Invent. math. Ill, 171 195 (1993) Inve l tio les matbematicae Springer-Vedag 1993
The size of Selmer groups for the congruent number problem
D.R. Heath-Brown Magdalen College, Oxford OX1 4AU, United Kingdom
Oblatum 18-1-1992 & 20-VII-1992
1 Introduction
The oldest problem in the theory of elliptic curves is to determine which positive integers D can be the common difference of a three term arithmetic progression of squares of rational numbers. Such integers D are known as congruent numbers. Equivalently one may ask which elliptic curves ED: y2=x3--D2x have positive rank, Clearly one may, and we shall, restrict attention to square- free numbers D. At present 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 provide such a procedure. One defines
LD(S)= M (1--app-S+p'-2~) -1, ap=p+l--Np, p~2D where Np is the number of solutions of the congruence y2= X 3 -- D2X (mod p). Then Lv(s) has an analytic continuation as an entire function on the complex plane. The conjecture of Birch and Swinnerton-Dyer then states, in particular, that the rank r(D) of Eo is equal to the order R(D) of LD(s) at s= 1, this being the so-called analytic rank. While we cannot at present find R(D) in all cases, we can at least determine whether or not Lo(1)=0, and hence, conjecturally, whether or not r(D)= 0. Moreover one has a functional equation for Lo(s) which relates its values at s and 2-s, via a sign change e,o= _+ 1. One may deduce that (- 1)m~ It would then follow from the conjecture of Birch and Swin- nerton-Dyer that the rank is positive whenever eo = - 1. According to the calcu- lations of Birch and Stephens E2] one has
+l, D = 1, 2, 3 (mod 8), ~D ~ f -- 1, D -= 5, 6, 7 (mod 8), which would imply 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 172 D.R. Heath-Brown
[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 approach to these questions is provided by the use of descents. We shall be concerned with the "full 2-descent", which can be done over Q for our curves. The process will be described in detail in the next section. However what is of interest for the present discussion is that the number of 2-descents is the order of the Selmer group S (2J. This is a power of 2, and will be a multiple of 4, on account of the rational points of order 2 on ED. We shall therefore write ~ S~2)= 2 2 +sID). The exponent s(D) has sometimes been refered to as the 'Selmer rank' of the curve E D. According to the Selmer conjec- ture, s(D) and r(D) should have the same parity. It therefore seems likely, in view of the conjecture of Birch and Swinnerton-Dyer, that s(D) and R(D) always have the same parity. It should be noted that our terminology differs from that of Birch and Swin- nerton-Dyer [1]. In their notation the 'number of first descents' is 2+21-2 which is often much larger than s(D). Indeed, 2+21-2 is 'usually' of order loglogD at least, whereas s(D) is 'usually' of order 1, as we shall see. For D=3.11.59 one may calculate that 2=4, 21=0 whereas s(D)=0. While it is known that 2 + 21 -2 must have the same parity as R (D), see Birch and Stephens [2] or Lagrange [9], the corresponding statement for s(D) has yet to be settled. The purpose of this paper is to investigate s(D) on average. We prove the following results. Theorem 1 For any odd integer h let
S(X, h) = {D = h (mod 8): 1 < D < X, D square-free}.
Then (1) 2stD)= 3 ~ S(X, h) + O(X (log X)- l/4(log log X)S). DES(X,h)
Of course +S(X,h) is of order X, so that we have an asymptotic 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 2r(D)<=3~S(X,h)+O(X(logX)-t/'~(loglogX)S). D~S(X,h)
When D- 1 or 3 (mod 8) we expect s(D) to be even, so that s(D) < 2 (2sip)_ 1).
Similarly when D = 5 or 7 (mod 8) we expect s(D) to be odd, and s(D)< + 1).
Without any assumption we can only say that s(D)< 2 ~D).
