ON THE 2-PART OF THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR QUADRATIC TWISTS OF ELLIPTIC CURVES

Transcription

1 ON THE 2-PART OF THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR QUADRATIC TWISTS OF ELLIPTIC CURVES LI CAI, CHAO LI, SHUAI ZHAI Abstract. In the present paper, we prove, for a large class of elliptic curves defined over Q, the existence of an explicit infinite faily of quadratic twists with analytic rank 0. In addition, we establish the 2-part of the conjecture of Birch and Swinnerton- Dyer for any of these infinite failies of quadratic twists. Recently, Xin Wan has used our results to prove for the first tie the full Birch Swinnerton-Dyer conjecture for soe explicit infinite failies of elliptic curves defined over Q without coplex ultiplication. Contents 1. Introduction 1 2. Modular sybols 5 3. Integrality at Non-vanishing results part of the Birch Swinnerton-Dyer conjecture Applications Quadratic twists of X 0 (14) More nuerical exaples Exaples satisfying the full Birch Swinnerton-Dyer conjecture 23 References Introduction Let E be an elliptic curve defined over Q with conductor C, and coplex L-series L(E, s). The Birch and Swinnerton-Dyer conjecture asserts that the rank of E(Q) is equal to its analytic rank r an := ord s=1 L(E, s). It furtherore predicts that the Tate-Shafarevich group X(E) is always finite, and that L (ran) (E, 1) (1.1) r an!ω(e)r(e) = l c l(e) X(E), E(Q) tor 2 where Ω(E) is the Taagawa factor at infinity, R(E) is the regulator fored with the Néron Tate pairing, E(Q) tor is the torsion subgroup of E(Q), and the c l (E) are the 2010 Matheatics Subject Classification. 11G05, 11G40. 1

2 Taagawa factors (see [19], for exaple). In fact, the finiteness of X(E) is only known at present when r an is at ost 1, in which case it is also known that r an is equal to the rank of E(Q) (see [11], for exaple). If p is any prie nuber, the equality of the powers of p occurring on the two sides of (1.1) is called the p-part of the exact Birch Swinnerton-Dyer forula (but we should reeber that the left hand side of (1.1) is only known at present to be a rational nuber when r an is at ost 1). We stress that, up until now, the full Birch Swinnerton-Dyer conjecture had never been proven for infinitely any elliptic curves without coplex ultiplication. Roughly speaking, our present knowledge of Iwasawa theory shows that for a given E, the p-part of the Birch Swinnerton-Dyer conjecture is valid for all sufficiently large pries p when r an 1. But there are real technical difficulties at present in using Iwasawa theory to prove, in particular, the 2-part of the Birch Swinnerton-Dyer conjecture. However, we can apply rather classical results on odular sybols to derive the precise 2-adic valuation of the algebraic part of the value of the coplex L-series at s = 1 in the faily of quadratic twists of certain optial elliptic curves E over Q with r an = 0 and E(Q)[2] = Z/2Z. In particular, for all of these twists, our results show that r an = 0, whence the Mordell Weil group and the Tate Shafarevich group of these twists are both finite by the celebrated theores of Gross Zagier and Kolyvagin. Moreover, we can prove the 2-part of exact Birch Swinnerton- Dyer forula for soe of these twists. Happily, Xin Wan has now used soe of our results in this paper, cobined with deep arguents fro Iwasawa theory to prove for the first tie the validity of the full Birch Swinnerton-Dyer conjecture for infinitely any elliptic curves over Q without coplex ultiplication (see [20, Appendix]). He eploys deep and coplicated arguents fro Iwasawa theory to establish the p-part of the Birch Swinnerton-Dyer conjecture for all odd pries p for the elliptic curves in these failies. However, it is still not known how to extend these Iwasawa-theoretic arguents to the prie p = 2, whereas our eleentary arguents work well for p = 2. For the current progress on the Birch and Swinnerton-Dyer conjecture, one can see the survey article by Coates [7]. We now denote the left-hand-side of (1.1) by L (alg) (E, 1). In particular, when r an = 0, L (alg) (E, 1) := L(E, 1)/Ω E, where Ω E is equal to Ω + E or 2Ω+ E, depending on whether or not E(R) is connected, and here Ω + E is the least positive real period of a Néron differential on the global inial Weierstrass equation for E. For each discriinant of a quadratic extension of Q, we write E () for the twist of E by this quadratic extension, and write L(E (), s) for its coplex L-series. Let ord 2 for the order valuation of Q at the prie 2, noralized by ord 2 (2) = 1, and with ord 2 (0) =. If q be any prie of good reduction for E, let a q be the trace of Frobenius at q on E, so that N q = 1 + q a q is the nuber of F q -points on the reduction of E odulo q. We shall always assue that E(Q)[2] = Z/2Z, and 2

