Heath-Brown, D. R.

Moroz, B. Z.

English

Mathematical Institute Eprints Archive

The Density of Rational Points on the Cubic
Surface X30 = X1X2X3
D.R. Heath-Brown
Magdalen College, Oxford
and
B.Z. Moroz
Max Planck Institute, Bonn
1 Introduction and Statement of Results
In this paper we are concerned with the problem of counting rational points of
bounded height on rational cubic surfaces. For most such surfaces this question
appears much too hard for current methods. We shall therefore examine a
particular example, namely the surface
X30 = X1X2X3, (1)
for which the problem is tractable, though by no means trivial. This surface
contains the lines X0 = Xi = 0, for i = 1, 2, 3, which we shall exclude from
consideration. We therefore define
N (H) = #{x ∈ N4 : x30 = x1x2x3, x primitive, xi ≤ H, (i = 0, 1, 2, 3)},
where an integer vector is said to be primitive if its components have no common
factor. The function N (H) counts points with the most naive height available.
This suffices, however, to bring out the key features of the problem. We may
observe at once that the condition x0 ≤ H is redundant, and that it suffices to
assume that (x1, x2, x3) is primitive.
Our principal result is the following.
Theorem As H tends to infinity we have
N (H) = H(logH)
6
4× 6!
∏
p
σp +O(H(logH)5),
where
σp = (1− 1
p
)7(1 + 7
1
p
+
1
p2
).
1
It should be stressed that the entire difficulty of this question lies in the
constraints imposed on the size of x1, x2 and x3. If we merely require that
x0 ≤ H, the problem is far easier. Indeed if we let F (x0) denote the number of
representations of x30 as a product of three coprime factors, then it is apparent
that F (n) is a multiplicative function, and it is an easy exercise to show that∑
n≤H F (n) is asymptotically cH(logH)
8, where c is an appropriate positive
constant.
The estimate given by the theorem may be viewed as an instance of a very
general conjecture of Manin (see Batyrev and Manin [1]), which predicts the
occurence of such asymptotic formulae, and describes the exponents of H and
logH which should appear. The particular surface (1) has attracted atten-
tion elsewhere, and versions of the asymptotic formula have been obtained by
Batyrev and Tschinkel [2], by Fouvry [4], and, in work still in preparation, by
Salberger and by de la Brete`che [3]. Batyrev and Tschinkel use an analytic
method, based on a related non-singular toric variety. The approach employed
by de la Brete`che is also analytic, but in the more traditional sense. Salberger’s
treatment is based on consideration of the ‘universal torsor’ of the surface (1).
Like that of Batyrev and Tschinkel, it has the advantage of providing a full
interpretation of the constants in the asymptotic formula. Technically it is ‘el-
ementary’ though couched in highly geometric language. Fouvry’s approach is
elementary, in the traditional sense, though certainly not simple. In contrast to
all these, we believe that our treatment, which is also elementary, has the advan-
tage of being both straightforward and natural. Moreover there is no obvious
obstacle to its generalization to other related counting problems.
2 Interpretation of the Theorem
The simplest way to interpret the theorem is to look at the density of solutions
of (1) in each of the completions of Q. In the case of a p-adic completion we
define
σ′p = lim
n→∞
1
p3n
#{x (mod pn) : x30 ≡ x1x2x3 (mod pn), p |/x}.
It is not difficult to see that there are φ(pn)3 solutions with p |/ x1, x2, x3, since
then x0, x1 and x2 determine x3. Similarly there are 3φ(pn)2pn−1 solutions in
which p divides exactly one of x1, x2 or x3, since in this case, if p|x3 for example,
x1, x2 and p−1x0 determine x3 modulo pn. Finally, if k+j is a multiple of three,
and is less than n, there are
φ(pn−k)φ(pn−j)φ(pn−(k+j)/3)pj+k
2
solutions in which pk||x1 and pj ||x2. It follows that there are
3p3n(1− 1
p
)3
∞∑
k,j=1
3|k+j
p−(k+j)/3 +O(n2p3n−[n/3])
= 3(2 + p−1)(1− p−1)p3n−1 +O(n2p3n−[n/3])
solutions in which p divides exactly two of x1, x2 or x3. We therefore see that
σ′p = (1− p−1)(1 + 7p−1 + p−2).