We therefore deduce the following average bound for s(D). The size of Selmer groups for the congruent number problem 173
Corollary 2 Assume that s(D) and R(D) always have the same parity. Then for any odd integer h we have
s(D)<~ @ S(X, h)+ O(X(log X)- t/4(log log X)8). DeS(X, h)
Hence s(D)=0 for at least one third of all O-t or 3 (rood 8), and s(D)=l for at least five-sixths of all D--5 or 7 (rood 8). Unconditionally we have
~" s(D)<~ #~S(X,h)+O(X(logX)-'/4(IoglogX)S). DeS(X, h)
We automatically deduce average bounds for r(D). Corollary 3 Assume that r(D) and R(D) always have the same parity. Then for any odd integer h we have
r(D)<=~ @S(X,h)+O(X(log X)- ~/4(loglog X)S). DeS(X, h)
Hence r(D)--0 for at least one third of all D-1 or 3 (rood 8), and r(D)=l for at least five-sixths of all D- 5 or 7 (mod 8). Unconditionally we have
r(D) < ~ ~- S (X, h) + O(X (log X)- '/4 (log log X)S). DeS(X,h)
A couple of remarks should be made. I. Our proof could be applied with only minor changes to the residue classes D=2 or 6 (rood 8). Other curves with 3 rationat points 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 important constant 3 on the right hand side of (1) remains the same. One might also ask whether a corre- sponding result can be obtained when there is only one rational point 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-part of the class group. 2. Averages of the analytic rank have been estimated for a number of classes of curves. Thus for example, Brumer and Heath-Brown [3] show that, for twists of any given modular elliptic curve, R(D) has average at most 2, providing that the corresponding L-functions satisfy the Riemann Hypothesis. Corollary 3 gives the same bound unconditionally, or a better bound under a far weaker hypothesis, but holds in a more restricted setting. It has been conjectured that the rank of an elliptic curve can be arbitrarily large, but it is not clear how frequent large ranks might be. Moreover it is unclear whether one should expect arbitrarily large ranks when one restricts attention to a family of twists of a fixed curve. For our curves the work of Gouv~a and Mazur [6] shows that
@ {DeS(X, 1): R(D)>=2} > X ~/2-~, for any e > 0. Indeed, it is evident that one can in fact obtain
@ {DeS(X, 1): R(D)>=2} >~X 1/2, 174 D.R. Heath-Brown but we expect that much more is true. The corresponding problem for s(D) is far more tractable however, and we prove the following estimate. Theorem 2 For any constant 0 < 1 there exists Co > 0 such that
{DeS(X, 1): s(D)> c0]/]og D} >>0 X~
Of course it is immediate from the work of w 2 that
log D s(D) < 2~ ~-log~og D , where o (D) is the number of prime 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 positive proportion of elliptic curves, for the Selmer rank this seems rather likely. Here we shall prove the following rather trivial result. Theorem 3 Let a non-negative integer n be given. Then
X # {DeS(X,h): s(D)=n} >>, logX' for h = 1 or 3, for n even, or h = 5 or 7, for n odd. This is certainly not the strongest result of its type, but to achieve a lower bound of order X appears to be beyond our reach at present. Such a bound would of course allow the constant 4/3 in the corollaries to Theorem 1 to be improved.
2 Counting 2-descents
We begin by describing the familiar descent process. Various accounts of this are available in the literature, see for example Serf [12], but the arguments for removing the contribution of the torsion points, and for dismissing the 2-adic conditions, seem to justify inclusion of a full description. We start from the well-known fact that the homomorphism
E~ -~G•215 0: 2ED(Q) given at the non-torsion points by
(x, y) --+ (x, x + D, x - D) (mod Q • 2) is injective. The only torsion points of Eo(ff)) are the points of order two. We now observe that the coset of a non-torsion point in E(~)/Tors (E(Q)), consisting of (_D2 D2y ( x+D -2D2y (x, y), (x, y) + (0, 0) = x ' x 2 }' (x, y) + (D, 0) = D x- D' (x-~)2} The size of Selmer groups for the congruent number problem 175 and
D--x-2D2y~ (x,y)+(--D,O)= D D+x' (D+x)2] ' contains exactly one element (x', y') for which x'> 0 and [x'12 4: 1. Consequently, if we restrict our points (x, y) to have x>0 and lxt24: 1, the image under the map 0 will have size 2 r~~ To analyze Im (0) we write x=r/s, y=t/u with (r,s)=(t, u)= 1 and where r, s, u > 0 and r, s have opposite parities. Then r(r +sD)(r--sO)uZ=tZs 3, and since (t, u) = 1 we have uZls3. Similarly, since (r, s) = 1 we see that s a is coprime to r(r+sD)(r-sD), so that s31u2. Thus s3=u z, and s=W z, u=W 3 for some integer W. It follows that r(r + sD)(r--sD)=t 2.
We now write (r,D)=Do and r=Dor', whence D3[tz and therefore D~[t, since D o must be square-free. Thus
(2) r'(r'+s D , D t 2
Since (r', sD/Do)= 1 and r'+ s D/D o is odd, because D and r + s are odd, it follows that the three factors on the left of (2) are coprime in pairs. We may therefore write Do=D~D2D3, tDo2=XyZ, and r'=D1X2, r'+sD/Do=D2Y2, r, sD/Do=D3Z2.
On setting D/Do = D4 we obtain the system
(3) D1X2+D4W2=D2Y 2, DIX2-D4W2=D3 Z2.
