On the Number of Rational Points of Bounded Height on Smooth Bilinear Hypersurfaces in Biprojective Space

HEIGHT ON SMOOTH BILINEAR HYPERSURFACES IN

BIPROJECTIVE SPACE

MARCELLO ROBBIANI

A

Asymptotic formulae for the number of rational points of bounded height on flag varieties have earlier been established. In the paper these asymptotic formulae are recovered by a new method for varieties in biprojective space defined over that are isomorphic to the flag variety of lines in hyperplanes.

The result is obtained by an application of Heath-Brown’s new form of the circle method. It serves as a pointer to the investigation of rational points of bounded height on varieties in multiprojective space.

1. Introduction

Let k be a number field. In the last few years a great deal of effort has been exerted to establish some asymptotic formulae for the number of k-rational points of bounded height on FanoŠarieties X. One expects results of the following form.

Suppose that the k-rational points are dense in X and that there exists a Zariski-open subset U of X defined over k, in which the k-rational points are in some precise sense ‘ homogeneous ’ and their density ‘ minimal ’. Let H be one of the multiplicative anticanonical heights on X and letρ be the rank of the Picard group Pic X. Then, as

Btends to infinity, the integer

N(U, B)l cardoP ? U(k) Q H(P) Bq

satisfies

N(U, B)" CB(log(B))ρ−",

where C is some constant that depends only on X and k.

Asymptotic formulae of this type have been established for certain classes of Fano varieties, for example in [1–3, 6, 12, 15, 16, 20]. These results are in agreement with general interpretations by Batyrev, Manin and Peyre, which state a deep relationship between geometry and arithmetic on Fano varieties. For precise statements we refer for example to [1, 6, 12].

For a summary of the research in this area we may also refer to [11, ‘ Anticanonical varieties and rational points ’].

2. On the method of Schanuel, Peyre and Salberger

A method that has been developed in [12, 16, 17] to tackle this type of problem is now presented.

Received 15 October 1998 ; final revision 18 April 2000. 2000 Mathematics Subject Classification 11G35.

Let be the ring of integers of k. One proves, under some additional conditions on X, that there exist a finite set of places S of k,_S-models Xg of X and Rg of the Ne! ron–Severi-torus of X (the dual of the Picard-group-scheme), and a finite family of

_S-schemesα, such that

pα:α,- Xg

are torsors ( principal homogeneous spaces) oŠer Xg under Rg, and such that as a disjoint union



α

pαα(_S)l Xg(_S). As X is projective one also has

X(k)l Xg(_S).

Thus, after choosing sets$αof representatives for the action of Rg(_S), one obtains a one-to-one parametrisation of the k-rational points on X by the_S-points in



α $

α_Eα_(

S).

For toric varieties eachαis the complement in some n

_Sof a finite union of

subvarieties of at least codimension 2. Hence, the initial problem reduces to that of counting points in n₍

S) that satisfy some primitiveness conditions and various inequalities. Such lattice point problems are classical and can be solved by methods from algebraic number theory and the geometry of numbers.

The next natural class of varieties to be considered is the class of smooth hypersurfaces X in a toric Fano variety Y. In this case, one can try to attack the problem by the following programme.

If Y is, say, of dimension greater than or equal to 3, then Pic X and Pic Y are, by a well known theorem of Grothendieck and Lefschetz (see for example [7, Chapter IV, Theorem 3.1 and Corollary 3.3]), isomorphic, and one can find torsors _Xα over X with the above properties by restricting similar torsors

pα:_Yα,- Y.

Therefore the _Xα are open subsets of hypersurfaces in affine spaces. Thus choosing appropriate _S-models one can parametrise the k-rational points on X by the_S-points on affine hypersurfaces that satisfy some primitiveness conditions and various inequalities. One hopes to solve this last problem, at least for certain classes of varieties, by the circle method.

This hope is justified by the fact that, when Y is the projective space, various results obtained by the circle method, for example in [4, 8, 9, 18], fit into this programme. For a discussion in detail we may refer to [6, 12].

