For given non-zero integers we investigate the density of solutions to the binary cubic congruence and use it to establish the Manin conjecture for a singular del Pezzo surface of degree defined over .
Inhomogeneous cubic congruences]Inhomogeneous cubic congruences and rational points on del Pezzo surfaces
2000 Mathematics Subject Classification. — 11D45 (11G05, 14G05).
The quantitative arithmetic of low degree del Pezzo surfaces has received a great deal of attention in recent years. The aim of the present investigation is to provide a new tool in the analysis of such questions and to show how it can be used to estimate the number of -rational points of bounded height on a del Pezzo surface of degree defined over . Such surfaces arise as subvarieties of weighted projective space and are given by equations of the shape
where is a quartic form. In the classification of del Pezzo surfaces it is those of degree and whose arithmetic remains the most elusive.
If and denotes the anticanonical height function then it is natural to study the counting function
as , for a Zariski open subset obtained by deleting the accumulating subvarieties. These are the exceptional divisors arising from the bitangents of the plane quartic curve . There are of these when is non-singular, producing exceptional curves on . A well-known conjecture of Manin  predicts the existence of constants and such that
as . Moreover, if denotes the minimal desingularisation of , with if it is non-singular, then it is expected that , where is the Picard group of . There is a prediction of Peyre  concerning the value of the constant . These refined conjectures have received a great deal of attention in the context of del Pezzo surfaces of degree at least , an account of which can be found in Browning’s treatise . Our success in degree has been rather limited however.
When is non-singular it follows from work of Broberg  that for any . This argument uses Siegel’s lemma to cover the rational points of height at most on with plane sections defined over . For each of these one is left with counting points of bounded height on the curves given by
with a binary form of degree . This one needs to do uniformly with respect to the coefficients of and is achieved via a modification of Heath-Brown’s determinant method . For generic the curve has genus and so one expects it to have very few rational points. Nevertheless it is difficult to demonstrate this with the requisite degree of uniformity.
In this paper we will consider split singular , by which we mean that the quartic form is singular and the singularities and exceptional curves of are all defined over . According to the classification of Alexeev and Nikulin  the minimal desingularisation can then be realised as the blow-up of along -points in “almost general position”. As such one verifies that , so that in the Manin conjecture. We will impose the condition that is reducible and we will assume that takes the shape
This defines a del Pezzo surface of degree with a unique exceptional curve and a unique singularity . According to the classification of Arnol’d  a hypersurface has a simple singularity if it can be put in the local normal form where is the dimension of the hypersurface. Dividing through by and making a change of variables one is directly led to this normal form, so that has a simple singularity. It follows from work of Ye [26, Lemma 4.6] that there is only one such surface up to isomorphism. We let be the Zariski open subset formed by deleting the curve from .
When one returns to the above argument involving plane sections, the situation is more favourable since the resulting curves generically define elliptic curves with rational -torsion. In this way one can show that for any . Our goal is to show how the full Manin conjecture (1.1) can be established for this particular surface using analytic number theory. We will establish the following result.
as , where is the constant predicted by Peyre.
The investigations of Derenthal and Loughran [10, 12] show that is neither toric nor an equivariant compactification of . Thus Theorem 1.1 is not a special case of  or . The proof of our theorem relies on a passage to the universal torsor above the minimal desingularisation of , which in this setting is a subset of the affine hypersurface , given by the equation
The idea is to establish a bijection between and a suitable subset of . This step underpins many proofs of the Manin conjecture, such as that found in work of la Bretèche, Browning and Derenthal  dealing with the split cubic surface of singularity type . Here, as there, it is useful to view the torsor equation as a congruence , with being thought of as the main variables. Regrettably one finds that the arguments developed in  no longer bear fruit in the present setting. Rather one is forced to consider in general terms the counting function
for , and non-zero integers such that . In our case of interest we have , , , and . One seeks an asymptotic formula for which is completely uniform in the relevant parameters. This will ultimately be achieved in §[, where it is recorded as Theorem 8.1. A substantially easier problem is to produce a good upper bound for the counting function in which the condition is dropped. Let us denote by this counting function. During the course of our argument we will be led to the following result in §[, which we record here for ease of use.