Since r>0 and D0>0 we automatically have D~, D4>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=DID2D3D4>O we see that D 3 must also be positive. We have therefore proved the following. Lemma 1 There are exactly 2r(~ systems (3) with non-trivial integer solutions. Moreover there are 2SIp) 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 p-adic solutions whenever pX2D. For primes p[D~ it is clearly necessary and sufficient for D4D 2 and --D4D3 to be squares modulo p. Similarly, when p[D4 we require D 1D 2 and DID 3 to be squares modulo p. In case p]D 2 we write (3) as D1XZ+D4W2=D2y2 , 2D1X2=D2y2+D3 Z2, 176 D.R. Heath-Brown which is solvable in ~v if and only if -D1 D4 and 2D1 D3 are squares modulo p. Finally, when p]D 3 the condition is that D~ D 4 and 2D1 D2 are squares modulo p. Fortunately, for the prime p=2 no further condition is required, as the following result shows. Lemma 2 If the system (3) has solutions in IR and in @v for every odd prime p, then there are also solutions in ~2. To prove this we observe that the equation 2D 1X2=D2 YZ+D3Z2 has solutions in ~, and in Qp for every odd prime p, by our hypothesis about the solvability of (3). By the product formula for the Hasse norm residue symbol there are also solutions in Q2. We may assume that such a solution involves 2-adic integers, at least one of which is a unit. Since D1, D2 and D3 are odd, we see that Yand Z are 2-adic units, and therefore we must have D2+D3==-2D1 or 0(mod 8). A similar argument applied to the equation 2D4 W 2 =D2 y2 __D3Z2 shows that D2--D3=_2D4 or 0(mod 8). It now follows that either Dz+D3=0(mod8) and Dz=-D4(mod4 ) or D 2 -D 3 (rood 8) and D 2 ~ D l (mod 4). Either possibility suffices for the 2-adic solu- bility of the system (3). For example, in the first case we can take W= 1 and X=0 or 2 according as Dz=D4 or 4+D4(mod 8). This ensures that D21 (D1 X 2 -~ D4 W 2) = D~ 1(D 1 X 2 _ D4 W e) ~ l (mod 8) so that Y and Z can be determined appropriately. A similar argument applies when D 2 ~ D 3 (rood 8) and D2 = D1 (rood 4). We are now in a position to write down our formula for 2 s(~ When plD~, for example, the expression 4{ 1 + (D2 DA)}{ 1 + (~))= I{1 +(D~)+ (-D3D4)+(--D;D3)} takes the values 1 or 0 according as DzD4 and -D3D 4 are both squares (mod p) or not. Thus, on setting
"2 = U (lff-(-DID4]q-(2DID3]q-(--2D3D4])
/73---- 1~ (1 +(2D1Dz]+(D1D4]+(2Dz~D4]] viol\ \ P / \ P / \ P // The size of Selmer groups for the congruent number problem 177 we see that the product 4- o,~D)H ~Hz H3 H4, where co(D) is the number of prime factors of D, will be l if the system (3) is everywhere locally solvable, and 0 otherwise. We can expand H~, for example, as /D z D~\/- D 3 D4\/-- D 2 D3\ where the sum is over all factorizations
D1 =DIoDI2D13D14.
For brevity we shall write the sum as ~f~. We shall expand the other factors H i in the same way, and write Hf/=f(D). Here D represents the 16-tuple of elements Dij with 1 D = ~I D~j, i.j and where g(D): -1 2 U 4-'~1761764-'~ I~ ]-](D~L) with o~=D12DI4D23D21, fl=D24DzlD34D31. 3 Averaging over D; Linked variables In this section we begin our estimation of 2 2s(D)" DeS(X, h) Instead of summing over D we sum over the 16 variables D~, subject to the conditions that each D~j is square-free, that they are coprime in pairs, and that their product D satisfies D We divide the range of each variable D~j into intervals (A~, 2AJ where A~j runs over powers of 2. This will give us O(logl6X) non-empty subsums, which 178 D.R. Heath-Brown we shall write as S(A), where A is the 16-tuple of numbers A~r Here we may suppose that (4) 1 ~ I-[ Air <" X. We shall describe the variables Dij and Dkl as being 'linked' if/+ k, and precisely one of the conditions l=~0, i or j=t=0, k holds. This means that exactly one of the Jacobi symbols \ Okl,I occurs in the expression for g(D). Let us suppose that the variables Dij and Dkt 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) = (~)a(D~j)b(Dkt), where the function a(D~j) depends on all the other variables Du~, say, as well as Dij, but is independent of Dkt, and similarly for the function b(Dkl). Moreover we have la(Dij)l, Ib(Dkt)l <=1. We can now write The conditions that Dij and Dkt should be coprime to each of the Duv may be expressed