Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves
aa r X i v : . [ m a t h . N T ] F e b GENERALIZED BIRCH LEMMA AND THE 2-PART OF THE BIRCH ANDSWINNERTON-DYER CONJECTURE FOR CERTAIN ELLIPTIC CURVES
JIE SHU AND SHUAI ZHAI
Abstract.
In the present paper, we generalize the celebrated classical lemma of Birch andHeegner on quadratic twists of elliptic curves over Q . We prove the existence of explicit infinitefamilies of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, anduse Heegner points to explicitly construct rational points of infinite order on the twists of rank1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birchand Swinnerton-Dyer conjecture when the original curve does. We also prove a new result inthe direction of the Goldfeld conjecture. Contents
1. Introduction 12. The arithmetic of the elliptic curve E Introduction
Let E be an elliptic curve defined over Q , and let N be the conductor of E . Then E ismodular by the theorem of Wiles et al [31] [3], and we let f : X ( N ) → E be an optimalmodular parametrization sending the cusp at infinity, which is denoted by [ ∞ ], to the zeroelement of E . We write [0] for the cusp of X ( N ) arising from the zero point in P ( Q ) under thecomplex uniformization of X ( N ). By the theorem of Manin–Drinfeld, f ([0]) is a torsion pointin E ( Q ). For any square-free integer D = 1, we write E ( D ) for the twist of E by the extension Q ( √ D ) / Q , and L ( E ( D ) , s ) for its complex L -series. We assume throughout this paper that thegroup E [2]( Q ) of rational 2-division points on E is cyclic of order 2, and we define the elliptic Mathematics Subject Classification.
Primary 11G05.Jie Shu is supported by NSFC-11701092. urve E ′ / Q to be the quotient curve E ′ = E/E [2]( Q ). In all that follows, we will make thefollowing assumption:( Tor ) E [2]( Q ) = E ′ [2]( Q ) = Z / Z . Thus Q ( E [2]) and Q ( E ′ [2]) are quadratic extensions of Q . We also remark that it is proven inProposition 2.3 that, for any elliptic curve E over Q with f ([0]) E ( Q ), condition (Tor) holdsfor E if and only if there exists an odd prime q of good reduction for E such that(1.1) a q ≡ − (cid:18) − q (cid:19) mod 4 , where a q is the trace of Frobenius on the reduction of E modulo q . We will use the followingexplicit set of primes, which, by Chebotarev’s theorem, has positive density in the set of allprimes. Definition 1.1.
A prime q is admissible for E if ( q, N ) = 1 and q is inert in both of thequadratic fields Q ( E [2]) and Q ( E ′ [2]) . Note also that, assuming that our elliptic curve E satisfies condition (Tor), we prove in Propo-sition 2.2 that an odd prime q of good reduction is admissible for E in the above sense if andonly if it satisfies (1.1).In what follows, p will always denote a prime > p ≡ K = Q ( √− p ). We say that p satisfies Heegner hypothesis for (
E, K ) if every prime ℓ dividing N splits in the field K . For any odd prime q , put q ∗ = (cid:16) − q (cid:17) q , where (cid:0) ·· (cid:1) denotes the Legendresymbol. We shall prove the following result, which generalizes an old lemma of Birch [2]. Theorem 1.2.
Let E be an elliptic curve over Q satisfying f ([0]) E ( Q ) and Condition (Tor) . Let p > be any prime ≡ satisfying the Heegner hypothesis for ( E, K ) , where K = Q ( √− p ) . For any integer r ≥ , let q , . . . , q r be distinct admissible primes which are notequal to p , and put M = q ∗ q ∗ · · · q ∗ r . Then we have ord s =1 L ( E ( M ) , s ) = rank E ( M ) ( Q ) = 0 , and ord s =1 L ( E ( − pM ) , s ) = rank E ( − pM ) ( Q ) = 1 . Moreover, the Shafarevich–Tate groups X ( E ( M ) ) and X ( E ( − pM ) ) are both finite. In view of the above remarks, we immediately obtain the following corollary.
Corollary 1.3.
Let E be any elliptic curve over Q such that f ([0]) E ( Q ) , and there existsan odd prime q of good reduction for E satisfying (1.1) . Then, for any integer r ≥ , thereexist infinitely many square-free integers M and M ′ , having exactly r prime factors, such that L ( E ( M ) , s ) does not vanish at s = 1 , and L ( E ( M ′ ) , s ) has a zero of order at s = 1 . Theorem 1.2 and Corollary 1.3 improve the results in [5, Theorem 1.1] and [6, Theorem 1.1],which were motivated by the remarkable progress on the congruent number problem made byTian [29]. Note, however, that, in these two papers, much stronger conditions are imposedon the prime factors of M and M ′ . In particular, the generalizations of the Birch lemma in[5, Corollary 2.6] and [6, Theorem 1.1] require that the prime factors q of M and M ′ satisfy q ≡ a q ≡ r +1 , where, as above, r denotes the number of prime factors of M or M ′ . Theorem 1.2 and Corollary 1.3, can be applied to many elliptic curves, especially toelliptic curves without complex multiplication. We present a wide range of examples in Section5. Note also that in Theorem 1.2, M will be negative precisely when the number of admissibleprimes q ≡ M is odd, whence − pM will be positive. In some cases, thisenables us to use Heegner points to construct rational points of infinite order on real quadratictwists of our elliptic curve E , as for example in Section 5.1, where E is a Neumann–Setzer curve.For the 2-part of the Birch and Swinnerton-Dyer conjecture for the quadratic twists of E occurring in Theorem 1.2, we prove the following result (see Theorem 4.10). heorem 1.4. Let E and M be as in Theorem 1.2. Assume that (i) the Manin constant of E is odd, and (ii) every prime ℓ dividing N splits in both K = Q ( √− p ) and Q ( √ M ) . Then, ifthe -part of the Birch and Swinnerton-Dyer conjecture holds for E , it also holds for both E ( M ) and E ( − pM ) . In Section 5.3, we give infinitely many examples of elliptic curves over Q , without complexmultiplication, which satisfy the full Birch and Swinnerton-Dyer conjecture, by combining thisresult with a recent theorem of Wan [30] on the p -part of the Birch and Swinnerton-Dyerconjecture for primes p >
2, whose proof uses deep methods from Iwasawa theory.We now present another application of Theorem 1.2. In 1979, Goldfeld [10] conjectured that,for every elliptic curve defined over Q , amongst the set of all its quadratic twists with rootnumber +1, there is a subset of density one where the central L -value of the twist is non-zero,and amongst the set of all its quadratic twists with root number −
1, there is a subset of densityone where the central L -value of the twist has a zero of order equal to 1. For an elliptic curve E over Q , define N r ( E, X ) = { square free D ∈ Z : | D | ≤ X, ord s =1 L ( E ( D ) , s ) = r } , where r = 0 ,
1. We prove the following result.
Theorem 1.5.
Let E be an elliptic curve over Q satisfying f ([0]) E ( Q ) , and Condition (Tor) . Then, as X → ∞ , we have N r ( E, X ) ≫ X log / X .
Previously, for any given elliptic curve E over Q , Ono and Skinner [21] showed that N ( E, X ) ≫ X log X , and later Ono [20] proved that, in particular for E without a rational 2-torsion point, N ( E, X ) ≫ X (log X ) − α for some 0 < α <
1. Perelli and Pomykala [23] proved that N ( E, X ) ≫ X − ε for every ε >
0. Much progress for various families of elliptic curves has been made towards Goldfeld’sconjecture. For example, see the early survey article by Silverberg [27], and very recent results in[28] and [13]. However, Theorem 1.5 improves previous results when E satisfies f ([0]) E ( Q )and Condition (Tor). The proof of Theorem 1.5 is a straightforward consequence of Theorem1.2. Indeed, we just count the number of integers M appearing in Theorem 1.2, say S ( X ) := X | M |≤ X , where M = q ∗ · · · q ∗ r as in Theorem 1.2. In view of Proposition 2.2, the Chebotarev densitytheorem shows that the admissible primes have natural density 1 / S ( X ) ∼ c X/ (log X ) − / ∼ c X/ (log X ) / , where c is a constant. Then Theorem 1.5 follows immediately since N r ( E, X ) ≫ S ( X ).2. The arithmetic of the elliptic curve E Throughout the rest of the paper, we shall always assume that E is an elliptic curve over Q satisfying f ([0]) E ( Q ) and Condition (Tor). In this section, we establish some of the basicarithmetic properties of these curves. .1. Condition (Tor) and admissible primes.
