Acta Mathematica Sinica, English Series 2000, April, Vol.16, No.2, p. 229{236

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves

ED y2 x3 , D2x :

=

Delang Li

Department of Mathematics, Sichuan Union University, Chengdu 610064; P. R. China E-mail: [email protected]

Ye Tian

Department of Mathematics, Columbia University, New York, NY 10027; USA E-mail: [email protected]

Abstract We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves. Keywords Elliptic curve, BSD conjecture, Graph, 2{component 1991MR Subject Classi cation 14H52 Let E=Q be an elliptic curve de ned over Q . The Birch-Swinnerton-Dyer conjecture (BSD conjecture for short) says that (with the notation of [1]): (1) LE (s) has a zero at s = 1 of order equal to the rank of E (Q ). (2) Let r = rankE (Q ); then

Q L E (s) = #X(E=Q )R(E=Q ) pj4 Cp : lim s!1 (s , 1)r j#Etors(Q )j2

(a)

Q

For the elliptic curves ED of our theorem, we have r = 0 (so R(E=Q ) = 1); pj4 Cp = 22n+1 (see [2]) and Etors (Q )  = Z=2Z  Z=2Z (see [1], p. 311). So the conjecture in this case reduces to LED (1)= = # (ED =Q )
22n,3 : (b)

X

X

The aim of the present paper is to prove that the 2-primary component of (ED =Q ) has order 4. Chunlai Zhao showed that the 2-power of LED (1)= is 2n , 1 (see [3]). Hence the 2-parts of both sides of (b) are equal. Received January 28, 1997, Accepted December 2, 1997 Project supported by the National Natural Science Foundation of P. R. China

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p

On the other hand, obviously the elliptic curves have complex multiplication by ,1. Rubin [4,5] proved, for elliptic curve E=Q with complex multiplication and LE (1) 6= 0; that the group (E=Q ) is nite and the p-parts of both sides of (a) are equal for each prime number p  5: He also proved that the 3-parts of both sides of (a) are also equal if E=Q has no complex p multiplication by ,3. So the BSD conjecture is true for the elliptic curves of our theorem. In 1994, Feng [6] veri ed the BSD conjecture on the elliptic curves ED for other series of integers D. In this paper, we use the notation of elliptic curve theory as in [1] and the notation of graph theory as in [7].

X

Lemma-De nition 1 Let p be a prime, p  1(mod 8): Then there exist integers a; b; u; v such that p = a + 16b = u + 8v : Let c; j be the p-adic integers such that c = 2; j = ,1: Then the following are equivalent: (i) 2 y p, + b; (ii) Either p  1(mod 16); ( ) = ,1 or p  9(mod 16); ( ) = 1; (iii) 2 y v; (iv) 1 + j is not p-adic square; (v) 1 + c is not p-adic 2

2

2

2

2

8

2 p 4

1

2

2 p 4

square. If a prime number p satis es the equivalent conditions above, we de ne (p) = 1; otherwise (p) = 0: For an integer D = p1


pn ; where pi is prime and pi  1(mod 8) for all i, P we de ne (D) = ni=1 (pi ):

Proof (i) () (ii). This follows from the theorem of Gauss giving the quartic character of 2 (see [1], p. 318, Proposition 6.6, for instance). (ii) () (iii). From p , u2 = 8v2 , we see that ( pu ) = ( p2 )4 ( pv ); so ( p2 )4 = ( up )
( vp ) = ( up )
1 = 2 ( 2u ) = (,1) u 8,1 : The equivalence follows. (ii) () (iv). This follows from the fact that (1+ j )2 = 2j; and j is a p-adic quartic number i p  1(mod 16). (iv) () (v), since c2 (1 + j )(1 + c) = (1 + j + c)2 :

Remark Making use of the results of [2], we can easily prove that LE (1)=  2(mod 4) for the elliptic curve E : y = x , p x, where p is a prime for which (p) = 1: Notice that 2

