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J. reine angew. Math. 775 (2021), 117–143 Journal für die reine und angewandte Mathematik DOI 10.1515/crelle-2021-0004 © De Gruyter 2021

Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves

By Jie Shu at Shanghai and Shuai Zhai at Cambridge

Abstract. In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over Q. We prove the existence of explicit inﬁnite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of inﬁnite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

1. Introduction

Let E be an elliptic curve defined over Q, and let N be the conductor of E. Then E is modular by the theorem of Wiles et al. [3, 31], and we let f X0.N / E be an optimal W ! modular parametrization sending the cusp at infinity, which is denoted by Œ , to the zero 1 1 element of E. We write Œ0 for the cusp of X0.N / arising from the zero point in P .Q/ under the complex uniformization of X0.N /. By the theorem of Manin–Drinfeld, f .Œ0/ is a torsion point in E.Q/. For any square-free integer D 1, we write E.D/ for the twist of E by the ¤ extension Q.pD/=Q, and L.E.D/; s/ for its complex L-series. We assume throughout this paper that the group EŒ2.Q/ of rational 2-division points on E is cyclic of order 2, and we define the elliptic curve E =Q to be the quotient curve E E=EŒ2.Q/. In all that follows, 0 0 D we will make the following assumption:

(Tor) EŒ2.Q/ E Œ2.Q/ Z=2Z: D 0 D

Thus Q.EŒ2/ and Q.E0Œ2/ are quadratic extensions of Q. We also remark that it is proven in Proposition 2.3 that, for any elliptic curve E over Q with f .Œ0/ 2E.Q/, condition (Tor) 62

The corresponding author is Jie Shu. Jie Shu is supported by the National Natural Science Foundation of China (Grant No. 11701092). 118 Shu and Zhai, Generalized Birch lemma holds for E if and only if there exists an odd prime q of good reduction for E such that Â 1Ã (1.1) aq 1 mod 4; Á q where aq is the trace of Frobenius on the reduction of E modulo q. We will use the following explicit set of primes, which, by Chebotarev’s theorem, has positive density in the set of all primes.

Deﬁnition 1.1. A prime q is admissible for E if .q; 2N / 1 and q is inert in both of D the quadratic ﬁelds Q.EŒ2/ and Q.E0Œ2/.

Note also that, assuming that our elliptic curve E satisfies condition (Tor), we prove in Proposition 2.2 that an odd prime q of good reduction is admissible for E in the above sense if and only if it satisfies (1.1). In what follows, p will always denote a prime > 3 such that p 3 mod 4, and we Á will write K Q.p p/. We say that p satisfies the Heegner hypothesis for .E; K/ if every D prime ` dividing N splits in the field K. For any odd prime q, put q . 1 /q, where . / D q denotes the Legendre symbol. 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 .Œ0/ 2E.Q/ and condi- 62 tion (Tor). Let p > 3 be any prime 3 mod 4 satisfying the Heegner hypothesis for .E; K/, Á where K Q.p p/. For any integer r 0, let q1; : : : ; qr be distinct admissible primes which D are not equal to p, and put M q q q . Then we have D 1 2 r .M / .M / ords 1L.E ; s/ rank E .Q/ 0; D D D . pM / . pM / ords 1L.E ; s/ rank E .Q/ 1: D D D .M / . pM / Moreover, the Shafarevich–Tate groups X.E / and X.E / 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 .Œ0/ 2E.Q/, and there 62 exists an odd prime q of good reduction for E satisfying (1.1). Then, for any integer r 1, there exist inﬁnitely many square-free integers M and M 0, having exactly r prime factors, such that L.E.M /; s/ does not vanish at s 1, and L.E.M 0/; s/ has a zero of order 1 at s 1. D D Theorem 1.2 and Corollary 1.3 improve the results in [8, Theorem 1.1] and [4, Theo- rem 1.1], which were motivated by the remarkable progress on the congruent number problem made by Tian [29]. Note, however, that, in these two papers, much stronger conditions are im- posed on the prime factors of M and M 0. In particular, the generalizations of the Birch lemma in [8, Corollary 2.6] and [4, Theorem 1.1] require that the prime factors q of M and M 0 satisfy r 1 q 1 mod 4 and aq 0 mod 2 , where, as above, r denotes the number of prime factors Á Á C of M or M 0. Theorem 1.2 and Corollary 1.3, can be applied to many elliptic curves, espe- cially to elliptic 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 admissible primes q 3 mod 4 dividing M is odd, whence pM will be positive. In Á some cases, this enables us to use Heegner points to construct rational points of inﬁnite order Shu and Zhai, Generalized Birch lemma 119 on real quadratic twists 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).

Theorem 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 ` dividing 2N splits in both K Q.p p/ and Q.pM/. D Then, if the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E, it also holds for .M / . pM / both E and E .

In Section 5.3, we give infinitely many examples of elliptic curves over Q, without complex multiplication, which satisfy the full Birch and Swinnerton-Dyer conjecture, by combining this result with a recent theorem of Wan [30] on the p-part of the Birch and Swinnerton-Dyer conjecture 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 root number 1, there is a subset of density one where the central L-value of the twist is non-zero, C and amongst the set of all its quadratic twists with root number 1, there is a subset of density one where the central L-value of the twist has a zero of order equal to 1. For an elliptic curve E over Q, define .D/ Nr .E; X/ # square free D Z D X; ords 1L.E ; s/ r ; D ¹ 2 W j j Ä D D º where r 0; 1. We prove the following result. D Theorem 1.5. Let E be an elliptic curve over Q satisfying f .Œ0/ 2E.Q/, and con- 62 dition (Tor). Then, as X , we have ! 1 X Nr .E; X/ : log3=4 X

Previously, for any given elliptic curve E over Q, Ono and Skinner [21] showed that X N0.E; X/ ; log X and later Ono [20] proved that, in particular for E without a rational 2-torsion point, X N0.E; X/ 1 ˛ .log X/ 1 " for some 0 < ˛ < 1. Perelli and Pomykała [23] proved that N1.E; X/ X for every " > 0. Much progress for various families of elliptic curves has been made towards Goldfeld’s conjecture. For example, see the early survey article by Silverberg [25], and very recent results in [15,28]. However, Theorem 1.5 improves previous results when E satisﬁes f .Œ0/ 2E.Q/ 62 and condition (Tor). The proof of Theorem 1.5 is a straightforward consequence of Theo- rem 1.2. Indeed, we just count the number of integers M appearing in Theorem 1.2, say X S0.X/ 1; WD M X j jÄ 120 Shu and Zhai, Generalized Birch lemma where M q q as in Theorem 1.2. In view of Proposition 2.2, the Chebotarev density D 1 r theorem shows that the admissible primes have natural density 1=4 in the set of all primes. Applying the Ikehara Tauberian theorem, it follows that c X c X S .X/ 0 0 ; 0 1 1=4 3=4 .log X/ .log X/ where c0 is a constant. Then Theorem 1.5 follows immediately since Nr .E; X/ S0.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 .Œ0/ 2E.Q/ and condition (Tor). In this section, we establish some of the 62 basic arithmetic properties of these curves.

2.1. Condition (Tor) and admissible primes. Under condition (Tor), the quotient map E E is an isogeny over Q of degree 2. Let E E be its dual isogeny. W ! 0 0 W 0 ! Proposition 2.1. Under condition (Tor), we have EŒ2 .Q/ EŒ2.Q/ Z=2Z. 1 D D

Proof. Suppose on the contrary that there is a point P1 in E.Q/ of exact order 4, it follows that the image of P1 in E0, namely .P1/, is rational by the deﬁnition of 2-isogeny. Since 2P1 generates EŒ2.Q/, we have .2P1/ 0, but P1 does not lie in the kernel of , D so .P1/ is of order 2, and .P1/ 2P1. Let P2 EŒ2 EŒ2.Q/ be a point of order 2. 0 ı D 2 n Then .P2/ must be rational of order 2 since .P2/ 0. Thus, the images of P1 and P2 0 ı D in E E=EŒ2.Q/ are different rational points of order 2 and they generate E Œ2, that is, 0 D 0 E Œ2.Q/ E Œ2 .Z=2Z/2, which does not satisfy condition (Tor). 0 D 0 D

Note Q.EŒ2/ Q.pE / and Q.E Œ2/ Q.pE / are quadratic extensions of Q, D 0 D 0 where and are the discriminants of E and E , respectively. For any prime q of good E E 0 0 reduction for E, let aq be the trace of the Frobenius on the reduction of E mod q.