3 we write E := E/E(Q)[2] for the 2-isogenous curve of E. For each integer n > 1, write E[n] for the Galois odule of n-division points on E. Let S be the set of pries S = {q 1 od 4 : q C, ord 2 (N q ) = 1}. Theore 1.1. Let E be an optial elliptic curve over Q with conductor C. Assue that (1) E has odd Manin constant; (2) E(Q)[2] = Z/2Z; (3) ord 2 (L (alg) (E, 1)) = 1. Let M = q 1 q 2 q r be a product of r distinct pries in S. Then L(E (M), 1) 0, and we have ord 2 (L (alg) (E (M), 1)) = r 1. In particular, E (M) (Q) and X(E (M) /Q) are both finite. Reark 1.2. This theore generalises [6, Theore 1.2] (where E = X 0 (49)) and [1, Theore 1.3] (where E = X 0 (36)) to a uch wider class of elliptic curves E, with no hypothesis of coplex ultiplication. It also generalises [21, Theore 1.5 and 1.7] and [15], where only prie twists are considered. For siilar results for E without rational 2-torsion, see [21] and [14]. In the presence of rational 2-torsion, the ethods of [21] and [15] cannot easily treat twists by non-prie quadratic discriinants, because the obvious induction arguent fails. We overcoe this difficulty by introducing a new integrality arguent to ake the induction work. Reark 1.3. If S is non-epty, we ust have E(Q)[2] = E (Q)[2] = Z/2Z, which is also equivalent to the assertion that q is inert in both the 2-division field Q(E[2]) and Q(E [2]) (see [15, Lea 4.1]), where as before E := E/E(Q)[2]. Thus, by Chebotarev s density theore, the set of pries S has positive density. Reark 1.4. We suppose that the Manin constant of E has to be odd, which will be fully discussed in Section 2. However, we can reove the Manin constant assuption when 4 C by the recent work of Česnavičius [4]. Moreover, the conjecture that the Manin constant is always ±1 has been proved by Creona for all optial elliptic curves of conductor less than (see [9]). Our second ain result is a proof of the 2-part of the Birch and Swinnerton-Dyer conjecture for any of the twists in Theore 1.1. As before, let E := E/E(Q)[2] be the 2-isogenous curve of E. Theore 1.5. Let E and M be as in Theore 1.1. Assue further that (1) X(E )[2] = 0; (2) all pries l which divide 2C split in Q( M); (3) the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E. 3

4 Then the 2-priary coponent of X(E (M) /Q) is zero, and the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E (M). Reark 1.6. In view of our assuption that #(E(Q)[2]) = 2, the 2-part of the Birch and Swinnerton-Dyer conjecture for E would show that our hypothesis that ord 2 (L (alg) (E, 1)) = 1 iplies that X(E)[2] = 0, but it is still not known how to prove this at present. However, if we assue that the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E, as well as the hypotheses of Theore 1.1, we will have X(E)[2] = 0. Moreover, if we assue two ore conditions on X(E )[2] and l, then we can copare the local conditions of the Seler groups of E and E (M), and get the triviality of X(E (M) /Q)[2]. Reark 1.7. The 2-part of Birch Swinnerton-Dyer conjecture for a single elliptic curve (of sall conductor) can be verified by nuerical calculation when r an = 0. Theore 1.5 then allows one to deduce the 2-part of Birch Swinnerton-Dyer conjecture for any of its quadratic twists (of arbitrarily large conductor). Reark 1.8. We ephasize that the theore applies to elliptic curves with various different reduction types at 2, such as X 0 (14) with non-split ultiplicative reduction at 2, 34A1 with split ultiplicative reduction at 2, and 99C1 with good ordinary reduction at 2 (we use Creona s label for each curve). We ephasize that it also applies to elliptic curves with potentially supersingular reduction at 2, such as X 0 (36) and 56B1. We will give a detailed descriptions of quadratic twists of X 0 (14) and soe nuerical exaples in Section 6. Of course, the theore could apply ore failies of elliptic curves, such as quadratic twists of 46A1, X 0 (49), 66A1, 66C1 and so on. Recently, a rearkable preprint of Sith [18] uses soe arithetic properties of elliptic curves at the prie 2 to establish soe deep results conjectured by Goldfeld (in particular, that the set of all square free congruent nubers congruent to 1, 2, 3 odulo 8 has natural density zero). However, Sith s analytic arguents at present see only valid for elliptic curves with full rational 2-torsion. We should ention that the non-vanishing result presented in this paper give a uch weaker result in the direction of Goldfeld s conjecture for the faily of elliptic curves in Theore 1.1. In a forthcoing work [3], we will cobine the Heegner points arguents (see [6]) and the explicit Gross Zagier forula (see [2]) to prove analogous results to those established here for rank one quadratic twists of elliptic curves. Acknowledgents. We would like to thank Jack Thorne for his useful coents and suggestions, and thank Jianya Liu and Ye Tian for their encourageent. The secondnaed author (CL) would also like to thank X. Wan and the Morningside Center of Matheatics for the hospitality during his visit. 4