3

X

2

LE (1)= = 41

2

2

2

"2x +y +8z =p X 1

(,1)z

=4 = 14

"

2x2 +y2 +8z2 =p x2 =4z2

X

y2 +16z2 =p x2 =4z2

 2(mod 4);

(,1)z +

(,1)z +

X 2x2 +y2 +8z2 =p x2 6=4z2 xz=0

 X

(,1)z +

2x2 +y2 +8z2 =p x2 6=4z2 xz6=0

X

(,1)z +

y2 +8z2 =p

X (,1)

x2 +y2 =p

2

0



(,1)z

+

# X

2x2 +y2 +8z2 =p x2 6=4z2 xz=0

(,1)z

#

where we are using (iii) in the de nition of (p) = 1: Also [2] points out that (p) = 1 is equivalent to h(,p)  4(mod 8), where h(,p) is the p class number of Q ( ,p):

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves

ED : y 2

= x3 , D2 x

231

De nition 2 Let D = p


pn ; where p ; : : : ; pn are distinct prime numbers such that pi  1

1

1(mod 8) for all i. De ne the simple (non-directed) graph G(D) in the following way: (1) The vertex set V = fp1 ; : : : ; pn g, (2) For distinct vertices pi and pj , there is an edge pi pj in G(D) i ( ppji ) = ,1: We denote by  (G) the number of spanning trees of G. The aim of the present paper is to prove the following theorem.

Theorem Let D = p


pn , where p ; : : : ; pn are distinct prime numbers such that pi  1

1

1(mod 8) for all i. If the following two conditions are satis ed: (I)  (G(D)) is odd, (II) (D) is odd, then for the elliptic curve ED =Q : y2 = x3 , D2 x one has rank(ED (Q )) = 0, and the 2-primary component of the Shafarevich-Tate group (ED =Q ) is isomorphic to Z=2Z  Z=2Z and has order 4.

X

From the theorem, we have (see the beginning of this paper):

Corollary 1 If an integer D satis es the conditions (I) and (II) of the theorem, then the BSD conjecture is true for the elliptic curve ED .

It is well-known that a positive integer D is a non-congruent number i rank (ED (Q )) = 0 (see [8], for instance). So we have:

Corollary 2 If an integer D satis es the conditions (I) and (II) of the theorem, then D is a non-congruent number.

Example Let D = 17  73  97; then  (G(D)) = (D) = 3, thus D satis es the conditions

(I) and (II). So the BSD conjecture is true for the elliptic curve ED and D is a non-congruent number. For the sake of completeness, we give a brief summary of the theory and method we need. (I) Graph theory. Let D be a directed graph, with V its vertex set and A its arc set. Let F be a eld of characteristic p. A circulation of D over F is a function f : A ! F; which satis es the conservation condition at each vertex: f + (v) = f , (v); for all v 2 V; where

X

f + (v) = a

a2A

has tail

f (a); v

X

f , (v ) = a

a2A

has head

f (a): v

The set of all circulations in D is a vector space over eld F . This vector space is called the cycle space of D and denoted by CF : A function g : A ! F is called a potential dierence of D over F , if there exists some function on the vertex set V such that g(a) = (v) , (v0 ); for all arc a = (v; v0 ) 2 A: The set of all potential dierences is a vector space over eld F . This vector space is called the bond space of D and denoted by BF : We will use the following theorem of Shank (see [7], Ex 12.2.4): \Let G be a connected (nondirected) graph and let D be an arbitrary orientation of G. Then dimF (BF \CF ) > 0 i pj (G):00