We may also calculate the real density of solutions as
σ∞ =
∫ H
0
∫ H
0
∫ H
0
dx1dx2dx3
3(x1x2x3)2/3
= 9H.
The product of the p-adic densities is divergent, and the standard procedure is
to introduce a ‘renormalising factor’ of (1− p−1)6, compensated for by a factor
(logH)6. This leads us to compare N (H) with
9H(logH)6
∏
p
(1− p−1)7(1 + 7p−1 + p−2),
and we see that the two differ by a factor which tends to 36× 6!.
The renormalization procedure above has a distinct air of arbitrariness about
it, and a more sophisticated interpretation of the theorem comes from consider-
ation of the ‘main term’ in the analysis of the Hardy-Littlewood circle method.
We shall not go into this in detail. However we point out that the ‘main term’
involves a partial sum of the ‘singular series’, taking the form∑
q≤Q
aq.
If this sum is evaluated via Perron’s formula, one needs to examine the residue
of Z(s)Qss−1 at s = 0, where Z(s) =
∑∞
1 aqq
−s. Since the function Z(s) turns
out to have a pole of order 7 at the origin, this process produces a ‘main term’
containing a factor (logQ)6/6!. This goes some way towards explaining the
presence of the factor 6! in our theorem. However there will inevitably be other
terms arising out of the circle method, in addition to what has been called here
the ‘main term’.
3 Proof of the Theorem
As has already been remarked, N (H) is the number of primitive integer vectors
x = (x1, x2, x3) with 1 ≤ xi ≤ H for i = 1, 2, 3, and such that x1x2x3 is a
3
cube. For such a vector we write xi = yiz3i where yi is cube-free. Then y1y2y3
will be a cube, y3, say, and the vectors y = (y1, y2, y3) and z = (z1, z2, z3) will
be primitive. Moreover if we set w1 = (y2, y3), and similarly w2 = (y1, y3),
w3 = (y1, y2), we must have (zi, wi) = 1 for i = 1, 2, 3. One may readily check,
conversely, that these conditions imply that the original vector x is primitive.
Notice that the integers w1, w2 and w3 must be coprime in pairs.
We proceed to count the number of possible vectors z corresponding to each
vector y. This will be achieved by means of the following estimate.
Lemma 1 Let w1, w2 and w3 be positive integers, which are coprime in pairs.
Let Z1, Z2 and Z3 be any positive real numbers. Then there are
Z1Z2Z3
ζ(3)
f(w1w2w3) +O(g(w1w2w3)Z1Z2Z3{maxZ−1/2i })
primitive integer vectors (z1, z2, z3) such that 1 ≤ zi ≤ Zi and (zi, wi) = 1 for
i = 1, 2, 3.
Here f(d) and g(d) are multiplicative functions defined by
f(pe) =
1− p−1
1− p−3 , g(p
e) = 1 + p−1/2, (2)
for every prime power pe > 1.
We shall prove this in the next section.
It is now apparent, on taking Zi = (H/yi)1/3, that
N (H) =M(H) +R(H)
where
M(H) = Hζ(3)−1
∑
y
f(w1w2w3)
y
, (3)
and
R(H)¿ H5/6
∑
y
g(w1w2w3)(y1y2y3)−1/3
∑
i
y
1/6
i .
Here y = (y1, y2, y3) runs over primitive integer vectors with 1 ≤ yi ≤ H,
subject to the restriction that y1y2y3 is a cube.
We now analyse further the conditions on y. Any prime factor of y1y2y3
must divide exactly two of y1, y2 and y3. Moreover it will divide one of these
with exponent 1 and the other with exponent 2. We therefore define yij (for
i 6= j) to be the product of those primes p for which p||yi and p2||yj . With
this notation we find that the numbers yij are square-free and coprime in pairs.