by taking the functions a and b to vanish at appropriate values. Moreover the Jacobi symbol is automatically zero if the Dij and Dkt are not coprime. The remaining conditions on these two variables may therefore be expressed by insisting that they are square-free and satisfy DijDkl=h'(mod 8), DijDkl<=X ', where h' and X' will depend on the other variables Du,,. We now call on the following estimate which we shall prove in w 6. Lemma 4 Let am, bn be complex numbers of modulus at most 1. Let an odd number h be given and let M, N, X ~ 1. Then ~., (~) ambn ~ MN {min(M, N)} -1/32, uniformly in X, where the sum is for square-free m, n satisfying M S(A) ~ (1-I Auv)AiiAkz {min(A,;, Akt)} - 1/32 ~ X {min (A,j, Akl)} - 1/32 uv The size of Selmer groups for the congruent number problem 179 by (4), and we deduce as follows. Lemma 5 We have S(A)~X(Iog X) -17 whenever there is a pair of linked variables with Aij, Akl>=log544 X. We now examine the case in which Aij>=logS44X, but every variable Dkl to which D~i is linked has Akt 4-~176 z(D,)c, where Z is a character modulo 8, which may depend on the variables D,,, other than D~j, and the where the remaining factor c is independent of Di i and satisfies Icl < 1. It follows that (5) IS(A)[< ~ I~ 4-o'(~ Duv D~j where the inner sum is restricted by the conditions that Di~ must be square-free and coprime to all the other variables Du,,, and that Dij= h'(mod 8), Aij Y, li2(n)4-~'~")z(n)~xd(r)exp(-c~gx) n 1 ~, ~(D,j)tp(h'). qt (mod 8) Taking q = 8 D' ~ (log 5.4 X)I 5 and r= liD, v, we conclude that S(A)~Aij exp(-c]/i~) y, d d(Dk~) ~ Ak~ log AkZ ~ Akt log X, whence (4) yields S(A),~ X(log X) is exp(- c~), providing that D' 4= 1. We can now summarize as follows, Lemma 7 There is an absolute constant K > 0 such that S(A) ~ X (log X)- ~7 whenever there are linked variables Dij and Dkt for which (6) Air > exp {to(log log X) 2} and Dkl > 1, We end this section with a straightforward estimate to handle the case in which at most three of the variables D~j lie in ranges satisfying (6). For brevity we shall write C = exp {~c(log log X) 2 } and assume that C is a power of 2. Then if ~' indicates the condition that at most three of the Azj satisfy Aij ~ C, we have ~' IS(A)I < ~ 4-~"'~... 4 -~1"'6~, Aij ill ,.,rll6~X where the n~ square-free and coprime in pairs, and at most three of the n~ have n~ > 2 C. We write m= I] nl, n= H ni, n~<2C nz>2C so that m=<(2C) t6 and n E' IS (A){ <~ E 4~ Z (]-)'~'")" Aij m n We now use the bound (7) 7 ~'t") ,~ N(log N) ~- 1, n X/m>> XC- ~6 >>X~/2, The size of Selmer groups for the congruent number problem 181 we have log X/m >>log X, and we therefore find that ~' IS (A)[ ~ X(log X)-1/,, ~. 4"'"m- ~. h,j m A second application of (7), together with partial summation, shows that &,l,,) m- t ,~ log4 M, m In view of Lemma 7 we may now summarize as follows. Lemma 8 We have ]S(A)I ~ X(log X)- 1/4 (log log X) s, A where the sum over A is for all sets in which either there are at most three elements Aij > C, or there are linked variables Dij and Dkl with Aij > C and Dkl > 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~j>C. If these include Alo and A2o, say, then we must have D13~-D14=D23=D24=D31 =932 =D34 =D41 =04. 2 =D43 = 1, since these variables are all linked to either Alo or A2o , or both. It follows that two or more of A12 , A21 , A3o or A4o must be at least C. If AI2~C, then D30=D40= 1, since these variables are linked to D12, and similarly if A12 >C. On the other hand, if A30 or A40 is at least C, then we will have D12=D2~ = 1. We therefore conclude that when Ato, A20>C, we must have either Ax2, A21~C and the remaining variables all equal to 1, or A3o, A4o~C and the remaining variables all equal to 1. Of course an analogous conclusion holds whenever Ago, Ajo > C. Now let us suppose that exactly one element Ai0 satisfies Aio>C. Let us take this to be A~o. Then D23=D24=D32 =D34 =D42 =D43 = 1, these variables being linked to A~o. If A12 , A21 ~ C, say, then also D13=D14=D3o=D31 = D40 =D41 = 1, and there is no fourth element A~ which can be greater than or equal to C. A similar argument applies if A~3, A3~>C or A14, A41>--_C. Hence we must have either A12, At3, A~4>C, or A21 , A31, A41~C, and in either case we 182 