3. On bilinear hypersurfaces in biprojectiŠe space Let n 2, and let X be a hypersurface in n

indefined by an equation F(xiy) l n i=! n j=! m_ijx_iy_jl 0,

where xiy or (x!:…:x_n)i(y!:…:y_n) are bihomogeneous coordinates, and [m_ij] is a matrix with entries in. Note that X is smooth if and only if [m_ij] is regular. The dimension of X is 2nk1.

Let Yg be the subscheme ofn

in, defined by x:y l 0. As [mij] is regular, one has

X% Y l Yg "_.

The condition x:y l 0 may be seen as an incidence relation. In fact X is isomorphic to the flag variety of lines in hyperplanes.

Recall thatn

in is a smooth toric Fano variety, that is a variety with ample

anticanonical class. By the adjunction formula the anticanonical sheaf of X can be obtained by restriction fromnin_{to X of the sheaf}(n, n). It follows that X is itself Fano. An embedding of X into projective space can be given by a choice of a set of generators for the space of global sectionsΓ(X, O(n, n)), that is by a set of monomials of bidegree (n, n). Such an embedding will be called anticanonical.

For any projective embeddingφ of X the notion of height can be defined. Let -be the set of all places of, and let Q Q₎be for all) ? - the usual )-adic norm (we write at infinity Q Q for the usual real norm). The height H(p) of a -rational point p in projective space represented by homogeneous coordinates ( p!:…:pi: … : pn) is defined by the formula H(p)l  )?-max i Qp_iQ).

This height is by the product formula 

)?-QqQ)l 1

for q? * independent on the choice of the representative coordinates. In particular if p is represented by relatively prime integral coordinates one has

H(p)l max i

Qp_iQ.

The height H(xiy) of a point xiy of X() attached to an embedding φ is defined by the formula

H_φ(xiy) l H(φ(xiy)).

As outlined in the introduction one is interested in the asymptotical behaviour of

counting functions

N(X,φ, B) l cardoxiy ? X() Q H_φ(xiy)  Bq.

Asymptotic formulae for counting functions of rational points on flag varieties have been established in [6, 20]. The purpose of this paper is to give an alternative proof of the following theorem.

T 1. Assume that n  3, and let φ be an embedding attached to the sheaf

O(1, 1). There exists a positiŠe constant C such that for B tending to infinity

N(X,φ, B) " CBn_{log (B).}

Cdepends on the embedding, however, the order of magnitude does not. In fact, by a theorem of Weil (see for example [10]), for any pair of embeddingsφ and φh attached to the sheafs O(1, 1) and O(n, n), respectively, there exists a positive constant

C, such that for any point xiy in X() one has

C−"H

φ(xiy)n Hφh(xiy)  CHφ(xiy)n.

It follows in particular, as the rank of Pic X is 2, that this result is in accordance with the general philosophy outlined in the introduction.

By similar arguments, one sees that for any embeddingφ attached to O(1, 1) the behaviour of N(X,φ, B) for B tending to infinity is similar to the one of N(Y, ψ, B), whereψ is the Segre embedding that maps a point xiy to

(x!y!:…:xiyj: … : xnyn). It suffices therefore to prove Theorem 1 for N(Y,ψ, B).

The method of Schanuel, Peyre and Salberger reduces in this case to the fact that each point in Y() can be represented uniquely by an integral point in its affine cone whose coordinates are relatively prime and contained in a fundamental domain for the four possible choices of signs. Indeed, let  be the regular subscheme of

n+"

 i n+ "associated to the affine cone of Y. One can show that is a torsor over

Ygunder #_m, the Ne! ron–Severi-torus of Yg. Hence, each point in Yg() is (up to the action of#_m()) in bijection with a point in ().

Define

H(xiy) l max i,j

Qx_iy_jQ,

where xiy or (x!,…,x_n)i(y!,…,y_n) are now affine coordinates. Then for integral points given by relatively prime coordinates one has (carrying on with this precarious notation) the identity

H(xiy) l H_ψ(xiy).

