Analytic Number Theory A Tribute to Gauss and Dirichlet Part 4 pdf

52 T.D. BROWNING But now (18) implies that Y 14  B 1/2 /(Y 1/2 1 Y 1/2 04 Y 24 Y 1/2 34 ), and (20) and (21) together imply that Y 03  Y 33 Y 34 /Y 04 . We therefore deduce that  Y 1 ,Y i3 ,Y i4 (20) holds NB 1/2  Y 03 ,Y 04 ,Y 33 Y 1 ,Y 23 ,Y 24 ,Y 34 Y 3/4 03 Y 3/4 04 Y 1/2 23 Y 1/2 24 Y 1/4 33 Y 1/4 34  B 1/2  Y 1 ,Y 04 ,Y 33 Y 23 ,Y 24 ,Y 34 Y 1/2 23 Y 1/2 24 Y 33 Y 34 . Finally it follows from (17) and (21) that Y 33  B 1/2 /(Y 1/2 23 Y 1/2 24 Y 34 ), whence  Y 1 ,Y i3 ,Y i4 (20) holds NB  Y 04 ,Y 13 ,Y 14 ,Y 23 ,Y 34 1  B(log B) 5 , which is satisfactory for the theorem. Next we suppose that (22) holds, so that (23) also holds. In this case it follows from (19), together with the inequality Y 1 Y 13 Y 14  Y 03 Y 04 ,that Y 13  min  Y 1/2 04 Y 14 Y 24 Y 1/2 34 Y 1/2 03 Y 23 Y 1/2 33 , Y 03 Y 04 Y 1 Y 14   Y 1/4 03 Y 3/4 04 Y 1/2 24 Y 1/4 34 Y 1/2 1 Y 1/2 23 Y 1/4 33 . On combining this with the inequality Y 14  B 1/2 /(Y 1/2 1 Y 1/2 04 Y 24 Y 1/2 34 ), that follows from (18), we may therefore deduce from (25) that  Y 1 ,Y i3 ,Y i4 (22) holds N  Y 1 ,Y i3 ,Y i4 (22) holds Y 1 Y 13 Y 14 Y 23 Y 24 Y 33 Y 34   Y 1 ,Y 03 ,Y 04 ,Y 33 Y 14 ,Y 23 ,Y 24 ,Y 34 Y 1/2 1 Y 1/4 03 Y 3/4 04 Y 14 Y 1/2 23 Y 3/2 24 Y 3/4 33 Y 5/4 34  B 1/2  Y 1 ,Y 03 ,Y 04 Y 23 ,Y 24 ,Y 33 ,Y 34 Y 1/4 03 Y 1/4 04 Y 1/2 23 Y 1/2 24 Y 3/4 33 Y 3/4 34 . Now it follows from (23) that Y 33  Y 03 Y 04 /Y 34 . We may therefore combine this with the first inequality in (17) to conclude that  Y 1 ,Y i3 ,Y i4 (22) holds NB 1/2  Y 1 ,Y 03 ,Y 04 Y 23 ,Y 24 ,Y 34 Y 03 Y 04 Y 1/2 23 Y 1/2 24  B(log B) 5 , which is also satisfactory for the theorem. Finally we suppose that (24) holds. On combining (19) with the fact that Y 33 Y 34  Y 03 Y 04 ,weobtain Y 33  min  Y 04 Y 2 14 Y 2 24 Y 34 Y 03 Y 2 13 Y 2 23 , Y 03 Y 04 Y 34   Y 04 Y 14 Y 24 Y 13 Y 23 . Summing (25) over Y 33 first, with min{Y 03 Y 04 ,Y 33 Y 34 }  Y 1/2 03 Y 1/2 04 Y 1/2 33 Y 1/2 34 ,we therefore obtain  Y 1 ,Y i3 ,Y i4 (24) holds N  Y 1 ,Y 03 ,Y 04 ,Y 13 Y 14 ,Y 23 ,Y 24 ,Y 34 Y 1 Y 1/2 03 Y 04 Y 1/2 13 Y 3/2 14 Y 1/2 23 Y 3/2 24 Y 1/2 34 . AN OVERVIEW OF MANIN’S CONJECTURE FOR DEL PEZZO SURFACES 53 But then we may sum over Y 03 ,Y 13 satisfying the inequalities in (17), and then Y 1 satisfying the second inequality in (18), in order to conclude that  Y 1 ,Y i3 ,Y i4 (24) holds NB 1/4  Y 1 ,Y 04 ,Y 13 Y 14 ,Y 23 ,Y 24 ,Y 34 Y 1 Y 1/2 04 Y 1/2 13 Y 3/2 14 Y 1/4 23 Y 5/4 24 Y 1/2 34  B 1/2  Y 1 ,Y 04 ,Y 14 Y 23 ,Y 24 ,Y 34 Y 1/2 1 Y 1/2 04 Y 14 Y 24 Y 1/2 34  B(log B) 5 . This too is satisfactory for Theorem 3, and thereby completes its proof. 4. Open problems We close this survey article with a list of five open problems relating to Manin’s conjecture for del Pezzo surfaces. In order to encourage activity we have deliberately selected an array of very concrete problems. (i) Establish (3) for a non-singular del Pezzo surface of degree 4. The surface x 0 x 1 −x 2 x 3 = x 2 0 + x 2 1 + x 2 2 −x 2 3 −2x 2 4 = 0 has Picard group of rank 5. (ii) Establish (3) for more singular cubic surfaces. Can one establish the Manin conjecture for a split singular cubic surface whose universal torsor has more than one equation? The Cayley cubic surface (8) is such a surface. (iii) Break the 4/3-barrier for a non-singular cubic surface. We have yet to prove an upper bound of the shape N U,H (B)=O S (B