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Let G be a simple graph, with V its vertex set and E its edge set. Let S; T be two subsets of V . We denote by [S; T ] the set of edges with one end in S and the other in T . We have: Lemma 2  (G) is even i the graph G has the following property: (P): There exists a non-trivial partition V = V0 [ V1 such that #[v0 ; V1 ] is even for all v0 2 V0 , and #[v1 ; V0 ] is even for all v1 2 V1 : Proof (For another proof see [6]) If G is not connected there is nothing to prove. Let G be connected and let D be an arbitrary orientation of G. If  (G) is even, then there exists a nonzero function g 2 BF2 \ CF2 by Shank's theorem. Let g be the potential dierence corresponding to function on V and V0 = fv 2 V j (v) = 0g; V1 = fv 2 V j (v) = 1g: From g 6= 0; we see that V = V0 [ V1 is a non-trivial partition and has property (P). On the other hand, suppose G has property (P). Consider the function : V ! F 2 de ned by: 8 < 0; v 2 V0 ;

(v) = : 1; v 2 V1 : Obviously, the potential dierence corresponding to is also a circulation, so dimF2 (BF2 \CF2 ) > 0: Moreover,  (G) is even by Shank's theorem. (II) Elliptic curve theory. Let E=K; E 0 =K be two elliptic curves de ned over a eld K , and let ' : E=K ! E 0 =K be an isogeny de ned over K . Then there is an exact sequence: 0 ! E 0 (K )='E (K ) ! S (') (E=K ) !

X(E=K )['] ! 0:

If : E 0 =K ! E=K is an arbitrary isogeny de ned over K , then one has the following commutative diagram of exact sequences: 0 0 ! E 0 (K )='E (K ) ! S (') (E=K ) !

#

0 ! E (K )= 'E (K ) ! S (

#

! # # 0 ! E (K )= E 0 (K ) ! S ( ) (E 0 =K ) ! # ') (E=K )

0

#

X(E=K )['] # X(E=K )[ '] # X(E0=K )[ ]

! 0 ! 0:

()

! 0

Let Se( ) (E 0 =K ) denote the image of S ( ') (E=K ) in S ( ) (E 0 =K ): By the exactness of (), one has S (') (E=K ) ; # (E 0 =K )[ ] = #S ( ) (E 0 =K ) : # (E=K )['] = ## 0 E (K )='E (K ) #E (K )= E 0 (K ) Moreover, (') (E=K )
#Se( ) (E 0 =K ) : # (E=K )[ '] = #E#0 (SK )='E (K )
#E (K )= E 0 (K )

X

X

X

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves

Similarly,

ED : y 2

= x3 , D2 x

233

( ) 0 )
#Se(') (E=K ) # (E 0 =K )[' ] = #E#(KS )= (EE0 (=K K )
#E 0 (K )='E (K ) :

X

From now on, we assume that the elliptic curves E=Q ; E 0 =Q have the following equations: E : y2 = x3 , D2 x; E 0 : y2 = x3 + 4D2 x, where D 2 Q : We have the standard isogeny of degree 2. ' : E ! E 0 ; '(x; y) = (y2 =x2 ; y(x2 + D2 )=x2 ) and the conjugate isogeny to ' : E 0 ! E; (x; y) = (y2 =4x2 ; y(x2 , 4D2 )=8x2 ): Using the 2-descent method, we can compute the Selmer group S (') (E=Q ) (similarly, for ( ) S (E 0 =Q )): (For details, see [1], p. 302). Let S = fprime factors of 2Dg [ f1g Q (S; 2)

= fb 2 Q  =Q 2 : ordp (b)  0(mod 2); for all p 62 S g:

Then one has the exact sequence:  0 ! E 0 (Q )='E (Q ) ! Q (S; 2) ! WC (E=Q )['];

0 7! 1; (0; 0) 7! 4D2 ; (x; y) 7! x;

d 7! fC =Q g;

where Cd =Q is the homogeneous space for E=Q by the equation Cd : dw2 = d2 + 4D2 z 4 : The -selmer group is then