Proposition 2.2. Let q be a prime with .q; 2N / 1. Under condition (Tor), we have D 8 ˆ0 mod 4 if q 1 mod 4 and q is inert in both Q.pE / and Q.pE /, ˆ Á 0 <ˆ0 mod 4 if q 3 mod 4 and q splits in Q.p / or Q.p /, E E 0 aq Á Á 2 mod 4 if q 1 mod 4 and q splits in Q.pE / or Q.pE /, ˆ 0 ˆ Á :2 mod 4 if q 3 mod 4 and q is inert in both Q.pE / and Q.pE /. Á 0 In particular, q is inert in both Q.p / and Q.p /, i.e., admissible for E if and only if E E 0 Â 1Ã (2.1) aq 1 mod 4: Á q

Proof. The elliptic curve E has good reduction at q. Now reduction modulo q gives the exact sequence 0 E.qb Zq/ E.Qq/ E.Fq/ 0; ! ! ! ! Shu and Zhai, Generalized Birch lemma 121 where E=b Zq denotes the formal group associated to E=Qq. Since q > 2, by [27, Chapter IV, Theorem 6.4], multiplication by 4 induces an isomorphism on E.qb Zq/, and hence we have EŒ4.Qq/ EŒ4.Fq/. If q splits in Q.pE / or Q.pE /, then either D 0 2 EŒ2.Fq/ EŒ2.Qq/ .Z=2Z/ D D or 2 E Œ2.Fq/ E Œ2.Qq/ .Z=2Z/ : 0 D 0 D Since E is isogenous to E , we have E.Fq/ E .Fq/ . In either case, 4 divides E.Fq/ . 0 j j D j 0 j j j If q is inert in both Q.p / and Q.p /, by [33, Lemma 2.1], we have E E 0

EŒ4.Fq/ EŒ4.Qq/ Z=2Z: D D In conclusion, we have ´ 2 mod 4 if q is inert in both Q.p / and Q.p /, E E 0 E.Fq/ j j Á 0 mod 4 if q splits in Q.p / or Q.p /. E E 0 Since q is an odd good prime for E, the assertion follows immediately from the injection E.Qq/Œ2 , E.Fq/ under reduction modulo q and the formula aq q 1 E.Fq/ . 1 ! D C j j Proposition 2.3. If f .Œ0/ 2E.Q/, then condition (Tor) is equivalent to the existence … of an odd good prime q with Â 1Ã aq 1 mod 4: Á q

Proof. Under condition (Tor), by Proposition 2.2, admissible primes are exactly those primes q − 2N satisfying (2.1). Suppose q − 2N satisﬁes (2.1). Then

E.Fq/ q 1 aq 2 mod 4; j j D C Á which implies that both EŒ2 .Q/ and E Œ2 .Q/ are less than or equal to 2 for q − 2N . j 1 j j 0 1 j Since f .Œ0/ 2E.Q/, we have EŒ2 .Q/ 0 and hence EŒ2 .Q/ Z=2Z. Then the quo- … 1 ¤ 1 D tient map E E is an isogeny over Q of degree 2, and we write E E for its dual W ! 0 0 W 0 ! isogeny. Then the kernel of 0 is rational over Q, and hence contains a rational point of order 2. Therefore E Œ2 .Q/ Z=2Z, and hence condition (Tor) holds. 0 1 D 2.2. Equivalent condition for the 2-part of the Birch and Swinnerton-Dyer conjecture. In the rest of this section, we assume that the Manin constant cE of E is always odd. Let C (respectively, ) be the minimal real (respectively, imaginary) period of E.

Proposition 2.4. For the elliptic curve E we have L.E; 1/ 0, and ¤ ÂL.E; 1/Ã ord2 1: C D

Proof. Let f X0.N / E be an optimal modular parametrization. We embed X0.N / W ! into its Jacobian J0.N / via the base point Œ . Then there exists a unique homomorphism 1 fe J0.N / E through which f factors. We consider the complex uniformization of fe. W ! 122 Shu and Zhai, Generalized Birch lemma

By the Abel–Jacobi theorem, we have 1 1 J0.N / H .X0.N /; R/=H1.X0.N /; Z/ and Jac.E/ H .E; R/=H1.E; Z/: D D Then f induces a homomorphism 1 J0.N / Jac.E/; f . /; H .X0.N /; R/: ! 7! 2 The elliptic curve E has a complex uniformization E C=L where L C is the period lattice D of E, and we have an isomorphism Z Jac.E/ E; ˛ !E ; ' 7! ˛ where !E is the Néron differential of E. Then the complex uniformization of fe is explicitly given as Z (2.2) fe J0.N / E; !E : W ! 7! f . /

In particular, we take Œ0 i to be the path on X0.N / joining Œ0 and Œ i: Let g be D ! 1 1 the newform associated to E. Since f !E cE 2ig.z/ dz, it follows that D Z Z 1 L.E; 1/ 2i g.z/dz cE !E : D D f . / Then from the complex uniformization (2.2), we see that

(2.3) f .Œ0/ cE L.E; 1/ mod L: D Let H1.E; Z/C be the submodule of H1.E; Z/ that is invariant under the complex conjugation. Then H1.E; Z/C is free of rank 1 and ˇZ ˇ ˇ ˇ C ˇ !E ˇ; D ˇ ˇ C where C is a generator of the submodule H1.E; Z/C. Since is a real path on X0.N /, it follows that f . / H1.E; R/ H1.E; Z/ Z R. Let C be a sufﬁciently large odd integer 2 C D C ˝ such that Cf .Œ0/ has order 2. Then by (2.3) we have L.E; 1/ L.E; 1/ CcE Z and 2CcE Z: C 62 C 2 Hence ÂL.E; 1/Ã ord2 1; C D as desired.

Denote Z !E D E.R/ j j the real period of E. Then or 2 according to that E.R/ is connected or not. The D C C period is the one appearing in the exact formula which is part of the Birch and Swinnerton- Dyer conjecture and we deﬁne L.E; 1/ Lalg.E; 1/ : D Shu and Zhai, Generalized Birch lemma 123

Proposition 2.5. The discriminant E < 0. In particular, we have alg ord2.L .E; 1// 1: D

Proof. Suppose, on the contrary, that E > 0. Then the period lattice is Z Z . C C Let q be an admissible prime. By Manin’s formula [16, Theorem 3.3], we have

q 1 X ˝® k ¯ ˛ E.Fq/ L.E; 1/ cE 0; ; g ; j j D q k 1 D where ; denotes the modular symbol, and 0; k ; g s t , with s ; t Z. ¹ º h¹ q º i D k C C k k k 2 Since ˝®0; k ¯; g˛ ˝®0; 1 k ¯; g˛; q D q we have q 1 2 X E.Fq/ L.E; 1/ 2 cE s : j j D C k k 1 D Then by Proposition 2.4, we have ord2. E.Fq/ / 2. Since q is admissible, that is, j j Â 1Ã aq 1 mod 4; D q it follows that E.Fq/ q 1 aq has 2-adic valuation 1, and we get a contradiction. Thus j j D C E < 0, whence E.R/ is connected and , and D C L.E; 1/ Lalg.E; 1/ D C has 2-adic valuation 1. Corollary 2.6. We have

rank E.Q/ ords 1L.E; s/ 0: D D D The 2-part of the Birch and Swinnerton-Dyer conjecture holds for the elliptic curve E if and only if Â X.E/ Y Ã ord2 j j2 c`.E/ 1: E.Q/tor D j j ` Proof. By Proposition 2.4, L.E; 1/ 0 and by the work of Gross and Zagier [12] and ¤ Kolyvagin [14], E.Q/ has rank zero and X.E/ is ﬁnite. The statement for the 2-part of the j j Birch and Swinnerton-Dyer conjecture for E follows immediately from Proposition 2.5.