θ ), with θ<4/3, for a single non-singular cubic surface S ⊂ P 3 . This seems to be hardest when the surface doesn’t have a conic bundle structure over Q.Thesurfacex 0 x 1 (x 0 + x 1 )=x 2 x 3 (x 2 + x 3 ) admits such a structure; can one break the 4/3-barrier for this example? (iv) Establish the lower bound N U,H (B)  B(log B) 3 for the Fermat cubic. The Fermat cubic x 3 0 + x 3 1 = x 3 2 + x 3 3 has Picard group of rank 4. (v) Better bounds for del Pezzo surfaces of degree 2. Non-singular del Pezzo surfaces of degree 2 take the shape t 2 = F (x 0 ,x 1 ,x 2 ), for a non-singular quartic form F .LetN (F ; B) denote the number of integers t, x 0 ,x 1 ,x 2 such that t 2 = F (x)and|x|  B. Can one prove that we always have N(F ; B)=O ε,F (B 2+ε )? Such an estimate would be essentially best possible, as consideration of the form F 0 (x)=x 4 0 +x 4 1 −x 4 2 shows. The best result in this direction is due to Broberg [Bro03a], who has established the weaker bound N(F ; B)=O ε,F (B 9/4+ε ). For certain quartic forms, such as F 1 (x)=x 4 0 + x 4 1 + x 4 2 , the Manin conjecture implies that one ought to be able to replace the exponent 2 + ε by 1 + ε. Can one prove that N(F 1 ; B)=O(B θ )forsomeθ<2? Acknowledgements. 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Swinnerton-Dyer – “Diophantine equations: progress and problems”, in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., vol. 226, Birkh¨auser Boston, Boston, MA, 2004, p. 3–35. [SD05] , “Counting points on cubic surfaces. II”, in Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkh¨auser Boston, Boston, MA, 2005, p. 303– 309. [Sko93] A. N. Skorobogatov – “On a theorem of Enriques-Swinnerton-Dyer”, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 3, p. 429–440. [SSD98] J. B. Slater & P. Swinnerton-Dyer – “Counting points on cubic surfaces. I”, Ast´erisque (1998), no. 251, p. 1–12, Nombre et r´epartition de points de hauteur born´ee (Paris, 1996). [Tsc02] Y. Tschinkel – “Lectures on height zeta functions of toric varieties”, in Geometry of toric varieties,S´emin. Congr., vol. 6, Soc. Math. France, Paris, 2002, p. 227–247. [Tsc03] , “Fujita’s program and rational points”, in Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud., vol. 12, Springer, Berlin, 2003, p. 283–310. [Wal80] C. T. C. Wall – “The first canonical stratum”, J. London Math. Soc. (2) 21 (1980), no. 3, p. 419–433. School of Mathematics, University of Bristol, Bristol BS8 1TW, UK E-mail address: t.d.browning@bristol.ac.uk Clay Mathematics Proceedings Volume 7, 2007 The density of integral solutions for pairs of diagonal cubic equations J¨org Br¨udern and Trevor D. Wooley Abstract. We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least thirteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least seven, we obtain a lower bound for the order of magnitude of the number of integral solutions consistent with the product of local densities associated with the system. 