5 2. Modular sybols Modular sybols were first used by Birch, and a little later by Manin [16], for the study of the conjecture of Birch and Swinnerton-Dyer. They subsequently becae the basic tool in Creona s construction of his rearkable tables of elliptic curves and their arithetic invariants [8]. We shall show in this paper that they are also very useful in studying the 2-part of the conjecture of Birch and Swinnerton-Dyer. For each integer C 1, let S 2 (Γ 0 (C)) be the space of cusp fors of weight 2 for Γ 0 (C). In what follows, f will always denote a noralized priitive eigenfor in S 2 (Γ 0 (C)), all of whose Fourier coefficients belong to Q. Thus f will correspond to an isogeny class of elliptic curves defined over Q, and we will denote by E the unique optial elliptic curve in the Q-isogeny class of E. The coplex L-series L(E, s) will then coincide with the coplex L-series attached to the odular for f. Moreover, there will be a non-constant rational ap defined over Q φ : X 0 (C) E, which does not factor through any other elliptic curve in the isogeny class of E. Let ω denote a Néron differential on the global inial Weierstrass equation for E. Then, writing φ (ω) for the pull back of ω by φ, there exists ν E Q such that (2.1) ν E f(τ)dτ = φ (ω). The rational nuber ν E is called the Manin constant. It is well known to lie in Z, and it is conjectured to always be equal to 1. Moreover, it is known to be odd whenever the conductor C of E is odd. Let H be the upper half plane, and put H = H P 1 (Q). Let g be any eleent of Γ 0 (C). Let α, β be two points in H such that β = gα. Then any path fro α to β on H is a closed path on X 0 (C) whose hoology class only depends on α and β. Hence it deterines an integral hoology class in H 1 (X 0 (C), Z), and we denote this hoology class by the odular sybol {α, β}. We can then for the odular sybol {α, β}, f := β α 2πif(z)dz. The period lattice Λ f of the odular for f is defined to be the set of these odular sybols for all such pairs {α, β}. It is a discrete subgroup of C of rank 2. If L E denotes the period lattice of a Néron differential ω on E, it follows fro (2.1) that (2.2) L E = ν E Λ f. Define define Ω + E (resp. iω E ) to be the least positive real (resp. purely iaginary) period of a Néron differential of the global inial equation for E, and Ω + f (resp. iω f ) to be the least positive real (resp. purely iaginary) period of f. Thus, by (2.2), we have (2.3) Ω + E = ν EΩ + f, Ω E = ν EΩ f. 5

