The size of Selmer groups for the congruent number problem

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1 The size of Selmer grous for the congruent number roblem D.R. Heath-Brown Magdalen College, Oxford OX1 4AU 1 Introduction The oldest roblem in the theory of ellitic curves is to determine which ositive integers D can be the common difference of a three term arithmetic rogression of squares of rational numbers. Such integers D are known as congruent numbers. Equivalently one may ask which ellitic curves E D : y 2 = x 3 D 2 x have ositive rank. Clearly one may, and we shall, restrict attention to squarefree numbers D. At resent there is no known algorithm for deciding whether or not a given integer is a congruent number. However the conjecture of Birch and Swinnerton-Dyer [1], if true, would rovide such a rocedure. One defines L D (s) = (1 a s + 1 2s ) 1, a = + 1 N, / 2D where N is the number of solutions of the congruence y 2 x 3 D 2 x (mod ). Then L D (s) has an analytic continuation as an entire function on the comlex lane. The conjecture of Birch and Swinnerton-Dyer then states, in articular, that the rank r(d) of E D is equal to the order R(D) of L D (s) at s = 1, this being the so-called analytic rank. While we cannot at resent find R(D) in all cases, we can at least determine whether or not L D (1) = 0, and hence, conjecturally, whether or not r(d) = 0. Moreover one has a functional equation for L D (s) which relates its values at s and 2 s, via a sign change ε D = ±1. One may deduce that ( 1) R(D) = ε D. It would then follow from the conjecture of Birch and Swinnerton-Dyer that the rank is ositive whenever ε D = 1. According to the calculations of Birch and Stehens [2] one has { +1, D 1, 2, 3 (mod 8), ε D = 1, D 5, 6, 7 (mod 8), which would imly that D is congruent whenever D 5, 6 or 7 (mod 8). We know from the work of Coates and Wiles [5], Gross and Zagier [7], and Rubin 1

2 [11] that, for our curves, r(d) = R(D) whenever R(D) = 0 or 1, but little can be said when R(D) 2. A straightforward aroach to these questions is rovided by the use of descents. We shall be concerned with the full 2-descent, which can be done over Q for our curves. The rocess will be described in detail in the next section. However what is of interest for the resent discussion is that the number of 2- descents is the order of the Selmer grou S (2). This is a ower of 2, and will be a multile of 4, on account of the rational oints of order 2 on E D. We shall therefore write #S (2) = 2 2+s(D). The exonent s(d) has sometimes been refered to as the Selmer rank of the curve E D. According to the Selmer conjecture, s(d) and r(d) should have the same arity. It therefore seems likely, in view of the conjecture of Birch and Swinnerton-Dyer, that s(d) and R(D) always have the same arity. It should be noted that our terminology differs from that of Birch and Swinnerton-Dyer [1]. In their notation the number of first descents is λ+λ 1 2 which is often much larger than s(d). Indeed, λ + λ 1 2 is usually of order log log D at least, whereas s(d) is usually of order 1, as we shall see. For D = one may calculate that λ = 4, λ 1 = 0 whereas s(d) = 0. While it is known that λ + λ 1 2 must have the same arity as R(D), see Birch and Stehens [2] or Lagrange [9], the corresonding statement for s(d) has yet to be settled. The urose of this aer is to investigate s(d) on average. We rove the following results. Theorem 1 For any odd integer h let S(X, h) = {D h (mod 8) : 1 D X, D square-free}. (1) Then D S(X,h) 2 s(d) = 3#S(X, h) + O(X(log X) 1/4 (log log X) 8 ). Of course #S(X, h) is of order X, so that we have an asymtotic formula, with a relative saving of O((log X) 1/4 (log log X) 8 ). We can immediately deduce the following. Corollary 1 For any odd integer h we have 2 r(d) 3#S(X, h) + O(X(log X) 1/4 (log log X) 8 ). D S(X,h) When D 1 or 3 (mod 8) we exect s(d) to be even, so that s(d) 2 3 (2s(D) 1). 2

3 Similarly when D 5 or 7 (mod 8) we exect s(d) to be odd, and s(d) 1 3 (2s(D) + 1). Without any assumtion we can only say that s(d) 1 2 2s(D). We therefore deduce the following average bound for s(d). Corollary 2 Assume that s(d) and R(D) always have the same arity. Then for any odd integer h we have D S(X,h) s(d) 4 3 #S(X, h) + O(X(log X) 1/4 (log log X) 8 ). Hence s(d) = 0 for at least one third of all D 1 or 3 (mod 8), and s(d) = 1 for at least five-sixths of all D 5 or 7 (mod 8). Unconditionally we have D S(X,h) s(d) 3 2 #S(X, h) + O(X(log X) 1/4 (log log X) 8 ). We automatically deduce average bounds for r(d). Corollary 3 Assume that r(d) and R(D) always have the same arity. Then for any odd integer h we have D S(X,h) r(d) 4 3 #S(X, h) + O(X(log X) 1/4 (log log X) 8 ). Hence r(d) = 0 for at least one third of all D 1 or 3 (mod 8), and r(d) = 1 for at least five-sixths of all D 5 or 7 (mod 8). Unconditionally we have D S(X,h) r(d) 3 2 #S(X, h) + O(X(log X) 1/4 (log log X) 8 ). A coule of remarks should be made. 1. Our roof could be alied with only minor changes to the residue classes D 2 or 6 (mod 8). Other curves with 3 rational oints of order two could be handled the same way. However a certain amount of extra work would be needed to determine whether or not the all imortant constant 3 on the right hand side of (1) remains the same. One might also ask whether a corresonding result can be obtained when there is only one rational oint of order 2. In this case the descent must be done over a quadratic number field, but one still has good control over the 2-art of the class grou. 3