2.3. Discriminants and periods.

Proposition 2.7. We have < 0 and > 0. E E 0

Proof. Moving the rational 2-torsion point on E to .0; 0/, we can ﬁnd a Weierstrass equation for E=Q of the form (2.4) Y 2 X 3 aX 2 bX; D C C 124 Shu and Zhai, Generalized Birch lemma where a; b Z. Dividing this curve by the subgroup generated by the point .0; 0/, we obtain a 2 Weierstrass equation for E0=Q of the form (2.5) y2 x3 2ax2 .a2 4b/x: D C Note that the discriminants of various Weierstrass equations of elliptic curves over Q are the same up to multiplication by non-zero rational twelfth powers. It is easy to calculate that the discriminant of equation (2.4) is 24b2.a2 4b/. It is proved in Proposition 2.5 that E < 0, and hence b > 0. A simple computation shows that the discriminant of equation (2.5) 8 2 2 is 2 b.a 4b/ , which is positive since b > 0. Therefore, E > 0. 0

Lemma 2.8. Let and 0 denote the least positive real periods of E and E0, respectively. Then 1 1 or : 0 D 2 Proof. In view of Proposition 2.7, and by [9, Theorem 1.2], we have ˇ ˇ ˇ ! ˇ ˇ ˇ; 0 D ˇ!0 ˇ where ! and !0 are the minimal differentials (unique up to signs) on E and E0, respectively. Suppose ! ˛! and ! ˇ! with ˛; ˇ Z. Since 0 D 0 D 0 2 ˛ˇ! . !/ Œ2 ! 2!; D 0 D D we see ˛ 1 or 2 and the lemma follows. D 2.4. Selmer groups. Recall that E E is the isogeny of degree 2 defined over Q W ! 0 with kernel EŒ EŒ2.Q/, and E E its dual isogeny. For any finite or infinite D 0 W 0 ! place v of Q, we write Qv for the completion of Q at v. Let v;2.E/ be the image of the Kummer map 1 v;2 E.Qv/=2E.Qv/, H .Qv; EŒ2/: W ! Similarly, we write

1 v;.E/ Im.E .Qv/=.E.Qv// , H .Qv; EŒ//; WD 0 ! 1 v; .E0/ Im.E.Qv/=0.E0.Qv// , H .Qv;E0Œ0//: 0 WD ! The 2-Selmer group over Q is then deﬁned by Â Ã 1 M 1 Sel2.E/ Ker H .Q; EŒ2/ H .Qv; EŒ2/=v;2.E/ : WD ! v The Shafarevich–Tate group X.E/ is deﬁned by Â Ã 1 M 1 X.E/ Ker H .Q;E/ H .Qv;E/ : WD ! v Then we have the following well-known exact sequence:

0 E.Q/=2E.Q/ Sel2.E/ X.E/Œ2 0: ! ! ! ! Shu and Zhai, Generalized Birch lemma 125

Similarly, we deﬁne the -Selmer group and 0-Selmer groups over Q by Â Ã 1 M 1 Sel.E/ Ker H .Q; EŒ/ H .Qv; EŒ/=v;.E/ ; WD ! v Â M Ã Sel .E / Ker H1.Q;E Œ / H1.Q ;E Œ /= .E / : 0 0 0 0 v 0 0 v;0 0 WD ! v

Proposition 2.9. Assume that the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E. Then we have

Sel2.E/ Sel2.E0/ Z=2Z; Sel.E/ Sel .E0/ Z=2Z: Š Š Š 0 Š Proof. By Proposition 2.4, we have L.E; 1/ 0. By the work of Gross and Zagier [12] ¤ and Kolyvagin [14], E.Q/ has rank zero and X.E/ is ﬁnite, and hence a square. By Corol- j j lary 2.6, we have Â X.E/ Y Ã ord2 j j2 c`.E/ 1; E.Q/tor D j j ` we then must have X.E/Œ2 0, and hence Sel2.E/ EŒ2.Q/ Z=2Z. By the invariance D D D of the Birch and Swinnerton-Dyer conjecture under isogeny [6, (Corollary to) Theorem 1.3], we have that E .Q/ 0 also has rank 0, and 0 D Â Ã Â Ã Â Ã X.E0/ Y X.E/ Y (2.6) ord2 j j2 c`.E0/ ord2 j j2 c`.E/ ord2 E .Q/tor D E.Q/tor C j 0 j ` j j ` 0 Â Ã 1 ord2 1 or 2: D C 0 D

The last equality follows from Lemma 2.8. In view of (2.6), we have ord2. X.E / / 1. Since j 0 j Ä the order of X.E / is a square, we conclude that X.E /Œ2 0, and hence 0 0 D Sel2.E / E Œ2.Q/ Z=2Z: 0 D 0 D For the second assertion, consider the following well-known exact sequence:

0 E .Q/Œ =.E.Q/Œ2/ Sel.E/ Sel2.E/ ! 0 0 ! ! Sel .E0/ X.E0/Œ0=.X.E/Œ2/ 0: ! 0 ! ! Since Sel2.E/ Sel2.E / Z=2Z, the non-trivial element in each Selmer group comes from D 0 D the rational 2-torsion point in E.Q/Œ2 and E0.Q/Œ2, respectively. Thus, we have X.E/Œ2 X.E /Œ2 0; D 0 D and hence the last term in the above exact sequence is zero. Since

E .Q/Œ =.E.Q/Œ2/ Z=2Z; 0 0 Š it follows from the above exact sequence and its dual form that

Sel.E/ Sel .E0/ Sel2.E/ Sel2.E0/ Z=2Z; Š 0 Š Š Š as desired. 126 Shu and Zhai, Generalized Birch lemma

3. Heegner points

Throughout this section, let E, p, and M be as in Theorem 1.2, and take K Q.p p/. . pM / D We will construct Heegner points lying in E .Q/. In the main result of this section (Theo- rem 3.7), we prove our Heegner points do indeed have inﬁnite order, and we also establish the exact 2-divisibility of these Heegner points. Then we show that Theorem 1.2 follows immediately from the non-triviality of Heegner points via the work of Gross–Zagier and Kolyvagin. The exact 2-divisibility of Heegner points is used to prove 2-part of the Birch and Swinnerton- Dyer conjecture for the related elliptic curves in the next section. The induction arguments on Heegner points given in this section strengthen those in the earlier papers [29], [8] and [4], 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 given in Theorem 3.2: t0 z z0 T; 0 C D where T is a torsion point. What is new in our approach is that, while the previous authors always make congruence restrictions on the prime factors of M to ensure the 2-primary part of 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 methods use norm relations on Heegner points in one way or another. We use the full strength of these norm relations of Heegner points in the induction process to make the required 2-adic valuation of the Hecke eigenvalues aq as small as possible.