Now, to determine the asymptotical behaviour of N(Y,ψ, B), one will proceed as follows. At a first stage one forgets about the primitiveness condition and establishes an asymptotic formula for N(B), the cardinality of the set

(B) l oxiy ? n+"in+" Q x

max
0, ymax
0, x:y l 0, H(xiy)  Bq,

where the index ‘ max ’ indicates the coordinate whose absolute value is maximal. At a second stage, one takes into account of the primality conditions by a sieve-argument, the so-called Mo$ bius-inversion, which is particularly simple in this case.

4. On the method of Heath-Brown We begin by defining ω!(x)l12 3 4 exp(k(1kx#)−"), QxQ 1, 0, QxQ  1, and c!l

&

R 2. h(x, y) is an infinitely differentiable real function defined on (0,_)i, which is non-zero only for

x max(1, 2QyQ),

and which satisfies for all y

Qh(x, y)Q 1

The following result is due to Duke, Friedlander and Iwaniec [5]. T 3. For any integer n let

δ_nl12

3

4

1, nl 0,

0, n 0.

Then for any Q
1 there is a positiŠe constant c_Q, which satisfies

c_Ql 1jO_N(Q−N₎

for any N
0, such that

δ_nl c_QQ−#  _ q="  d(q) * e_q(dn) h

0

1

Here the notation e(x) for exp(2πix), e_q(x) for e(x\q), and d(q) for d mod q, has been introduced. Moreover* indicates a summation for residues d with (d, q) l 1. Let F(x) be a polynomial in n variables with coefficients in , and let w(x) be a bounded function well-defined on the integral points of a compact subset ofn_{. The} aim is to estimate

N(F, w)l  w(x), the sum being taken over all x? n _{for which F(x)}l 0.

Observe that

N(F, w)l  w(x) δ_F(x₎.

Hence Theorem 3 leads immediately to the following corollary. C 4. For any constant Q
1

N(F, w)l  x c_QQ−#  _ q="  d(q) * w(x) e_q(dF(x)) h

0

1

The starting point of the method of Heath-Brown is the following result, which can be derived from Corollary 4.

T 5. If, moreoŠer, w is a ‘smooth weight function’, that is an infinitely

differentiable function with compact support, then N(F, w)l  c?n _ q=" q−n_S q(c) Iq(c), where S_q(c)l  d(q) * b_(q) e_q(dF(b)jc:b)), and I_q(c)l

&

0

1

R 6. For a generalisation along the lines of [13] of the classical circle method and its application to hypersurfaces of the former type we refer to [14].

5. On smooth weights Let 0 ε 1\4n. Define the weights

ωε(x)l c−_{! ε}" −"

&

x−ε

−_ω!(ε −"y) dy,

and ω`_ε(x) by replacing ‘ xkε’ by ‘xjε’. Assume now that, say,

x!lmax_i QxiQ, y!lmax

i Qy_iQ.

This is case A. Case B, in which the maximal indices differ, can be treated similarly, and is in fact slightly simpler. Some details are given in Section 6 and in Section 7.

Set

xl (x",…,x_n), yl (y",…,y_n). Define w_{ε, B, (}

!,!)l wε,B(x!,y!,x,y) as the product of

w_{ε,B(x!,y!)lωε(x!k1)ωε}(Bkx!y!)ω_{ε( y!kNB),} and w_{ε(x!,x)l } i! ωε(1kQxiQ\x!)we(x!,x), with w_e(x!,x)l1k

0

1

!,!)l wε,B(x!,y!,x,y) in a similar way.

This choice is justified on the one hand by the fact that by [9, Theorem 5] the number of points whose coordinates are bounded byNB is O(Bn_{). On the other hand} in Section 7 it will be shown that there exists a positive constant C!,!and a function

B(ε) such that for B  B(ε)

QN(F, wε,B(x!,y!,x,y))kC!,!Bnlog(B)Q  O(ε) Bnlog(B),

where

F_{(x!,y!,x,y)lx!y!jx:y.}

It will also be shown that a similar inequality with the same constant C!,!holds if w

is replaced by w. Similar results, with constants C_i,j, hold for all other choices of pairs of indexes.