4 2. Averages of the analytic rank have been estimated for a number of classes of curves. Thus for examle, Brumer and Heath-Brown [3] show that, for twists of any given modular ellitic curve, R(D) has average at most 3 2, roviding that the corresonding L-functions satisfy the Riemann Hyothesis. Corollary 3 gives the same bound unconditionally, or a better bound under a far weaker hyothesis, but holds in a more restricted setting. It has been conjectured that the rank of an ellitic curve can be arbitrarily large, but it is not clear how frequent large ranks might be. Moreover it is unclear whether one should exect arbitrarily large ranks when one resticts attention to a family of twists of a fixed curve. For our curves the work of Gouvêa and Mazur [6] shows that #{D S(X, 1) : R(D) 2} X 1/2 ε, for any ε > 0. Indeed, it is evident that one can in fact obtain #{D S(X, 1) : R(D) 2} X 1/2, but we exect that much more is true. The corresonding roblem for s(d) is far more tractable however, and we rove the following estimate. Theorem 2 For any constant θ < 1 there exists c θ > 0 such that #{D S(X, 1) : s(d) > c θ log D} θ X θ. Of course it is immediate from the work of 2 that s(d) 2ω(D) log D log log D, where ω(d) is the number of rime factors of D. Thus the correct maximum order for s(d) is still in some doubt. While it is uncertain whether each fixed rank occurs for a ositive roortion of ellitic curves, for the Selmer rank this seems rather likely. Here we shall rove the following rather trivial result. Theorem 3 Let a non-negative integer n be given. Then #{D S(X, h) : s(d) = n} n for h = 1 or 3, for n even, or h = 5 or 7, for n odd. X log X, This is certainly not the strongest result of its tye, but to achieve a lower bound of order X aears to be beyond our reach at resent. Such a bound would of course allow the constant 4/3 in the corollary to Theorem 1 to be imroved. The resent aer was reared while the author was enjoying the hositality and financial suort of the University of Hong Kong. This assistance is gratefully acknowledged. 4

5 2 Counting 2-Descents We begin by describing the familiar descent rocess. Various accounts of this are available in the literatutre, see for examle Serf [12], but the arguments for removing the contribution of the torsion oints, and for dismissing the 2-adic conditions, seem to justify inclusion of a full descrition. We start from the well-known fact that the homomorhism θ : given at the non-torsion oints by E D (Q) Q G G G (G = 2E D (Q) Q 2 ) (x, y) (x, x + D, x D) (mod Q 2 ) is injective. The only torsion oints of E D (Q) are the oints of order two. We now observe that the coset of a non-torsion oint in E(Q)/Tors(E(Q)), consisting of (x, y), (x, y) + (0, 0) = ( D2 x and, D2 y x 2 (x, y) + ( D, 0) = (D D x D + x, x + D ), (x, y) + (D, 0) = (D x D, 2D 2 y (D + x) 2 ), 2D 2 y (x D) 2 ) contains exactly one element (x, y ) for which x > 0 and x 2 1. Consequently, if we restrict our oints (x, y) to have x > 0 and x 2 1, the image under the ma θ will have size 2 r(d). To analyze Im(θ) we write x = r/s, y = t/u with (r, s) = (t, u) = 1 and where r, s, u > 0 and r, s have oosite arities. Then r(r + sd)(r sd)u 2 = t 2 s 3, and since (t, u) = 1 we have u 2 s 3. Similarly, since (r, s) = 1 we see that s 3 is corime to r(r+sd)(r sd), so that s 3 u 2. Thus s 3 = u 2, and s = W 2, u = W 3 for some integer W. It follows that r(r + sd)(r sd) = t 2. We now write (r, D) = D 0 and r = D 0 r, whence D 3 0 t 2 and therefore D 2 0 t, since D 0 must be square-free. Thus r (r + s D )(r s D ) = D 0 ( t D 0 D 0 D0 2 ) 2. (2) Since (r, sd/d 0 ) = 1 and r + sd/d 0 is odd, because D and r + s are odd, it follows that the three factors on the left of (2) are corime in airs. We may therefore write D 0 = D 1 D 2 D 3, td 2 0 = XY Z, 5

6 and r = D 1 X 2, r + sd/d 0 = D 2 Y 2, r sd/d 0 = D 3 Z 2. On setting D/D 0 = D 4 we obtain the system D 1 X 2 + D 4 W 2 = D 2 Y 2, D 1 X 2 D 4 W 2 = D 3 Z 2. (3) Since r > 0 and D 0 > 0 we automatically have D 1, D 4 > 0 and therefore we see that D 2 > 0 if the first of the equations (3) is to have non-trivial real solutions. Then, as D = D 1 D 2 D 3 D 4 > 0 we see that D 3 must also be ositive. We have therefore roved the following. Lemma 1 There are exactly 2 r(d) systems (3) with non-trivial integer solutions. Moreover there are 2 s(d) systems (3) which are everywhere locally solvable. Of course the second assertion is just the definition of s(d). Our insistence that D j > 0 for j = 1,..., 4 already ensures that (3) have real solutions. Moreover it is an easy exercise to show that there are -adic solutions whenever / 2D. For rimes D 1 it is clearly necessary and sufficient for D 4 D 2 and D 4 D 3 to be squares modulo. Similarly, when D 4 we require D 1 D 2 and D 1 D 3 to be squares modulo. In case D 2 we write (3) as D 1 X 2 + D 4 W 2 = D 2 Y 2, 2D 1 X 2 = D 2 Y 2 + D 3 Z 2, which is solvable in Q if and only if D 1 D 4 and 2D 1 D 3 are squares modulo. Finally, when D 3 the condition is that D 1 D 4 and 2D 1 D 2 are squares modulo. Fortunately, for the rime = 2 no further condition is required, as the following result shows. Lemma 2 If the system (3) has solutions in R and in Q for every odd rime, then there are also solutions in Q 2. To rove this we observe that the equation 2D 1 X 2 = D 2 Y 2 + D 3 Z 2 has solutions in R and in Q for every odd rime, by our hyothesis about the solvability of (3). By the roduct formula for the Hasse norm residue symbol there are also solutions in Q 2. We may assume that such a solution involves 2-adic integers, at least one of which is a unit. Since D 1, D 2 and D 3 are odd, we see that Y and Z are 2-adic units, and therefore we must have D 2 + D 3 2D 1 or 0 (mod 8). A similar argument alied to the equation 2D 4 W 2 = D 2 Y 2 D 3 Z 2 6