S (') (E=Q )  = fd 2 Q (S; 2) : Cd (Q p ) 6= ; for all p 2 S g: Finally, the map  : Cd ! E 0 ; (z; w) = (d=z 2 ; dw=z 3 ) has the property that if P 2 Cd (Q ) then ((P ))  d(mod Q 2 ): There is no general method known for determining E 0 (Q )='E (Q ); but Se(') (E=Q ) (similarly, for Se( ) (E 0 =Q )) can be computed with the help of a result from [9] (Lemma 10, p. 95). Let d 2 S (') (E=Q ); and let ; ;  be rational integers with no common divisors such that d2 = d2  2 +4D2 2 (the existence of ; ;  come from the Hasse{Minkowski theorem, see [10], Chapter IV, Theorem 8). Let Mb corresponding to b 2 Q  =Q 2 be given by

Mb : dw2 = d2 + 4D2 z4 ; dw , d2  , 4D2 z2 = bu2 : Then d 2 Se(') (E=Q ) i there is some b 2 Q (S; 2) for which Mb is everywhere locally solvable. Now, using the theory and method above, we prove our theorem. Let p1 ; : : : ; pn be distinct prime numbers such that pi  1(mod 8) for all i . Let D = p1


pn and #E 0 (Q )='E (Q ) = 201 ; #S (') (E=Q ) = 21 ; #Se(') (E=Q ) = 21 ; #E (Q )= E 0 (Q ) = 20 ; #S ( ) (E 0 =Q ) = 2 ; #Se( ) (E 0 =Q ) = 2 :

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Then one has

X

X

 0 0  0 0 # (E=Q )[2] = 2 ,
21 ,1 ; # (E 0 =Q )[2] = 21 ,1
2,

and rank(E (Q )) = rank(E 0 (Q )) = 0 + 01 , 2 (see [9]).

Lemma 3 If Condition (I) of the theorem is satis ed, then  = 0 =  =  = 2; S ' (E=Q ) = f1; 2; D; 2Dg and E (Q )= E 0 (Q )  = S (E 0 =Q ) = f1; Dg: 1

( )

( )

Proof Using the 2-descent method, we rst compute the '{selmer group S (') (E=Q ). Now S = f1; 2; p1 ; : : : ; pn g; Q (S; 2) = f1; 2; pi ; 2pi ;


; p1 : : : pn ; 2p1


pn g; Cd =Q : dw2 = d2 + 4D2 z4 : It is obvious that Cd (R ) 6=  () d > 0:

d = 2 C2 : w2 = 2 + 2D2 z4 (i) Since pi  1(mod 8) for all i; D2

a = C2 (Q 2 ): 2

1+ 2

D2

 1(mod 8): There is a 2-adic integer a such that by Hensel's Lemma (see [10], p. 14, Theorem 1, for instance), so (z; w) = (1; 2a) 2 1+ 2

p2 ,1

(ii) For every i, since pi  1(mod 8); we have ( p2i ) = (,1) i8 = 1: By Hensel's Lemma, there exists a pi -adic integer ai such that a2i = 2: Let bi be a pi -adic integer such that b2i = 1+D2 : Then (z; w) = (1; ai bi ) 2 C2 (Q pi ): So z 2 S (') (E=Q ).

d = D Cd : w2 = D(1 + 4z4 ) (i) (z; w) = (0; a) 2 Cd (Q 2 ); where the 2-adic integer a satis es a2 = D: (ii) Since ( p2i ) = 1; let ai be the pi -adic integer for which a2i = 2: Since ( ,pi1 )4 = 1; let bi be the pi -adic integer for which b4i = D , 1: One has (z; w) = (bi a,i 1 ; D) 2 Cd (Q pi ); (i = 1; 2; : : : ; n): So D 2 S (') (E=Q ). Also since 2 2 S (') (E=Q ); one has 2D 2 S (') (E=Q ): d = pi1


pik (1  k < n) Cd : w2 = pi1


pik (1 + 4p2ik+1


p2in z4 ) Suppose d = pi1


pik 2 S (') (E=Q ): For every p 2 fpi1 ; : : : ; pik g; let (z; w) 2 Cd (Q p ): Considering the p-valuation of both sides of Cd , we see that z is a p-adic unit and the exponential 2 valuation vp ( pi1
w

pik ) > 0: So 1 + 4p2ik+1


p2in z 4  0(mod p); 2pik+1


pin z 2 = j (mod p) (j 2 = ,1):