Hence, one concludes for B tending to infinity that

N(B)" 2  i,j

N(F, w_{ε,B, (i, j)}" 2  i,j

C_i,jBn_log(B),

where it makes no difference if w is underlined or overlined. Note that the asymmetries of the weights are corrected by the factor 2.

The identity

x!y!jx:yl0

implies the extra condition

Qx_iQ x!

n (1)

for at least one index i 0. Thus the extra factor w_ehas no influence on the counting problem. As will become clear in Section 7, its role is purely technical.

R 7. In case B no extra factor is needed.

Finally, one is led to evaluate N(F, w_{ε,B(x!,y!,x,y)). Drop all the indices and} subscripts and write this cardinal as

 x_!,y_!  x_,y w_{(x!,y!,x,y)δF(x} !,y_!,x_,y₎. Split the inner sum into

 x_!,y_! w_(x!,y!) x w_(x!,x) y w_{( y!,y)δF(x} !,y_!,x_,y₎, and apply to the innermost sum Corollary 4 with

Ql Nx!y!.

Then Theorem 5 states that the innermost sum can be replaced by  c?n _ q=" q−n_S q(x!,y!,x)(c)Iq(x!,y!,x)(c), where S_{q(x!,y!,x)(c) is equal to}

 d(q)

* b_(q)

e_{q(dF(x!,y!,x,b)jc:b),}

and I_{q(x!,y!,x)(c) is up to a factor c(x!,y!) equal to}

&

w_{( y!,y)}

x!y! h

0

q

Nx!y!,x!y!jx:yx!y!

1

)

x!y!

)

by Remark 2 these integrals are non-zero only if

q Nx!y!. (2)

Therefore this condition will be set through this paper as a general assumption. Note that despite having defined weights on all x_i and y_j Theorem 5 is in fact applied only to y. Hence the preceding step may be viewed as an application of the method of Heath-Brown to an inhomogeneous linear equation.

6. On the error term

P 8. The expression

)

)

To handle I_{q(x!,y!,x)(c) write}

yl y!v. Then I_{q(x!,y!,x)(c) transforms up to the factor x}−"

! yn−! " (and the negligible factor

c_{(x!,y!)) into}

&

w(v) h

0

Nx!y!, 1jx\x!:v

1

where w(v) is the corresponding transformed weight. Set

ω"(s)lω!

0

1

Then, as in [9, Section 7], one may use the Fourier transform to write

ω"(1jx\x!:v)h

0

1

&

&

0

1

On supp w(v) (and for the x! and x one is interested in) the expression Q1jx\x!:vQ

is bounded by 2n, and therefore the expression

ω"(1jx\x!:v)

is strictly greater than some positive constant. Hence integral (3) may be expressed as

&

&

One needs now several results on oscillatory integrals. One could invoke [19, Chapter VIII] or follow [9, Section 6, Section 7]. In what follows the second alternative has been chosen.

L 9. For all c  0, for all sufficiently small ε
0, and for all positiŠe integers

N one has

)&

&

p(t)ψ(v) e(φ(v, t)) dv dt

)

! y−N! . Set, say,

Qc"Qlmax_i Qc_iQ.

To prove Lemma 9 use now the fact that inside suppψ and for QtQ  1

2n QcQ

qy!

the partial derivative ofφ with respect to Š" satisfies

)

)

recall thatQx"Q\x!1jε.

On repeated integration by parts and estimating q by (2) Lemma 9 follows from

&

&

&

Df(Š")l(iφh(Š"))−" f h(Š")

and Dt _{for its transpose}

Dt_f(Š")lk

0

iφh(Š")

1

and from the fact that Remark 2 yields

&

L 10. For all sufficiently small ε
0, for all c with QcQ

0

1

and for all positiŠe integers N
1\ε one has

)&

&

p(t)ψ(v) e(φ(v, t)) dv dt

)

! y−N! .