Under Condition (Tor), the quotient map φ : E → E ′ is an isogeny over Q of degree 2. Let φ ′ : E ′ → E be its dual isogeny. Proposition 2.1.
Under Condition (Tor) , we have E [2 ∞ ]( Q ) = E [2]( Q ) = Z / Z .Proof. Suppose on the contrary that there is a point P in E ( Q ) of exact order 4, it followsthat the image of P in E ′ , namely φ ( P ), is rational by the definition of 2-isogeny. Since 2 P generates E [2]( Q ), we have φ (2 P ) = 0, but P does not lie in the kernel of φ , so φ ( P ) is oforder 2, and φ ′ ◦ φ ( P ) = 2 P . Let P ∈ E [2] \ E [2]( Q ) be a point of order 2. Then φ ( P ) must berational of order 2 since φ ′ ◦ φ ( P ) = 0. Thus, the images of P and P in E ′ = E/E [2]( Q ) aredifferent rational points of order 2 and they generate E ′ [2], that is, E ′ [2]( Q ) = E ′ [2] = ( Z / Z ) ,which does not satisfy Condition (Tor). (cid:3) Note Q ( E [2]) = Q ( √ ∆ E ) and Q ( E ′ [2]) = Q ( √ ∆ E ′ ) are quadratic extensions of Q , where ∆ E and ∆ E ′ are the discriminants of E and E ′ , respectively. For any prime q of good reduction for E , let a q be the trace of the Frobenius on the reduction of E mod q . Proposition 2.2.
Let q be a prime with ( q, N ) = 1 . Under Condition (Tor) , we have that a q ≡ if q ≡ and q is inert in both Q ( √ ∆ E ) and Q ( √ ∆ E ′ ) ; if q ≡ and q splits in Q ( √ ∆ E ) or Q ( √ ∆ E ′ ) ; if q ≡ and q splits in Q ( √ ∆ E ) or Q ( √ ∆ E ′ ) ; if q ≡ and q is inert in both Q ( √ ∆ E ) and Q ( √ ∆ E ′ ) .In particular, q is inert in both Q ( √ ∆ E ) and Q ( √ ∆ E ′ ) , i.e., admissible for E if and only if (2.1) a q ≡ − (cid:18) − q (cid:19) mod 4 . Proof.
The elliptic curve E has good reduction at q . Now reduction modulo q , gives the exactsequence 0 → b E ( q Z q ) → E ( Q q ) → E ( F q ) → , where b E/ Z q denotes the formal group associated to E/ Q q . Since q >
2, by [25, Chapter IV,Theorem 6.4 ], multiplication by 4 induces an isomorphism on b E ( q Z q ), and hence we have E [4]( Q q ) = E [4]( F q ). If q splits in Q ( √ ∆ E ) or Q ( √ ∆ E ′ ), then either E [2]( F q ) = E [2]( Q q ) = ( Z / Z ) or E ′ [2]( F q )[2] = E ′ [2]( Q q )[2] = ( Z / Z ) . Since E is isogenous to E ′ , | E ( F q ) | = | E ′ ( F q ) | . In either case, 4 divides | E ( F q ) | . If q is inert inboth Q ( √ ∆ E ) and Q ( √ ∆ E ′ ), by [33, Lemma 2.1], we have E [4]( F q ) = E [4]( Q q ) = Z / Z . In conclusion, we have | E ( F q ) | ≡ (cid:26) q is inert in both Q ( √ ∆ E ) and Q ( √ ∆ E ′ );0 mod 4 if q splits in Q ( √ ∆ E ) or Q ( √ ∆ E ′ ).Since q is an odd good prime for E , the assertion follows immediately from the injection E ( Q q )[2 ∞ ] ֒ → E ( F q ) under reduction modulo q and the formula a q = q + 1 − | E ( F q ) | . (cid:3) Proposition 2.3. If f ([0]) / ∈ E ( Q ) , then condition (Tor) is equivalent to the existence of anodd good prime q with a q ≡ − (cid:18) − q (cid:19) mod 4 . roof. Under Condition (Tor), by Proposition 2.2, admissible primes are exactly those primes q ∤ N satisfying (2.1).Suppose q ∤ N satisfies (2.1). Then | E ( F q ) | = q + 1 − a q ≡ , which implies that both | E [2 ∞ ]( Q ) | and | E ′ [2 ∞ ]( Q ) | are less than or equal to 2 for q ∤ N . Since f ([0]) / ∈ E ( Q ), we have E [2 ∞ ]( Q ) = 0 and hence E [2 ∞ ]( Q ) = Z / Z . Then the quotient map φ : E → E ′ is an isogeny over Q of degree 2, and we write φ ′ : E ′ → E for its dual isogeny. Thenthe kernel of φ ′ is rational over Q , and hence contains a rational point of order 2. Therefore E ′ [2 ∞ ]( Q ) = Z / Z , and hence Condition (Tor) holds. (cid:3) Equivalent condition for the -part of the Birch and Swinnerton-Dyer conjec-ture. In the rest of this section, we assume that the Manin constant c E of E is always odd.Let Ω + (respectively, Ω − ) be the minimal real (respectively, imaginary) period of E . Proposition 2.4.
For the elliptic curve E we have L ( E, = 0 , and ord (cid:18) L ( E, + (cid:19) = − . Proof.
Let f : X ( N ) → E be an optimal modular parametrization. We embed X ( N ) into itsJacobian J ( N ) via the base point [ ∞ ]. Then there exists a unique homomorphism e f : J ( N ) → E through which f factors. We consider the complex uniformization of e f . By the Abel–Jacobitheorem, we have J ( N ) = H ( X ( N ) , R ) / H ( X ( N ) , Z ) and Jac( E ) = H ( E, R ) / H ( E, Z ) . Then f induces a homomorphism J ( N ) → Jac( E ) , γ f ( γ ) , γ ∈ H ( X ( N ) , R ) . The elliptic curve E has a complex uniformization E = C /L where L ⊂ C is the period latticeof E , and we have an isomorphismJac( E ) ≃ E, α Z α ω E , where ω E is the N´eron differential of E . Then the complex uniformization of e f is explicitlygiven as(2.2) e f : J ( N ) → E, γ Z f ( γ ) ω E . In particular, we take γ = [0 → ∞ i ] to be the path on X ( N ) joining [0] and [ ∞ i ] . Let g be thenewform associated to E . Since f ∗ ω E = c E πig ( z ) dz , it follows that L ( E,
1) = 2 πi Z γ g ( z ) dz = c − E Z f ( γ ) ω E . Then from the complex uniformization (2.2), we see that(2.3) f ([0]) = c E L ( E,
1) mod L. Let H ( E, Z ) + be the submodule of H ( E, Z ) that is invariant under the complex conjugation.Then H ( E, Z ) + is free of rank 1 and Ω + = (cid:12)(cid:12)(cid:12)(cid:12)Z γ + ω E (cid:12)(cid:12)(cid:12)(cid:12) , where γ + is a generator of the submodule H ( E, Z ) + . Since γ is a real path on X ( N ), it followsthat f ( γ ) ∈ H ( E, R ) + = H ( E, Z ) + ⊗ Z R . Let C be a sufficiently large odd integer such that Cf ([0]) has order 2. Then by (2.3) we have Cc E L ( E, + Z and 2 Cc E L ( E, + ∈ Z . ence ord (cid:18) L ( E, + (cid:19) = − , as desired. (cid:3) Denote Ω = Z E ( R ) | ω E | the real period of E . Then Ω = Ω + or 2Ω + according to that E ( R ) is connected or not. Theperiod Ω is the one appearing in the exact formula which is part of the Birch and Swinnerton-Dyer conjecture and we define L alg ( E,
1) = L ( E, . Proposition 2.5.
The discriminant ∆ E < . In particular, we have ord ( L alg ( E, − . Proof.