Since 1; 2; j are all p-adic square, we have ( pik+1 p


pin ) = 1: So the number of ,1 among ( pikp+1 ); : : : ; ( ppin ) is even. On the other hand, since D; pi1


pik 2 S (') (E=Q ); we have pik+1


pin 2 S (') (E=Q ): In the same way, we know that the number of ,1 among ( ppi1 ); : : : ; ( ppik ) is even, for every

p 2 fpik+1 ; : : : ; pin g: So  (G(D)) is even by Lemma 2. This contradicts condition (I) of the theorem. So pi1


pik 62 S (') (E=Q ): Since z 2 S (') (E=Q ); one has 2pi1


pik 62 S (') (E=Q ): Therefore, S (') (E=Q ) = f1; 2; D; 2Dg; and 1 = 2:

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves

ED : y 2

= x3 , D2 x

235

We compute the -selmer group S ( ) (E 0 =Q ) now. At present, S and Q (S; 2) are the same as the preceding respectively. Cd0 =Q : dw2 = d2 , D2 z 4 : For d = 2; 2pi ;


; 2p1


pn , considering the 2-valuation of both sides of Cd0 , we see Cd0 (Q 2 ) = : So 2; 2pi ; : : : ; 2p1 : : : pn 62 S ( ) (E 0 =Q ): Because O; (0; 0); (D; 0) 2 E (Q ); one has 1; D 2 S ( ) (E 0 =Q ): So the order of E (Q )= E 0 (Q ) is not less that 4. The same method as that for computing S (') (E=Q ) yields that pi1


pik 62 S ( ) (E 0 =Q ); for any pi1


pik (1  k < n): Hence, ,pi1


pik 62 S ( ) (E 0 =Q ) because of ,1 2 S ( ) (E 0 =Q ): So E (Q )= E 0 (Q )  = S ( ) (E 0 =Q ) = f1; Dg; 0 =  = 2: By the exactness of (), one has (E 0 =Q )[ ] = 0;  = 2: Lemma 4 If the conditions (I) and (II) of the theorem are satis ed then 1 = 0. Proof The existence of the Cassels' alternating, bilinear pairing on implies that the dierence 1 , 1 must be even (see [11]). Therefore, it is enough to show that 1 < 2, and then 1 = 0 will follow. It will be shown that Mb with d = 2 is not locally trivial for any b 2 Q (S; 2): Take  = 2D;  = D;  = 1: Obviously, it is only necessary to show that for any b 2 f1; pi ; pi pj ; : : : ; p1


pn g; the equations  2 w = 2t4 + 2D2 z4 ; (c) w , t2 , Dz2 = bu2 have no solution (z; w; t 6= 0) in Zpi for some i. Suppose that there be some b for which (c) have solution (zi ; wi ; ti 6= 0) in Zpi for all i. Take p 2 fp1 ; : : : ; pn g and let (z; w; t 6= 0) is a p-adic integral solution to (c). Then it can be assumed that there is at least one p-adic unit among them, It is obvious that t is a p-adic unit i w is so. There are two cases: (1) w; t are all p-adic units. Then  2 w  2t4 (mod p) (=) w  ct2 (mod p); where c2 = 2): w , t2  bu2 (mod p) Hence (1  c)t2  b(ju)2 (mod p); and b must be a p-adic unit. (2) z is a p-adic unit. Neither w nor t is a p-adic unit. Let w = pw1 ; t = pt1 , where w1 is a p-adic unit and t1 is a p-adic integer. One has the following:  2 w1 = 2p21