D.R. Heath-Brown see that the remaining variables D~i must all be equal to 1, since each one will be linked to a variable Dkl with Akl> C. Finally we examine the case in which all of the variables A~o are below C. If, say A~2, A13>=C, then D20= D21 = D23 = D24 = D30= D3~ =D32 = D34 = D40= D 41 = I. If also A 14 > C then D42 = D43 = 1, so that there cannot be four elements Aij > C. We must therefore have A42 , A43 =>C, whence all the remaining variables Dij will be 1. An analogous argument applies whenever Aij, A~k>C with i, j, k distinct. There remains the possibility that the four elements for which ,4ij> C have four different values for i. If one of these is A ~z, say, then D2o=D23=D24=1, since these are linked to O~2 , and so A21~=C. The only variables linked to neither of D12, D2~ are Dlo, D20, D34 and D43. It follows that A34 , A43~C , and hence that all remaining variables D~j 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 AIj>C, and the remaining variables Dkl must take the value 1. The possible sets of indices ij are 10, 20, 30, 40, iO,jO, ij, ji, iO, ij, ik, il, iO, ji, ki, li, ij, ik, lj, Ik, and ij, ji, kl, lk, where i, j, k, I denote different non-zero indices. It remains to handle these 24 types of sum. We shall rename the variables Di~ which occur non-trivially as n 1 .... , n4, and write N1, ..., N4 for the corre- sponding A,j. We shall describe the variables Ni, Nj as being 'joined' if both Jacobi symbols occur in the definition of g(D). Thus Dij , Dkz are joined if i=~ k and j, 14: i, k, 0. If two variables are not joined we shall say they are 'independent'. By abuse of terminology we shall also refer to the indices ij and kl as being joined or independent, as appropriate. For each A occurring in Lemma 9 we may now write S(A) in the form (8) ~ xI(nO...z4(n4)PQ, Q=4 -'~ ...... ), nl,...,n4 The size of Selmer groups for the congruent number problem 183 where the variables are square-free, coprime in pairs, and satisfy ~ (9) ~ I Z z3(n3)z4(n4)PQI nl,n2 n3,n4 with n 3, n 4 independent. It follows that P can be written as a product of charac- ters ~/3(n3), ~4(n4) modulo 4, depending on nl, r/E, together with a factor depend- ing on hi, n2 alone. We claim that, except for the indices 10,20,30,40; 40,41,42,43; iO, ji, ki, li, we can choose the labeling so that ~3=t/,4 and Z3+Z4- To justify the first of these conditions we shall arrange that nl, n3 are joined if and only if nl, n 4 are joined, and similarly for n2, n 3 and n2, n4. To justify our claim we first observe that the character Z corresponding toD~2, D14andD23is(~),thecharactercorrespondingtoDz4, D31 and D34 is (2), and the character corresponding to D21 is (~-~). The remaining variables have the trivial character. We begin by considering sums with indices iO, jO, ij, ji. Here one or other of ij or ji, say ij, automatically corresponds to a nontrivial character X. We may then take nl=D~o, n2-=Dji, n3=Djo, n4=Dij since every pair of variables here is independent. Next we examine sums with indices i0, ij, ik, il. If i~4 then at least one of ij, ik, il, say ij, corresponds to a non-trivial character g. Again each pair of variables is independent, and we can take nl =Dik, nz=Dil, n3=Dio, n4=Dij. For sums with indices ij, ji, kl, lk, we observe that ij, ji necessarily correspond to different characters Z, and have independent variables associated to them. Moreover Dij is joined to both Dkt and Dik, as is Dj~. In this case we may therefore take nl = Dkt, nz = Dtk, n3 = Di j, n4 = Djl. Finally, for sums with indices ij, ik, lj, lk we observe that we can assume D~j, Dik to correspond to different characters Z. This is clearly true if i--2, or by 184 D.R. Heath-Brown interchanging the labels i and l, if 1= 2. We may therefore suppose that j, say is 2. Now, if i=3, then D3z and D3k will have different associated characters Z, whether k = 1 or 4. A similar argument applies if l= 3, so we may take k = 3, whence O12 and D13 will be variables with different associated characters Z. Finally, if Dij, Dig correspond to different characters Z, we can take nl = Dlj, n2 = Dtk, n3 = Dij, n4---- Dik, since ij, ik are independent, whereas ij and ik are both joined to lj and lk. We have now verified that, for the sums in question, (9) may be put into the shape (10) ~ [2 IP3z3(n3)~/4z4(n4)QI, hi,n2 n3,n4 with ~/3)~3:#~/4)~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 Z is (2). We may then choose nl = Dki, n2 = Dsi, n3-= Dio, n4 = Djl, which makes n3 and n4 independent. Moreover, since ~3 and ~'4 are characters modulo 4, we automatically have @3.