in PN K × PK .   1 X˜ = ([x0 ; . . . ; xN ]; [a; b]) ∈ PN ax0 = bx1 K × PK and the blown up is the first projection p1 : X˜ → PN K . The canonical line bundle of X˜ is ωX˜ = p1∗ (OPN (−(N + 1)) ⊗ OX˜ (E), where E is the ˜ An easy calculation gives OX˜ (E) = p1∗ (OPN (1))⊗ exceptional divisor of X. p2∗ (OP1 (−1)) (p2 : X˜ → P1 is the second projection); so we obtain T˜ ' p1∗ (OPN (N )) ⊗ p2∗ (OP1 (1)).

(2)

We fix the canonical Fubini–Study metric on OPN (1) and on OP1 (1) and 1 the canonical models PN OK and POK of the P’s over Spec(OK ); in this way ˜ of T˜ and a metric on T˜ . we fixed a model of X, Now we are ready to start the computation: it is well known that, by using ˜ the closed points of P1K parametrize the hyperplanes of PN X, K containing F . We remark also that two of such hyperplanes intersect only on F . If Q ∈ P1K (K), we will denote by LQ the corresponding hyperplane in PN K; because of our normalization, the height of Q is the same (up to some constant factor) of the height of LQ . We will denote NQ the strict transform

474

C. Gasbarri

of LQ and HQ the tautological ample line bundle of LQ (with the induced metric and model); by abuse of notation, we will denote again HQ the pull back of HQ to NQ . Because of (2), we see that, if p ∈ NQ , then hT˜ (p) = N hHQ (p) + hOP1 (1) (Q), so we see that o n  p ∈ X˜ hT˜ (p) ≤ T =

[ hO

P1 (1)



p ∈ NQ



hT˜ (p) ≤ T



(Q)≤T

and the NQ ’s are disjoint. This gives ˜ K; T ) N(X; n o  = Card p ∈ X˜ hT˜ (p) ≤ T   X  T − OP1 (1) (Q) . = Card p ∈ NQ (K) hHQ (p) ≤ N h (Q)≤T O 1 P (1)

We apply Theorem 3.6 to the NQ ’s and we obtain ( ) T − hOP1 (1) (Q)  Card p ∈ NQ (K) hHQ (p) ≤ N ∼ b exp(−hOP1 (1) (Q) ) exp N ·

T − hOP1 (1) (Q)

!!

N

= b exp(T ) exp(−2hOP1 (1) (Q)); so we obtain

X

˜ K; T ) ∼ b exp(T ) N(X; hO

P1 (1)

(Q)≤T

exp(−2hOP1 (1) (Q)).

Now, since we suppose T ∈ N, we have ˜ K; T ) N(X; ∼ b exp(T )

T X

X

i=0 i−1≤hO

∼ b exp(T )

T X i=0

P1 (1)

(Q)≤i

exp(−2hOP1 (1) (Q))

n o  exp(−2i) Card Q ∈ P1K (K) i −1 ≤ hOP1 (1) (Q) ≤ i .

On the number of points of bounded height on arithmetic projective spaces

475

We apply Theorem 3.6 to P1K and we obtain that n o  Card Q ∈ P1K (K) i − 1 ≤ hOP1 (1) (Q) ≤ i ∼ c exp(2i) so ˜ K; T ) ∼ b exp(T ) N (X;

T X

c

i=0

∼ aT exp(T ).

t u

Remark. If we blow up a linear subspace of codimension bigger than two, then the NQ ’s will not be isomorphic to hyperplanes (but just birationally equivalent to hyperplanes) so we cannot simply apply Theorem 3.6. Remark. If we want to deal with PN K blown up at more than one linear subspace, we need a theorem similar to Theorem 3.6 for metrics which are not Fubini–Study. References [A]

Arakelov, S.Ju.: Intersection Theory of Divisors on a Arithmetic Surface. Math. USSR Isvestija Vol. 8, 6 (1974) [BM] Batyrev, V., Manin, Y.: Sur le nombre de points rationnels de hauteur bornée des variétés algebriques. Math. Ann. 286, 27–43 (1990) [BoGS] Bost, J., Gillet, J., Soulé, C.: Heights of projective varieties and positive Green forms. J. of the AMS, 71, 903–1027 (1994) [Ca] Cassels, J.S.: An introduction to the theory of numbers. Berlin–Heidelberg–New York: Springer Verlag, 1971 [Fa] Faltings, G.: Diophantine approximations on abelian varieties. Ann. of Math. 133, 549–596 (1991) [FMT] Franke, J., Manin, Y., Tschinkel, Y.: Rational points of bounded heigh on Fano varietes. Inv. Math. 95, 421–435 (1989) [Ga] Gasbarri, C.: Heights and geometric invariant theory. Preprint, Rennes (1997) [GS] Gillet, H., Soulé, C.: Arithmetic Intersection Theory. Publications Math. IHES, vol. 72 (1990) [La] Lang, S.: Algebraic Number Theory. Berlin–Heidelberg–New York: SpringerVerlag, 1971 [Pe] Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79, 101–218 [Sc] Schanuel, S.: Heights in number fields. Bull. Soc. Math. France, 107, 433–449 (1979) [SABK] Soulé, C., Abramovich, D., Burnol, J.F., Kramer, J.: Lectures in Arakelov geometry. Cambridge Studies Adv. Math., Vol. 33, Cambridge: Cambridge Univ. Press, 1992 [Sz] Szpiro, L.: Degrés, Intersections, Hauteurs. Séminaire sur les Pinceaux Arithmétiques: la Conjecture de Mordell, L. Szpiro ed., Exposé 1, Asterisque 127, Paris, (1985) [T] Thunder, J.: An asymptotic estimete for heights of algebraic subspaces. Trans. A.M.S. 331, 395–424 (1992) [We] Weil, A.: Number theory and algebraic geometry. Proc Internat. Math. Congress, Vol II, Cambridge MA: 1950, pp. 90–100