Suppose, on the contrary, that ∆ E >
0. Then the period lattice is Z Ω + + Z Ω − . Let q be an admissible prime. By Manin’s formula [16, Theorem 3.3], we have | E ( F q ) | L ( E,
1) = c E q − X k =1 h{ , k/q } , g i , where {· , ·} denotes the modular symbol, and h{ , k/q } , g i = s k Ω + + t k Ω − , with s k , t k ∈ Z .Since h{ , k/q } , g i = h{ , − k/q } , g i , we have | E ( F q ) | L ( E,
1) = 2Ω + c E q − X k =1 s k . Then by Proposition 2.4, we have ord ( | E ( F q ) | ) ≥
2. Since q is admissible, that is a q = 1 − (cid:18) − q (cid:19) mod 4 , it follows that | E ( F q ) | = q + 1 − a q has 2-adic valuation 1, and we get a contradiction. Thus∆ E <
0, whence E ( R ) is connected and Ω = Ω + , and L alg ( E,
1) = L ( E, + has 2-adic valuation − (cid:3) Corollary 2.6.
We have rank E ( Q ) = ord s =1 L ( E, s ) = 0 . The -part of the Birch and Swinnerton-Dyer conjecture holds for the elliptic curve E if andonly if ord | X ( E ) || E ( Q ) tor | Y ℓ c ℓ ( E ) ! = − . Proof.
By Proposition 2.4, L ( E, = 0 and by the work of Gross and Zagier [12] and Kolyvagin[15], E ( Q ) has rank zero and | X ( E ) | is finite. The statement for the 2-part of the Birch andSwinnerton-Dyer conjecture for E follows immediately from Proposition 2.5. (cid:3) .3. Discriminants and periods.Proposition 2.7.
We have ∆ E < and ∆ E ′ > .Proof. Moving the rational 2-torsion point on E to (0 , E/ Q of the form(2.4) Y = X + aX + bX, where a, b ∈ Z . Dividing this curve by the subgroup generated by the point (0 , E ′ / Q of the form(2.5) y = x − ax + ( a − b ) x. Note that the discriminants of various Weierstrass equations of elliptic curves over Q are thesame up to multiplication by non-zero rational twelfth powers. It is easy to calculate that thediscriminant of equation (2.4) is 2 b ( a − b ). It is proved in Proposition 2.5 that ∆ E <
0, andhence b >
0. A simple computation shows that the discriminant of equation (2.5) is 2 b ( a − b ) ,which is positive since b >
0. Therefore, ∆ E ′ > (cid:3) Lemma 2.8.
Let Ω and Ω ′ denote the least positive real periods of E and E ′ , respectively. Then ΩΩ ′ = 1 or . Proof.
In view of Proposition 2.7, and by [9, Theorem 1.2], we haveΩΩ ′ = (cid:12)(cid:12)(cid:12)(cid:12) ωφ ∗ ω ′ (cid:12)(cid:12)(cid:12)(cid:12) , where ω and ω ′ are the minimal differentials (unique up to signs) on E and E ′ , respectively.Suppose φ ∗ ω ′ = αω and φ ′∗ ω = βω ′ with α, β ∈ Z . Since αβω = φ ∗ ( φ ′∗ ω ) = [2] ∗ ω = 2 ω, we see α = 1 or 2 and the lemma follows. (cid:3) Selmer groups.
Recall that φ : E → E ′ is the isogeny of degree 2 defined over Q withkernel E [ φ ] = E [2]( Q ), and φ ′ : E ′ → E its dual isogeny. For any finite or infinite place v of Q ,we write Q v for the completion of Q at v . Let κ v, ( E ) be the image of the Kummer map κ v, : E ( Q v ) / E ( Q v ) ֒ → H ( Q v , E [2]) . Similarly, we write κ v,φ ( E ) := Im (cid:0) E ′ ( Q v ) /φ ( E ( Q v )) ֒ → H ( Q v , E [ φ ]) (cid:1) ; κ v,φ ′ ( E ′ ) := Im (cid:0) E ( Q v ) /φ ′ ( E ′ ( Q v )) ֒ → H ( Q v , E ′ [ φ ′ ]) (cid:1) . The 2-Selmer group over Q is then defined bySel ( E ) := Ker H ( Q , E [2]) → M v H ( Q v , E [2]) /κ v, ( E ) ! . The Shafarevich–Tate group X ( E ) is defined by X ( E ) := Ker H ( Q , E ) → M v H ( Q v , E ) ! . Then we have the following well-known exact sequence0 → E ( Q ) / E ( Q ) → Sel ( E ) → X ( E )[2] → . Similarly, we define the φ -Selmer group and φ ′ -Selmer groups over Q bySel φ ( E ) := Ker H ( Q , E [ φ ]) → M v H ( Q v , E [ φ ]) /κ v,φ ( E ) ! ; el φ ′ ( E ′ ) := Ker H ( Q , E ′ [ φ ′ ]) → M v H ( Q v , E ′ [ φ ′ ]) /κ v,φ ′ ( E ′ ) ! . Proposition 2.9.
Assume that the -part of the Birch and Swinnerton-Dyer conjecture holdsfor E . Then we have Sel ( E ) ∼ = Sel ( E ′ ) ∼ = Z / Z ; Sel φ ( E ) ∼ = Sel φ ′ ( E ′ ) ∼ = Z / Z . Proof.
By Proposition 2.4, we have L ( E, = 0. By the work of Gross and Zagier [12] andKolyvagin [15], E ( Q ) has rank zero and | X ( E ) | is finite, and hence a square. By Corollary 2.6,we have ord | X ( E ) || E ( Q ) tor | Y ℓ c ℓ ( E ) ! = − , we then must have X ( E )[2] = 0, and hence Sel ( E ) = E [2]( Q ) = Z / Z . By the invariance ofthe Birch and Swinnerton-Dyer conjecture under isogeny [4, (Corollary to) Theorem 1.3], wehave that E ′ ( Q ) = 0 also has rank 0, andord | X ( E ′ ) || E ′ ( Q ) tor | Y ℓ c ℓ ( E ′ ) ! = ord | X ( E ) || E ( Q ) tor | Y ℓ c ℓ ( E ) ! + ord (cid:18) ΩΩ ′ (cid:19) (2.6) = − (cid:18) ΩΩ ′ (cid:19) = − − . The last equality follows from Lemma 2.8. In view of (2.6), we have ord ( | X ( E ′ ) | ) ≤
1. Sincethe order of X ( E ′ ) is a square, we conclude that X ( E ′ )[2] = 0, and henceSel ( E ′ ) = E ′ [2]( Q ) = Z / Z . For the second assertion, consider the following well-known exact sequence:0 → E ′ ( Q )[ φ ′ ] /φ ( E ( Q )[2]) → Sel φ ( E ) → Sel ( E ) → Sel φ ′ ( E ′ ) → X ( E ′ )[ φ ′ ] /φ ( X ( E )[2]) → . Since Sel ( E ) = Sel ( E ′ ) = Z / Z , the non-trivial element in each Selmer group comes from therational 2-torsion point in E ( Q )[2] and E ′ ( Q )[2], respectively. Thus, we have X ( E )[2] = X ( E ′ )[2] = 0 , and hence the last term in the above exact sequence is zero. Since E ′ ( Q )[ φ ′ ] /φ ( E ( Q )[2]) ∼ = Z / Z , it follows from the above exact sequence and its dual form thatSel φ ( E ) ∼ = Sel φ ′ ( E ′ ) ∼ = Sel ( E ) ∼ = Sel ( E ′ ) ∼ = Z / Z , as desired. (cid:3) Heegner points
Throughout this section, we let E , p and M be as in Theorem 1.2, and take K = Q ( √− p ). Wewill construct Heegner points lying in E ( − pM ) ( Q ). In the main result of this section (Theorem3.7), we prove our Heegner points do indeed have infinite order, and we also establish the exact2-divisibility of these Heegner points. Then we show that Theorem 1.2 follows immediatelyfrom the non-triviality of Heegner points via the work of Gross–Zagier and Kolyvagin. Theexact 2-divisibility of Heegner points is used to prove 2-part of the Birch and Swinnerton-Dyerconjecture for the related elliptic curves in the next section.The induction arguments on Heegner points given in this section strengthen those in theearlier papers [29], [5] and [6], in two significant ways. Firstly, in all such arguments, the non-triviality of Heegner points relies on the crucial Galois identity of the genus Heegner point givenin Theorem 3.2: z σ t + z = T, here T is a torsion point. What is new in our approach is that, while the previous authorsalways make congruence restrictions on the prime factors of M to ensure the 2-primary partof T is non-trivial, we make no such congruence restrictions, allowing the 2-primary part of T to be zero in the induction process (see Corollary 3.5 and (3.5)). Secondly, all these methodsuse norm relations on Heegner points in one way or another. We use the full strength of thesenorm relations of Heegner points in the induction process to make the required 2-adic valuationof the Hecke eigenvalues a q as small as possible.3.1. Non-triviality of Heegner points.