6 In this section, we will carry out all of our coputations with the period lattice Λ f, but whenever we subsequently translate the into assertions about the conjecture of Birch and Swinnerton-Dyer for the elliptic curve E, we ust switch to the period lattice L E by aking use of (2.2). More generally, if α, β are any two eleents of H, and g is any eleent of S 2 (Γ 0 (C)), we put {α, β}, g := β 2πig(z)dz. This linear functional defines an eleent of α H 1 (X 0 (C), R), which we also denote by {α, β}. The following theore was proven by Manin and Drinfeld [16]: Theore 2.1. (Manin Drinfeld). For all pairs of cusps α, β H, we have {α, β} H 1 (X 0 (C), Q). Let be a positive integer satisfying (, C) = 1. According to Birch, Manin [16, Theore 4.2] and Creona [8, Chapter 3], we have the following forulae:- (2.4) ( l l a )L(E, 1) = here l runs over all positive divisors of ; and (2.5) L(E, χ, 1) = g( χ) k od l k od l {0, k }, f ; l χ(k) {0, k }, f ; here χ is any priitive Dirichlet character odulo, and g( χ) = k od χ(k)e2πi k. For each odd square-free positive integer, we define r() to be the nuber of prie factors of. Also, in what follows, we shall always only consider the positive divisors of, and define χ to be the priitive quadratic character odulo. Define S := {0, k }, f, S := (k,)=1 {0, k }, f, T := χ (k) {0, k }, f. Recall that (see [21, Lea 2.2]), for each odd square-free positive integer > 1, we have (2.6) r() S l = 2 r() d l d=1 n S n. r(n)=d We repeatedly use the above identity to prove the following lea. Lea 2.2. Let E be the optial elliptic curve over Q attached to f. Let be any integer of the for = q 1 q 2 q r(), with (, C) = 1, r() 1, and q 1,..., q r() 6

7 arbitrary distinct odd pries. Then we have N q1 N q2 N qr() L(E, 1) = r() d=1 n r(n)=d b n S n, where b n = ( 1) r() q (1 q), here q runs over the prie factors of. n n Proof. We give the proof of the lea by induction on r(), the nuber of prie factors of. The assertion is true for r() = 1 by (2.4). Assue next that r() = 2. Note that N q1 N q2 = ((1 + q 1 )(1 + q 2 ) (1 + q 1 N q1 )(1 + q 2 N q2 )) + (1 + q 2 )N q1 + (1 + q 1 )N q2 = ((1 + q 1 )(1 + q 2 ) a q1 a q2 ) + (1 + q 2 )N q1 + (1 + q 1 )N q2, we then have that N q1 N q2 L(E, 1) = l q 1 q 2 S l ((1 + q 2 )S q1 + (1 + q 1 )S q2 ) = (1 q 2 )S q1 + (1 q 1 )S q2 + S q 1 q 2, as required. Now assue that r() > 2, and that the lea is true for all divisors n > 1 of with n. We then consider the case = q 1 q 2 q r(). First note that N q1 N q2 N qr() =( 1) r() 1 ((1 + q 1 )(1 + q 2 ) (1 + q r() ) a q1 a q2 a qr() ) r() + ( 1) r() 2 i=1 r() (1 + q k ) + ( 1) r() 3 N qi r() k i r() + + ( 1) (1 + q i )(1 + q j ) i,j=1 r() k i,j i,j=1 r() N qk + (1 + q i ) i=1 r() N qi N qj r() k i k i,j N qk. (1 + q k ) Without loss of generality, here we can just consider the coefficients of S q 1, S q 1 q 2,..., S q 1 q 2 q r() in the identity of the lea, i.e. b q1, b q1 q 2,..., b q1 q 2 q r(). By our assuption, we conclude that r() r() r() b q1 = ( 1) r() 1 2 r() 1 + ( 1) r() 1 (1 + q i ) + ( 1) r() 1 (1 q i ) (1 + q k ) r() + ( 1) r() 1 (1 q i )(1 q j ) i,j=2 i j i=2 r() k=2 k i,j 7 i=2 k=2 k i r() r() (1 + q k ) + + ( 1) r() 1 (1 + q i ) (1 q k ). i=2 k=2 k i