Let , let and assume that . Then we have
The implied constant in this estimate is allowed to depend on the choice of small parameter , a convention that we adhere to in all of our estimates. Our estimate for is sharpest when has small square-free kernel or when is small compared with . It is the cornerstone of our entire investigation. It should be noted that when is square-free and Theorem 1.2 does not give anything sharper than what is available through the work of Pierce .
We proceed to give a simplified description of the method behind Theorem 1.2. The appearing in (1.4) are constrained to lie in a certain region and our first task will be to cover this region by small boxes and to approximate their characteristic functions by smooth weights. This facilitates an application of the Poisson summation formula, which we invoke after breaking the sums over and into residue classes modulo . This transformation leads to expressions involving exponential sums of the form
for , where . If one were now to estimate these sums directly one would retrieve an estimate of the sort obtained by Pierce . In the setting of Theorem 1.1, however, this would only yield a final upper bound of the form . A key point in the method is to evaluate the sums explicitly for power-full moduli , a situation that explicitly arises in the application to Theorem 1.1 since then contains high powers. This is carried out in §[, where an easy multiplicativity property renders it sufficient to study for primes and suitable . It turns out that considerable labour is required to deal with the primes and and the reader may be inclined to take the results of this section on faith at a first reading.
After explicitly evaluating the exponential sums at power-full moduli we encounter terms of the shape
for such that and where is the multiplicative inverse of modulo . We need to sum these up non-trivially. Our second key innovation is to flip the numerator and denominator, using the familiar identity
for any non-zero integers such that . This has the desired effect of reducing the size of the denominator drastically. Next, we break up the summation over into residue classes modulo the new denominator and use Poisson summation in again. This time we encounter new cubic exponential sums and cubic exponential integrals which we need to estimate non-trivially. This leads to an asymptotic estimate for the number of solutions of the congruence in small boxes. The final step is to sum up all these contributions. In fact this summation is also carried out non-trivially, with parts of the averaging process replaced by an integration, in order to take advantage of extra cancellations. While this leads to substantial extra work it is nonetheless essential for obtaining the asymptotic formula in Theorem 1.1.
Aside from its intrinsic utility in the proof of Theorem 1.1 we can use Theorem 1.2 and its refinement Theorem 8.1 to tackle other questions in Diophantine geometry. Firstly, an inspection of the various universal torsors calculated by Derenthal  arising in the theory of split singular del Pezzo surfaces, shows that the underlying counting function precisely matches (1.4) whenever its singularity type is maximal. Although we will not present details here, it is possible to provide independent proofs of the Manin conjecture for such cases, albeit with weaker error terms than are already available. This includes the cubic surface considered by la Bretèche, Browning and Derenthal , the degree del Pezzo surface considered by la Bretèche and Browning  and the degree del Pezzo surface. The latter two surfaces are equivariant compactifications of by  and so are covered by the investigation of Chambert-Loir and Tschinkel .
A second and rather different application of Theorem 1.2 lies in the theory of elliptic curves over . Such curves may be brought into Weierstrass form
for with non-zero discriminant , say. It is presently unknown whether there are infinitely many for which is prime. An easier questions concerns the square-freeness of . Let be the exponential height of and let be the Möbius function. We maintain the conventions and . A measure of the density of elliptic curves with square-free discriminant is achieved by studying the quantity
For let . Then for any we have
For comparison Wong [25, Proposition 6] has shown a similar asymptotic formula but with only a logarithmic saving in the error term. In fact, as we shall see in §Acknowledgements. —, an adaptation of an argument due to Estermann  would permit a power saving but only leads to an error term of order . Through Möbius inversion the problem is to count solutions to the congruence for square-free integers . Estermann’s approach helps us to deal with the contribution from both small and large . Theorem 1.2 is the key ingredient in the treatment of medium .
Some of this work was done while the authors were visiting the Institute for Advanced Study in Princeton, the hospitality and financial support of which is gratefully acknowledged. While working on this paper the authors were supported by EPSRC grant number EP/E053262/1. The authors are grateful to Ulrich Derenthal for drawing their attention to the del Pezzo surface of degree and to both Pierre Le Boudec and the referee for useful comments on an earlier version.
where . We will estimate this inner sum differently according to the size of . For parameters let us write for the overall contribution to from in the interval , with
Finally for we recall the definition of from Theorem 1.3. This is a multiplicative function of and it is easy to see that for any prime . Hence we have for any square-free .