7 shows that D 2 D 3 2D 4 or 0 (mod 8). It now follows that either D 2 +D 3 0 (mod 8) and D 2 D 4 (mod 4) or D 2 D 3 (mod 8) and D 2 D 1 (mod 4). Either ossibility suffices for the 2-adic solubility of the system (3). For examle, in the first case we can take W = 1 and X = 0 or 2 according as D 2 D 4 or 4 + D 4 (mod 8). This ensures that D 1 2 (D 1X 2 + D 4 W 2 ) D 1 3 (D 1X 2 D 4 W 2 ) 1 (mod 8) so that Y and Z can be determined aroriately. A similar argument alies when D 2 D 3 (mod 8) and D 2 D 1 (mod 4). We are now in a osition to write down our formula for 2 s(d). When D 1, for examle, the exression 1 4 {1 + (D 2D 4 )}{1 + ( D 3D 4 )} = 1 4 {1 + (D 2D 4 ) + ( D 3D 4 ) + ( D 2D 3 )} takes the values 1 or 0 according as D 2 D 4 and D 3 D 4 are both squares (mod ) or not. Thus, on setting Π 1 = (1 + ( D 2D 4 D 1 Π 2 = (1 + ( D 1D 4 D 2 Π 3 = (1 + ( 2D 1D 2 D 3 Π 4 = (1 + ( D 1D 2 D 4 we see that the roduct ) + ( D 3D 4 ) + ( 2D 1D 3 ) + ( D 1D 4 ) + ( D 1D 3 4 ω(d) Π 1 Π 2 Π 3 Π 4, ) + ( D 2D 3 )) ) + ( 2D 3D 4 )) ) + ( 2D 2D 4 )) ) + ( D 2D 3 )), where ω(d) is the number of rime factors of D, will be 1 if the system (3) is everywhere locally solvable, and 0 otherwise. We can exand Π 1, for examle, as Π 1 = ( D 2D 4 )( D 3D 4 )( D 2D 3 ), D 13 D 12 D 14 where the sum is over all factorizations D 1 = D 10 D 12 D 13 D 14. For brevity we shall write the sum as f 1. We shall exand the other factors Π i in the same way, and write f i = f(d). Here D reresents the 16-tule of elements D ij with 1 i 4, 0 j 4 and i j. According to Lemma 1, if we now sum over all quadrules D 1, D 2, D 3, D 4 with D = D 1 D 2 D 3 D 4 we will get a total of 4 ω(d) 2 s(d). We therefore conclude as follows. 7

8 Lemma 3 We have 2 s(d) = D g(d), where the sum is taken over all factorizations D = i,j D ij, and where with g(d) = ( 1 α )( 2 β ) i 4 ω(di0) j 0 4 ω(dij) k i,j α = D 12 D 14 D 23 D 21, β = D 24 D 21 D 34 D 31. l ( D kl D ij ) 3 Averaging over D; Linked Variables In this section we begin our estimation of 2 s(d). D S(X,h) Instead of summing over D we sum over the 16 variables D ij, subject to the conditions that each D ij is square-free, that they are corime in airs, and that their roduct D satisfies D X, D h (mod 8). We divide the range of each variable D ij into intervals (A ij, 2A ij ] where A ij runs over owers of 2. This will give us O(log 16 X) non-emty subsums, which we shall write as S(A), where A is the 16-tule of numbers A ij. Here we may suose that 1 A ij X. (4) We shall describe the variables D ij and D kl as being linked if i k, and recisely one of the conditions l 0, i or j 0, k. holds. This means that exactly one of the Jacobi symbols ( D kl D ij ), ( D ij D kl ) occurs in the exression for g(d). Let us suose that the variables D ij and D kl are linked, and that it is the first of the above Jacobi symbols which occurs. We can then write g(d) in the form g(d) = ( D kl D ij )a(d ij )b(d kl ), 8

9 where the function a(d ij ) deends on all the other variables D uv, say, as well as D ij, but is indeendent of D kl, and similarly for the function b(d kl ). Moreover we have a(d ij ), b(d kl ) 1. We can now write S(A) D uv D ij,d kl ( D kl D ij )a(d ij )b(d kl ). The conditions that D ij and D kl should be corime to each of the D uv may be exressed by taking the functions a and b to vanish at aroriate values. Moreover the Jacobi symbol is automatically zero if the D ij and D kl are not corime. The remaining conditions on these two variables may therefore be exressed by insisting that they are square-free and satisfy D ij D kl h (mod 8), D ij D kl X, where h and X will deend on the other variables D uv. We now call on the following estimate which we shall rove in 6. Lemma 4 Let a m, b n be comlex numbers of modulus at most 1. Let an odd number h be given and let M, N, X 1. Then ( n m )a mb n MN{min(M, N)} 1/32, m,n uniformly in X, where the sum is for square-free m, n satisfying M < m 2M, N < n 2N, mn X, and mn h (mod 8). It immediately follows that S(A) ( uv A uv )A ij A kl {min(a ij, A kl )} 1/32 X{min(A ij, A kl )} 1/32, by (4), and we deduce as follows. Lemma 5 We have S(A) X(log X) 17 whenever there is a air of linked variables with A ij, A kl log 544 X. We now examine the case in which A ij log 544 X, but every variable D kl to which D ij is linked has A kl < log 544 X. We write D for the roduct of these variables D kl. Using the law of quadratic recirocity we can now ut g(d) into the shae 4 ω(d ij) ( D ij D )χ(d ij)c, 9