1. Introduction This paper is concerned with the solubility in integers of the equations (1.1) a 1 x 3 1 + a 2 x 3 2 + + a s x 3 s = b 1 x 3 1 + b 2 x 3 2 + + b s x 3 s =0, where (a i ,b i ) ∈ Z 2 \{0} are fixed coefficients. It is natural to enquire to what extent the Hasse principle holds for such systems of equations. Cook [C85], refining earlier work of Davenport and Lewis [DL66], has analysed the local solubility problem with great care. He showed that when s ≥ 13 and p is a prime number with p =7, then the system (1.1) necessarily possesses a non-trivial solution in Q p . Here, by non-trivial solution, we mean any solution that differs from the obvious one in which x j =0for1≤ j ≤ s. No such conclusion can be valid for s ≤ 12, for there may then be local obstructions for any given set of primes p with p ≡ 1(mod3); see [BW06] for an example that illuminates this observation. The 7-adic case, moreover, is decidedly different. For s ≤ 15 there may be 7-adic obstructions to the solubility of the system (1.1), and so it is only when s ≥ 16 that the existence of non-trivial solutions in Q 7 is assured. This much was known to Davenport and Lewis [DL66]. Were the Hasse principle to hold for systems of the shape (1.1), then in view of the above discussion concerning the local solubility problem, the existence of 2000 Mathematics Subject Classification. Primary 11D72, Secondary 11L07, 11E76, 11P55. Key words and phrases. Diophantine equations, exponential sums, Hardy-Littlewood method. First author supported i n part by NSF grant DMS-010440. The authors are grateful to the Max Planck Institut in Bonn for its generous hospitality during the perio d in which this paper was conceived. c  2007 J¨org Br¨udern and Trevor D. Wooley 57 58 J ¨ ORG BR ¨ UDERN AND TREVOR D. WOOLEY integer solutions to the equations (1.1) would be decided in Q 7 alone whenever s ≥ 13. Under the more stringent hypothesis s ≥ 14, this was confirmed by the first author [B90], building upon the efforts of Davenport and Lewis [DL66], Cook [C72], Vaughan [V77] and Baker and Br¨udern [BB88] spanning an interval of more than twenty years. In a recent collaboration [BW06]wehavebeenableto add the elusive case s = 13, and may therefore enunciate the following conclusion. Theorem 1. Suppose that s ≥ 13. Then for any choice of coefficients (a j ,b j ) ∈ Z 2 \{0} (1 ≤ j ≤ s), the simultaneous equations (1.1) possess a non-trivial solution in rational integers if and only if they admit a non-trivial solution in Q 7 . Now let N s (P ) denote the number of solutions of the system (1.1) in rational integers x 1 , ,x s satisfying the condition |x j |≤P (1 ≤ j ≤ s). When s is large, ana¨ıve application of the philosophy underlying the circle method suggests that N s (P ) should be of order P s−6 in size, but in certain cases this may be false even in the absence of local obstructions. This phenomenon is explained by the failure of the Hasse principle for certain diagonal cubic forms in four variables. When s ≥ 10 and b 1 , ,b s ∈ Z \{0}, for example, the simultaneous equations (1.2) 5x 3 1 +9x 3 2 +10x 3 3 +12x 3 4 = b 1 x 3 1 + b 2 x 3 2 + + b s x 3 s =0 have non-trivial (and non-singular) solutions in every p-adic field Q p as well as in R, yet all solutions in rational integers must satisfy the condition x i =0(1≤ i ≤ 4). The latter must hold, in fact, independently of the number of variables. For such examples, therefore, one has N s (P )=o(P s−6 )whens ≥ 9, whilst for s ≥ 12 one may show that N s (P )isoforderP s−7 . For more details, we refer the reader to the discussion surrounding equation (1.2) of [BW06]. This example also shows that weak approximation may fail for the system (1.1), even when s is large. In order to measure the extent to which a system (1.1) may resemble the pathological example (1.2), we introduce the number q 0 , which we define by q 0 =min (c,d)∈Z 2 \{0} card{1 ≤ j ≤ s : ca j + db j =0}. This important invariant of the system (1.1) has the property that as q 0 becomes larger, the counting function N s (P ) behaves more tamely. Note that in the example (1.2) discussed above one has q 0 = 4 whenever s ≥ 8. Theorem 2. Suppose that s ≥ 13, and that (a j ,b j ) ∈ Z 2 \{0} (1 ≤ j ≤ s) satisfy the condition that the system (1.1) admits a non-trivial solution in Q 7 .Then whenever q 0 ≥ 7, one has N s (P )  P s−6 . The conclusion of Theorem 2 was obtained in our recent paper [BW06]for all cases wherein q 0 ≥ s − 5. This much suffices to establish Theorem 1; see §8of [BW06] for an account of this deduction. Our main objective in this paper is a detailed discussion of the cases with 7 ≤ q 0 ≤ s −6. We remark that the arguments of this paper as well as those in [BW06] extend to establish weak approximation for the system (1.1) when s ≥ 13 and q 0 ≥ 7. In the special cases in which s =13and q 0 is equal to either 5 or 6, a conditional proof of weak approximation is possible by invoking recent work of Swinnerton-Dyer [SD01], subject to the as yet unproven finiteness of the Tate-Shafarevich group for elliptic curves over quadratic fields. Indeed, equipped with the latter conclusion, for these particular values of q 0 one may relax the condition on s beyond that addressed by Theorem 2. When s =13 PAIRS OF DIAGONAL CUBIC EQUATIONS 59 and q 0 ≤ 4, on the other hand, weak approximation fails in general, as we have already seen in the discussion accompanying the system (1.2). The critical input into the proof of Theorem 2 is a certain arithmetic variant of Bessel’s inequality established in [BW06]. We begin in §2 by briefly sketching the principal ideas underlying this innovation. In §3 we prepare the ground for an application of the Hardy-Littlewood method, deriving a lower bound for the major arc contribution in the problem at hand. Some delicate footwork in §4 establishes a mean value estimate that, in all circumstances save for particularly pathological situations, leads in §5 to a viable complementary minor arc estimate sufficient to establish Theorem 2. The latter elusive situations are handled in §6 via an argument motivated by our recent collaboration [BKW01a] with Kawada, and thereby we complete the proof of Theorem 2. Finally, in §7, we make some remarks concerning the extent to which our methods are applicable to systems containing fewer than 13 variables. Throughout, the letter ε will denote a sufficiently small positive number. We use  and  to denote Vinogradov’s well-known notation, implicit constants de- pending at most on ε, unless otherwise indicated. In an effort to simplify our analysis, we adopt the convention that whenever ε appears in a statement, then we are implicitly asserting that for each ε>0 the statement holds for sufficiently large values of the main parameter. Note that the “value” of ε may consequently change from statement to statement, and hence also the dependence of implicit constants on ε. Finally, from time to time we make use of vector notation in order to save space. Thus, for example, we may abbreviate (c 1 , ,c t )toc. 2. An arithmetic variant of Bessel’s inequality The major innovation in our earlier paper [BW06] is an arithmetic variant of Bessel’s inequality that sometimes provides good mean square estimates for Fourier coefficients averaged over sparse sequences. Since this tool plays a crucial role