3.1. Non-triviality of Heegner points. Let OK be the ring of integers of K. Since .E; K/ satisfies the Heegner hypothesis, there exists an ideal N OK such that OK =N Z=N Z: ' 2 For any integer c coprime to pN , let Oc OK be the order of discriminant pc . Define the Heegner points 1 yc .C=Oc C=.Oc N / / X0.N /.Hc/; D ! \ 2 where Hc denotes the ring class field over K of conductor c. For any abelian group G denote

Gb G Z bZ; WD ˝ Q ab where bZ Zp. Let K Kb Gal.K =K/ be the Artin reciprocity map. Let t0 Kb D p W n ! 2 be the idéle such that t0OK Nc. Let w be the Atkin–Lehner operator on X0.N /. We have D the following proposition from [11, Section 5]:

Proposition 3.1. The Heegner point yc satisfies w t 1 y yc 0 : b c D Let q1; : : : ; qr be distinct admissible primes for E, and set M q q . Let D 1 r q p Á H0 K q ;:::; q HM D 1 r be the genus field. We define the genus Heegner point

z0 tr f .yM / E.H0/: WD HM =H0 2 Shu and Zhai, Generalized Birch lemma 127

Theorem 3.2. The genus Heegner point z0 satisﬁes

t0 z z0 T; 0 C D where T ŒHM H0f .Œ0/ E.Q/. D W 2 Proof. By Proposition 3.1 we have X T f .Œ0/ D Gal.HM =H0/ 2 X w .f .y / f .yM // D M C Gal.HM =H0/ 2 X t 1 .f .yM 0 / f .yM // D C Gal.HM =H0/ 2 t 1 z0 0 z0: D C t The theorem follows by acting t on the above equality and noting that T 0 T . 0 D

Lemma 3.3. We have E.H0/Œ2 E.Q/Œ2: 1 D

Proof. Suppose the contrary and P E.H0/Œ2 E.Q/Œ2. Since E has good reduc- 2 1 n tion outside N , it follows that Q.P / is unramified outside 2N . Then we have Q.P / Q by p p D noting H0 Q.p p; q ;:::; q /, which is a contradiction. D 1 r Definition 3.4. An admissible prime q is of first kind if q 1 mod 4 is inert in K or if Á q 3 mod 4 splits in K; otherwise, q is of second kind. Á Corollary 3.5. If all prime factors of M are of first kind, then the two primary component T2 of T is of order 2 and z0 is of infinite order; otherwise T2 0. D Proof. Just note that the degree Y q 1 Y q 1 ŒHM H0 C : W D 2 2 q M q M inertj in K splitj in K

It follows that ŒHM H0 is odd if all prime factors of M are of first kind. By Proposition 2.1, W EŒ2 .Q/ EŒ2.Q/ and f .Œ0/ 2E.Q/, we see that T2 is of order two. Otherwise, T2 0. 1 D 62 D In the case T2 is of order 2, we suppose z0 is torsion, and choose sufficiently large odd C so that C z0 E.H0/Œ2 E.Q/Œ2 and CT T2. Then 2 1 D D t0 (3.1) T2 CT C z C z0 C z0 0; D D 0 C D D which is a contradiction. Hence in this case z0 is of infinite order.

We only consider divisors of M which are congruent to 1 mod 4 so that pd H0. 2 For any such d M , let Gal.H0=K/ 2 be the quadratic character associated to the j d W ! quadratic extension K.pd/=K through class field theory. Then the character d is explicitly 128 Shu and Zhai, Generalized Birch lemma given by ./ .pd/ 1. Define the twisted Heegner points d D X X (3.2) z ./f .yM / ./z E.K.pd// ; d D d D d 0 2 Gal.HM =K/ Gal.H0=K/ 2 2 where the superscript “ ” means the eigenspace that the generator of Gal.K.pd/=K/ acts as 1. Proposition 3.6. We have z z t0 0; d C d D and X r z 2 z0: d D d M d 1jmod 4 Á Proof. The proof of the first assertion is similar to that of Theorem 3.2. By Proposi- tion 3.1 we have

X X w ./f .Œ0/ ./.f .y / f .yM // d D d M C Gal.HM =K/ Gal.HM =K/ 2 2 X t 1 ./.f .yM 0 / f .yM // D d C Gal.HM =K/ 2 1 z t0 z : D d C d On the other hand, we have X ÂX Ã ./f .Œ0/ ./ f .Œ0/ 0: d D d D Gal.HM =K/ 2 The second assertion is standard and straight-forward.

Theorem 3.7. The Heegner point zM has exact 2-divisibility index r 1: zM lies in r 1 p p r p p 2 E.Q. pM // E.Q. pM //tor 2 E.Q. pM // E.Q. pM //tor ; C n C where the superscript “ ” means the eigenspace that the generator of Gal.Q.p pM /=Q/ acts as 1. Remark 3.1. If r 0, i.e., M 1, the statement in the theorem exactly means D D 2z1 E.Q.p p// but z1 E.Q.p p// E.Q.p p// : 2 62 C tor Proof. In the following, without loss of generality we may replace S f .Œ0/, if nec- D essary, by its two primary component S2 E.Q/Œ2 for simpliﬁcation. Indeed, we may do this 2 as in (3.1) by multiplying all points by a sufﬁciently large odd integer. Let r.M / denote the number of distinct prime factors of M . We proceed by induction on r r.M / to prove D r t0 (3.3) zM 2 xM for some xM E.H0/ with xM x S. D 2 C M D Then we deduce the assertion in the theorem from this property (3.3). Shu and Zhai, Generalized Birch lemma 129

First we consider the initial case r 0, i.e., M 1. Then H0 K and zM z0 and D D D D take xM zM E.K/. By Theorem 3.2 we have D 2 t (3.4) zM zM 0 S: C D We prove the theorem in this case M 1 as follows. First, note that 2zM 2zM and hence D D 2z1 E.Q.p p// . Suppose the contrary that zM y s with y E.Q.p p// and 2 D C 2 s E.Q.p p// . By Lemma 3.3, we can choose C a sufficiently large odd integer such that 2 tor C s E.Q/Œ2. Then from (3.4) we have 2 S CS C.zM zM / C.y y/ 2C s 2C s 0; D D C D C C D D which is a contradiction. Next assume r > 0. For any proper d M with d 1 mod 4, let d M=d and d j Á 0 D 0C (respectively, d 0) be the product of prime factors of d 0 that split (respectively, are inert) in K. By Euler system property [14], we have Â Y Y Ã z tr f .yM / .aq v v/ aq z ; d D HM =Hd D d0 q d q d j 0C j 0 where X z ./f .y /: d0 D d d Gal.Hd =K/ 2 Here v and v are the two places of K above q if q d , and v and v are the corresponding j 0C Frobenius automorphisms, respectively. Note that we have Âd Ã .v/ .v/ 1; d D d D q D ˙ and v.z / .v/z and v.z / .v/z : d0 D d d0 d0 D d d0 By condition (2.1) on Hecke eigenvalues, we know for all q M we have j Â 1Ã aq 1 mod 4: Á q For any proper divisor d M with d 1 mod 4, we put j Á Â Â Â ÃÃ Ã .d / Y d Y A 2 0 aq 2 aq : d D q q d q d j 0C j 0 Let M1 (respectively, M2) be the product of prime factors of M which are of first (respectively, second) kind. From condition (2.1) for Hecke eigenvalues and Definition 3.4, we see the following two facts:

(i) If some prime factor of d 0 is of ﬁrst kind, i.e., M1 − d, then Ad is an even integer.