10 where χ is a character modulo 8, which may deend on the variables D uv other than D ij, and the where the remaining factor c is indeendent of D ij and satisfies c 1. It follows that S(A) D uv D ij 4 ω(dij) ( D ij D )χ(d ij), (5) where the inner sum is restricted by the conditions that D ij must be square-free and corime to all the other variables D uv, and that D ij h (mod 8), A ij < D ij min(2a ij, X ), where h and X deend on the variables D uv other than D ij. We now aly the following result, which we shall rove in 6. Lemma 6 Let N > 0 be given. Then for arbitrary ositive integers q, r and any non-rincial character χ (mod q), we have µ 2 (n)4 ω(n) χ(n) xd(r) ex( c log x) n x, (n,r)=1 with a ositive constant c = c N, uniformly for q log N x. To use this result we remove the condition D ij h (mod 8) from the inner sum on the right of (5) and insert instead a factor Taking 1 4 and r = D uv, we conclude that ψ(mod 8) ψ(d ij )ψ(h ). q = 8D (log 544 X) 15 S(A) A ij ex( c log A ij ) D uv d(r), roviding that D 1. Since the variables D uv are corime in airs we have d(r) = d(d uv ). Moreover for a single variable D kl we will have D kl d(d kl ) A kl log A kl A kl log X, whence (5) yields S(A) X(log X) 15 ex( c log A ij ), roviding that D 1. We can now summarize as follows. 10

11 Lemma 7 There is an absolute constant κ > 0 such that S(A) X(log X) 17 whenever there are linked variables D ij and D kl for which and D kl > 1. A ij ex{κ(log log X) 2 } (6) We end this section with a straightforward estimate to handle the case in which at most three of the variables D ij lie in ranges satisfying (6). For brevity we shall write C = ex{κ(log log X) 2 } and assume that C is a ower of 2. Then if indicates the condition that at most three of the A ij satisfy A ij C, we have S(A) 4 ω(n1)... 4 ω(n16), A ij n 1...n 16 X where the n i are square-free and corime in airs, and at most three of the n i have n i 2C. We write m = n i, n = n i, n i<2c n i 2C so that m (2C) 16 and n X/m. Moreover we see that each value of m can arise at most 16 ω(m) times, and each value of n can arise at most ( 16 3 )3ω(n) times. We may therefore deduce that S(A) 4 ω(m) ( 3 4 )ω(n). m n A ij We now use the bound γ ω(n) N(log N) γ 1, (7) n N which is valid for any fixed γ > 0. Since X/m XC 16 X 1/2, we have log X/m log X, and we therefore find that S(A) X(log X) 1/4 4 ω(m) m 1. m A ij 11

12 A second alication of (7), together with artial summation, shows that 4 ω(m) m 1 log 4 M, m M whence S(A) X(log X) 1/4 log 4 C X(log X) 1/4 (log log X) 8. A ij In view of Lemma 7 we may now summarize as follows. Lemma 8 We have S(A) X(log X) 1/4 (log log X) 8, A where the sum over A is for all sets in which either there are at most three elements A ij C, or there are linked variables D ij and D kl with A ij C and D kl > 1. 4 Averaging over D; Characters modulo 8 We must now identify those sums S(A) which are not eliminated by Lemma 8. There must be four or more elements A ij C. If these include A 10 and A 20, say, then we must have D 13 = D 14 = D 23 = D 24 = D 31 = D 32 = D 34 = D 41 = D 42 = D 43 = 1, since these variables are all linked to either A 10 or A 20, or both. It follows that two or more of A 12, A 21, A 30 or A 40 must be at least C. If A 12 C, then D 30 = D 40 = 1, since these variables are linked to D 12, and similarly if A 12 C. On the other hand, if A 30 or A 40 is at least C, then we will have D 12 = D 21 = 1. We therefore conclude that when A 10, A 20 C, we must have either A 12, A 21 C and the remaining variables all equal to 1, or A 30, A 40 C and the remaining variables all equal to 1. Of course an analogous conclusion holds whenever A i0, A j0 C. Now let us suose that exactly one element A i0 satisfies A i0 C. Let us take this to be A 10. Then D 23 = D 24 = D 32 = D 34 = D 42 = D 43 = 1, these variables being linked to A 10. If A 12, A 21 C, say, then also D 13 = D 14 = D 30 = D 31 = D 40 = D 41 = 1, 12

13 and there is no fourth element A ij which can be greater than or equal to C. A similar argument alies if A 13, A 31 C or A 14, A 41 C. Hence we must have either A 12, A 13, A 14 C, or A 21, A 31, A 41 C, and in either case we see that the remaining variables D ij must all be equal to 1, since each one will be linked to a variable D kl with A kl C. Finally we examine the case in which all of the variables A i0 are below C. If, say A 12, A 13 C, then D 20 = D 21 = D 23 = D 24 = D 30 = D 31 = D 32 = D 34 = D 40 = D 41 = 1. If also A 14 C then D 42 = D 43 = 1, so that there cannot be four elements A ij C. We must therefore have A 42, A 43 C, whence all the remaining variables D ij will be 1. An analogous argument alies whenever A ij, A ik C with i, j, k distinct. There remains the ossibility that the four elements for which A ij C have four different values for i. If one of these is A 12, say, then D 20 = D 23 = D 24 = 1, since these are linked to D 12, and so A 21 C. The only variables linked to neither of D 12, D 21 are D 10, D 20, D 34 and D 43. It follows that A 34, A 43 C, and hence that all remaining variables D ij must be 1. We summarize our conclusions as follows. Lemma 9 A sum S(A) which is not considered by Lemma 8 must have exactly four elements A ij C, and the remaining variables D kl must take the value 1. The ossible sets of indices ij are 10, 20, 30, 40, i0, j0, ij, ji, i0, ij, ik, il, i0, ji, ki, li, ij, ik, lj, lk, and ij, ji, kl, lk, where i, j, k, l denote different non-zero indices. It remains to handle these 24 tyes of sum. We shall rename the variables D ij which occur non-trivially as n 1,..., n 4, and write N 1,..., N 4 for the corresonding A ij. We shall describe the variables N i, N j as being joined if both Jacobi symbols ( N i N j ), ( N j N i ) 13