Moreover we have
y1 = y12y13(y21y31)2, y2 = y21y23(y12y32)2, and y3 = y31y32(y13y23)2.
4
It follows that w1 = y23y32 and similarly for w2 and w3, whence
∏
i wi =
∏
i,j yij .
We note also that
∏
i,j yij = y, where
∏
i yi = y
3.
We are now ready to estimate R(H). We have
R(H) ¿ H5/6
∑
yij : yi≤H
∗ g(w1w2w3)(y1y2y3)−1/3
∑
i
y
1/6
i
¿ H5/6
∑
yij : yi≤H
∗ g(w1w2w3)y
−1/3
1 y
−1/3
2 y
−1/6
3 ,
by symmetry. Here we have used Σ∗ to denote restriction to variables yij which
are square-free and coprime in pairs. It follows that
R(H)¿ H5/6
∑
yij
∗
∏
i,j g(yij)
y12y21(y13y23)2/3(y31y32)5/6
,
where the summation is subject to the conditions
y12y13(y21y31)2 ≤ H, y21y23(y12y32)2 ≤ H, and y31y32(y13y23)2 ≤ H. (4)
If we substitute y31y32 = u, this leads to the estimate
R(H)¿ H5/6
∑
yij≤H, (i 6=3)
∏
i,j: (i 6=3) g(yij)
y12y21(y13y23)2/3
∑
u≤U
G(u)u−5/6,
in which U = H(y13y23)−2 and
G(u) =
∑
u=u1u2
g(u1u2).
Since the Dirichlet series generating function for G(u) has a double pole at
s = 1, a standard Tauberian argument shows that∑
u≤U
G(u)u−5/6 ¿ U1/6 logU ¿ H1/6(logH)(y13y23)−1/3,
whence
R(H) ¿ H(logH)
∑
yij≤H, (i 6=3)
∏
i,j: (i 6=3) g(yij)
y12y21y13y23
¿ H(logH){
∑
h≤H
g(h)/h}4.
The Dirichlet series generating function for g(h) has a simple pole at s = 1 and
the usual Tauberian argument therefore shows that∑
h≤H
g(h)/h¿ logH.
5
Hence we conclude that
R(H)¿ H(logH)5. (5)
We turn now to the leading term M(H). We shall write (3) in the form
M(H) = Hζ(3)−1
∑
yij
∗∗y−1
∏
i 6=j
µ2(yij)f(yij),
where Σ∗∗ denotes restriction to variables yij which are coprime in pairs, and
subject to the inequalities (4), but are not necessarily square-free. We introduce
the multiplicative function h(d) defined by
h(pe) =
 f(p)− 1, e = 1,−f(p), e = 2,0, e ≥ 3, (6)
so that
µ2(n)f(n) =
∑
d|n
h(d).
We may now write
M(H) = Hζ(3)−1
∑
dij
∏
i,j
h(dij)
∑
yij : dij |yij
∗∗y−1,
where the variables dij (for i 6= j) run over all positive integers, without restric-
tion.
In order to remove the constraint that the yij are coprime in pairs we intro-
duce a factor ∏
i,j,k,l
{
∑
δijkl|yij ,ykl
µ(δijkl)},
where (i, j, k, l) runs over quadruples of integers in the range 1 ≤ i, j, k, l ≤ 3,
subject to the conditions that i 6= j, k 6= l and that the ordered pairs (i, j) and
(k, l) are distinct. It may be observed that the above product contains factors
corresponding to both the conditions (yij , ykl) = 1 and (ykl, yij) = 1. This
however has no material effect on the argument.
We may now write
M(H) = Hζ(3)−1
∑
dij
∏
i,j
h(dij)
∑
δijkl
∏
i,j,k,l
µ(δijkl)
∑
yij : rij |yij
y−1,
where rij is the lowest common multiple of dij and the various δijkl for (i, j) 6=
(k, l). The variables yij are still subject to the inequalities (4). We now call on
the following estimate, which will be proved in the next section.
