* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download On the Equation Y2 = X{ X2 + p - American Mathematical Society
Elementary algebra wikipedia , lookup
System of linear equations wikipedia , lookup
Quartic function wikipedia , lookup
Factorization wikipedia , lookup
History of algebra wikipedia , lookup
Quadratic equation wikipedia , lookup
Quadratic form wikipedia , lookup
MATHEMATICS OF COMPUTATION VOLUME 42, NUMBER 165 JANUARY 1984, PAGES 257-264 On the Equation Y2 = X{ X2 + p) By A. Bremner and J. W. S. Cassels Abstract. Generators are found for the group of rational points on the title curve for all primes p = 5 (mod 8) less than 1,000. The rank is always 1 in accordance with conjectures of Seltner and Mordell. Some of the generators are rather large. 1. Let p be a positive prime, (1) /7 = 5(mod8). It is easy to see that the Mordell-Weil rank of (2) Y2 = X(X2 + p) is at most 1 (e.g. Section 5 of Birch and Swinnerton-Dyer [1]); and the Selmer conjecture [5] predicts rank exactly 1. As we shall note below, this is equivalent to a conjecture of Mordell [3], [4]. We shall verify this conjecture for all p < 1,000. Table 1 gives for each p a point P0 which, together with the point (0,0) of order 2, generates the entire Mordell-Weil group. Some of the generators P0 are rather large, the most startling being that for p = 877, namely 3 _ 37 5494528127162193105504069942092792346201 621598777687 1505425463220780697238044100 ' 2562562679889268093887768340455 13089648669153204356603464786949 490078023219787588959802933995928925096061616470779979261000 ' A rational point (X,Y)on (4) for integers R, S, T with (3) is of the shape X=R/S2, (5) R>0, The height H(X, Y) is by definition (6) Y=T/S3 (R,S)=l. H(X,Y) = max(R,S2), so that the height of (3) is - 3.75 x 1041.It is of interest to compare this with the discriminant |A| = 4p3 of (2), which for p = 877 is - 2.70 x IO9.Hence H(P0) |A|441.Lang observed to us that in tables of elliptic curves and generators published to date the heights of generators are never much greater than |A|2. This, in our view, is almost certainly because these tables cover only curves with relatively small ReceivedDecember 28, 1982. 1980 MathematicsSubject Classification.Primary 10B10, 14K07. ©1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 257 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 258 A BREMNER ANDJ. W. S.CASSELS discriminant. Further, unless one has strong reason to believe that a rational point exists, there is a marked reluctance to persevere in a search once the numbers cease to be small. 2. In (2) clearly either X or pX is square. Since (7) (X,Y) + (0,0) = (p/X,-pY/X2). we may suppose that X is a square. Then (8) Y = rt/s 1.3 X=r2/s2, for integers r, s, t with r,s coprime and (9) r4+ps4 = t2. It was for this equation that Mordell made his conjecture mentioned above. Clearly (10) rmO, t*0(modp), and (1) implies that (11) r = t= l, s = 0(mod2). We choose the sign of t so that t = 1 (mod4), and then (12) /sl(mod8) by (9) and (11). Since (9) can be written as (13) (t + r2)(t - r2) = ps\ we have t + r2 = 2a4 or 2pa4 for some odd a. The first alternative leads to a point Q on (2) with (X,Y)= 2Q. Hence by the "infinite descent" argument we may suppose that there is a coprime pair of integers a, b with ( 14) r2 = pa4 - 4b4. a = b=l (mod2) and (15) s = 2ab, t=pa4-r4b4. We can write ( 14) in the shape (16) (r + 2ib2)(r- 2ib2) = pa4 and consider factorization in Z[i]. By (1) (17) p = u2 + 4v2, where u, v are odd and without loss of generality (18) v= 1 (mod 4). The sign of r may be chosen so that r + 2ib2 is divisible by u + 2iv, and then (16) implies (19) r + lib1 = (u + 2iv) • unit ■(c + id)4 for some c, d with (20) c2 + d2 = a. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON THE EQUATION Y2 = X( X2 + p ) 259 On considering (19) modulo 8 and using (18), we find that the unit is necessarily 1. Hence on equating real and imaginary parts, (21) b2 = u(/2 - m2) + ulm, with (22) l = c2-d2, m = 2cd. If there are solutions of (2), then (21) must have rational solutions. This turns out to be the case for all the/7 under consideration, and so every solution of (21) is given by one of a finite number of parametrizations (23) / = fi(M). m = q2(6,x¡,), b = q,(6,xp). Here qx, q2, <73are known quadratic forms with rational integer coefficients and 0, xp are integers to be found. It turns out that only one parametrization is compatible with the other conditions. We have now (24) qx(6,xp) + iq2(6,xp) = (c + id)2. Arguing as before, but in Z[i], we have (25) 6 = Qx(X,p), xp= Q2(X,p), c + id=Q3(X,p), where Qx, Q2, Q3 are known quadratic forms with coefficients in Z[i] and where X, p are elements of Z[i] to be found. Again, only one parametrization turns out to be compatible with the other conditions. The condition that 6, xpare real leads to a pair of simultaneous homogeneous quadratic equations in the four variables Re\, Im\, Re/i, Imp. These were the equations searched for solutions, though most of the entries in Table 1 could be spotted at an earlier stage in the process. The primes 317, 797, 877, 997 required an HP67, but the other solutions were found by hand. At the suggestion of the referee we illustrate the last part of the argument and take (26) p - 877, u = 29, v-3. Then (21) is (27) b2 + 3/2 - 3w2 - 29/m = 0. The left-hand side vanishes for (b,l,m) = (7,2,1) and so, by a standard algorithm, (27) is equivalent to (28) -LM + 877/V2= 0, where the forms (29,) L= 14/3- 17/-64m, (292) M = 50746-6242/- (293) N = 9b-lll-4\m 23035m, are unimodular. On taking -b for b if need be, we have 877 + M, so (28) implies (30) ±L = 87702, ±M = <b2, ±N = 6tt> for some integers 6, (/>and some choice of sign. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 260 A. BREMNER AND J. W. S. CASSELS On solving for b, l, m and putting (31) <t> = -t^ + 5630, we obtain (32,) ±b= -7xp2 - llxp6 + 2162, (322) ±/= -2f- + 6xpe- 302, (323) ±m= -xp2 -4xpe - IO2. The lower sign is incompatible 2-adically with (22), so we must take the upper sign. By (22) we have (33) / + /« = (c + /¿)2 = Y2(say); and so (34) y2 + (2 4- i)xp2 + (-6 + 4i)xpe + (3 + li)02 = 0 by (322), (323). This is equivalent to (35) (6 -29i)S2 + RT=0, where (36,) S = (l + i)y + (-4 + i)0- ixp, (362) R = (3 + 2/)y + (-10 + 5i)0-2ixp, (363) 7= (-15 + 6i)y + (8-45i)« + (14 + 4i)^. Hence (37,) 20 = (-4 - 3i)R + (22 + 10/)5 + iT, (372) 2xp= (-l (373) 2y = (-15 + 6i)Ä + (70 - 46/)S + (3 + 2i)T. +2i)R+ (6- \6i)S + T, By (35), on taking - y for y if need be, there are Gaussian integers X, p such that (38) R = (l±i)X2, r=(6-29/)(l ±i)p2, S = (1+/)a/i, for either the upper or the lower signs. Put (39) X = x + iy, p = u + iv, where x, y, u, v are rational integers. On substituting (38), (39) in (37, ), (372) we get (40,) 21?= Fx(x,y,u,v) + iF2(x,y,u,v), (402) 2t/>= Gx(x,y,u,v) + iG2(x,y,u,v), where Fx, F2,GX,G2 are quadratic forms with rational integer coefficients. [There is a set of forms for each choice of sign in (38).] Hence (41) F2(x,y,u,v) = G2(x,y,u,v) = 0. If the lower signs hold in (39), the simultaneous equations (41) turn out to be 2-adically incompatible, so we must take the upper signs. A search yields (42) X = 324 - 385L p = 136 + 145/. It may be noted that (41) implies congruence conditions on x, y, u, v to various small moduli, and these greatly facilitate the search. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON THE EQUATION Y2 = X( X2 + p ) 261 3. Having indicated how the rational points P0 were obtained, we must now show that they are generators. This requires consideration of heights. Let (4) be a rational point P on (2). The A'-coordinate of 2/* is (43) Xx^{Rl-pS*)2452r2 (*2-'S4)2 4RS2(R2 + pS4)' We consider only points of the type (8), (9), so p \ R. It is then easy to see that the numerator and denominator in (43) are coprime. Hence (44) //(2/>) = max{(/v2-/>S4)2, 4RS2(R2 + pS4)}. We distinguish three cases: (i) 0 « S2/R < 1/2px>/2.Then H(2P) >(R2(ii)l/2^'/2< pS4)2 > (9/16)*4 = (9/l6)H(P)\ S2/R*i l.Then H(2P) > 4RS2(R2 + pS4) > {2/px/2)R4 * {2/px/2)H(P)\ (iii) 1 < S2/R. Then H(2P) > (R2 - pS4)2 >(p- 1)V = (p - l)2/7(/>)4. Hence in any case (45) H(2P)>(2/px/2)H(P)\ . Similarly, but more simply, (46) "', ". H(2P) < p2H(P)\ With the usual notation