Beginning with the contribution from small , the idea is to break the sum over into congruence classes modulo . Beginning with the sum over one sees that
where we recall that . For and square-free let denote the number of incongruent solutions modulo of . It is trivial to see that for any . We may now write
where the error term comes from . Next we claim that
for any square-free . Once armed with this it is then straightforward to see that
on extending the summation over to infinity. We establish the claim using a simple argument involving exponential sums.
Breaking the sum over into residue classes modulo we find that
The contribution from is clearly
which is satisfactory. Likewise one finds that the contribution from non-zero is
The exponential sum satisfies a basic multiplicativity property in rendering it sufficient to understand when is a prime power. In this way one easily concludes that . Thus the contribution from non-zero is also seen to be satisfactory for (2.1), after redefining .
Turning to the large values of we write the congruence as an equation and note that
To estimate the summand we fix and consider it as a problem about counting the representations of by the binary quadratic form . The classical argument of Estermann  shows that there are solutions for any , whence
At this point we can recover a preliminary estimate for by taking . This gives a version of Theorem 1.3 with the weaker error term , as remarked in the introduction.
We now come to the treatment of the middle range for . Thus we have
For given let us write where , and is coprime to . It readily follows that and in the summand. Making the change of variables and we deduce that
where and . In particular it follows that and . We will need to account for possible common factors of and . Drawing out the greatest common divisor of and we write and , with . It easily follows from the square-freeness of that and we can write with . Hence
in the notation of §[, with .
Everything is now in place for an application of Theorem 1.2. On noting that , we deduce that
Inserting this into our bound for we conclude that
There are numerous ways in which one can approximate the characteristic function of intervals using smooth weights. In our work, which will involve repeated applications of the Poisson summation formula, it will be useful to have weights which transform well under the Fourier transform. We are naturally drawn to construct weight functions from the Gaussian
Note that the usual Gamma function does not occur anywhere in our work and so we trust that this choice of notation doesn’t cause confusion. Let denote the characteristic function of the interval . We will approximate this using the weights
for any , where and Clearly, and are smooth (infinitely differentiable) functions that have rapid decay at and . Thus as for any fixed . We proceed to show how they approximate in the following result.
Assume that . Then for all . Moreover, we have
The equation (3.2) is easily proved as follows. We have
Let . Next, we want to show that for all . If then obviously . If then
Hence we have established that for all .
Finally we want to show that for all . If then
If then for all we have
By taking logarithms one easily verifies that the last line is if and
which is true for all . This completes the proof of for all and therefore the proof of the lemma. ∎
From Lemma 3.1 one can easily deduce a similar result for the characteristic function of a general interval by making the change of variables , which maps the interval bijectively onto . We include this result here because it may be useful for future applications.
Let be real numbers with . Denote the characteristic function of the interval by . Suppose that and set
Then we have for all and
Moreover and are infinitely differentiable functions that have rapid decay at .
Recall the definition (3.1) of the Gaussian function. We begin with a general upper bound for exponential integrals weighted by the Gaussian.
Let be a smooth function and suppose that there exists and such that for every . Then we have
It will be sufficient to prove the corresponding bound for the integral over since the treatment of the integral over is similar. For any it follows from integration by parts that
Now it follows from Lemma 8.10 in  and the hypotheses of the lemma that
for any . Hence it readily follows that
Taking the limit we are easily led to the desired bound. This completes the proof. ∎
The special case will feature quite heavily in our work, corresponding to weighted Airy–Hardy integrals. For any such that we set
We begin by recording the trivial estimate
Furthermore, applying Lemma 3.3 with , we deduce that
Moreover, if , then for all we have
An application of Lemma 3.3 with now gives
We henceforth assume that . In this case we have stationary points which give a main term contribution. We write
where we take into account that our function is even. We make a change of variables , getting
To estimate the integral on the right-hand side we use Theorem 2.2 in , which we record here for convenience.
Lemma 3.4 (Stationary phase with weights).
Let , be two functions of the complex variable and be a real interval such that the following hold.
For the function is real and .
For a certain positive differentiable function , defined on , and are analytic for , .
There exist positive functions , defined on such that for , we have
and the implied constants are absolute.
Let and if has a zero in denote it by . Let the values of , and so on, at , , and be characterised by the suffixes , and , respectively. Then, for some absolute constant , we have
If has no zero in , then the terms involving are to be omitted.