Lemma 10 is an immediate consequence of the second statement of [9, Lemma 17] (note that the function p in [9] contains an additional multiplicative factor q\Nx!y!).

L 11. For all positiŠe integers N one has Qp(t)Q 

0

qt

1

N . By (2) there exists a positive constant c! such that

q c!Nx!y!.

Following [9] a point x? n _{such that grad}φ is small at x is called a ‘bad point’. Precisely, x is said to be ‘ bad ’, if its distance from the line segment

xlx!y! qt c, with QtQ  1 2n QcQ qy!, is smaller than

0

1

c

0

q

1

ε

,

for all positiŠe integers N, and for all x that are not ‘bad’ one has

)&

&

p(t)ψ(v) e(φ(v, t)) dv dt

)

! y−N! .

If x is ‘ bad ’ then one has instead

)&

&

p(t)ψ(v) e(φ(v, t)) dv dt

)

q xN! y−N! .

The first statement in Lemma 12 is obtained on the one hand, following the proof of Lemma 9, on repeated integration by parts with respect to the variableŠ_ifor which φh(Š_i) is maximal at x. On the other hand one makes use of the estimate of Lemma 11 as in the proof of Lemma 10.

The second statement in Lemma 12 is weaker, as integration by parts is no longer available. It follows as Lemma 10 immediately from Lemma 11.

Putting Lemmas 9, 10 and 12 together one is led to evaluate the expressions  x w_{(x!,x) } c!  qNx_!y_! q−n_S q(x!,y!,x)(c)QcQ−NxN−! "y−N+n−! " (4) and  x_‘bad’ w_{(x!,x) } c_!qNx_!y_! q−n−"S q(x!,y!,x)(c)QcQ−NNx!y!xN−! "y−N+n−! ". (5) Note that in the first expression one could limit the summation to the x which are not ‘ bad ’.

L 13. For all c  0 one has

)

w_{(x!,x)Sq(x!,y!,x)(c)}

)

)

)

and observe that by elementary properties of Gauss sums this is equal to

)

* e_{q(dx!y!) } x_−d*c_(q)

w_(x!,x)qn

)

 x−d*c_(q)

w_(x!,x)max

(

qn

*

Applying Lemma 13 to expression (4) one is finally led to evaluate  x_!,y_! w_(x!,y!)xN−" ! y−N+n−! "c! QcQ−N  qNx_!y_! q"−n_maxoqn_{, x}n !q, which is O(Bn_{) for an appropriate choice of N.}

To tackle expression (5) one makes use of the observation that the maximal number of ‘ bad points ’ contained in supp w(x) is bounded by a multiple of

x!

0

1

ε(n−")

. With the trivial estimate

QS_{q(x!,y!,x)(c)Qq}n+", one is finally led to evaluate

 x_!,y_! w_{(x!,y!)Nx!y!"}+ε(n−")_xN ! y−N+n−! "_qN_x !y_! q−ε(n−") c! QcQ−N_,

which is O(Bn/#+") for an appropriate choice of N and ε. This concludes the proof of

Proposition 8.

R 14. In case B assume that, say,

x"lmax_i QxiQ, y!lmax i Qy_iQ. Choosing Ql Nx"y!, and writing y_il y!Š_i,

one has to turnφ(v) in the previous computations into

t

0

x"jŠ"jx#x"Š#j…j

x_n

This slight modification, however, implies no significant modification in the proofs of Lemmas 9, 10 and 12. Hence Proposition 8 remains valid if (x!,y!) is replaced by (x",y!).

Recalling Theorem 1, it makes sense to speak about such expressions as error

terms.

7. On the main term

Proposition 8 suggests that the main contribution to N(B) comes from the summands with cl 0. One is therefore led to evaluate

 x_!y_! w_(x!,y!) x w_{(x!,x) } qNx_!y_! q−n_S q(x!,y!,x)(0)Iq(x!,y!,x)(0). (6) A necessary condition for

S_{q(x!,y!,x)(0)l }

d(q) *

b_(q)

e_{q(dF(x!,y!,x,b))}

to be different from zero is

x 0(q).