Let O K be the ring of integers of K . Since ( E, K )satisfies the Heegner hypothesis, there exists an ideal N ⊂ O K such that O K / N ≃ Z /N Z .For any integer c coprime to pN , let O c ⊂ O K be the order of discriminant − pc . Define theHeegner points y c = ( C / O c → C / ( O c ∩ N ) − ) ∈ X ( N )( H c ) , where H c denotes the ring class field over K of conductor c .For any abelian group G denote b G := G ⊗ Z b Z , where b Z = Q p Z p . Let σ : K × \ b K × → Gal( K ab /K ) be the Artin reciprocity map. Let t ∈ b K × be the id´ele such that t d O K = c N . Let w be the Atkin–Lehner operator on X ( N ). We havethe following proposition from [11, § Proposition 3.1.
The Heegner point y c satisfies y wc = y cσ t − . Let q , · · · , q r be distinct admissible primes for E , and set M = q ∗ · · · q ∗ r . Let H = K ( p q ∗ , · · · , p q ∗ r ) ⊂ H M be the genus field. We define the genus Heegner point z := tr H M /H f ( y M ) ∈ E ( H ) . Theorem 3.2.
The genus Heegner point z satisfies z σ t + z = T, where T = [ H M : H ] f ([0]) ∈ E ( Q ) .Proof. By Proposition 3.1 we have T = X σ ∈ Gal( H M /H ) f ([0]) σ = X σ ∈ Gal( H M /H ) ( f ( y wM ) + f ( y M )) σ = X σ ∈ Gal( H M /H ) ( f ( y M σ t − ) + f ( y M )) σ = z σ t − + z . The theorem follows by acting σ t on the above equality and noting that T σ t = T . (cid:3) Lemma 3.3.
We have E ( H )[2 ∞ ] = E ( Q )[2] . Proof.
Suppose the contrary and P ∈ E ( H )[2 ∞ ] \ E ( Q )[2]. Since E has good reduction outside N , it follows that Q ( P ) is unramified outside 2 N . Then we have Q ( P ) = Q by noting H = Q ( √− p, p q ∗ , · · · , √ q ∗ r ), which is a contradiction. (cid:3) Definition 3.4.
An admissible prime q is of first kind, if q ≡ is inert in K , or q ≡ splits in K ; otherwise, q is of second kind. Corollary 3.5.
If all prime factors of M are of first kind, then the two primary component T of T is of order and z is of infinite order; otherwise T = 0 . roof. Just note that the degree[ H M : H ] = Y q | M inert in K q + 12 · Y q | M split in K q − . It follows that [ H M : H ] is odd, if all prime factors of M are of first kind. By Proposition 3.3, E [2 ∞ ]( Q ) = E [2]( Q ) and f ([0]) E ( Q ), we see that T is of order two. Otherwise, T = 0.In the case T is of order 2, we suppose z is torsion, and choose sufficiently large odd C sothat Cz ∈ E ( H )[2 ∞ ] = E ( Q )[2] and CT = T . Then(3.1) T = CT = ( Cz σ t + Cz ) = 2 Cz = 0 , which is a contradiction. Hence in this case z is of infinite order. (cid:3) We only consider divisors of M which are congruent to √ d ∈ H . For anysuch d | M , let χ d : Gal( H /K ) → µ be the quadratic character associated to the quadraticextension K ( √ d ) /K through class field theory. Then the character χ d is explicitly given by χ d ( σ ) = ( √ d ) σ − . Define the twisted Heegner points(3.2) z d = X σ ∈ Gal( H M /K ) χ d ( σ ) f ( y M ) σ = X σ ∈ Gal( H /K ) χ d ( σ ) z σ ∈ E ( K ( √ d )) − , where the superscript “ − ” means the eigenspace that the generator of Gal( K ( √ d ) /K ) acts as − Proposition 3.6.
We have z d + z σ t d = 0 , and X d | Md ≡ z d = 2 r z . Proof.
The proof of the first assertion is similar to that of Theorem 3.2. By Proposition 3.1 wehave X σ ∈ Gal( H M /K ) χ d ( σ ) f ([0]) σ = X σ ∈ Gal( H M /K ) χ d ( σ )( f ( y wM ) + f ( y M )) σ = X σ ∈ Gal( H M /K ) χ d ( σ )( f ( y M σ t − ) + f ( y M )) σ = z dσ t − + z d . On the other hand, we have X σ ∈ Gal( H M /K ) χ d ( σ ) f ([0]) σ = X σ χ d ( σ ) ! f ([0]) = 0 . The second assertion is standard and straight-forward. (cid:3)
Theorem 3.7.
The Heegner point z M has exact -divisibility index r − : z M ∈ (cid:16) r − E ( Q ( p − pM )) − + E ( Q ( p − pM )) tor (cid:17) /(cid:16) r E ( Q ( p − pM )) − + E ( Q ( p − pM )) tor (cid:17) , where the superscript “ − ” means the eigenspace that the generator of Gal( Q ( √− pM ) / Q ) actsas − .Remark . If r = 0, i.e., M = 1, the statement in the theorem exactly means2 z ∈ E ( Q ( √− p )) − but z (cid:0) E ( Q ( √− p )) − + E ( Q ( √− p )) − tor (cid:1) . roof. In the following, without loss of generality we may replace S = f ([0]), if necessary, byits two primary component S ∈ E ( Q )[2] for simplification. Indeed, we may do this as in (3.1)by multiplying all points by a sufficiently large odd integer. Let r ( M ) denote the number ofdistinct prime factors of M . We proceed by induction on r = r ( M ) to prove(3.3) z M = 2 r x M for some x M ∈ E ( H ) with x M + x σ t M = S .And then we deduce the assertion in the theorem from this property (3.3).First we consider the initial case r = 0 i.e., M = 1. Then H = K and z M = z and take x M = z M ∈ E ( K ). By Theorem 3.2 we have(3.4) z M + z M σ t = S. We prove the theorem in this case M = 1 as follows. First, note that 2 z M = − z M andhence 2 z ∈ E ( Q ( √− p )) − . Suppose the contrary that z M = y + s with y ∈ E ( Q ( √− p )) − and s ∈ E ( Q ( √− p )) − tor . By Lemma 3.3, we can choose C a sufficiently large odd integer such that Cs ∈ E ( Q )[2]. Then from (3.4) we have S = CS = C ( z M + z M ) = C ( y + y ) + 2 Cs = 2 Cs = 0 , which is a contradiction.Next assume r >
0. For any proper d | M with d ≡ d ′ = M/d and d ′ + (respectively, d ′− ) be the product of prime factors of d ′ that split (respectively, are inert) in K .By Euler system property [15], we have z d = tr H M /H d f ( y M ) = Y q | d ′ + ( a q − σ v − σ v ) · Y q | d ′− a q z ′ d , where z ′ d = X σ ∈ Gal( H d /K ) χ d ( σ ) f ( y d ) . Here v and v are the two places of K above q if q | d ′ + , and σ v and σ v are the correspondingFrobenius automorphisms, respectively. Note that we have χ d ( σ v ) = χ d ( σ v ) = (cid:18) dq (cid:19) = ± , and σ v ( z ′ d ) = χ d ( σ v ) z ′ d and σ v ( z ′ d ) = χ d ( σ v ) z ′ d . By the condition (2.1) on Hecke eigenvalues, we know for all q | M we have a q ≡ − (cid:18) − q (cid:19) mod 4 . For any proper divisor d | M with d ≡ A d = 2 − µ ( d ′ ) Y q | d ′ + ( a q − (cid:18) dq (cid:19) ) · Y q | d ′− a q . Let M (respectively, M ) be the product of prime factors of M which are of first (respectively,second) kind. From the condition (2.1) for Hecke eigenvalues and Definition 3.4, we see thefollowing two facts:(i) If some prime factor of d ′ is of first kind, i.e., M ∤ d , then A d is an even integer.