~3 =~=@4-)~4" We may proceed to apply the following lemma which we shall establish inw Lemma 10 Let X>0 and M, N>_C>O be given. Then for an arbitrary positive integer r, any odd integer h, and any distinct characters Zl, Z2 (rood 8), we have p2(m)p2(n)4-~ (m)Zz(n)~ d(r)X exp(- cl/log C)log X, m, n for some positive absolute constant c, where the sum is over coprime variables satisfying the conditions M It follows that the sums S(A) in question are all O(X(logX)-17), since the constant x in Lemma 7 may be taken sufficiently large. The total contribution of these sums is therefore O(X(log X)- l), which is satisfactory. We summarize as follows. 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 corresponding to indices 10,20,30,40; 40,41,42,43; 20,12,32,42; 30,13,23,43. The size of Selmer groups for the congruent number problem 185 5 The leading terms For sums with indices 10, 20, 30, 40 or 40, 41, 42, 43 the function g(D) merely reduces to 4 ,,,(D), where D is the product of the variables Dij. The contribution of all sums with indices 10, 20, 30, 40 and Aio > C, is therefore Dto where the sum is subject to the conditions Dio>C, D Precisely the same argument applies to sums with indices 40, 41, 42, 43. The situation for sums with indices 20, 12, 32, 42 is slightly more complicated. Here we can compute, using the law of quadratic reciprocity that g(D) reduces to 2[ \D12D32] \DI2D42/ \D32 D,2,/J " The term involving ~ for example, may be handled using Lemma 10 as before, by putting the relevent part of the sum into the form (10) with nl=D20, n2=D12, H3=D32, n4=D34. The terms containing (-~) and (-~2-D~2) may be handled in precisely the same way, while the leading term, when summed over all appropriate vectors A, yields (11) 4# S(X, h)+ O(X(log X)-1/*(log log X)2), 186 D.R. Heath-Brown by exactly the same argument as above. Finally we observe that sums with indices 30, 13, 23, 43 behave in precisely 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 proof of Lemmas 4, 6 and 10. We start with Lemma 4. Results of this type appear to have their origins in work of Heilbronn [8]. We begin by writing our sum as (n) Z E (~mo mambn, i,j(mod8) m~i(modS) n=_j dS) where m, n are square-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 reciprocity. We may therefore suppose that N > M, whence it suffices to prove that E am(~) "~NM31/32" N< =2N M Here we have dropped all the conditions on n, but we have to retain the restric- tions on m. By Cauchy's inequality the sum on the left is at most N 1/2 a m . M \m}lJ ' and on expanding the sum, and inverting the order of summations we get at most N1/2 {ml~m2 ~n \mlm~]]. ) ' with ram, m2 still restricted to be square-free. The innermost sum is therefore trivial only when ml = m2, contributing N 1/2 {MN} 1/2, which is satisfactory. When ml + m2 we may estimate the inner sum as E (?1 t<~N1/2(mlm2)3/16+~ N We turn now to Lemma 6. For the proof we introduce the Dirichlet series f(s)= I-I /{1 + Z(P)]= ~. ,,2(n)4-~,,.,z(n) n vz~" 4P~] ~.,,)=l and pl, \ -4p7,1J" The products for f(s) and g(s) converge absolutely for ~(s)> 1 and 91(st> respectively. Moreover, since f(s)4 = g(s) L(s, Z), the function f(s) has an analytic continuation into any region er > ao > ftr < T free of zeros of L(s, Z). We now recall that there are constants c l > 0 and c2(e)> 0 such that L(s, Z) has no com- plex zeros for a>l cl log q Y' [tl _-< T, for T>2, and, by Siegel's Theorem, no real zeros for a> 1 -c2(0 q-q On taking e=l/2N and T=exp( lol/i~x ), the condition q<(logx) N gives us a zero-free region . C3 R={s:o>l-a, ltl L(s, X)~(1 +(qT) I1 -a)/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 apply Perron's formula (see Titchmarsh [13, Lemma 3. l 9]) to give 1 Z ,u2(n)O-~ f(s)xsdS+o +O(1), n< x,(n,r)= l where 1 ~=l+-- logx' and T= exp( 1]//1]/~ x) as before. We shall replace the path of integration by three line segments from e- iT to 1 -- &- i T to 1 - 6 + i T to ~ + iT. From the first 188 D.R. Heath-Brown and third of these we get a contribution O(xd(r)/T), and from the second, a contribution O(x 1 -Od(r) log2x). It follows that 1~2(n) 4-~'~")Z(n) <~x--l~f + l + x~-) + xl -~ d(r) log2 x n 1 4 ~ ~b (m n) ~b(h). O(mod 8) We shall estimate individually the sums corresponding to each character q/. Since ~b)~l :t:~bZ2, we may suppose that ~bZl, say, is non-principal. Then the double sum under consideration is at most ~1~ #2(m) 4-"t") @(m) Z1 (m)l. n m The inner sum here is subject to the conditions (m, n r)= 1 and M < m < min (2 M, X/n). Thus Lemma 6 provides an estimate M d (r) d (n) exp ( - c 1/log M) for each of the inner sums. On summing over n we now obtain a bound md(r) N(log N) exp(- cl/logg m), which is satisfactory. This completes the proof of the lemma. 7 Proof of Theorem 2 In order to prove Theorem 2 we shall construct numbers D = Pl ... Pk with distinct prime factors Pi - 1 (rood 8) for which According to Lemma 1 we will then have s(D)=2k. We shall take P to be a sufficiently large parameter and restrict the prime factors Pi to lie in the range P/2 ~)~(S) ~a~'15 13(S)I = We shall also say that D is 'good' if every real character to modulus 8D is good. We shall define 5f(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 5f(P, k) which are multiples of q. We shall put ro(q)=j. Using induction on k, we shall show that 1 / \P k (12) #J(P, k)> l---1 2 -3k keg-,,/2 = k! \8 log P/ provided that (13) 2 a < pt/40, and 1 / p \k (14) #J(P,k+j,q)__<,,K!\![~L~] g 22k-k2 k'k ')/2, provided that q~Sr~(P,j) and (15) 2k+J<=PU4~ We begin by observing that when k=0 we will have r176 0)={l], and 5f(P,j, q)= {q}, so that (12) and (14) are certainly true. This establishes the base step for our induction. In getting from the case k to the case k+ 1 we shall first prove (14) and then (12). While doing this we shall use the fact that P>P0, say, is sufficiently large, and we must be careful to ensure that Po is independent of k and q. To prove (12) and (14) we shall take DeSf(R k) or D~Sf(P, k+j, q) respective- ly, and write l= k or k+j as appropriate. In both cases we begin by counting the number of primes p in the set @(D)={P/2 If we let Z run over all real characters modulo 8 D we see that 2 2-1~ Z(p) Z takes the value 1 if p- 1 (mod 8) and 190 D.R. Heath-Brown and 0 otherwise. We now define A(Z)= ~ Z(P)(P--14p--3PI)logp P/2 B(Z)= ~ z(n)(P--14n-3P])A(n). P/2 Then if p runs over ~ (D) we will have (t6) (PlogP)#~(D)>~.(P-14p--3P])logp=2 2-t~A(z ). p z We observe moreover that (17) A (Z) = B (Z) + 0 (p3/2) = B (Z*) + 0 (lP log P) + O (p3/2), where Z* is the primitive character which induces ;6 When the conductor of Z* does not divide 8, we handle B(Z* ) by means of the integral representation 1 2+iao( E Z.)'lps+lw(s)ds, B(Z*)= 2~/~/2-~/oo ~ -z(S' where 4(1 +(1/2) s+l - 2(3/4) s+ 1) w(s)- s(s + 1) We move the line of integration to 9t (s)= --1, where the integral is 0 (p1/2 log D), and obtain the formula (18) B (X*) = -- ~ po + 1 w (p) + O (IP1/2 log P). p Here p runs over all non-trivial zeros of L(s, g*). Since Z is 'good', all such zeros satisfy either 9t(p)__<~ or [3(p)] >P. The symmetry of the zeros about the critical line then ensures that IP] > ~6 for each zero. We can then conclude that ~ [w(p)[- 1 .~log D, and Iw(p)l-' <~ P-' log D, I-~(p)l__>P since there are O(Tlog DT) zeros up to height T. It now follows from (18) that B(Z*) ~ 1P 31/16 log P. The size of Selmer groups for the congruent number problem 191 If the conductor of Z* divides 8, the Prime Number Theorem for arithmetic progressions modulo 8 yields p2 B(z,)=e ~_+ O(p2(log p)-1), where e= 1 if Z* is identically 1, and ~,=0 otherwise. A comparison with (16) and (17) now reveals that @~(D)> 2-2-t P(log p)-a +O(lP15/16)+O(2-tp(logP)-2). There is therefore an absolute constant Ca such that P (19) +~;3(D)>_2 -5-~- log P' providing that 21 This last condition is a consequence of (13) or (15), if P is large enough. A precisely similar argument, based on the fact that (P logP/2)4e~(D)<= ~ (2P-14p-3Pl)logp, P/4 P (20) +~(D)<__2 -'-t- log P' subject similarly to the conditions (l 3) and (15). We are now ready to prove (14) in the case k+ I. Each D'e,~(P, k+j+ 1, q) can be written in exactly k+ 1 ways as Dp with qpD. Moreover we must have pe~(D). Not all products Dp with DecJ(R k+j, q) and pe~(D) will be good. However it certainly follows that 1 4eJ(P,k+j+l,q)< ~ 4~(D) =k+l D~Sf(P,k+j,q) <--. 