pb2


p2n z4 + 2p2 t41 ; w1 , p1


pb


pn z2 , pt21 = (b=p)u2 : Hence (1  c)z 2  (b=p1


pn )(ju)2 (mod p); and b=p must be a p-adic unit. By Condition (II), there is at least one pi for which (pi ) = 1: By Lemma 1, it is impossible that b is equal to 1 or D. So let b = pi1


pik (1  k < n); then (i) For any p 2 fpik+1 ; : : : ; pin g; case (1) must be satis ed. By Lemma 1, one has the number of ,1 among ( ppi1 ); : : : ; ( ppik ) is even (odd) when (p) = 0(1): (ii) For any p 2 fpi1 ; : : : ; pik g, case (2) must be satis ed. By Lemma 1, one has the number of ,1 among ( pikp+1 ); : : : ; ( ppin ) is even (odd) when (p) = 0(1):

X

X

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Delang Li et al.

Combining these gives a non-trivial partition V = V1 [ V2 ; where V is the vertex set of G(D); V1 = fpi1 ; : : : ; pik g and V2 = fpik+1 ; : : : ; pin g: From (i), we have that #[V1 ; V2 ]  From (ii), we have that So

#[V1 ; V2 ] 

 (D ) =

X pi 2V1

X

pi 2V2

X pi 2V1

(pi ) +

(pi ) (mod 2): (pi ) (mod 2):

X pi 2V 2

(pi )  0(mod 2);

which contradicts condition (II) of the theorem. Thus we have proved that 1 = 0: By the exactness of (), we see that 01 = 0; then 0 =  =  = 1 = 2; 01 = 1 = 0: So rank(E (Q )) = rank(E 0 (Q )) = 0 + 01 , 2 = 0: From (), we have

X(E=Q )[2] = X(E=Q )['] = Z=2Z  Z=2Z; X(E0=Q )[2] = X(E0=Q )[ ] = (0):

Noting the arbitrariness of the isogenies of the commutative diagram of exact sequences (), we have the following exact sequences: (1) 0 ! (E 0 =Q )[ ' ] ! (E 0 =Q )[' ' ] = (E 0 =Q )[4]; (2) 0 ! (E=Q )['] ! (E=Q )[ ' '] = (E=Q )[4] ! (E 0 =Q )[ ' ]: Since (E 0 =Q )[2] = 0; the 2-primary component of (E 0 =Q ) is trivial. Then (E 0 =Q )[4] = 0. By (1), (E 0 =Q )[ ' ] = 0; and by (2), we have (E=Q )[4]  = (E=Q )  = Z=2Z  Z=2Z , which has no element of order 4. So the 2-primary component of (E=Q ) is isomorphic to Z=2Z  Z=2Z , and has order 4. The theorem is proved.

X X

X X

X

X

X X X X

X

X X

X

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

J H Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986 J Tunell. A classical Diophantine problem and modular form of weight 3=2. Invent Math, 1983, 72:323{334 C Zhao. For Elliptic curves with second lowest two-power in L(1). Communication K Rubin. Tate-Shafarevich group and L-function of elliptic curves with complex multiplication. Invent Math, 1987, 89:527{560 K Rubin. The main conjecture for imaginary quadratic elds. Invent Math, 1991, 103:25{68 K Feng. Noz-congruent numbers, odd graphs and the Birch-Swinnerton-Dyer conjecture. Acta Arithmetic, 1996, XXV 1 J A Bondy, U S R Murty. Graph theory with applications. The Macmillan Press LTD, 1976 W Koblit. Introduction to Elliptic Curves and Modular Forms. Springer-Verlag, 1984 B Birch, H P F Swinnerton-Dyer. Notes on elliptic curves (II). J Reine Angew Math, 1965, 218:79{108 J P Serre. A course in Arithmetic. Springer-Verlag, 1973 J W S Cassels. Arithmatic on curves of genus 1, (IV) proof of the Hauptvermutung. J Reine Angew Math, 1962, 211:95{112 M J Razar. A relation between the two-component of the Tate-S^afarevic group and L(1) for certain elliptic curves. American Journal of Math, 1974, 96:127{144