14 occur in the definition of g(d). Thus D ij, D kl are joined if i k and j, l i, k, 0. If two variables are not joined we shall say they are indeendent. By abuse of terminology we shall also refer to the indices ij and kl as being joined or indeendent, as aroriate. For each A occuring in Lemma 9 we may now write S(A) in the form n 1,...,n 4 χ 1 (n 1 )... χ 4 (n 4 )P Q, Q = 4 ω(n1...n4), (8) where the variables are square-free, corime in airs, and satisfy N i < n i 2N i. Here the characters χ i are to modulus 8, and are exactly those arising from the terms ( 1 α ) and ( 2 β ) in the definition of g(d). The factor P is the result of alying the law of quadratic recirocity, to roduce a roduct of exressions ( 1) {n i 1}{n j 1}/4, one for each air of joined variables. For each tye of sum in Lemma 9 there is at least one air of indeendent variables. Thus, by re-labeling the variables n i as necessary we can estimate the exression (8) as χ 3 (n 3 )χ 4 (n 4 )P Q (9) n 1,n 2 n 3,n 4 with n 3, n 4 indeendent. It follows that P can be written as a roduct of characters ψ 3 (n 3 ), ψ 4 (n 4 ) modulo 4, deending on n 1, n 2, together with a factor deending on n 1, n 2 alone. We claim that, excet for the indices 10, 20, 30, 40; 40, 41, 42, 43; i0, ji, ki, li, we can choose the labeling so that ψ 3 = ψ 4 and χ 3 χ 4. To justify the first of these conditions we shall arrange that n 1, n 3 are joined if and only if n 1, n 4 are joined, and similarly for n 2, n 3 and n 2, n 4. To justfy our claim we first observe that the character χ corresonding to D 12, D 14 and D 23 is ( 1 ), the character corresonding to D 24, D 31 and D 34 is ( 2 ), and the character corresonding to D 21 is ( 2 ). The remaining variables have the trivial character. We begin by considering sums with indices i0, j0, ij, ji. Here one or other of ij or ji, say ij, automatically corresonds to a nontrivial character χ. We may then take n 1 = D i0, n 2 = D ji, n 3 = D j0, n 4 = D ij since every air of variables here is indeendent. Next we examine sums with indices i0, ij, ik, il. If i 4 then at least one of ij, ik, il, say ij, corresonds to a non-trivial character χ. Again each air of variables is indeendent, and we can take n 1 = D ik, n 2 = D il, n 3 = D i0, n 4 = D ij. 14

15 For sums with indices ij, ji, kl, lk, we observe that ij, ji necessarily corresond to different characters χ, and have indeendent variables associated to them. Moreover D ij is joined to both D kl and D lk, as is D ji. In this case we may therefore take n 1 = D kl, n 2 = D lk, n 3 = D ij, n 4 = D ji. Finally, for sums with indices ij, ik, lj, lk we observe that we can assume D ij, D ik to corresond to different characters χ. This is clearly true if i = 2, or by interchanging the labels i and l, if l = 2. We may therefore suose that j, say is 2. Now, if i = 3, then D 32 and D 3k will have different associated characters χ, whether k = 1 or 4. A similar argument alies if l = 3, so we may take k = 3, whence D 12 and D 13 will be variables with different associated characters χ. Finally, if D ij, D ik corresond to different characters χ, we can take n 1 = D lj, n 2 = D lk, n 3 = D ij, n 4 = D ik, since ij, ik are indeendent, whereas ij and ik are both joined to lj and lk. We have now verified that, for the sums in question, (9) may be ut into the shae ψ 3 χ 3 (n 3 )ψ 4 χ 4 (n 4 )Q, (10) n 1,n 2 n 3,n 4 with ψ 3 χ 3 ψ 4 χ 4. In fact this is also true for sums with indices i0, ji, ki, li when i = 1, 4. Here there is always at least one variable, ji, say, whose associated character χ is ( 2 ). We may then choose n 1 = D ki, n 2 = D li, n 3 = D i0, n 4 = D ji, which makes n 3 and n 4 indeendent. Moreover, since ψ 3 and ψ 4 are characters modulo 4, we automatically have ψ 3 χ 3 ψ 4 χ 4. We may roceed to aly the following lemma which we shall establish in 6. Lemma 10 Let X > 0 and M, N C > 0 be given. Then for an arbitrary ositive integer r, any odd integer h, and any distinct characters χ 1, χ 2 (mod 8), we have µ 2 (m)µ 2 (n)4 ω(m) ω(n) χ 1 (m)χ 2 (n) d(r)x ex( c log C) log X, m,n for some ositive absolute constant c, where the sum is over corime variables satisfying the conditions M < m 2M, N < n 2N, mn X, mn h (mod 8), (mn, r) = 1. It follows that the sums S(A) in question are all O(X(log X) 17 ), since the constant κ in Lemma 7 may be taken sufficiently large. The total contribution of these sums is therefore O(X(log X) 1 ), which is satisfactory. We summarize as follows. 15