8 Note that hence we have r() 2 r() 1 = ((1 q i ) + (1 + q i )), i=2 r() b q1 = ( 1) r() (1 q i ). Siilar arguents hold for b q1 q 2,..., b q1 q 2 q r() 1, and it is easy to see that i=2 b q1 q 2 q r() = ( 1) r(). The proof of the lea is coplete. Lea 2.3. Let E be the optial elliptic curve over Q with analytic rank zero attached to f. Let be any integer of the for = q 1 q 2 q r(), with (, C) = 1, r() 1, and q 1,..., q r() arbitrary distinct odd pries congruent to 1 odulo 4. If ord 2 (N qi ) = 1 holds for any 1 i r(), then we have ord 2 (S /Ω + f ) = ord 2(N q1 N q2 N qr() L(E, 1)/Ω + f ). Proof. We give the proof of the lea by induction on r(). The assertion is obviously true for r() = 1 according as (2.4). When r() = 2, say = q 1 q 2, by Lea 2.2, we have that N q1 N q2 L(E, 1) = (1 q 2 )S q 1 + (1 q 1 )S q 2 + S q 1 q 2. The assertion for r() = 2 then follows by noting that q i 1 od 4. Now assue that r() > 2, and that the lea is true for all divisors n > 1 of with n. We then consider the case = q 1 q 2 q r(). According to Lea 2.2, we have that N q1 N q2 N qr() L(E, 1) = r() 1 d=1 n r(n)=d By our assuption, it is not difficult to see that ord 2 ( q n ( 1) r() q n (1 q)s n + ( 1) r() S. (1 q)s n/ω + f ) > ord 2(N q1 N q2 N qr() L(E, 1)/Ω + f ) holds for all divisors n > 1 of with n. Then the assertion for = q 1 q 2 q r() follows iediately. This copletes the proof of the lea. 8

9 3. Integrality at 2 Let E be the optial elliptic curve defined over Q with discriinant E and conductor C, which is attached to our odular for f. In this section, we will prove soe results of integrality at 2, and apply the to get the non-vanishing results for soe certain quadratic twists of elliptic curves, provided L(E, 1) 0. Recall that we then have Ω + E = ν EΩ + f, ord 2 (L(E, 1)/Ω + E ) = ord 2(L(E, 1)/Ω + f ) ord 2(ν E ) = ord 2 (L(E, 1)/Ω + f ), under our assuption on the Manin constant. When the coplex L-series of E does not vanish at s = 1, for every prie nuber p, the strong Birch Swinnerton-Dyer conjecture predicts the following exact forula ord p (L (alg) (E, 1)) = ord p (#(X(E))) + ord p ( l C c l (E)) 2ord p (#(E(Q))), We begin by establishing soe preliinary results, which will be needed for the proof of the desired results. Throughout this section, we will always assue 1 od 4. Since the for of the period lattice of a Néron differential on E is different, according as the sign of the discriinant of E. We first consider the case when the discriinant of E is negative. Recall that when the discriinant of E is negative, then E(R) has only one real coponent, and so the period lattice L of a Néron differential on E has a Z-basis of the for [ Ω + E, Ω+ E + ] iω E, 2 where Ω + E and Ω E are both real, and the period lattice Λ f of f has a Z-basis of the for [ ] where Ω + f and Ω f Ω + f, Ω+ f + iω f 2 are also both real. We can then write (3.1) {0, k }, f = (s k,ω + f + it k,ω f )/2 for any integer coprie to C, where s k,, t k, are integers of the sae parity. Moreover, by the basic property of odular sybols, {0, k k }, f and {0, }, f are coplex conjugate periods of f. Thus we obtain ( 1)/2 (3.2) S /Ω + f = (k,)=1 9, s k,.

10 Siilarly, when 1 od 4, we have ( 1)/2 (3.3) T /Ω + f = (k,)=1 χ (k)s k,. Moreover, in this case, by (3.2), we always have that ord 2 (N q L(E, 1)/Ω + f ) 0, for any prie q with (q, C) = 1. We define T d, = χ d (k) {0, k }, f, k (Z/Z) then we have the following theore of integrality at 2. Theore 3.1. Let E be an optial elliptic curve over Q with E < 0. Let be any integer of the for = q 1 q 2 q r(), with (, C) = 1, r() 1, and q 1,..., q r() arbitrary distinct odd pries in S. Then T d,/ω + f = 2r() Ψ, d where Ψ is an integer with ord 2 (Ψ ) 0. Proof. It is easy to see that T d, = d k (Z/Z) d χ d (k) {0, k }, f = 2 r() {0, k }, f, k (Z/Z) where eans that k runs over all the eleents in (Z/Z) such that χ qi (k) = 1 for all 1 i r(). Since 1 od 4, if k is of an eleent in the above suation, so is k. Then by (3.1), we have that {0, k }, f = k (Z/Z) ( 1)/2 (k,)=1 Then the arguent follows iediately if we define s k, Ω + f. which is an integer. Ψ = ( 1)/2 (k,)=1 10 s k,,