also in our current excursion, we briefly sketch the principal ideas. When P and R are real numbers with 1 ≤ R ≤ P , we define the set of smooth numbers A(P, R)by A(P, R)={n ∈ N ∩[1,P]:p prime and p|n ⇒ p ≤ R}. The Fourier coefficients that are to be averaged arise in connection with the smooth cubic Weyl sum h(α)=h(α; P, R), defined by (2.1) h(α; P, R)=  x∈A(P,R) e(αx 3 ), wherehereandlaterwewritee(z)forexp(2πiz). The sixth moment of this sum has played an important role in many applications in recent years, and that at hand is no exception to the rule. Write ξ =( √ 2833 − 43)/41. Then as a consequence of the work of the second author [W00], given any positive number ε, there exists a positive number η = η(ε) with the property that whenever 1 ≤ R ≤ P η , one has (2.2)  1 0 |h(α; P, R)| 6 dα  P 3+ξ+ε . We assume henceforth that whenever R appears in a statement, either implicitly or explicitly, then 1 ≤ R ≤ P η with η a positive number sufficiently small in the context of the upper bound (2.2). 60 J ¨ ORG BR ¨ UDERN AND TREVOR D. WOOLEY The Fourier coefficients over which we intend to average are now defined by (2.3) ψ(n)=  1 0 |h(α)| 5 e(−nα) dα. An application of Parseval’s identity in combination with conventional circle method technology readily shows that  n ψ(n) 2 is of order P 7 . Rather than average ψ(n)in mean square over all integers, we instead restrict to the sparse sequence consisting of differences of two cubes, and establish the bound (2.4)  1≤x,y≤P ψ(x 3 − y 3 ) 2  P 6+ξ+4ε . Certain contributions to the sum on the left hand side of (2.4) are easily es- timated. By Hua’s Lemma (see Lemma 2.5 of [V97]) and a consideration of the underlying Diophantine equations, one has  1 0 |h(α)| 4 dα  P 2+ε . On applying Schwarz’s inequality to (2.3), we therefore deduce from (2.2) that the estimate ψ(n)=O(P 5/2+ξ/2+ε ) holds uniformly in n. We apply this upper bound with n = 0 in order to show that the terms with x = y contribute at most O(P 6+ξ+2ε ) to the left hand side of (2.4). The integers x and y with 1 ≤ x, y ≤ P and |ψ(x 3 − y 3 )|≤P 2+ξ/2+2ε likewise contribute at most O(P 6+ξ+4ε )within the summation of (2.4). We estimate the contribution of the remaining Fourier coefficients by dividing into dyadic intervals. When T is a real number with (2.5) P 2+ξ/2+2ε ≤ T ≤ P 5/2+ξ/2+2ε , define Z(T ) to be the set of ordered pairs (x, y) ∈ N 2 with (2.6) 1 ≤ x, y ≤ P, x = y and T ≤|ψ(x 3 − y 3 )|≤2T, and write Z(T )forcard(Z(T )). Then on incorporating in addition the contributions of those terms already estimated, a familiar dissection argument now demonstrates that there is a number T satisfying (2.5) for which (2.7)  1≤x,y≤P ψ(x 3 − y 3 ) 2  P 6+ξ+4ε + P ε T 2 Z(T ). An upper bound for Z(T ) at this point being all that is required to complete the proof of the estimate (2.4), we set up a mechanism for deriving such an upper bound that has its origins in work of Br¨udern, Kawada and Wooley [BKW01a] and Wooley [W02]. Let σ(n) denote the sign of the real number ψ(n) defined in (2.3), with the convention that σ(n)=0whenψ(n) = 0, so that ψ(n)=σ(n)|ψ(n)|. Then on forming the exponential sum K T (α)=  (x,y)∈Z(T ) σ(x 3 − y 3 )e(α(y 3 − x 3 )), we find from (2.3) and (2.6) that  1 0 |h(α)| 5 K T (α) dα ≥ TZ(T ). PAIRS OF DIAGONAL CUBIC EQUATIONS 61 An application of Schwarz’s inequality in combination with the upper bound (2.2) therefore permits us to infer that (2.8) TZ(T)  (P 3+ξ+ε ) 1/2   1 0 |h(α) 4 K T (α) 2 |dα  1/2 . Next, on applying Weyl’s differencing lemma (see, for example, Lemma 2.3 of [V97]), one finds that for certain non-negative numbers t l , satisfying t l = O(P ε ) for 0 < |l|≤P 3 , one has |h(α)| 4  P 3 + P  0<|l|≤P 3 t l e(αl). Consequently, by orthogonality,  1 0 |h(α) 4 K T (α) 2 |dα  P 3  1 0 |K T (α)| 2 dα + P 1+ε K T (0) 2  P ε (P 3 Z(T )+PZ(T ) 2 ). Here we have applied the simple fact that when m is a non-zero integer, the number of solutions of the Diophantine equation m = x 3 −y 3 with 1 ≤ x, y ≤ P is at most O(P ε ). Since T ≥ P 2+ξ/2+2ε , the upper bound Z(T )=O(T −2 P 6+ξ+2ε )now follows from the relation (2.8). On substituting the latter estimate into (2.7), the desired conclusion (2.4) is now immediate. Note that in the summation on the left hand side of the estimate (2.4), one may restrict the summation over the integers x and y to any subset of [1,P] 2 without affecting the right hand side. Thus, on recalling the definition (2.3), we see that we have proved the special case a = b = c = d = 1 of the following lemma. Lemma 3. Let a, b, c, d denote non-zero integers. Then for any subset B of [1,P] ∩ Z, one has  1 0  1 0 |h(aα)h(bβ)| 5     x∈B e((cα + dβ)x 3 )    2 dα dβ  P 6+ξ+ε . This lemma is a restatement of Theorem 3 of [BW06]. It transpires that no great difficulty is encountered when incorporating the coefficients a, b, c, d into the argument described above; see §3of[BW06]. We apply Lemma 3 in the cosmetically more general formulation provided by the following lemma. Lemma 4. Suppose that c i ,d i (1 ≤ i ≤ 3) are integers satisfying the condition (c 1 d 2 − c 2 d 1 )(c 1 d 3 − c 3 d 1 )(c 2 d 3 − c 3 d 2 ) =0. Write λ j = c j α + d j β (j =1, 2, 3). Then for any subset B of [1,P] ∩ Z, one has  1 0  1 0 |h(λ 1 )h(λ 2 )| 5     x∈B e(λ 3 x 3 )    2 dα dβ  P 6+ξ+ε . Proof. The desired conclusion follows immediately from Lemma 3 on making a change of variable. The reader may care to compare the situation here with that occurring in the estimation of the integral J 3 in the proof of Theorem 4 of [BW06] (see §4 of the latter).  [...]... , and su that o of su−1 , and with concommitant adjustments to the associated indices throughout In this second situation we ultimately arrive at a scenario in which u = 3 and su−1 = 5, and in these circumstances the constraints (4. 2) imply that necessarily (s1 , s2 , , su ) = (5, 5, 1) On recalling (4. 1) and (4. 4), and making use of a trivial inequality for |G0 (α, β)|, we may conclude thus far... consideration This completes the proof of Theorem 2 for the latter systems, and so we may turn our attention in the next section to systems of types I and II 6 An exceptional approach to systems of types I and II Systems of type II split into two almost separate diagonal cubic equations linked by a single variable Here we may apply the main ideas from our recent collaboration with Kawada [BKW0 1a] in... order to show that this linked cubic variable is almost always simultaneously as often as expected equal both to the first and to the second residual diagonal cubic A lower bound for Ns (P ) of the desired strength follows with ease Although systems of type I are accessible in a straightforward fashion to the modern theory of cubic smooth Weyl sums (see, for example, [V89] and [W00]), we are able to avoid... 