16 Lemma 11 We have S(A) X(log X) 1/4 (log log X) 8, A where the sum over A is for all sets other than those corresonding to indices 10, 20, 30, 40; 40, 41, 42, 43; 20, 12, 32, 42 : 30, 13, 23, The leading terms For sums with indices 10, 20, 30, 40 or 40, 41, 42, 43 the function g(d) merely reduces to 4 ω(d), where D is the roduct of the variables D ij. The contribution of all sums with indices 10, 20, 30, 40 and A i0 C, is therefore D i0 4 ω(d), where the sum is subject to the conditions D i0 > C, D X, D h (mod 8), D square-free. We can remove the condition D i0 > C with an error µ 2 (abcd)4 ω(a) ω(b) ω(c) ω(d) = abcd X, a C ae X, a C a C 4 ω(a) µ 2 (ae)4 ω(a) ( 3 4 )ω(e) e X/a X(log X) 1/4 a C ( 3 4 )ω(e) 4 ω(a) a 1 X(log X) 1/4 (log log X) 2, by (7). Since D is square-free it factorizes as D 10 D 20 D 30 D 40 in exactly 4 ω(d) different ways. We therefore obtain 1+O(X(log X) 1/4 (log log X) 2 ) = S(X, h)+o(x(log X) 1/4 (log log X) 2 ). D X Precisely the same argument alies to sums with indices 40, 41, 42, 43. The situation for sums with indices 20, 12, 32, 42 is slightly more comlicated. Here we can comute, using the law of quadratic recirocity that g(d) reduces to 1 2 {1 + ( ) + ( ) ( )}4 ω(d). D 12 D 32 D 12 D 42 D 32 D 42 16

17 1 The term involving ( D 12D 32 ), for examle, may be handled using Lemma 10 as before, by utting the relevent art of the sum into the form (10) with n 1 = D 20, n 2 = D 12, n 3 = D 32, n 4 = D The terms containing ( D 12 D 42 ) and ( D 32 D 42 ) may be handled in recisely the same way, while the leading term, when summed over all aroriate vectors A, yields 1 2 S(X, h) + O(X(log X) 1/4 (log log X) 2 ), (11) by exactly the same argument as above. Finally we observe that sums with indices 30, 13, 23, 43 behave in recisely the same way, and again contribute a total of the form (11). Theorem 1 now follows. 6 Lemmas on character sums It remains to give the roof of Lemmas 4, 6 and 10. We start with Lemma 4. Results of this tye aear to have their origins in work of Heilbronn [8]. We begin by writing our sum as ( n m )a mb n, 1 i,j(mod 8) m i(mod 8) n j(mod 8) where m, n are sqare-free, and i, j are restricted to satisfy ij h (mod 8). The restrictions on m and n mean that the summand is now essentially symmetrical between m and n, by the law of quadratic recirocity. We may therefore suose that N M, whence it suffices to rove that a m ( n m ) NM31/32. N<n 2N M<m 2M Here we have droed all the conditions on n, but we have to retain the restrictions on m. By Cauchy s inequality the sum on the left is at most N 1/2 { a m ( n m ) 2 } 1/2, n M<m 2M and on exanding the sum, and inverting the order of summations we get at most N 1/2 { n ( ) } 1/2, m 1 m 2 m 1,m 2 n with m 1, m 2 still restricted to be square-free. The innermost sum is therefore trivial only when m 1 = m 2, contributing N 1/2 {MN} 1/2, 17

18 which is satifactory. When m 1 m 2 we may estimate the inner sum as N<n 2N n ( ) N 1/2 (m 1 m 2 ) 3/16+ε m 1 m 2 for any ε > 0, by Burgess bound [4]. Taking ε = 1/32, we get a total contribution N 1/2 {M 2 N 1/2 M 7/16 } 1/2 NM 31/32, since M N, and this also is satisfactory. This roves Lemma 4. We turn now to Lemma 6. For the roof we introduce the Dirichlet series f(s) = (1 + χ() 4 s ) = / r (n,r)=1 µ 2 (n)4 ω(n) χ(n) and g(s) = (1 χ() s ) {(1 χ() χ() s )(1 + 4 s )4 }. r / r The roducts for f(s) and g(s) converge absolutely for R(s) > 1 and R(s) > 1 2 resectively. Moreover, since f(s) 4 = g(s)l(s, χ), the function f(s) has an analytic continuation into any region σ σ 0 > 1 2, t T free of zeros of L(s, χ). We now recall that there are constants c 1 > 0 and c 2 (ε) > 0 such that L(s, χ) has no comlex zeros for σ 1 c 1, t T, log qt for T 2, and, by Siegel s Theorem, no real zeros for σ 1 c 2 (ε)q ε. On taking ε = 1/2N and T = ex( log x), the condition q (log x) N gives us a zero-free region R = {s : σ 1 δ, t T }, δ = c 3 log T, for an aroriate c 3 = c 3 (N) > 0. Moreover for such s we have L(s, χ) (1 + (qt ) (1 σ)/2 ) log T log T. We also have the trivial bound g(s) d(r) in the region R. We may therefore conclude that f(s) has an analytic continuation in the region R and satisfies f(s) d(r) log x there. We now aly Perron s formula (see Titchmarsh [13: Lemma 3.19]) to give n x, (n,r)=1 µ 2 (n)4 ω(n) χ(n) = 1 α+it f(s)x s ds 2πi α it s log x + O(x ) + O(1), T 18

19 where α = log x, and T = ex( log x) as before. We shall relace the ath of integration by three line segments from α it to 1 δ it to 1 δ + it to α + it. From the first and third of these we get a contribution O(xd(r)/T ), and from the second, a contribution O(x 1 δ d(r) log 2 x). It follows that n x, (n,r)=1 µ 2 (n)4 ω(n) χ(n) x log x T xd(r) T xd(r) ex( c log x) + x 1 δ d(r) log 2 x for an aroriate constant c > 0. This comletes the roof of Lemma 6. Finally we consider Lemma 10. This is in fact a straightforward deduction from Lemma 6. We shall suose that MN X, for otherwise the sum in question is emty. We remove the condition mn h (mod 8) from the summation and instead introduce the factor 1 ψ(mn)ψ(h). 4 ψ(mod 8) We shall estimate individually the sums corresonding to each character ψ. Since ψχ 1 ψχ 2, we may suose that ψχ 1, say, is non-rincial. Then the double sum under consideration is at most µ 2 (m)4 ω(m) ψ(m)χ 1 (m). n m The inner sum here is subject to the conditions (m, nr) = 1 and Thus Lemma 6 rovides an estimate M < m min(2m, X/n). Md(r)d(n) ex( c log M) for each of the inner sums. On summing over n we now obtain a bound Md(r)N(log N) ex( c log M), which is satisfactory. This comletes the roof of the lemma. 7 Proof of Theorem 2 In order to rove Theorem 2 we shall construct numbers D = 1... k with distinct rime factors i 1 (mod 8) for which ( i j ) = 1, i j. 19