If q
x! this implies that xl0. However, as the origin is not contained in supp w(x!,x), this possibility may be excluded, and one may assume that qx!. Expression (6) can now be rewritten as

 x_!,y_! c_(x!,y!) x!y! w(x!,y!) qx_!  a_(q) q−n_S q(x!,y!,a)(0)I(x!,y!,x), (7) where I_{(x!,y!,x)l } xa(q) w_(x!,x)

&

0

1

R 15. In case B a necessary condition for S_{q(x!,y!,x)(0) to be different} from zero is x_max 0(q). Thus, one may assume that q  x_max.

Now, to ensure that condition (1) may be applied, a partition of the domain of integration is needed. One defines the weights

w_{i(x!,x)lw(x!,x)(1kωε}(1\nkQx_iQ\x!))i−" j=" ωε(1\nkQxjQ\x!), for il 1,…, nk1, and w_{n(x!,x)l }n−" j=" ωε(1\nkQxjQ\x!). Then one has

w_(x!,x)ln

j="

w_j(x!,x),

and I(x!,y!,x) can be written as the sum of the

I_{i(x!,y!,x)l } xa(q) w_i(x!,x)

&

0

1

Replace now y_i by tl 1jx x!: y y! to obtain I_{i(x!,y!,x)lx!y!}

&

0

1

&

Here the ‘ hat ’ stands for the fact that the ith variable has been dropped, and

w_{i( y!,t,y#) denotes the corresponding transformed weight function.}

Next, one wants to make use of the fact that, by [9, Lemma 9], for small values of x the function h(x, y) acts very much like a delta-function. Precisely, one has the following lemma.

L 16. Let f be an infinitely differentiable function with compact support.

Suppose that on supp f

Q f(k)_(t)Q  1, k l 0, 1, 2,….

Then, for s 1, and for any N
0

&

Condition (1) ensures that w( y!,t,y#) and, as a consequence, Ji(t) and the corresponding higher derivatives may be bounded by constants that depend only on

ε on supp w_{i(x!,x)w(y!,t,y#). Hence, the hypotheses of Lemma 16 are fulfilled, and}

one concludes that I_{i(x!,y!,x) can be written as}

x!y!

0

&

00

1

11

Note, that by (2) the second term of (8) is O_N(1). Its contribution to (7) is therefore bounded up to a multiplicative constant by

 x_!,y_! w_{(x!,y!) } "qx_!  a_(q) q−n_S q(x!,y!,a)(0)(x!\q)n. (9)

Here one uses that (x!\q)n 1. The fact that this expression is O

N(Bn/#+"), and therefore belongs to the error term, follows withβ l 1 from Lemma 17. This lemma will also be used later in the paper.

L 17. For all x!, y!, and β1, one has 

βq

q−#n

a_(q)

S_{q(x!,y!,a)(0)lO(β}−n+").

In case B a similar result holds, except for nl 3, and x!y!0(q). In that case one

has  βq q−#n−"  a_(q) S_{q(x!,y!,a)(0)lO(β}−n+").

The first statement of Lemma 17 is a direct consequence of [9, Lemma 25] applied to the quadratic form x:y.

The second statement follows from [9, Lemma 25] in a similar elementary way. Note that the first statement is also a direct consequence of the precedent observation that S_{q(x!,y!,a)(0)l} 1 2 3 4 qn _{if a} 0, 0 if a) 0.

R 18. The treatment of the main term in case B is simpler, as no partition of the domain of integration is needed. If, say,

x"lmax_i QxiQ, y!lmax

i Qy_iQ, then replace y" by

tlx!

x"jy"y!jx#y#x"y!j…j x_ny_n

x"y!,

and proceed as in case A. Note, that for nl 3 one has to replace (x!\q)n

by (x!\q)n+" in (9). The upper bound for (9) will be slightly weaker, but still sufficient.

Next, one wants to replace the sums over the x_iby integrals. Asε  1\4n one has (1kω_ε(1\nkQx_iQ\x!))ω_ε(1\2nkQx_iQ\x!)l0.