(ii) If all prime factors of d ′ are of second kind, i.e., M | d , then A d is an odd integer.By the induction hypothesis, we have z ′ d = 2 µ ( d ) x ′ d ∈ µ ( d ) E ( H ) , with x ′ d + x ′ σ t d = S . Hence z d = 2 r A d x ′ d ∈ r E ( H ) . hen z M = 2 r z − X M ∤ dd = M A d x ′ d − X M | dd = M A d x ′ d ∈ r E ( H ) . Denote x M = z − X M ∤ dd = M A d x ′ d − X M | dd = M A d x ′ d ∈ E ( H ) . If M ∤ d , then A d is even and A d ( x ′ d + x ′ dσ t ) = 0 . If M | d , then A d is odd and A d ( x ′ d + x ′ dσ t ) = S. Then x M + x σ t M = z + z σ t + X M | dd = M A d ( x ′ d + x ′ dσ t ) . If all prime factors of M are of first kind, i.e., M = M and M = 1, then there are no termswith M | d and by Corollary 3.5, we have x M + x σ t M = z + z σ t = S. If some prime factors of M are of second kind, again by Corollary 3.5, we have(3.5) z + z σ t = 0 , and hence x M + x σ t M = X M | dd = M A d ( x ′ d + x ′ dσ t ) = X M | dd = M S = S. In conclusion, we have(3.6) z M = 2 r x M , x M + x σ t M = S. Next we use a descent argument (as in [5, page 365]) to prove the theorem when r >
0. Weembed E ( K ( √ M )) / r E ( K ( √ M )) into H ( K ( √ M ) , E [2 r ]) by the Kummer map and considerthe inflation-restriction exact sequence:0 → H ( H /K ( √ M ) , E ( H )[2 r ]) → H ( K ( √ M ) , E [2 r ]) → H ( H , E [2 r ]) . Since z M ∈ r E ( H ), the image of z M in H ( H , E [2 r ]) under the Kummer map is zero, so theimage of z M in H ( K ( √ M ) , E [2 r ]) lies in H ( H /K, E ( H )[2 r ]). Because E ( H )[2 ∞ ] = E ( Q )[2] , the image of z M in H ( K ( √ M ) , E [2 r ]) must be of order 2, that is, 2 z M ∈ r E ( K ( √ M )).Next we prove(3.7) 2 z M ∈ r E ( Q ( p − pM )) − . Let σ be the generator of Gal( K ( √ M ) /K ), and by definition (3.2) we have(3.8) σ ( z M ) = − z M . By [33, Theorem 5.2], we already know that the L ( E ( M ) , = 0 and hence its root number (cid:16) M − N (cid:17) = 1. Note χ M ( σ t ) = (cid:0) MN (cid:1) . Then χ M ( σ t )sgn( M ) = (cid:18) M − N (cid:19) = 1 . Here we divide into two cases: χ M ( σ t ) = sgn( M ) = +1 . Then2 z M = − χ M ( σ t )2 z M = − z M . Since
M >
0, together with (3.8) we conclude2 z M ∈ r E ( Q ( p − pM )) − . Here we note σ is a generator of Gal( Q ( √− pM ) / Q ). • χ M ( σ t ) = sgn( M ) = − . Then2 z M = − χ M ( σ t )2 z M = 2 z M . Again since
M <
0, we have 2 z M ∈ r E ( Q ( p − pM )) − . Consequently we conclude (3.7) and hence z M ∈ r − E ( Q ( p − pM )) − + E ( Q ( p − pM )) tor . Assume the contrary that z M = 2 r y + t, y ∈ E ( K ( p − pM )) − , t ∈ E ( K ( p − pM )) tor . From (3.6), we have x M = y + t ′ with t ′ ∈ E ( K ( √− pM )) tor and by Lemma 3.3 and the factthat a a σ t is also a generator of Gal( Q ( √− pM ) / Q ). S = x M + x σ t M = y + y σ t + 2 t ′ = 2 t ′ will have trivial 2-primary component which is a contradiction. (cid:3) Explicit Gross–Zagier formulae.
Let π be the automorphic representation on GL ( A )associated to E , let π χ M be the representation on GL ( A ) constructed from the quadraticcharacter χ M on A × K through Weil–Deligne representations. Since ( E, K ) satisfies Heegnerhypothesis, the Rankin–Selberg L -series L ( π × π χ M , s ) has sign − L ( π × π χ M , s ) = L ( E ( M ) , s ) L ( E ( − pM ) , s ) . Here we normalize the Rankin–Selberg L -series with the central point at s = 1.Recall f : X ( N ) → E is the optimal modular parametrization of E sending [ ∞ ] to the zeroelement of E , and g is the newform associated to E . Denote the Peterson norm( g, g ) Γ ( N ) := Z Z Γ ( N ) \H | g ( z ) | dxdy, z = x + iy, where H is the Poincar´e upper half plane. Invoking the general explicit Gross–Zagier formulaestablished in [8], the Heegner point z M ∈ E ( − pM ) ( Q ) satisfies the following explicit heightformula. Theorem 3.8.
The Heegner point z M ∈ E ( − pM ) ( Q ) satisfies L ′ ( π × π χ M ,
1) = 16 π ( g, g ) Γ ( N ) √ p | M | · b h Q ( z M )deg f , where b h Q ( · ) denotes the N´eron–Tate height on E over Q .Proof. Since the pair (
E, K ) satisfies the Heegner hypothesis, we just apply [8, Theorem 1.1]. (cid:3)
Now we can give the proof of Theorem 1.2.
Theorem 3.9 (Theorem 1.2) . Let E , p and M be as in Theorem 1.2. Then we have ord s =1 L ( E ( M ) , s ) = rank E ( M ) ( Q ) = 0 and ord s =1 L ( E ( − pM ) , s ) = rank E ( − pM ) ( Q ) = 1 . Moreover, the Shafarevich–Tate groups X ( E ( M ) ) and X ( E ( − pM ) ) are finite. roof. By Theorem 3.7, z M has infinite order and hence L ( π × π χ M , s ) has vanishing order 1 at s = 1. From the decomposition in (3.9) we must haveord s =1 L ( E ( M ) , s ) = 0 and ord s =1 L ( E ( − pM ) , s ) = 1 . Then Theorem 1.2 follows from the work of Gross–Zagier and Kolyvagin. (cid:3)
Let Ω + E ( D ) be the minimal real period of E ( D ) . Let Ω ( D ) denote the period of E ( D ) , which isequal to Ω + E ( D ) or 2Ω + E ( D ) according to that E ( R ) is connected or not. Proposition 3.10.
We have the following relation between periods and Peterson norm of g : Ω ( M ) Ω ( − pM ) = 8 π ( g, g ) Γ ( N ) deg f √ p | M | Proof.
Since the discriminant ∆ E < E ( R ) has only one connected component. The periodlattice L = Z Ω + Z (Ω / − / π ( g, g ) Γ ( N ) deg f √ p | M | = 2 √ p | M | Z C /L dxdy = ΩΩ − √− p | M | = Ω ( M ) Ω ( − pM ) , where the last equality follows from [22]. (cid:3) Combining Theorem 3.8, Proposition 3.10 and the decomposition (3 . Corollary 3.11.
The Heegner point z M ∈ E ( − pM ) ( Q ) satisfies L ( E ( M ) , L ′ ( E ( − pM ) , ( M ) Ω ( − pM ) = 2 b h Q ( z M ) . The -part of the Birch and Swinnerton-Dyer conjecture for the quadratictwists Throughout this section, we assume E has odd Manin constant. We shall compare all thearithmetic invariants of E with those of E ( M ) and E ( − pM ) .4.1. Tamagawa numbers.Proposition 4.1.
Let q be any prime dividing D with ( D, N ) = 1 . We have c q ( E ( D ) ) = (cid:26) if q is inert in Q ( √ ∆ E ) ; if q splits in Q ( √ ∆ E ) .Proof. Since ( q, N ) = 1, reduction modulo q on E gives an isomorphism E ( Q q )[2] ∼ = E ( F q )[2] . It follows from [7, Lemma 36 and Lemma 37] thatord ( c q ( E ( D ) )) = ord ( | E ( D ) ( Q q )[2] | ) = ord ( | E ( Q q )[2] | ) = ord ( | E ( F q )[2] | ) , which is equal to 1 if q is inert in Q ( E [2]), and equal to 2 if q splits in Q ( E [2]). Note that thecurve E ( D ) has additive reduction at q , so we have c q ( E ( D ) ) ≤
4. The assertion of the lemmathen follows. (cid:3)
Proposition 4.2.