20 According to Lemma 1 we will then have s(d) = 2k. We shall take P to be a sufficiently large arameter and restrict the rime factors i to lie in the range P/2 < i P. We shall be interested in a certain subset, S(P, k) say, of these numbers D. We shall say that a character χ is good if either its conductor is a divisor of 8, or if the Dirichlet L-function L(s, χ) has no zeros in the region R(s) 15, I(s) P. 16 We shall also say that D is good if every real character to modulus 8D is good. We shall define S(P, k) to be the set of numbers D, of the form already described, which are good. We shall also write S(P, k, q) for those elements of S(P, k) which are multiles of q. We shall ut ω(q) = j. Using induction on k, we shall show that rovided that and rovided that q S(P, j) and #S(P, k) 1 k! ( P 8 log P )k 2 3k k(k 1)/2 (12) 2 k P 1/40, (13) #S(P, k + j, q) 1 k! ( P 8 log P )k 2 2k kj k(k 1)/2. (14) 2 k+j P 1/40. (15) We begin by observing that when k = 0 we will have S(P, 0) = {1}, and S(P, j, q) = {q}, so that (12) and (14) are certainly true. This establishes the base ste for our induction. In getting from the case k to the case k + 1 we shall first rove (14) and then (12). While doing this we shall use the fact that P P 0, say, is sufficiently large, and we must be careful to ensure that P 0 is indeendent of k and q. To rove (12) and (14) we shall take D S(P, k) or D S(P, k + j, q) resectively, and write l = k or k + j as aroriate. In both cases we begin by counting the number of rimes in the set D(D) = {P/2 < P : 1 (mod 8), ( i ) = 1 for 1 i l}. If we let χ run over all real characters modulo 8D we see that 2 2 l χ χ() takes the value 1 if 1 (mod 8) and ( i ) = 1, 1 i l, 20

21 and 0 otherwise. We now define A(χ) = χ()(p 4 3P ) log P/2< P and B(χ) = χ(n)(p 4n 3P )Λ(n). P/2<n P Then if runs over D(D) we will have (P log P )#D(D) (P 4 3P ) log = 2 2 l χ A(χ). (16) We observe moreover that A(χ) = B(χ) + O(P 3/2 ) = B(χ ) + O(lP log P ) + O(P 3/2 ), (17) where χ is the rimitive character which induces χ. When the conductor of χ does not divide 8, we handle B(χ ) by means of the integral reresentation where B(χ ) = 1 2πi 2+i 2 i { L L (s, χ )}P s+1 w(s)ds, w(s) = 4(1 + (1/2)s+1 2(3/4) s+1 ). s(s + 1) We move the line of integration to R(s) = 1 2, where the integral is O(P 1/2 log D), and obtain the formula B(χ ) = ρ P ρ+1 w(ρ) + O(lP 1/2 log P ). (18) Here ρ runs over all non-trivial zeros of L(s, χ ). Since χ is good, all such zeros satisfy either R(ρ) or I(ρ) P. The symmetry of the zeros about the critical line then ensures that ρ 1 16 for each zero. We can then conclude that w(ρ) 1 log D, and w(ρ) 1 P 1 log D, I(ρ) P since there are O(T log DT ) zeros u to height T. It now follows from (18) that B(χ ) lp 31/16 log P. 21

22 If the conductor of χ divides 8, the Prime Number Theorem for arithmetic rogressions modulo 8 yields B(χ ) = ε P O(P 2 (log P ) 1 ), where ε = 1 if χ is identically 1, and ε = 0 otherwise. A comarison with (16) and (17) now reveals that #D(D) 2 2 l P 4 (log P ) 1 + O(lP 15/16 ) + O(2 l P (log P ) 2 ). There is therefore an absolute constant C 1 such that roviding that #D(D) 2 5 l P log P, (19) 2 l C 1 P 1/17. This last condition is a consequence of (13) or (15), if P is large enough. A recisely similar argument, based on the fact that (P log P/2)#D(D) (2P 4 3P ) log, shows that P/4< 5P/4 #D(D) 2 1 l P log P, (20) subject similarly to the conditions (13) and (15). We are now ready to rove (14) in the case k +1. Each D S(P, k +j +1, q) can be written in exactly k + 1 ways as D with q D. Moreover we must have D(D). Not all roducts D with D S(P, k + j, q) and D(D) will be good. However it certainly follows that #S(P, k + j + 1, q) = 1 k + 1 D S(P,k+j,q) #D(D) 1 k k! ( P 8 log P )k 2 2k kj k(k 1)/2.2 1 k j P log P 1 (k + 1)! ( P 8 log P )k+1 2 2(k+1) (k+1)j k(k+1)/2, by means of (20) and the case k of (14). This establishes (14) for k + 1. For the roof of (12) we observe that each D S(P, k + 1) can be written in exactly k + 1 ways as D = D, and in each such reresentation we have 22