Thus, for il 1,…, nk1, the weight w_{i(x!,x) can be written as} n

j="

wg_j(x!,xj),

with factors that depend on i. For il n the weight may be written as a difference of two such products, and the subsequent argument is similar.

Write  xa_(q) w_i(x!,x) Qx_iQ l xa_(q) wg_i(x!,xi) Qx_iQ _ji wg_j(x!,xj) (10) or  z n j=" wg_j(x!,aj(q)jqz_j) Qa_i(q)jqz_iQ . For  zj wg _j(x!,aj(q)jqz_j), j i,

Euler’s summation formula yields

&

&

cz_j(x!,aj(q)jqzj) dzj. Sincecwg_j\cz_jis O_ε(qx−"

! ), and is non-vanishing on a set of measure Oε(x!q−"), the last

expression is

1

Similarly  zi wg_i(x!,aj(q)jqz_j) Qa_i(q)jqz_iQ can be evaluated by 1 q

&

&

&

Qx_iQ dxjl Oε(1), expression (10) may be replaced by

q−n

&

n

w_i(x!,x)

Qx_iQ dxjOε(xn−_! #q−n+").

Observe that the second summand belongs to an error term. Indeed, plugging this summand into (7) and treating the cases q Nx! and qNx! separately, one is led on the one hand to evaluate after multiplication with an extra x"/#

! q−"-factor  x_!,y_! w_(x!,y!)xn−$/# ! yn−! "_qN_x !  a_(q) q−#n_S q(x!,y!,a)(0), where one makes use of

&

w_{( y!,0,y#)dy#lOε}( yn−" ! ).

Lemma 17 then ensures that this expression is O_ε(Bn_{). On the other hand one has to} evaluate after multiplication with an extra x!q−"-factor

 x_!,y_! w_(x!,y!)xn−" ! yn−! "_Nx !qx_!  a_(q) q−#n_S q(x!,y!,a)(0). By Lemma 17 this expression is bounded by

 x_!, y_!

w_(x!,y!)xn/#−"/#

! yn−! l" Oε(Bn).

R 19. To be able to proceed in a similar way for n l 3 in case B, one has to split the summation in a slightly different way : q x"/$

! , x"! /$ q x#!/$, x#! /$

q x!.

Concerning the main term one is left to evaluate, after the transformations xl x!u and y#ly!v#, the expressions

 x_!,y_! c_{(x!,y!)w(x!,y!)x}n−" ! yn−! "_qx !  a_(q) q−#n_S q(x!,y!,a)(0)

&

Since  x_!,y_! c_{(x!,y!)w(x!,y!)x}n−" ! yn−! "_qx !  a_(q) q−#n_S q(x!,y!,a)(0) is by Lemma 17 bounded up to a multiplicative constant by

 x_!,y_!

c_{(x!,y!)w(x!,y!)y}n−" ! ,

which is O(Bn_{), one may sum in (11) over all q} NB. By the same argument one may reduce the summation to q log(B)"/#−ε_.

R 20. To make this argument work in case B and for n l 3 one has, as for previous contributions to the error term, to introduce an extra x!\q-term.

Note at this stage that the integral in (11) contributes to the Tamagawa number

at infinity. To see the link with the treatment of the constant in [9], one has to make

use of [9, Theorem 3]. A ‘ light ’ version of this theorem is reproduced next.

T 21. Let G(s",…,s_{n) be a real function, and w(s",…,sn}) be a smooth

weight function. Suppose that the gradient of G does notŠanish on the closure of the

support of w. Suppose that on the same set firstly s_i is uniquely determined by the

condition G(s)l 0, and secondly cG\cs_i is positiŠe and bounded from below. Then

lim δ _! 1 2δ

&

&

0

1

In the present case set

G(u, v)lp(1ju:v) depending onpu_i
0. As

cG

cŠ_ilpuilpxi\x!1

the hypotheses of Theorem 21 are satisfied on the two disjunct subsets of supp w_i(u) w(v) characterised by pu_i
0. Hence the inner integral in the above expression may be replaced by

1

2δ

&

w_i(u) w(v) dv dujO(δ),

forδ arbitrarily small.