Let D ≡ be a square-free integer with ( D, N ) = 1 . Then ord Y ℓ | N c ℓ ( E ( D ) ) = 1 . roof. By Proposition 2.6, we haveord | X ( E ) || E ( Q ) tor | Y ℓ c ℓ ( E ) ! = − , Since ord ( | E ( Q ) tor | ) = 2 and | X ( E ) | must be a square, it follows that X ( E )[2] = 0 and(4.1) ord Y ℓ | N c ℓ ( E ) = 1 . We now prove the assertion in several cases according to the reduction types of E .(i) If E , and hence E ( M ) , have additive reduction at ℓ , from the usual table of special fibersthe 2-primary part of the connected component group is killed by 2. Again, by [7, Lemma 36and Lemma 37], we have(4.2) ord ( c ℓ ( E )) = ord ( | E ( Q ℓ )[2] | ) = ord ( c ℓ ( E ( M ) )) . (ii) If E has multiplicative reduction at ℓ , and ℓ | N splits in Q ( √ D ). Then E ( D ) is isomorphicto E over Q ℓ , we immediately have c ℓ ( E ( D ) ) = c ℓ ( E ).(iii) If E has split multiplicative reduction at ℓ , and ℓ | N is inert in Q ( √ D ). Then E ( M ) hasnonsplit multiplicative reduction at ℓ . In this case E ( M ) Q ℓ and E Q ℓ are isomorphic and have thesame Tamagawa number equal to c ℓ ( E ). In view of [26, page 366], if c ℓ ( E ) is odd (respectively,even), then c ℓ ( E ( M ) ) = 1 (respectively, 2), and henceord ( c ℓ ( E )) = ord ( c ℓ ( E ( M ) )) . (iv) If E has nonsplit multiplicative reduction at ℓ , and ℓ | N is inert in Q ( √ D ). Then E ( M ) has split multiplicative reduction at ℓ . If c ℓ ( E ) = 1, then both c ℓ ( E ) and c ℓ ( E ( M ) ) are odd, i.e.,ord ( c ℓ ( E )) = ord ( c ℓ ( E ( M ) )) = 0 . If c ℓ ( E ) = 2, then E is semi-stable. Indeed from (4.2), the Tamagawa number is even for placesof additive reduction. But there is only one bad place of E with even Tamagawa number. By(2.6) and (4.1), the 2-adic valuation of the product of all Tamagawa numbers of E (respectively, E ′ ) is 1 (respectively, ≤ E is semi-stable of root number +1 and c ℓ ( E ) = 2, weconclude from [9, Theorem 6.1] that c ℓ ( E ′ ) = 1. Again, we haveord ( c ℓ ( E ′ )) = ord ( c ℓ ( E ′ ( M ) )) = 0 . Now φ ( M ) : E ( M ) → E ′ ( M ) is an isogeny of degree 2 of elliptic curves of split multiplication at ℓ . Since c ℓ ( E ′ ( M ) ) is odd, by [9, Theorem 6.1], ord ( c ℓ ( E ( M ) )) = 1. Therefore, for any ℓ | N ,we always have ord ( c ℓ ( E )) = ord ( c ℓ ( E ( M ) )) . This completes the proof of this proposition. (cid:3)
Selmer groups.
Let E , p and M be as in Theorem 1.2. Recall that E ( M ) is the quadratictwist of E by Q ( √ M ). Let φ ( M ) : E ( M ) → E ′ ( M ) be an isogeny of degree 2 with kernel E ( M ) [2]( Q ) and φ ′ ( M ) its dual isogeny. There is a natural identification of Galois modules E [ φ ] = E ( M ) [ φ ( M ) ]. This allows us to view both Sel φ ( E ) and Sel φ ( M ) ( E ( M ) ) as subgroupsof H ( Q , E [ φ ]) and enable us to compare the Selmer groups Sel φ ( E ), Sel φ ( M ) ( E ( M ) ) by localKummer conditions in H ( Q ℓ , E [ φ ]) for various places ℓ of Q . Lemma 4.3.
If all the prime factors of N split in Q ( √ M ) , then κ ℓ,φ ( E ) = κ ℓ,φ ( M ) ( E ( M ) ) and κ ℓ,φ ′ ( E ′ ) = κ ℓ,φ ′ ( M ) ( E ′ ( M ) ) for all places ℓ . roof. First consider the case ℓ = ∞ . We have the following Weierstrass equations E ( M ) : y = x + aM x + bM x, E ′ ( M ) : y = x − aM x + ( a − b ) M x. By Proposition 2.5, ∆ E ( M ) = M ∆ E < E ′ M = M ∆ E ′ >
0. Hence both E ( R ) and E ( M ) ( R ) are connected while both E ′ ( R ) and E ′ ( M ) ( R ) have two connected components. Then κ ∞ ,φ ( E ) = κ ∞ ,φ ( M ) ( E ( M ) ) = Z / Z and κ ∞ ,φ ′ ( E ′ ) = κ ∞ ,φ ′ ( M ) ( E ′ ( M ) ) = 0 . Suppose ℓ | N , then E and E ( M ) are isomorphic over Q ℓ . In particular, κ ℓ,φ ( E ) = κ ℓ,φ ( M ) ( E ( M ) ) and κ ℓ,φ ′ ( E ′ ) = κ ℓ,φ ′ ( M ) ( E ′ ( M ) )for ℓ | N . We now compare the local conditions at each place ℓ ∤ N ∞ and divide it into twocases.(i) For ℓ ∤ N M ∞ , we have both E and E ′ have good reduction at ℓ and ℓ is unramified in Q ( √ M ) / Q .(ii) For ℓ | M , we have both E and E ′ have good reduction at ℓ and ℓ is ramified in Q ( √ M ) / Q ,whence we have E ( Q ℓ )[2] ∼ = E ′ ( Q ℓ )[2] ∼ = Z / Z .Applying [14, Lemma 6.8], we have κ ℓ,φ ( E ) = κ ℓ,φ ( M ) ( E ( M ) ) and κ ℓ,φ ′ ( E ′ ) = κ ℓ,φ ′ ( M ) ( E ′ ( M ) )for ℓ ∤ N ∞ . Then the lemma follows. (cid:3) Lemma 4.4.
If all the prime factors of N split in Q ( √ M ) , then Sel φ ( E ( M ) ) ∼ = Sel φ ′ ( E ′ ( M ) ) ∼ = Z / Z . Proof.
By Lemma 4.3, we see that κ ℓ,φ ( E ) = κ ℓ,φ ( M ) ( E ( M ) ) holds for any places. Therefore, wehave Sel φ ( M ) ( E ( M ) ) = Sel φ ( E ) = Z / Z . The result for E ′ and E ′ ( M ) follows similarly. (cid:3) Proposition 4.5.
If all the prime factors of N split in Q ( √ M ) , then Sel ( E ( M ) ) = Z / Z and X ( E ( M ) )[2] = 0 . Proof.
The first assertion follows from Lemma 4.4, and the following exact sequence:0 → E ′ ( M ) ( Q )[ φ ′ ( M ) ] /φ ( M ) ( E ( M ) ( Q )[2]) → Sel φ ( M ) ( E ( M ) ) → Sel ( E ( M ) ) → Sel φ ′ ( M ) ( E ′ ( M ) ) , Noting Sel ( E ( M ) ) = E ( M ) ( Q )[2] = Z / Z , we have X ( E ( M ) )[2] = 0. (cid:3) Lemma 4.6.
The prime p is inert in Q ( √ ∆ E ) , and split in Q ( √ ∆ E ′ ) .Proof. Since (
E, K = Q ( √− p )) satisfies the Heegner hypothesis, for all prime ℓ | N , we have (cid:18) ℓp (cid:19) = (cid:18) − pℓ (cid:19) = +1 . By Proposition 2.5, ∆ E <
0. Then, noting, modulo squares, N and ∆ E have the same primefactors, it follows that (cid:18) ∆ E p (cid:19) = (cid:18) − · | ∆ E | p (cid:19) = (cid:18) − p (cid:19) = − . Hence p is inert in Q ( √ ∆ E ). Similarly, the second assertion is also true since ∆ E ′ > (cid:3) The following result is due to Cassels [4]. emma 4.7. We have | Sel φ ( E ) || Sel φ ′ ( E ′ ) | = Y ℓ | κ ℓ,φ ( E ) | . Proposition 4.8.