23 D(D). We shall show that at least half of the numbers D formed in this way are good. We will then have #S(P, k + 1) = 1 2k + 2 D S(P,k) #D(D) 1 2k k! ( P 8 log P )k 2 3k k(k 1)/2.2 5 k P log P 1 (k + 1)! ( P 8 log P )k+1 2 3(k+1) k(k+1)/2, by means of (19) and the case k of (12). This establishes (12) for k + 1. We must now see how many values of D fail to be good. There is then some character χ to modulus 8D which is not good. If the conductor of χ is q, say, then q / 8D, since D is good. Thus q, and q = q, 4q or 8q with q D. It follows that, to each such q with ω(q ) = j k, there corresond at most j + 1 choices for, and at most choices for D, where #S(P, k, q ) 1 (k j)! ( P 8 log P )k j 2 e e = 2(k j) (k j)j (k j)(k j 1)/2 3(k + 1) k(k + 1)/2 (j + 1) 2 /2. We therefore see that each conductor q will divide at most (k + 1) j+2 2 (j+1)2 ( 8 log P P numbers 8D. Since q 8P j+1, we have roviding that (k + 1) j+2 2 (j+1)2 ( 8 log P P ) j+1 1 (k + 1)! ( P 8 log P )k+1 2 3(k+1) k(k+1)/2 ) j {(k + 1)2 j+1 32 log P 2 } j+1 P (k + 1) 2 2 k {P 2/3 } j+1 q 2/3, P 32 log P. This condition follows from (13) if P is sufficiently large. According to the zero-density theorem of Montgomery [10] the total number of zeros in R(s) σ, I(s) T, 23

24 for all rimitive L-functions of conductor at most Q, is (Q 2 T ) 2(1 σ)/σ (log QT ) 13, 4 5 σ 1. For σ = 15 16, T = P and Q P, the number of conductors which can occur is therefore O(Q 1/2 ). Since the relevent conductors q are all at least P/2, we see that the number of roducts D which are not good is at most q>p/2 q 2/3 1 (k + 1)! ( P 8 log P )k+1 2 3(k+1) k(k+1)/2 P 1/6 1 (k + 1)! ( P 8 log P )k+1 2 3(k+1) k(k+1)/2. It follows that at least half the available values of D are good, roviding that 2 6(k+1) C 2 P 1/6, with a suitable constant C 2. This inequality follows from (13) if P is large enough. This comletes our roof of (12). It remains to deduce Theorem 2 from the estimate (12). We shall take [ ] (1 θ) log X k = log 4 and P = X 1/k. It follows that both k and P tend to infinity with X. We will then have a set S(P, k) of numbers D X, for which s(d) = 2k log X as required. If θ is sufficiently close to 1, we will have 2 k2 X (1 θ)/2 X 1/40, whence (13) will hold. We will then have since P #S(P, k) ( 8k2 3+(k 1)/2 log P )k (P θ ) k = X θ, 8k2 3+(k 1)/2 2 k X (1 θ)/2k = P (1 θ)/2 P 1 θ log P, for sufficiently large X. This yields the estimate claimed in Theorem 2. 24

25 8 Proof of Theorem 3 Theorem 3 follows from the corollary to Theorem 1 when n = 0 or 1, and so we shall assume in this section that n 2. We begin by writing n/2 1, h = 1, k = n/2, h = 3, (n 1)/2, h = 5 or 7. We then use the construction of the revious section to roduce a roduct D 0 = 1... k with i 1 (mod 8) and ( i j ) = 1, i j. We shall kee D 0 fixed 1, and consider numbers of the form D = D 0, with h (mod 8) and ( i ) = 1, 1 i k. The system (3) now has -adic solutions for each i. Moreover, if D 1 there will be -adic solutions if and only if 1 is a quadratic residue of. For D 2 the condition is that 1 and 2 are both quadratic residues, but for D 3 we only need 2 to be a quadratic residue. Finally if D 4 the system automatically has solutions. Thus for h = 1 each of the systems (3) is everywhere locally solvable. For h = 3 only one quarter of them are everywhere locally solvable, and for h = 5 or 7 one half of them are admissable. Lemma 1 now shows that s(d) = n in each case, in view of our choice of k. It remains to observe that our conditions on can be achieved by requiring to lie in a suitable congruence class modulo 8D 0, so that the number of available rimes is X/ log X. The theorem then follows. References [1] B.J.Birch and H.P.F. Swinnerton-Dyer, Notes on ellitic curves. II, J. Reine Angew. Math., 218 (1965), [2] B.J. Birch and N.M. Stehens, The arity of the rank of the Mordell-Weil grou, Toology, 5 (1966), [3] A. Brumer and D.R. Heath-Brown, Average ranks of ellitic curves, II, to aear. [4] D.A. Burgess, On character sums and L-series. II, Proc. London Math. Soc. (3), 13 (1963), By extending the argument, to allow D 0 to vary, one could roduce more admissable values of D, thereby imroving the bound in Theorem 3 somewhat. 25

26 [5] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math., 39 (1977), [6] F. Gouvêa and B. Mazur, The square-free sieve and the rank of ellitic curves, J. Amer. Math. Soc., 4 (1991), [7] B.H. Gross and D. Zagier, Heegner oints and derivatives of L-series, Invent. Math., 84 (1986), [8] H.A. Heilbronn, On the averages of some arithmetic functions of two variables, Mathematika, 5 (1958), 1-7. [9] J. Lagrange, Nombres congruents et courbes ellitiques, Sém. Delange- Pisot-Poitou (Théorie des nombres), 16e année 1974/75, no 16. [10] H.L. Montgomery, Zeros of L-functions, Invent. Math., 8 (1969), [11] K. Rubin, Tate-Shafarevich grous and L-functions of ellitic curves with comlex multilication, Invent. Math., 89 (1987), [12] P. Serf, Congruent numbers and ellitic curves, (Comutational number theory, ) (Walter der Gruyter, Berlin, 1991). [13] E.C. Titchmarsh, The theory of the Riemann Zeta-function, 2nd Edition, revised by D.R. Heath-Brown, Clarendon Press, Oxford,