6
Lemma 2 Let rij be arbitrary positive integers, and write R =
∏
rij. Then∑
yij : rij |yij
y−1 =
1
4
(logH)6
6!R
+O(R−3/4(logH)5),
the summation being over integers yij subject to the inequalities (4).
This result enables us to write
M(H) = κH(logH)6 +O(λH(logH)5), (7)
where
κ = (4ζ(3)6!)−1
∑
dij
∏
i,j
h(dij)
∑
δijkl
∏
i,j,k,l
µ(δijkl)
∏
i,j
r−1ij ,
and
λ =
∑
dij
∏
i,j
|h(dij)|
∑
δijkl
∏
i,j,k,l
µ2(δijkl)
∏
i,j
r
−3/4
ij .
It remains to evaluate κ, and to prove that the infinite sum λ is convergent. We
tackle the latter problem first. Since the terms involved are all multiplicative
functions, it suffices to consider the corresponding Euler factors, in which we
choose a prime p, and restrict the variables to be powers of p. The summand
will then vanish unless dij is 1, p or p2, and each δijkl is 1 or p. It follows
that there are at most 3× 230 terms, since there are 30 admissable quadruples
(i, j, k, l). Moreover, we see from (2) and (6) that |h(p)| ≤ p−1 and |h(p2)| ≤ 1.
Thus, if δijkl = p for any index (i, j, k, l), then p|rij , rkl, whence∏
i,j
|h(dij)|r−3/4ij ≤ p−3/2. (8)
The above estimate holds indeed whenever p2|∏ rij . On the other hand, if∏
rij = p, so that each variable δijkl is 1, there must be some index ij such
that dij = p. In the latter case |h(dij)| ≤ p−1, whence (8) still holds. We
therefore see that (8) holds except when each of the variables dij and δijkl is 1.
We may now conclude that the Euler factor for the prime p is 1+O(p−3/2). The
corresponding product is convergent, and hence the sum λ is also convergent.
Since the sum λ is absolutely convergent, the sum in the definition of κ
is also absolutely convergent. We may therefore evaluate it by considering its
Euler factors Ap, say. This will produce
κ = (4ζ(3)6!)−1
∏
p
Ap. (9)
To evaluate Ap we note that
Ap =
∑
dij
∏
i,j
h(dij)
∑
δijkl
∏
i,j,k,l
µ(δijkl)
∏
i,j
r−1ij ,
7
where the variables are restricted to run over powers of p. As observed above,
this is, in effect, a finite sum. Since rij must divide p2, we may write
p12
∏
i,j
r−1ij = #{n (mod p2) : rij |nij},
where n runs over 6-tuples of integers nij . It follows that
Ap = p−12
∑
dij
∏
i,j
h(dij)
∑
δijkl
∏
i,j,k,l
µ(δijkl)#{n (mod p2) : rij |nij}
= p−12
∑
n(mod p2)
∑
dij |nij
∏
i,j
h(dij)
∑
δijkl|nij ,nkl
∏
i,j,k,l
µ(δijkl),
since rij |nij if and only if dij and each δijkl divides nij . Since the variables δijkl
run over powers of p only, we find that∑
δijkl|nij ,nkl
∏
i,j,k,l
µ(δijkl)
vanishes unless at most one nij is a multiple of p. Otherwise the sum is 1. This
observation shows that
Ap = p−12
∑
n(mod p2)
∑
dij |nij
∏
i,j
h(dij),
the sum over n being restricted to sets of values for which at most one nij is a
multiple of p. In the same way, we find that∑
dij |nij
∏
i,j
h(dij)
is
∏
f((p2, nij)) unless some nij is divisible by p2, in which case the sum van-
ishes. The contribution to Ap arising from terms in which none of the nij is a
multiple of p is therefore p−12(p2− p)6. Similarly the contibution from terms in
which exactly one of the nij is a multiple of p is 6p−12(p2 − p)5(p− 1)f(p). In
view of (2) we therefore find that
Ap =
(1− p−1)7
1− p−3 (1 + 7p
−1 + p−2).
The theorem now follows from (5), (7) and (9).