Let E , p and M be as in Theorem 1.2. If all the prime factors of N splitin Q ( √ M ) and p ≡ − , then Sel ( E ( − pM ) ) = ( Z / Z ) and X ( E ( − pM ) )[2] = 0 .Proof. The proof is similar to that of Proposition 4.5. Since (
E, K = Q ( √− p )) satisfies theHeegner hypothesis, we have (cid:0) − pℓ (cid:1) = +1 for all ℓ | N . Moreover, since p ≡ − (cid:0) − p (cid:1) = +1. Then we have (cid:16) − pMℓ (cid:17) = +1 for all ℓ | N , i.e., any prime ℓ | N is split in Q ( √− pM ). Then for any place ℓ = p , as in Lemma 4.3, we have κ ℓ,φ ( E ) = κ ℓ,φ ( M ) ( E ( M ) ) and κ ℓ,φ ′ ( E ′ ) = κ ℓ,φ ′ ( M ) ( E ′ ( M ) ) . It suffices to compare the local Kummer conditions at p . By Lemma 4.6, p is inert in Q ( √ ∆ E )and splits in Q ( √ ∆ E ′ ). Then E [2]( Q p ) = E [2]( Q ) and E ′ [2]( Q p ) = E ′ [2] . By [14, Lemma 6.7], κ p,φ ( − pM ) ( E ( − pM ) ) (respectively, κ p,φ ′ ( − pM ) ( E ′ ( − pM ) )) has dimension 2 (re-spectively, 0) over Z / Z , i.e., κ p,φ ( − pM ) ( E ( − pM ) ) = H ( Q p , E [ φ ]) and κ p,φ ′ ( − pM ) ( E ′ ( − pM ) ) = 0 . On the other hand, since p is good for E and E ′ and p is prime to the degrees of φ and φ ′ , wehave κ p,φ ( E ) = H ( Q p , E [ φ ]) and κ p,φ ′ ( E ′ ) = H ( Q p , E ′ [ φ ′ ]) . By comparing the local Kummer conditions for φ ′ and φ ′ ( − pM ) , we seeSel φ ′ ( − pM ) ( E ′ ( − pM ) ) ⊂ Sel φ ′ ( E ′ ) = Z / Z . From Z / Z = E ( − pM ) ( Q ) tor /φ ′ ( − pM ) ( E ′ ( − pM ) ( Q ) tor ) ⊂ Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) , we conclude Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) = Sel φ ′ ( E ′ ) = Z / Z . By Lemma 4.7 and Proposition 2.9, we have | Sel φ ( − pM ) ( E ( − pM ) ) || Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) | = Y ℓ | κ ℓ,φ ( − pM ) ( E ( − pM ) ) | . and | Sel φ ( E ) || Sel φ ′ ( E ′ ) | = Y ℓ | κ ℓ,φ ( E ) | . Since κ ℓ,φ ( − pM ) ( E ( − pM ) ) = κ ℓ,φ ( E ) for ℓ = p , it follows that | Sel φ ( − pM ) ( E ( − pM ) ) || Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) | = Y ℓ | κ ℓ,φ ( − pM ) ( E ( − pM ) ) || κ ℓ,φ ( E ) | = | κ p,φ ( − pM ) ( E ( − pM ) ) || κ p,φ ( E ) | = 2 . Thus Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) = Z / Z and Sel φ ( − pM ) ( E ( − pM ) ) ≃ ( Z / Z ) . Note E ( − pM ) ( Q ) / E ( − pM ) ( Q ) = ( Z / Z ) . Considering the exact sequence0 → E ′ ( − pM ) ( Q )[ φ ′ ( − pM ) ] /φ ( − pM ) ( E ( − pM ) ( Q )[2]) → Sel φ ( − pM ) ( E ( − pM ) ) → Sel ( E ( − pM ) ) → Sel φ ′ ( − pM ) ( E ′ ( − pM ) ) , we concludeSel ( E ( − pM ) ) = E ( − pM ) ( Q ) / E ( − pM ) ( Q ) = ( Z / Z ) and X ( E ( − pM ) )[2] = 0 , s desired. (cid:3) Proposition 4.9.
Let E , p and M be as in Theorem 1.2 and suppose that all the prime factorsof N split in Q ( √ M ) and p ≡ − . We have ord | X ( E ( M ) ) || E ( M ) ( Q ) tor | · Y ℓ c ℓ ( E ( M ) ) ! = r − , and ord | X ( E ( − pM ) ) || E ( − pM ) ( Q ) tor | · Y ℓ c ℓ ( E ( − pM ) ) ! = r. Proof.
Note that E ( M ) ( Q )[2 ∞ ] = E ( − pM ) ( Q )[2 ∞ ] = E ( Q )[2 ∞ ] = Z / Z , and by Proposition 4.5 and 4.8 we have X ( E ( M ) )[2 ∞ ] = X ( E ( − pM ) )[2 ∞ ] = X ( E )[2 ∞ ] = 0 . Since, by assumption, any prime q | M and, by Lemma 4.6, the prime p are inert in Q ( √ ∆ E ),it follows from Proposition 4.1 that c q ( E ( M ) ) = c q ( E ( − pM ) ) = 2 and c p ( E ( − pM ) ) = 2 . Then the assertion of the proposition follows from Proposition 4.2. (cid:3)
The -part of the Birch and Swinnerton-Dyer conjecture. Recall that, for anyelliptic curve defined over Q , the Birch and Swinnerton-Dyer conjecture predicts that(4.3) L ( r an ) ( E, r an !Ω E R ( E ) = Q ℓ c ℓ ( E ) · | X ( E ) || E ( Q ) tor | , where r an := ord s =1 L ( E, s ), and R ( E ) is the regulator formed with the N´eron–Tate pairing. Itis known that R ( E ) = 1 when r an = 0, and R ( E ) = b h Q ( P ) when r an = 1, where P is a freegenerator of E ( Q ), and b h Q ( P ) is the N´eron–Tate height on E over Q . At present, the finitenessof X ( E ) is only known when r an is at most 1, in which case it is also known that r an is equalto the rank of E ( Q ). Suppose the 2-part of (4.3) holds for E , we shall show that the 2-part of(4.3) holds for both E ( M ) and E ( − pM ) under mild assumptions. Theorem 4.10.
Let E , p and M be as in Theorem 1.2 and suppose that all the prime factorsof N split in Q ( √ M ) , p ≡ − and E has odd Manin constant. Then we have ord s =1 L ( E ( M ) , s ) = rank E ( M ) ( Q ) = 0 , and ord s =1 L ( E ( − pM ) , s ) = rank E ( − pM ) ( Q ) = 1;ord ( L ( E ( M ) , / Ω E ( M ) ) = r − , and ord ( L ′ ( E ( − pM ) , / Ω E ( − pM ) R ( E ( − pM ) )) = r. Moreover, the Shafarevich–Tate groups X ( E ( M ) ) and X ( E ( − pM ) ) are both finite of odd cardi-nalities. If the -part of the Birch and Swinnerton-Dyer conjecture holds for E , then the -partof the Birch and Swinnerton-Dyer conjecture holds for both E ( M ) and E ( − pM ) .Proof. The rank part of the Birch and Swinnerton-Dyer conjecture for the two elliptic curves E ( M ) and E ( − pM ) has been established in Theorem 3.9. In particular, by the work of Gross–Zagier and Kolyvain, the Shafarevich–Tate groups X ( E ( M ) ) , X ( E ( − pM ) ) are finite. Moreover,the cardinalities of both X ( E ( M ) ) and X ( E ( − pM ) ) are odd by Proposition 4.5 and Proposition4.8.First consider the quadratic twists E ( M ) . The full Birch and Swinnerton-Dyer conjecturepredictsBSD(E (M) ) L ( E ( M ) , ( M ) = ? | X ( E ( M ) ) || E ( M ) ( Q ) tor | · Y ℓ c ℓ ( E ( M ) ) . y Proposition 4.9 and [33, Theorem 5.2], both sides of BSD(E (M) ) have 2-adic valuation r − E ( M ) follows.Next we consider the quadratic twists E ( − pM ) . The full Birch and Swinnerton-Dyer conjecturepredictsBSD(E ( − pM) ) L ′ ( E ( − pM ) , ( − pM ) = ? b h Q ( P ) | X ( E ( − pM ) ) || E ( − pM ) ( Q ) tor | · Y ℓ c ℓ ( E ( − pM ) ) , where P is a free generator of E ( − pM ) ( Q ). By the explicit height formula in Corollary 3.11,BSD(E (M) ) and BSD(E ( − pM) ) amount to(4.4)ord b h Q ( z M ) b h Q ( P ) ! = ? ord − · | X ( E ( M ) ) || X ( E ( − pM ) ) || E ( M ) ( Q ) tor | | E ( − pM ) ( Q ) tor | · Y ℓ (cid:16) c ℓ ( E ( M ) ) c ℓ ( E ( − pM ) ) (cid:17)! . By Proposition 4.9, the RHS is 2( r − z M in Theorem 3.7. Since the 2-part of BSD(E (M) ) isproved, the 2-part of BSD(E ( − pM) ) follows. (cid:3) Applications
In this final section, we shall illustrate our general results for the family of quadratic twistsboth of the Neumann–Setzer elliptic curves, and also some elliptic curves of small conductor.5.1.