h(P) = logH(P) it follows that (47) 4h(P) - ilog(/7/4) < h(2P) < 4h(P) + 21ogp. Now (43) is a perfect square, so the argument applies to 2P instead of P. By induction 4"h(P)- (l/6)(4" - l)log(/>/4) < h(2"P) . . ' ' < 4"h(P) + (2/3)(4" - l)log/7 for every n > 1. The Täte height is (48) h(P) = Urn 4-"rj(2nP), , n-"x> SO (49) h(P) - (l/6)log(/>/4) « h(P) < h(P) + (2/3)log/7. Let P0 be one of the points on (2) listed in Table 1. The descent argument shows that neither P0 nor P0 + (0,0) is divisible by 2. Suppose that P0 is divisible by 3, say P0 = 3Q. Let q be a prime distinct from 2, p. Then (2) has good reduction modulo q and'the reduced points P0, Q modq would satisfy P0- 3Q. For each j?0Table 1 gives a prime q such that P0 is not divisible by 3 in the group of points on (2) over the finite field Fq. Hence if PQ is not a generator we have PQ= kQ for some odd License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use o\ (^ ^h fO m <j\ r** ^ m CO (N CO ■"* s » ffi h en ~i r- oo en vo 1*1 I— en r~ o o ro o en r~ 00 en oo vo vo h (N CO in (N en r» en m (M en o r~ in rs oo vo po m (N rvi vo in m o eh m en •«r vcpo CM (N cu vo o r» oo .H ro o en in en en vo CO VO m « « o o N en *h o o co o m T r~ r» r*» ^ en m r~ o ^ n in -• in r- cnen^^cn^^pn^H^^in^H^Hr^cnin oinvccovO'Hi-icn-Hnvo-Hr»' h n « in oo -Hen co oo o oo co r*. r* vo o rvi en po in po r*. ^ r- rt vo rvi ci f-l r* *H m 'H in in CN in •h r- h en in rvi m m oo inPoer,r*.po^^cncptr^PO*Hr*.chchr^ ^<fMPOinvooo^iinr~cocnfMvor-cri'H HHHrtrtHHNfNINHn License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use *h T -* in en in m TT in en •» vo o en m ci co o m en en vo co co T m in o VO (N n r~ r- en en en oo en IS rn N r- po i-i H IN en vo in co CN oo o en r~ 00 in co en m (N en vo r~ o en cn m vo en en en co en ro ro vo vo vo ■>* vo vo r~ ^h o rP0 Tf i-i po cm «h rH '«T in in o .-i ro r» en ID rH rH vo in CN vo vo co vo oo id cn PO en cn CM in in r- oo o in oo vo PO o rr- oo o PO en in ■H (N *>> *r ** in m en o in pi ID ih CN en tN VO (N 00 in en o ■» rvi in r» po co ro ro ro vo vo 00 in <N >* vo PO Tf in ■<r CM oo PO CN oo TJ. ID o en 8en r- co r- vo PO CO CN o 00 cn CN VO CN 00 O 00 co vo o en i—i r- rn in in o o co CN co ro ih r^ O O r- -< en rCN TT o rH TT rH en m 3 O O CO en co oo vo o id co m CO in cn CN cn co VD cn in n in ro ^3 CN o n S CI c o Vj CN VO CN oo oo r» LU -J 03 < en o o ro ro po CT. O rI- r^ rH o co oo r^ <h co r- ^ ci ■H o in r- rn rH ro \0 CN in IC TT o> ro CO CN PO O co in in co oo TT in in co TT o ,H o rH TT o ID CT\ rH CO O CO in O VD CN CO O CN 0O r~ CO CO en rH CO i-t r* en m in m cn -h o co rr PO VD ID CN rH •-< en vo en ro en en in CN oo rH ro rH in 00 in CO 00 rH m m in oo oo in ^ oo ro r» ro en og PO in ID cn o vo CN rH in in m vo id r- vo r- VD vo ^ o r- en o r- License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use co ren rr- r- CN rH C. CO CN CN m CO CO co r- h o vo r~ o en in oo rH in •q- CO r- CN in T O r- in rH ^ co en 264 A. BREMNERAND J. W. S. CASSELS Ac3s 5 and some rational point Q. By the quadratic property of heights (50) h(Po) = k2h(Q) (e.g. Cassels[2, p. 262]).From this and (49) it followsthat Mo) < {\/k2)h(P0) + (2/3k2)logp + (l/6)log(p/4) < (1/25)A(>?0)+ (2/75)log/> + (l/6)log(/>/4). Even in the extreme case/? = 877 this implies H(Q) < 136, which in turn implies a solution of (9) with 0 < r, s < 12; a contradiction. Department of Pure Mathematics and Mathematical Statistics Universityof Cambridge CambridgeCB2 ISB, England 1. B. J. BIRCH& H. P. F. Swinnerton - Dyer, "Notes on elliptic curves II." J. Reine Angew. Math., v. 218, 1965,pp. 79-108. 2. J. W. S. Cassels. "Diophantine equations with special reference to elliptic curves." J. London Math. Soc.,v. 41. 1966,pp. 193-291. 3. L. J. MORDELL, "The diophantine equation ,x4 + mv* = :2." Quart. J. Math. (2). v. 18, 1967.pp. 1-6. 4. L. J. Mordell, y = Dx* + I, Number Theory Colloquium, János Bolyai Math. Soc., Debrecen, 1968.pp. 141-145(North-Holland,Amsterdam,1970). 5. E. Selmer, "A conjecture concerning rational points on cubic curves," Math. Scand., v. 2, 1954,pp. 49-54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use