4 Proof of Lemmas 1 and 2
To establish Lemma 1 we begin with the observation that
#{v ∈ N : v ≤ V, (v, w) = 1} =
∑
d|w
µ(d)[
V
d
]
8
for w ∈ N and V ≥ 0. We now employ the estimate
[θ] = θ +O(θ1/2),
which is valid for all θ ≥ 0. The astute reader will realize that, had the alter-
native error O(1) been used at this point, we would be unable to handle the
later errors purely in terms of convergent infinite sums. The approach used here
considerably reduces the complications involved in dealing with these errors.
We now find that
#{v ∈ N : v ≤ V, (v, w) = 1} = V φ(w)
w
+O(V 1/2g(w)),
where g(w) =
∑
d|w |µ(d)|d−1/2, whence
#{v ∈ N3 : vi ≤ Vi, (vi, wi) = 1, (i = 1, 2, 3)} =
∏
i
{Viφ(wi)
wi
+O(V 1/2i g(wi))}.
At this point we observe that if 0 ≤ ai, bi ≤ ci for i = 1, 2, 3 then∏
i
{ai +O(bi)} =
∏
i
ai +O(c1c2c3{max bi
ci
}).
If each of the Vi is at least 1 we may apply this with
ai = Vi
φ(wi)
wi
, bi = V
1/2
i g(wi), ci = Vig(wi),
whence
#{v ∈ N3 : vi ≤ Vi, (vi, wi) = 1 (i = 1, 2, 3)}
= V1V2V3
φ(w1w2w3)
w1w2w3
+O(V1V2V3g(w1w2w3){maxV −1/2i }). (10)
If Vi, say, is less than 1, the left hand side will vanish, and the estimate remains
valid, since V1V2V3 ¿ V1V2V3V −1/2i .
The number of primitive integer vectors (z1, z2, z3) such that 1 ≤ zi ≤ Zi
and (zi, wi) = 1 for i = 1, 2, 3, may now be found as
∞∑
d=1
µ(d)#{z ∈ N3 : zi ≤ Zi, (zi, wi) = 1, d|zi, (i = 1, 2, 3)}.
Using (10) we see that this is∑
d: (d,w1w2w3)=1
µ(d)#{v ∈ N3 : vi ≤ Zi/d, (vi, wi) = 1, (i = 1, 2, 3)}
9
=
∑
d: (d,w1w2w3)=1
µ(d)
Z1Z2Z3
d3
φ(w1w2w3)
w1w2w3
+O
(
Z1Z2Z3g(w1w2w3){maxZ−1/2i }
∞∑
d=1
d−5/2
)
=
Z1Z2Z3
ζ(3)
φ(w1w2w3)
w1w2w3
∏
p|w1w2w3
(1− p−3)−1
+O(Z1Z2Z3g(w1w2w3){maxZ−1/2i })
=
Z1Z2Z3
ζ(3)
f(w1w2w3) +O(Z1Z2Z3g(w1w2w3){maxZ−1/2i }),
as required.
We turn now to Lemma 2. We begin by observing that it suffices to establish
the result for the case in which the divisors rij are all 1. To see this we set
Σ(H) =
∑
yij
y−1,
the variables being subject to the inequalities (4). In general, if we put yij =
rijzij , we then see that the new variables zij lie in a region which is included in
the set
z12z13(z21z31)2 ≤ H, z21z23(z12z32)2 ≤ H, and z31z32(z13z23)2 ≤ H.
It follows that ∑
yij : rij |yij
y−1 ≤ R−1Σ(H).
In the same way, the zij include all those which satisfy
z12z13(z21z31)2 ≤ HR−2, z21z23(z12z32)2 ≤ HR−2,
and
z31z32(z13z23)2 ≤ HR−2,
whence ∑
yij : rij |yij
y−1 ≥ R−1Σ(H/R2).
However, if we know that
Σ(H) =
1
4
(logH)6
6!