The Neumann–Setzer elliptic curves.
Recall that the Neumann–Setzer elliptic curveshave prime conductor p (see [18], [19] and [24]), where p is any prime of the form p = u + 64for some integer u ≡ p = 73 , , . . . . Then, up to isomorphism, it isknown that there are just two elliptic curves of conductor p with a rational 2-division point,namely, A : y + xy = x + u − x + 4 x + u, (5.1) A ′ : y + xy = x − u − x − x. (5.2)The curves A and A ′ are 2-isogenous, and both have Mordell–Weil groups Z / Z . A simplecomputation shows that ∆ A = − p and ∆ A ′ = p , it follows that Q ( A [2]) = Q ( i ) , Q ( A ′ [2]) = Q ( √ p ) . Let X ( p ) be the modular curve of level p , and there is a non-constant rational map X ( p ) → A, making the modular parametrization Γ ( p )-optimal by Mestre and Oesterl´e [17]. Since theconductor of A is odd, the Manin constant of A is odd by the work of Abbes and Ullmo [1]. Inparticular, we have the following result on applying our main theorem. Theorem 5.1.
Assume that p is a prime of the form u + 64 with u ≡ , and let A be the Neumann–Setzer curve (5.1) . Let q , · · · , q r be distinct primes congruent to modulo which are inert in Q ( √ p ) . Let p ≡ be a prime greater than which splits in Q ( √ p ) .Denote M = q ∗ · · · q ∗ r . Then we have ord s =1 L ( A ( M ) , s ) = rank A ( M ) ( Q ) = 0 and ord s =1 L ( A ( − pM ) , s ) = rank A ( − pM ) ( Q ) = 1 . In particular, X ( A ( M ) ) and X ( A ( − pM ) ) are both finite of odd cardinality. Moreover, the -partof the Birch and Swinnerton-Dyer conjecture is valid for both A ( M ) and A ( − pM ) . roof. The primes q i , 1 ≤ i ≤ r , are admissible for A , since they are inert in both Q ( A [2])and Q ( A ′ [2]). The condition on p implies that p splits in K = Q ( √− p ). Hence, the Heegnerhypothesis for ( A, K ) holds. When u ≡ f ([0]) is precisely the non-trivial torsion point of order 2. Thus, allthe assumptions in Theorem 1.2 are satisfied.Moreover, the Manin constant of A is odd. The 2-part of the Birch and Swinnerton-Dyerconjecture of A is verified in [32, Proposition 5.13]. However, here we will not apply Theorem1.4, since a classical 2-descent has been carried out in [32, Section 5], which shows that underthe assumptions in Theorem 5.1, we have X ( A ( M ) )[2] = X ( A ( − pM ) )[2] = 0 . By Proposition 4.1 and Proposition 4.2, we haveord Y ℓ c ℓ ( A ( M ) ) ! = r + 1 and ord Y ℓ c ℓ ( A ( − pM ) ) ! = r + 2 . Note that, by [33, Theorem 5.2], we haveord ( L ( A ( M ) , / Ω A ( M ) ) = r − . Then combining with equation (4.4), it follows that the 2-part of the Birch and Swinnerton-Dyerconjecture is valid for both A ( M ) and A ( − pM ) . (cid:3) Note that for u ≡ ( L alg ( A, −
1. Forexample, we can take p = 73. Here is the beginning of an infinite set of primes which arecongruent to 3 modulo 4 and inert in Q ( √ S = { , , , , , , , , , , , , , , , , , . . . } . For more examples, there is a nice table presenting primes of the form u + 64 in [24].5.2. More numerical examples.
The theorem can be applied on the family of quadratictwists of many elliptic curves E/ Q , we include a table here when the conductor of E is less than100. Table. E/ Q satisfying f ([0]) E ( Q ) and Condition (Tor) with conductor N < E/ Q satisfying f ([0]) E ( Q ) and Condition (Tor). E Q ( E [2]) Q ( E ′ [2]) admissible primes q p a Q ( √− Q ( √
2) 3 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
5) 3 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
3) 5 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
2) 5 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
7) 5 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
13) 7 , , , , , , , , . . . , , , , , , . . . b Q ( √− Q ( √
2) 3 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
3) 5 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
73) 7 , , , , , , , , . . . , , , , , , . . . c Q ( √− Q ( √
11) 3 , , , , , , , , . . . , , , , , , . . . b Q ( √− Q ( √
5) 3 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
7) 5 , , , , , , , , . . . , , , , , , . . . b Q ( √− Q ( √
7) 5 , , , , , , , , . . . , , , , , , . . . b Q ( √− Q ( √
89) 3 , , , , , , , , . . . , , , , , , . . . a Q ( √− Q ( √
2) 5 , , , , , , , , . . . , , , , , , . . . .3. Examples of the full Birch and Swinnerton-Dyer conjecture.
Let A be the ellipticcurve “69 a
1” with the minimal Weierstrass equation given by A : y + xy + y = x − x − . We have a = 1, a = 1 and a = −
1. Moreover, A ( Q ) = Z / Z and L ( alg ) ( A,
1) = 1 /
2. Thediscriminant of A is − ·
23. The Tamagawa factors c = 2, c = 1. Also, a simple computationshows that Q ( A [2]) = Q ( √−
23) and Q ( A ′ [2]) = Q ( √ q with good ordinary reduction which are inert in both the fields Q ( √−
23) and Q ( √ S = { , , , , , , , , , , , , , , . . . } . Let M = q ∗ · · · q ∗ r be a product of r distinct primes in S . By Theorem 4.10, we have L ( A ( M ) , = 0 , and ord ( L alg ( A ( M ) , r − . If we carry out a classical 2-descent on A ( M ) , one shows easily that the 2-primary componentof X ( A ( M ) / Q ) is zero and ord ( c q i ) = 1 for 1 ≤ i ≤ r , and therefore the 2-part of the Birchand Swinnerton-Dyer conjecture holds for A ( M ) . Alternatively, we can just apply Theorem 1.4,take M ≡ Q ( √ M ) will hold, whencewe can also verify the 2-part of the Birch and Swinnerton-Dyer conjecture. For the full Birchand Swinnerton-Dyer conjecture, in order to apply Theorem 9.3 in Wan’s celebrated paper [30],we need to check the conditions in the theorem. To verify the third one, since A has non-splitmultiplicative reduction at 23, we could consider A (5) , which has split multiplicative reductionat 23, and the Tamagawa number is 1 at 23, hence the A [ p ] | G q ( q = 3 or 23) is a ramifiedrepresentation for any odd prime p . Other conditions are easy to verify, so the full Birchand Swinnerton-Dyer conjecture is valid for A ( M ) . Hence the full Birch and Swinnerton-Dyerconjecture is verified for infinitely many elliptic curves. The full Birch and Swinnerton-Dyerconjecture of rank 1 twists are also accessible in the future work. More examples are given inWan’s paper. Acknowledgements.
We would like to thank John Coates for encouragement, useful discus-sions and polishings on the manuscript, thank Ye Tian and Xin Wan for helpful advice andcomments, and thank Yongxiong Li for helpful comments and carefully reading the manuscript.We also thank the referee for helpful advice.
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Email address : [email protected] Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cam-bridge CB3 0WB, UK.