(ii) If all prime factors of d are of second kind, i.e., M1 d, then A is an odd integer. 0 j d By the induction hypothesis, we have

.d/ .d/ z 2 x 2 E.H0/; d0 D d0 2 130 Shu and Zhai, Generalized Birch lemma

t with x x0 0 S. Hence d0 C d D r r z 2 A x 2 E.H0/: d D d d0 2 Then Â Ã r X X r zM 2 z0 A x A x 2 E.H0/: D d d0 d d0 2 M1−d M1 d d M d Mj ¤ ¤ Denote X X xM z0 A x A x E.H0/: D d d0 d d0 2 M1−d M1 d d M d Mj ¤ ¤ If M1 − d, then Ad is even and A .x x t0 / 0: d d0 C d0 D If M1 d, then Ad is odd and j t A .x x 0 / S: d d0 C d0 D Then t0 t0 X t0 xM x z0 z A .x x /: C M D C 0 C d d0 C d0 M1 d d Mj ¤ If all prime factors of M are of ﬁrst kind, i.e., M1 M and M2 1, then there are no terms D D with M1 d and by Corollary 3.5, we have j t0 t0 xM x z0 z S: C M D C 0 D If some prime factors of M are of second kind, again by Corollary 3.5, we have

t0 (3.5) z0 z 0; C 0 D and hence t0 X t0 X xM x A .x x / S S: C M D d d0 C d0 D D M1 d M1 d d Mj d Mj ¤ ¤ In conclusion, we have

r t0 (3.6) zM 2 xM ; xM x S: D C M D Next we use a descent argument (as in [8, p. 365]) to prove the theorem when r > 0. We embed E.K.pM //=2r E.K.pM // into H1.K.pM /; EŒ2r / by the Kummer map and consider the inﬂation-restriction exact sequence

1 r 1 r 1 r 0 H .H0=K.pM /; E.H0/Œ2 / H .K.pM /; EŒ2 / H .H0; EŒ2 /: ! ! ! r 1 r Since zM 2 E.H0/, the image of zM in H .H0; EŒ2 / under the Kummer map is zero, 2 1 1 r so the image of zM in H .K.pM // lies in H .H0=K; E.H0/Œ2 /. Because

E.H0/Œ2 E.Q/Œ2; 1 D 1 r the image of zM in H .K.pM // must be of order 2, that is, 2zM 2 E.K.pM //. 2 Shu and Zhai, Generalized Birch lemma 131

Next we prove

r p (3.7) 2zM 2 E.Q. pM // : 2

Let 0 be the generator of Gal.K.pM /=K/, and by deﬁnition (3.2) we have

(3.8) 0.zM / zM : D By [33, Theorem 5.2], we already know that the L.E.M /; 1/ 0 and hence its root number M M ¤ . N / 1. Note M .t0 / . N /. Then D D Â M Ã M .t /sgn.M / 1: 0 D N D Here we divide into two cases.

Case (1): .t / sgn.M/ 1. Then M 0 D DC

2zM M .t /2zM 2zM : D 0 D Since M > 0, together with (3.8) we conclude

r p 2zM 2 E.Q. pM // : 2

Here we note 0 is a generator of Gal.Q.p pM /=Q/.

Case (2): .t / sgn.M/ 1. Then M 0 D D

2zM M .t /2zM 2zM : D 0 D Again since M < 0, we have

r p 2zM 2 E.Q. pM // : 2 Consequently, we conclude (3.7) and hence

r 1 p p zM 2 E.Q. pM // E.Q. pM //tor: 2 C Assume the contrary that

r p p zM 2 y t; y E.K. pM // ; t E.K. pM //tor: D C 2 2

From (3.6), we have xM y t 0 with t 0 E.K.p pM //tor and by Lemma 3.3 and the fact D C 2 that a a t0 is also a generator of Gal.Q.p pM /=Q/, 7! t0 t S xM x y y 0 2t 2t D C M D C C 0 D 0 will have trivial 2-primary component which is a contradiction.

3.2. Explicit Gross–Zagier formulae. Let be the automorphic representation on

GL2.A/ associated to E, let M be the representation on GL2.A/ constructed from the quadratic character M on AK through Weil–Deligne representations. Since .E; K/ satisﬁes 132 Shu and Zhai, Generalized Birch lemma the Heegner hypothesis, the Rankin–Selberg L-series L. ; s/ has sign 1 in its func- M tional equation. From the Artin formalism, we have

.M / . pM / (3.9) L. ; s/ L.E ; s/L.E ; s/: M D Here we normalize the Rankin–Selberg L-series with the central point at s 1. D Recall f X0.N / E is the optimal modular parametrization of E sending Œ to the W ! 1 zero element of E, and g is the newform associated to E. Denote the Peterson norm ZZ 2 .g; g/0.N / g.z/ dx dy; z x iy; WD 0.N / H j j D C n where H is the Poincaré upper half plane. Invoking the general explicit Gross–Zagier formula . pM / established in [5], the Heegner point zM E .Q/ satisﬁes the following explicit height 2 formula.

. pM / Theorem 3.8. The Heegner point zM E .Q/ satisﬁes 2 2 16 .g; g/0.N / bhQ.zM / L0. ; 1/ ; M D pp M deg f j j where bhQ. / denotes the Néron–Tate height on E over Q. Proof. Since the pair .E; K/ satisﬁes the Heegner hypothesis, we just apply [5, Theo- rem 1.1].

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

.M / .M / ords 1L.E ; s/ rank E .Q/ 0; D D D . pM / . pM / ords 1L.E ; s/ rank E .Q/ 1: D D D .M / . pM / Moreover, the Shafarevich–Tate groups X.E / and X.E / are ﬁnite.

Proof. By Theorem 3.7, zM has inﬁnite order and hence L. ; s/ has vanishing M order 1 at s 1. From the decomposition in (3.9) we must have D .M / . pM / ords 1L.E ; s/ 0 and ords 1L.E ; s/ 1: D D D D Then Theorem 1.2 follows from the work of Gross–Zagier and Kolyvagin.

Let be the minimal real period of E.D/. Let .D/ denote the period of E.D/, EC.D/ which is equal to or 2 according to that E.R/ is connected or not. EC.D/ EC.D/

Proposition 3.10. We have the following relation between periods and Peterson norm of g: 2 .M / . pM / 8 .g; g/0.N / : D deg f pp M j j Shu and Zhai, Generalized Birch lemma 133

Proof. Since the discriminant E < 0, E.R/ has only one connected component. The period lattice L Z Z.=2 =2/. We have the following period relation: D C C 2 Z 8 .g; g/0.N / 2 .M / . pM / dx dy ; deg f pp M D pp M C=L D p p M D j j j j j j where the last equality follows from [22].

Combining Theorem 3.8, Proposition 3.10 and the decomposition (3.9), we have:

. pM / Corollary 3.11. The Heegner point zM E .Q/ satisﬁes 2 L.E.M /; 1/L .E. pM /; 1/ 0 2h .z /: .M / . pM / bQ M D

4. The 2-part of the Birch and Swinnerton-Dyer conjecture for the quadratic twists

Throughout this section, we assume E has odd Manin constant. We shall compare all the .M / . pM / arithmetic invariants of E with those of E and E .

4.1. Tamagawa numbers.

Proposition 4.1. Let q be any prime dividing D with .D; 2N / 1. We have D ´ .D/ 2 if q is inert in Q.pE /, cq.E / D 4 if q splits in Q.pE /.

Proof. Since .q; 2N / 1, reduction modulo q on E gives an isomorphism D E.Qq/Œ2 E.Fq/Œ2: Š It follows from [7, Lemma 36 and Lemma 37] that .D/ .D/ ord2.cq.E // ord2. E .Qq/Œ2 / D j j ord2. E.Qq/Œ2 / D j j ord2. E.Fq/Œ2 /; D j j which is equal to 1 if q is inert in Q.EŒ2/, and equal to 2 if q splits in Q.EŒ2/. Note that the .D/ .D/ curve E has additive reduction at q, so we have cq.E / 4. The assertion of the lemma Ä then follows.

Proposition 4.2. Let D 1 mod 4 be a square-free integer with .D; N / 1. Then Á D Â Ã Y .D/ ord2 c .E / 1: ` D ` N j Proof. By Proposition 2.6, we have Â X.E/ Y Ã ord2 j j2 c`.E/ 1; E.Q/tor D j j ` 134 Shu and Zhai, Generalized Birch lemma

2 Since ord2. E.Q/tor / 2 and X.E/ must be a square, it follows that X.E/Œ2 0 and j j D j j D ÂY Ã (4.1) ord2 c .E/ 1: ` D ` N j We now prove the assertion in several cases according to the reduction types of E.