71 and this permits economies later in this section Much improvement is possible in the estimate for card(Et (P )) even when t = 6 (see Br¨ dern, Kawada and Wooley u [BKW0 1a] for the ideas necessary to save a relatively large power of P ) Here we briefly sketch a proof of Theorem 9 that employs a straightforward approach to the problem Proof Let B ⊆ [0, 1) be a measurable set, and consider a natural number. .. (Λ13 − a1 3 α) ∈ M(Q3 /4 ), whence (α, β) ∈ N (see the 13 proof of Lemma 10 in §6 of [BW06] for details of a similar argument) But the latter contradicts the hypothesis β ∈ F(α), in view of the definition of F(α) Thus we conclude that Λ13 ∈ k (mod 1), and so a standard application of Weyl’s inequality (see Lemma 2 .4 of [V97]) in combination with available major arc estimates (see Theorem 4. 1 and Lemma 4. 6... of any exponent as indicating that the associated exponential sum is deleted from the product In this way we obtain an upper bound of the shape (4. 4) in which the exponents su−2 and su−1 = su−2 − ν are replaced by su−2 + 1 and su−1 − 1, respectively, or else by su−2 − ν − 1 and su−1 + ν + 1 By relabelling if necessary, we derive an upper bound of the shape (4. 4), subject to the constraints (4. 2) and. .. transformations that ease the analysis of the singular integral, and here we follow the pattern of our earlier work [BW06] First, PAIRS OF DIAGONAL CUBIC EQUATIONS 63 by taking suitable integral linear combinations of the equations (1.1), we may suppose without loss that (3.3) b 1 = a2 = 0 and bi = 0 (8 ≤ i ≤ 12) Since we may suppose that a1 b2 = 0, the simultaneous equations (3 .4) L1 (θ) = L2 (θ) = 0 possess a solution... Lemma 4 is applicable to each of the mean values I12 (g13 ), I12 (h3 ) and I13 (h2 ), and so we see from (4. 6) that 1 1 ˜ ˜ ˜ g13 hs1 hs2 hs3 dα dβ 1 2 3 0 P 6+ξ+ε 0 The conclusion of Lemma 7 is now immediate on substituting the latter estimate into (4. 5) 5 Minor arcs, with some pruning Equipped with the mean value estimate provided by Lemma 7, an advance on the minor arc bound complementary to the major... η be a positive number with (c1 + c2 )η < 1 /4 sufficiently small in the context of the estimate (2.2), and put ν = 16(c1 + c2 )η Finally, recall from (5.3) that τ = (1 /4 − ξ)/3 > 10 4 Theorem 9 Suppose that t is a natural number with t ≥ 6, and let c1 , , ct be natural numbers satisfying (c1 , , ct ) = 1 Then for each natural number d there is a positive number ∆, depending at most on c and d,... the available generating functions We begin our account of the minor arcs by defining a set of auxiliary arcs to be employed in the pruning process Given a parameter X with 1 ≤ X ≤ P , we define M(X) to be the set of real numbers α with α ∈ [0, 1) for which there exist a ∈ Z and q ∈ N satisfying 0 ≤ a ≤ q ≤ X, (a, q) = 1 and |qα − a| ≤ XP −3 We then define sets of major arcs M = M(P 3 /4 ) and K = M(Q1/4 . we conclude that Λ 13 ∈ k (mod 1), and so a standard application of Weyl’s inequality (see Lemma 2 .4 of [V97]) in combination with available major arc estimates (see Theorem 4. 1 and Lemma 4. 6 of [V97]). all circumstances save for particularly pathological situations, leads in §5 to a viable complementary minor arc estimate sufficient to establish Theorem 2. The latter elusive situations are handled. efforts of Davenport and Lewis [DL66], Cook [C72], Vaughan [V77] and Baker and Br¨udern [BB88] spanning an interval of more than twenty years. In a recent collaboration [BW06]wehavebeenableto add the