+O((logH)5), (11)
the upper and lower bounds will both be
1
4
(logH)6
6!R
+O(R−3/4(logH)5),
10
providing that logR ¿ logH. On the other hand, if any of the rij exceeds H,
then ∑
yij : rij |yij
y−1 = 0,
and Lemma 2 holds automatically, since we must now have logH ¿ R1/4.
It therefore remains to establish (11). To do this, we observe that∫ y12+1
y12
. . .
∫ y32+1
y32
dy12
y12
. . .
dy32
y32
≤ 1
y
≤
∫ y12
y12−1
. . .
∫ y32
y32−1
dy12
y12
. . .
dy32
y32
,
since y =
∏
yij . It follows that Σ(H) ≥ I(H), where
I(H) =
∫
. . .
∫
dy12
y12
. . .
dy32
y32
,
the integral being over the region (4), subject to yij ≥ 1. Similarly we have∑
yij≥2
(
∏
yij)−1 ≤ I(H),
the sum being subject to (4). However
Σ(H) ≤
∑
yij≥2
(
∏
yij)−1 +
∑
yij=1 for some ij
(
∏
yij)−1,
in which we may relax the conditions on the variables yij in the final sum to
require merely that yij ≤ H. It then follows that
Σ(H) ≤
∑
yij≥2
(
∏
yij)−1 + 6{
∑
y≤H
y−1}5 ≤
∑
yij≥2
(
∏
yij)−1 +O((logH)5).
We therefore deduce that Σ(H) = I(H) +O((logH)5).
To evaluate the integral I(H) we begin by substituting yij = Htij , which
shows that I(H) = (logH)6V , where
V =
∫
. . .
∫
dt12 . . . dt32,
the final integral being over the 6-dimensional polyhedron P , given by
t12 + t13 + 2(t21 + t31) ≤ 1, t21 + t23 + 2(t12 + t32) ≤ 1,
t31 + t32 + 2(t13 + t23) ≤ 1,
with tij ≥ 0.
We have failed to find a straightforward way of interpreting the volume V ,
which is clearly a rational number. The following method, suggested to us by
11
G. Mersmann, and carried out by Th. Kleinjung, does at least determine its
value. Our original calculation was considerably more involved, and we are very
grateful for their permission to describe their method.
We observe that the polyhedron 12P has coordinates in ZZ6. It follows that
if
N(t) = #{n ∈ ZZ6 : n ∈ 12tP}
for non-negative integers t, then N(t) interpolates a polynomial,
∑6
k=0 ckt
k, say
(see Fulton [5; pages 110-114], for example). On taking t → ∞ one sees that
c6 must be the volume of 12P . A straightforward computer calculation to find
N(t) for t = 0, 1, . . . , 6 then allows one to evaluate the coefficients ck. In this
way one obtains c6 = 5184/5, and hence V = (4 × 6!)−1, as required. This
completes the proof of Lemma 2.
5 Acknowledgement
The second author wishes to thank the Royal Society of London for its financial
support, and the Mathematical Institute of Oxford University for its hospitality
during his visit to England, when this paper was written.
References
[1] V.V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de
hauteur borne´ des varie´te´s alge´briques, Math. Ann., 286 (1990), 27-43.
[2] V.V. Batyrev and Yu. Tschinkel, Height zeta functions of toric varieties,
Algebraic Geometry, 5, Journal of Mathematical Sciences, 82 (1996), 3220-
3239.
[3] R. de la Brete`che, Sur le nombre de points de hauteur borne´e d’une certaine
surface cubique singulie`re, to appear.
[4] E. Fouvry, Sur la hauteur des points d’une certaine surface cubique sin-
gulie`re, to appear.
[5] W. Fulton, Introduction to toric varieties, (Princeton University Press,
1993).
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The density of rational points on the cubic surface $X_{0}^{3}=X_{1}X_{2}X_{3}$

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The density of rational points on the cubic surface X03=X1X2X3X_{0}^{3}=X_{1}X_{2}X_{3}X03​=X1​X2​X3​

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The density of rational points on the cubic surface $X_{0}^{3}=X_{1}X_{2}X_{3}$