Case (i): E, and hence E .M/, have additive reduction at `. From the usual table of special ﬁbers the 2-primary part of the connected component group is killed by 2. Again, by [7, Lemma 36 and Lemma 37], we have

.M / (4.2) ord2.c .E// ord2. E.Q /Œ2 / ord2.c .E //: ` D j ` j D ` Case (ii): E has multiplicative reduction at `, and ` N splits in Q.pD/. Then j E.D/ is isomorphic to E over Q , we immediately have c .E.D// c .E/. ` ` D ` Case (iii): E has split multiplicative reduction at `, and ` N is inert in Q.pD/. .M / j .M / Then E has non-split multiplicative reduction at `. In this case E and EQ 2 and are Q`2 ` isomorphic and have the same Tamagawa number equal to c`.E/. In view of [26, p. 366], if c .E/ is odd (respectively, even), then c .E.M // 1 (respectively, 2), and hence ` ` D .M / ord2.c .E// ord2.c .E //: ` D ` Case (iv): E has non-split multiplicative reduction at `, and ` N is inert in Q.pD/. j Then E.M / has split multiplicative reduction at `. If c .E/ 1, then both c .E/ and c .E.M // ` D ` ` are odd, i.e., .M / ord2.c .E// ord2.c .E // 0: ` D ` D If c .E/ 2, then E is semi-stable. Indeed from (4.2), the Tamagawa number is even for ` D places of 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, 1). Since E is semi-stable of root number 1 and c .E/ 2, 0 Ä C ` D we conclude from [9, Theorem 6.1] that c .E / 1. Again, we have ` 0 D .M / ord2.c .E // ord2.c .E // 0: ` 0 D ` 0 D .M / .M / .M / Now E E0 is an isogeny of degree 2 of elliptic curves of split multiplica- W !.M / .M / tion at `. Since c .E / is odd, by [9, Theorem 6.1], ord2.c .E // 1. Therefore, for ` 0 ` D an ` N , we always have j .M / ord2.c .E// ord2.c .E //: ` D ` This completes the proof of this proposition.

4.2. Selmer groups. Let E, p and M be as in Theorem 1.2. Recall that E.M / is the .M / .M / .M / quadratic twist of E by Q.pM/. Let E E0 be an isogeny of degree 2 with .M / .M / W ! kernel E Œ2.Q/ and 0 its dual isogeny. There is a natural identiﬁcation of Galois mod- .M / .M / .M / ules EŒ E Œ . This allows us to view both Sel.E/ and Sel.M / .E / as sub- D1 .M / groups of H .Q; EŒ/ and enable us to compare the Selmer groups Sel.E/, Sel.M / .E / 1 by local Kummer conditions in H .Q`; EŒ/ for various places ` of Q. Shu and Zhai, Generalized Birch lemma 135

Lemma 4.3. If all the prime factors of 2N split in Q.pM/, then

.M / .M / `;.E/ `;.M / .E / and `; .E0/ `; .M / .E0 / D 0 D 0 for all places `.

Proof. First consider the case ` . We have the following Weierstrass equations: D 1 E.M / y2 x3 aM x2 bM 2x W D C C and E .M / y2 x3 2aM x2 .a2 4b/M 2x: 0 W D C 6 6 By Proposition 2.5, .M / M < 0 and M M > 0. Hence both E.R/ and E E E E 0 .M / D 0.M /D E .R/ are connected while both E0.R/ and E0 .R/ have two connected components. Then .M / ;.E/ ;.M / .E / Z=2Z 1 D 1 D and .M / .E / .M / .E / 0: ;0 0 ; 0 1 D 1 0 D Suppose ` 2N , then E and E.M / are isomorphic over Q . In particular, j ` .M / .M / `;.E/ `;.M / .E / and `; .E0/ `; .M / .E0 / D 0 D 0 for ` N . We now compare the local conditions at each place ` − 2N and divide it into j 1 two cases.

Case (i). For ` − 2NM , we have both E and E have good reduction at ` and ` is 1 0 unramiﬁed in Q.pM /=Q.

Case (ii). For ` M , we have both E and E have good reduction at ` and ` is ramiﬁed j 0 in Q.pM /=Q, whence we have E.Q /Œ2 E .Q /Œ2 Z=2Z. ` Š 0 ` Š Applying [13, Lemma 6.8], we have

.M / .M / `;.E/ `;.M / .E / and `; .E0/ `; .M / .E0 / D 0 D 0 for ` − 2N . Then the lemma follows. 1 Lemma 4.4. If all the prime factors of 2N split in Q.pM/, then

.M / .M / Sel.E / Sel .E0 / Z=2Z: Š 0 Š .M / Proof. By Lemma 4.3, we see that .E/ .M / .E / holds for any places. `; D `; Therefore, we have .M / Sel .M / .E / Sel.E/ Z=2Z: D D .M / The result for E0 and E0 follows similarly.

Proposition 4.5. If all the prime factors of 2N split in Q.pM/, then

.M / .M / Sel2.E / Z=2Z and X.E /Œ2 0: D D 136 Shu and Zhai, Generalized Birch lemma

Proof. The ﬁrst assertion follows from Lemma 4.4, and the following exact sequence: .M / .M / .M / .M / .M / .M / 0 E .Q/Œ = .E .Q/Œ2/ Sel .M / .E / Sel2.E / ! 0 0 ! ! .M / Sel .M / .E0 /; ! 0 .M / .M / .M / Noting Sel2.E / E .Q/Œ2 Z=2Z, we have X.E /Œ2 0. D D D Lemma 4.6. The prime p is inert in Q.p /, and split in Q.p /. E E 0

Proof. Since .E; K Q.p p// satisﬁes the Heegner hypothesis, for all prime ` N , D j we have Â ` Ã Â p Ã 1: p D ` DC By Proposition 2.5, E < 0. Then, noting, modulo squares, N and E have the same prime factors, it follows that Â Ã Â Ã Â Ã E 1 E 1 j j 1: p D p D p D Hence p is inert in Q.p /. Similarly, the second assertion is also true since > 0 by E E 0 Proposition 2.7.

The following result is due to Cassels [6].

Lemma 4.7. We have

Sel.E/ Y `;.E/ j j j j: Sel .E / D 2 j 0 0 j ` Proposition 4.8. Let E, p and M be as in Theorem 1.2. If all the prime factors of 2N . pM / 2 . pM / split in Q.pM/ and p 1 mod 8, then Sel2.E / .Z=2Z/ and X.E /Œ2 0. Á D D Proof. The proof is similar to that of Proposition 4.5. Since .E; K Q.p p// satis- p D ﬁes the Heegner hypothesis, we have . ` / 1 for all ` N . Moreover, since p 1 mod 8, p DCpM j Á it follows that . / 1. Then we have . / 1 for all ` 2N , i.e., any prime ` 2N 2 DC ` DC j j is split in Q.p pM /. Then for any place ` p, as in Lemma 4.3, we have ¤ .M / .M / `;.E/ `;.M / .E / and `; .E0/ `; .M / .E0 /: D 0 D 0 It sufﬁces to compare the local Kummer conditions at p. By Lemma 4.6, p is inert in Q.pE / and splits in Q.p /. Then E 0

EŒ2.Qp/ EŒ2.Q/ and E Œ2.Qp/ E Œ2: D 0 D 0 . pM / . pM / By [13, Lemma 6.7], p;. pM / .E / (respectively, . pM / .E0 /) has dimen- p;0 sion 2 (respectively, 0) over Z=2Z, i.e.,

. pM / 1 . pM / p;. pM / .E / H .Qp; EŒ/ and p; . pM / .E0 / 0: D 0 D On the other hand, since p is good for E and E0 and p is prime to the degrees of and 0, we have 1 1 p;.E/ H .Qp; EŒ/ and p; .E0/ H .Qp;E0Œ0/: D ur 0 D ur Shu and Zhai, Generalized Birch lemma 137

. pM / By comparing the local Kummer conditions for 0 and 0 , we see . pM / Sel . pM / .E0 / Sel .E0/ Z=2Z: 0 0 D From