2 -kj-k(k- l)/2. P =k+l El., log P -(k+ 1)! \81~] 22(k+1)-(k+l)j-k(k+l)/2' by means of (20) and the case k of (14). This establishes (14) for k+ 1. For the proof of (12) we observe that each D'eY(P,, k+ 1) can be written in exactly k+ 1 ways as D'=Dp, and in each such representation we have 192 D.R. Heath-Brown pe~(D). We shall show that at least half of the numbers Dp formed in this way are good. We will then have 1 @SP(P,k+ 1)> 2k+2 ~ :~(D) D~Aa(P,k) > 1 1-[ p~]k2-3k-k(k-l)/2 .2- 5-k__P =2k+2 k!\81ogP] log P I [ P ~k+l - (k + 1)! \~} 2-3(k+ l)-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 Dp fail to be good. There is then some character Z to modulus 8Dp which is not good. If the conductor of Z is q, say, then q~8D, since D is good. Thus plq, and q=pq', 4pq' or 8pq' with q'lD. It follows that, to each such q with e)(q')=j ~5~(R k 'q):(k-j~., < 1 (~)/ p \k-j 2~ choices for D, where e = 2(k-j)- (k-j)j- (k-j) (k-j- 1)/2 <3(k+ 1)- k(k + 1)/2 + (j + 1)1/2. We therefore see that each conductor q will divide at most (k+ 1)J+ 2o ~ 1)~(k + \~] 23 (k+l) i+22(j+l'2(81~ < 321~~+lP P-} <= <=q-2/3, providing that p1/3 (k+1)22 k+~_-< 32 logP' 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 ~(s)__>,r, [.~(s)l__ For cr = ~z,i s T= P and Q > P, the number of conductors which can occur is there- fore 0(Q1/2). Since the relevent conductors q are all at least P/2, we see that the number of products Dp which are not good is at most ~. q-2/3 1 / P ~k+x + .>P/2 1}~\~J(k+ 23(h+l)-k(k 1}/2 1)~(k+ \~] 23~+ l'-k~k+')/2" It follows that at least half the available values of Dp are good, providing that 26(k+ t1~C2 pl/6, with a suitable constant C 2. This inequality follows from (13) ifP is large enough. This completes our proof of (12). It remains to deduce Theorem 2 from the estimate (12). We shall take k=[~ (1-O)l~ 4 ] and p= xl/k. It follows that both k and P tend to infinity with X. We will then have a set 5:(R k) of numbers D #5'~(P,,k)>-(8k-- 2 3 +(k-P 1)/2 log tif >-(P~- ~ since p1-0 8k 23+tk- 1)/2 ~2 k <~X(1 -o)/2k=p(a --0)/2 <__ =logP' for sufficiently large X. This yields the estimate claimed in Theorem 2. 194 D.R. Heath-Brown 8 Proof of Theorem 3 When h = 1, n = 0 we observe that s (3 p) = 0 for any prime p = 3 (rood 8). Otherwise we write [ n/2 - 1, h = I, / n/Z h:3, [(n- 1)/2, h=5or 7. Wc then use the construction of the previous section to produce a product Do=p1 ...P~ with pl =l(mod 8) and We shall keep D o fixed ~, and consider numbers of the form D=D o p*, with p*- h(mod 8) and (PpT)=l, l The system (3) now has p-adic solutions for each Pi. Moreover, if p*lD 1 there will be p*-adic solutions if and only if - 1 is a quadratic residue of p*. For p*[D z the condition is that - 1 and 2 are both quadratic residues, but for p*ID3 we only need 2 to be a quadratic residue. Finally if p*ID4 the system automati- cally 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 p* can be achieved by requiring p* to lie in a suitable con- gruence class modulo 8D0, so that the number of available primes p* is X/log X. The theorem then follows. Acknowledgement. The present paper was prepared while the author was enjoy- ing the hospitality and financial support of the University of Hong Kong. This assistance is gratefully acknowledged. References 1. Birch, B.J., Swinnerton-Dyer, H.P.E: Notes on elliptic curves, II. J. Reine Angew. Math. 218, 79-108 (1965) 2. Birch, B.J., Stephens, N.M.: The parity of the rank of the Mordell-Weil group. Topology 5, 295 -299 (1966) 3. Brumer, A., Heath-Brown, D.R.: Average ranks of elliptic curves, II (to appear) 4. Burgess, D.A.: On character sums and L-series, II. Proc. Lond. Math. Soc., III. Ser. 13, 524-536 (1963) t By extending the argument, to allow Do to vary, one could produce more admissable values of D, thereby improving the bound in Theorem 3 somewhat. The size of Selmer groups for the congruent number problem 195 5. Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39, 223-251 [1977) 6. 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