Let w(u) be the sum of the w_i(u). Set

σ_(!,!)(w)l lim δ _! 1 2δ

&

!,!)l σ(!,!)(χ), where χ is the characteristic function of the 1-box at the

origin.

R 22. In a similar way one obtains constants σ#_(i,j) for the remaining pairs (i, j).

As the third sum in  qlog_(B)"/#−ε  a_!(q),a_!(q) a_(q) q−#n_S q(a!,a!,a)(0)  x_!a_!(q) y_!a_!(q) c_{(x!,y!)w(x!,y!)x}n−" ! yn−! " is 1 q#

0

1

Following the final part of [9, Section 11], one begins with the observation that the corresponding infinite sum is absolutely convergent, and that one can show that

 qlog_(B)"/#−ε q−(#n+#)_S q(0)l  _ q=" q−(#n+#)_S q(0)jO(log(B)("−n)/#+ε). Moreover q−(#n+#)_S q(0) is multiplicative. Therefore _ q=" q−(#n+#)_S q(0)l  pprime σ_p, where σ_pl _ t=! p−(#n+#)t_S pt(0) or, as in the usual analysis of the singular series,

σ_pl lim k _

p−(#n+")k_cardo(x, y) (pk₎Q xiy  0(pk₎q. The various findings can be summarised as follows.

P 23. One has  i,j N(F, w_ε,B,(i,j))l 1 2n_p σp

0

σ#_(i,j)

1

and in particular  i,j N(F, w_ε,B,(i,j))"  i,j N(F, w_ε,B,(i,j)). The following corollary is an immediate consequence. C 24. One has N(B)l1 n_p σpσ_Bnlog(B) (1jo(1)), where σ_l  i,j σ#_(i,j).

8. On the constant To derive Theorem 1 from Corollary 24 one writes

N(X,φ, B) l  _ l _ k µ(l) µ(k) N(B\kl),

whereµ denotes the standard MoWbius function. Therefore

N(X,φ, B) l1

n_p σpσpσ_Bnlog(B) (1jo(1)), where

σ_pl (1kp−n)#.

This concludes the proof of Theorem 1. Finally some considerations should be made on the constant. Observe that

σ_pσ_pl lim k _

p−(#n+")k_cardo(x, y) (pk₎Q xiy  0(pk_{), (x, p)}l (y, p) l 1q. The model has good reduction. Hence the limit stabilises at kl 1, and

σ_pσ_plcardo(p)q p#n+" l cardo#_m(_p)q p# cardoYg(_p)q p#n−" , with_pl \p.

Asσ_{_}is the Tamagawa factor at infinity,σ_{_}_pσ_pσ_padmits an interpretation as the Tamagawa number of the variety X, which is in accordance with [12, 16].

In [12] it is conjectured that, beside the Tamagawa number of the variety, an additional constant should appear in the asymptotical formula. It should be equal to a particular measure of a domain inρ, whereρ is the rank of Pic X. For a precise statement see [12, Definition 2.4].

In the present case the 1-dimensional domain in# that has to be considered is given with respect to the anticanonical sheaf by

z",z#0, nz"jnz#l1,

and the constant turns out to be equal to 1\n#.

Passing from O(n, n) to O(1, 1) involves a factor n. Therefore the present result is in accordance with Peyre’s conjecture, and the asymptotical formula is in accordance with the general philosophy on the interplay of geometry and arithmetics on Fano varieties.

Acknowledgements. The author would like to express his gratitude to D. R.

Heath-Brown, for the invitation to Oxford, for many helpful discussions, and for his patience in reading an earlier, far from flawless draft of this paper.

The author is also indebted to P. Salberger, for having suggested the problem and [9] as a key to its solution, and to D. Coray, for his constant interest in this work.

It is also a pleasure to record the logistical support of the Mathematical Institute, Oxford, where this work was carried out, and the financial support of the Swiss National Research Foundation, which was a conditio sine qua non for this research.

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