. pM / . pM / . pM / . pM / Z=2Z E .Q/tor=0 .E0 .Q/tor/ Sel . pM / .E0 /; D 0 we conclude . pM / Sel . pM / .E0 / Sel .E0/ Z=2Z: 0 D 0 D By Lemma 4.7 and Proposition 2.9, we have

. pM / . pM / Sel. pM / .E / Y `;. pM / .E / j j j j: . pM / D 2 Sel . pM / .E0 / j 0 j ` and Sel.E/ Y `;.E/ j j j j 1: Sel .E / D 2 D j 0 0 j ` . pM / Since `;. pM / .E / `;.E/ for ` p, it follows that D ¤ . pM / . pM / Sel. pM / .E / Y `;. pM / .E / j j j j . pM / D .E/ Sel . pM / .E0 / `; j 0 j ` j j . pM / p;. pM / .E / j j 2: D p;.E/ D j j Thus

. pM / . pM / 2 Sel . pM / .E0 / Z=2Z and Sel. pM / .E / .Z=2Z/ : 0 D ' Note E. pM /.Q/=2E. pM /.Q/ .Z=2Z/2. Considering the exact sequence D 0 E . pM /.Q/Œ . pM /=. pM /.E. pM /.Q/Œ2/ ! 0 0 . pM / . pM / . pM / Sel. pM / .E / Sel2.E / Sel . pM / .E0 /; ! ! ! 0 we conclude . pM / . pM / . pM / 2 Sel2.E / E .Q/=2E .Q/ .Z=2Z/ D D and X.E. pM //Œ2 0; D as desired.

Proposition 4.9. Let E, p and M be as in Theorem 1.2 and suppose that all the prime factors of 2N split in Q.pM/ and p 1 mod 8. We have Á Â X.E.M // Y Ã ord c .E.M // r 1 2 j .M / j2 ` E .Q/tor D j j ` and Â X.E. pM // Y Ã ord c .E. pM // r: 2 j . pM / j2 ` E .Q/tor D j j ` 138 Shu and Zhai, Generalized Birch lemma

Proof. Note that E.M /.Q/Œ2 E. pM /.Q/Œ2 E.Q/Œ2 Z=2Z; 1 D 1 D 1 D and by Proposition 4.5 and 4.8 we have X.E.M //Œ2 X.E. pM //Œ2 X.E/Œ2 0: 1 D 1 D 1 D Since, by assumption, any prime q M and, by Lemma 4.6, the prime p are inert in Q.pE /, j it follows from Proposition 4.1 that .M / . pM / . pM / cq.E / cq.E / 2 and cp.E / 2: D D D Then the assertion of the proposition follows from Proposition 4.2.

4.3. The 2-part of the Birch and Swinnerton-Dyer conjecture. Recall that, for any elliptic curve deﬁned over Q, the Birch and Swinnerton-Dyer conjecture predicts that

.ran/ Q L .E; 1/ ` c`.E/ X.E/ (4.3) j 2 j; ranŠE R.E/ D E.Q/tor j j where ran ords 1L.E; s/, and R.E/ is the regulator formed with the Néron–Tate pairing. It WD D is known that R.E/ 1 when ran 0, and R.E/ bhQ.P / when ran 1, where P is a free D D D D generator of E.Q/, andbhQ.P / is the Néron–Tate height on E over Q. At present, the ﬁniteness of X.E/ is only known when ran is at most 1, in which case it is also known that ran is equal to the rank of E.Q/. Suppose the 2-part of (4.3) holds for E, we shall show that the 2-part .M / . pM / of (4.3) holds for both E and E under mild assumptions.

Theorem 4.10. Let E, p and M be as in Theorem 1.2 and suppose that all the prime factors of 2N split in Q.pM/, p 1 mod 8 and E has odd Manin constant. Then we have Á .M / .M / ords 1L.E ; s/ rank E .Q/ 0; D D D . pM / . pM / ords 1L.E ; s/ rank E .Q/ 1 D D D and .M / ord2.L.E ; 1/= .M / / r 1; E D . pM / . pM / ord2.L0.E ; 1/=E . pM / R.E // r: D .M / . pM / Moreover, the Shafarevich–Tate groups X.E / and X.E / are both ﬁnite of odd cardinalities. If the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E, then the .M / . pM / 2-part of the Birch and Swinnerton-Dyer conjecture holds for both E and E .

Proof. The rank part of the Birch and Swinnerton-Dyer conjecture for the two elliptic .M / . pM / curves E and E has been established in Theorem 3.9. In particular, by the work of .M / . pM / Gross–Zagier and Kolyvain, the Shafarevich–Tate groups X.E /; X.E / are ﬁnite. .M / . pM / Moreover, the cardinalities of both X.E / and X.E / are odd by Proposition 4.5 and Proposition 4.8. First consider the quadratic twists E.M /. The full Birch and Swinnerton-Dyer conjecture predicts .M / .M / L.E ; 1/ ‹ X.E / Y (BSD.E.M //) c .E.M //: .M / j .M / j2 ` D E .Q/tor j j ` Shu and Zhai, Generalized Birch lemma 139

By Proposition 4.9 and [33, Theorem 5.2], both sides of BSD.E.M// have 2-adic valuation r 1, and hence the 2-part of the Birch and Swinnerton-Dyer conjecture for E.M / follows. . pM / Next we consider the quadratic twists E . The full Birch and Swinnerton-Dyer conjecture predicts

. pM / . pM / L .E ; 1/ ‹ X.E / Y (BSD.E. pM //) 0 h .P / c .E. pM //; . pM / bQ j . pM / j2 ` D E .Q/tor j j ` . pM / where P is a free generator of E .Q/. By the explicit height formula in Corollary 3.11, .M / . pM / (BSD.E /) and (BSD.E /) amount to Â Ã bhQ.zM / (4.4) ord2 bhQ.P / Â .M / . pM / Ã ‹ X.E / X.E / Y ord 2 1 c .E.M //c .E. pM // : 2 j.M / 2jj . pM / j 2 ` ` D E .Q/tor E .Q/tor j j j j ` By Proposition 4.9, the right-hand side is 2.r 1/, and therefore the above equality (4.4) follows from the exact 2-divisibility of the Heegner point zM in Theorem 3.7. Since the 2-part of .M / . pM / (BSD.E /) is proved, the 2-part of (BSD.E /) follows.

5. Applications

In this ﬁnal section, we shall illustrate our general results for the family of quadratic twists both 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 curves have prime conductor p0 (see [18], [19] and [24]), where p0 is any prime of the form 2 p0 u 64 for some integer u 1 mod 4, for example, p0 73; 89; : : : . Then, up to iso- D C Á D morphism, it is known that there are just two elliptic curves of conductor p0 with a rational 2-division point, namely, u 1 (5.1) A y2 xy x3 x2 4x u; W C D C 4 C C u 1 A y2 xy x3 x2 x: 0W C D 4

The curves A and A0 are 2-isogenous, and both have Mordell–Weil groups Z=2Z. A simple 2 computation shows that A p and A p0, it follows that D 0 0 D Q.AŒ2/ Q.i/; Q.A Œ2/ Q.pp0/: D 0 D Let X0.p0/ be the modular curve of level p0, and there is a non-constant rational map

X0.p0/ A; ! making the modular parametrization 0.p0/-optimal by Mestre and Oesterlé [17]. Since the conductor of A is odd, the Manin constant of A is odd by the work of Abbes and Ullmo [1]. In particular, we have the following result on applying our main theorem. 140 Shu and Zhai, Generalized Birch lemma

2 Theorem 5.1. Assume that p0 is a prime of the form u 64 with u 5 mod 8, and C Á let A be the Neumann–Setzer curve (5.1). Let q1; : : : ; qr be distinct primes congruent to 3 modulo 4 which are inert in Q.pp0/. Let p 3 mod 4 be a prime greater than 3 which splits Á in Q.pp0/. Denote M q q . Then we have D 1 r .M / .M / ords 1L.A ; s/ rank A .Q/ 0; D D D . pM / . pM / ords 1L.A ; s/ rank A .Q/ 1: D D D .M / . pM / In particular, X.A / and X.A / are both ﬁnite of odd cardinality. Moreover, the .M / . pM / 2-part of the Birch and Swinnerton-Dyer conjecture is valid for both A and A .

Proof. The primes qi , 1 i r, are admissible for A, since they are inert in both Ä Ä Q.AŒ2/ and Q.A Œ2/. The condition on p implies that p0 splits in K Q.p p/. Hence, the 0 D Heegner hypothesis for .A; K/ holds. When u 5 mod 8, it follows from [32, Lemma 5.11]) Á that the modular parametrization f .Œ0/ is precisely the non-trivial torsion point of order 2. Thus, all the assumptions in Theorem 1.2 are satisﬁed. Moreover, the Manin constant of A is odd. The 2-part of the Birch and Swinnerton-Dyer conjecture of A is veriﬁed in [32, Proposition 5.13]. However, here we will not apply Theorem 1.4, since a classical 2-descent has been carried out in [32, Section 5], which shows that under the assumptions in Theorem 5.1, we have

X.A.M //Œ2 X.A. pM //Œ2 0: D D By Proposition 4.1 and Proposition 4.2, we have Â Ã Y .M / ord2 c .A / r 1 ` D C ` and Â Ã Y . pM / ord2 c .A / r 2: ` D C ` Note that, by [33, Theorem 5.2], we have

.M / ord2.L.A ; 1/= .M / / r 1: A D Then combining with (4.4), it follows that the 2-part of the Birch and Swinnerton-Dyer con- .M / . pM / jecture is valid for both A and A .

alg Note that for u 1 mod 8, the same result holds if we assume ord2.L .A; 1// 1. Á D For example, we can take p0 73. Here is the beginning of an inﬁnite set of primes which are D congruent to 3 modulo 4 and inert in Q.p73/:

S 7; 11; 31; 43; 47; 59; 83; 103; 107; 131; 139; 151; 163; 167; 179; 191; 199; : : : : D ¹ º For more examples, there is a nice table presenting primes of the form u2 64 in [24]. C 5.2. More numerical examples. The theorem can be applied on the family of quadratic twists of many elliptic curves E=Q, we include a table here when the conductor of E is less than 100 (see Table1). Shu and Zhai, Generalized Birch lemma 141

E=Q satisfying f .Œ0/ 2E.Q/ and condition (Tor) 62 E Q.EŒ2/ Q.E0Œ2/ admissible primes q p 14a1 Q.p 7/ Q.p2/ 3; 5; 13; 19; 59; 61; 83; 101; : : : 7; 31; 47; 103; 167; 199; : : : 20a1 Q.p 1/ Q.p5/ 3; 7; 23; 43; 47; 67; 83; 103; : : : 31; 71; 79; 151; 191; 199; : : : 36a1 Q.p 3/ Q.p3/ 5; 17; 29; 41; 53; 89; 101; 113; : : : 23; 47; 71; 167; 191; 239; : : : 46a1 Q.p 23/ Q.p2/ 5; 11; 19; 37; 43; 53; 61; 67; : : : 7; 79; 103; 191; 199; 263; : : : 49a1 Q.p 7/ Q.p7/ 5; 13; 17; 41; 61; 73; 89; 97; : : : 19; 31; 47; 59; 83; 103; : : : 52a1 Q.p 1/ Q.p13/ 7; 11; 19; 31; 47; 59; 67; 71; : : : 23; 79; 103; 127; 191; 199; : : : 56b1 Q.p 7/ Q.p2/ 3; 5; 13; 19; 59; 61; 83; 101; : : : 31; 47; 103; 167; 199; 223; : : : 69a1 Q.p 23/ Q.p3/ 5; 7; 17; 19; 43; 53; 67; 79; : : : 11; 83; 107; 191; 227; 251; : : : 73a1 Q.p 1/ Q.p73/ 7; 11; 31; 43; 47; 59; 83; 103; : : : 19; 23; 67; 71; 79; 127; : : : 77c1 Q.p 7/ Q.p11/ 3; 13; 17; 31; 41; 47; 59; 61; : : : 19; 83; 131; 139; 167; 227; : : : 80b1 Q.p 1/ Q.p5/ 3; 7; 23; 43; 47; 67; 83; 103; : : : 31; 71; 79; 151; 191; 199; : : : 84a1 Q.p 3/ Q.p7/ 5; 11; 17; 23; 41; 71; 89; 101; : : : 47; 59; 83; 131; 167; 227; : : : 84b1 Q.p 3/ Q.p7/ 5; 11; 17; 23; 41; 71; 89; 101; : : : 47; 59; 83; 131; 167; 227; : : : 89b1 Q.p 1/ Q.p89/ 3; 7; 19; 23; 31; 43; 59; 83; : : : 11; 47; 67; 71; 79; 107; : : : 94a1 Q.p 47/ Q.p2/ 5; 11; 13; 19; 29; 43; 67; 107; : : : 23; 31; 127; 151; 167; 199; : : : Table 1. E=Q satisfying f .Œ0/ 2E.Q/ and condition (Tor) with conductor N < 100. 62

5.3. Examples of the full Birch and Swinnerton-Dyer conjecture. Let A be the elliptic curve “69a1” with the minimal Weierstrass equation given by

A y2 xy y x3 x 1: W C C D .alg/ We have a2 1, a3 1 and a23 1. Moreover, A.Q/ Z=2Z and L .A; 1/ 1=2. D D 2 D D D The discriminant of A is 3 23. The Tamagawa factors c3 2, c23 1. Also, a simple D D computation shows that Q.AŒ2/ Q.p 23/ and Q.A Œ2/ Q.p3/. Here is the beginning D 0 D of an infinite set of primes q with good ordinary reduction which are inert in both the fields Q.p 23/ and Q.p3/: S 5; 7; 17; 19; 43; 53; 67; 79; 89; 103; 113; 137; 149; 199; : : : : D ¹ º Let M q q be a product of r distinct primes in S. By Theorem 4.10, we have D 1 r L.A.M /; 1/ 0; ¤ and alg .M / ord2.L .A ; 1// r 1: D If we carry out a classical 2-descent on A.M /, one shows easily that the 2-primary compo- .M / nent of X.A =Q/ is zero and ord2.cq / 1 for 1 i r, and therefore the 2-part of the i D Ä Ä Birch and Swinnerton-Dyer conjecture holds for A.M /. Alternatively, we can just apply Theo- rem 1.4, take M 1 mod 8, then the assumption that 2, 3 and 23 all split in Q.pM/ will hold, Á 142 Shu and Zhai, Generalized Birch lemma whence we can also verify the 2-part of the Birch and Swinnerton-Dyer conjecture. For the full Birch and 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-split multiplicative reduction at 23, we could consider A.5/, which has split multiplicative reduction at 23, and the Tamagawa number is 1 at 23, hence the AŒp G (q 3 or 23) is a ram- j q D ified representation for any odd prime p. Other conditions are easy to verify, so the full Birch and Swinnerton-Dyer conjecture is valid for A.M /. Hence the full Birch and Swinnerton-Dyer conjecture is verified for infinitely many elliptic curves. The full Birch and Swinnerton-Dyer conjecture of rank 1 twists are also accessible in the future work. More examples are given in Wan’s paper.

Acknowledgement. We would like to thank John Coates for encouragement, useful dis- cussions and polishings on the manuscript, thank Ye Tian and Xin Wan for helpful advice and comments, and thank Yongxiong Li for helpful comments and carefully reading the manuscript. We also thank the referee for helpful advice.

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Jie Shu, School of Mathematical Sciences, Tongji University, Shanghai 200092, P.R. China https://orcid.org/0000-0001-7386-6778 e-mail: [email protected]

Shuai Zhai, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom https://orcid.org/0000-0001-9072-7227 e-mail: [email protected]

Eingegangen 8. August 2020, in revidierter Fassung 13. Januar 2021