11 When the discriinant of E is positive, then E(R) has two real coponents, and so the period lattice L of a Néron differential on E has a Z-basis of the for [Ω + E, iω E ], with Ω + E and Ω E real nubers, and the period lattice Λ f of f has a Z-basis of the for [Ω + f, iω f ], with Ω + f and Ω f real nubers too. We can then write (3.4) {0, k }, f = s k,ω + f + it k,ω f for any integer coprie to C, where s k,, t k, are integers. Siilarly, we can obtain ( 1)/2 (3.5) S /Ω + f = 2 and when 1 od 4, we have (k,)=1 ( 1)/2 (3.6) T /Ω + f = 2 (k,)=1 s k,, χ (k)s k,. Moreover, in this case, by (3.5), we always have that ord 2 (N q L(E, 1)/Ω + f ) 1, for any prie q with (q, C) = 1. integrality at 2. We then have the following parallel theore of Theore 3.2. Let E be an optial elliptic curve over Q with E > 0. Let be any integer of the for = q 1 q 2 q r(), with (, C) = 1, r() 1, and q 1,..., q r() arbitrary distinct odd pries in S. Then T d,/ω + f = 2r()+1 Ψ, d where Ψ is an integer with ord 2 (Ψ ) 0. Proof. The proof of the above theore is siilar to Theore 3.1. As usual, we have T d, = χ d (k) {0, k }, f d k (Z/Z) d = 2 r() {0, k }, f, k (Z/Z) 11

12 where eans that k runs over all the eleents in (Z/Z) such that χ qi (k) = 1 for all 1 i r(). Since 1 od 4, if k is of an eleent in the above suation, so is k. But when the discriinant is positive, by (3.4), we have k (Z/Z) {0, k ( 1)/2 }, f = 2 s k, Ω + f. (k,)=1 Then the arguent follows iediately if we define which is an integer. Ψ = ( 1)/2 (k,)=1 s k,, 4. Non-vanishing results The ai of this section is to apply the results of integrality at 2 in the previous section to obtain the corresponding non-vanishing results of quadratic twists of elliptic curves. Specifically, we prove the precise 2-adic valuation of the algebraic central value of these L-functions attached to soe certain failies of quadratic twists of elliptic curves. Moreover, one can use these non-vanishing theores to verify the 2-part of the Birch and Swinnerton-Dyer conjecture. Throughout this section, we will always assue > 0 and 1 od 4. Before proving our non-vanishing results, we will first prove the following lea, in which the action of Hecke operator on odular sybols is involved. For each prie p not dividing the conductor C, the Hecke operator T p acts on odular sybols {α, β} via T p ({α, β}) = {pα, pβ} + { α + k, β + k }. p p In particular, we have since T p f = a p f. k od p T p ({α, β}), f = {α, β}, T p f = a p {α, β}, f, Lea 4.1. Let be any integer of the for = q 1 q 2 q r(), with (, C) = 1, r() 2, and q 1,..., q r() arbitrary distinct odd pries. Let d > 1 be a positive integer dividing and q be a prie dividing, then we have d T d, = (a q 2χ d (q))t d,. q 12

13 Proof. Recall that T d, = χ d (k) {0, k }, f. k (Z/Z) By the Chinese reainder theore, we have (Z/Z) = (Z/qZ) (Z/ q Z). So we can write (4.1) T d, = χ d (k ) {0, k (Z/ q Z) k Z/qZ q k + k }, f χ d (k q) {0, k q }, f. k (Z/ q Z) Let the Hecke operator T q act on the odular sybol {0, k Hence, k Z/qZ T q ({0, {0, k k }) = {0, /q } + q k + k }, f = a q {0, k od q {0, k /q + k q /q } k }, f + {0, k /q q k Z/qZ }, we get that k od q {0, k q }. k }, f {0, }, f. Then the first ter of the right-hand side of (4.1) becoes χ d (k k )(a q {0, }, f + {0, k k }, f {0, }, f ), /q q k (Z/ q Z) k Z/qZ that is (4.2) k (Z/ q Z) χ d (k)(a q {0, k /q }, f {0, k }, f ), since Then (4.2) becoes a q k (Z/ q Z) χ d (k) {0, k (Z/ q Z) χ d (k ) = 0. k }, f /q k (Z/ q Z) χ d (k q) {0, k q }, f if we substitute k = k q in the second ter. We then have T d, k = a q χ d (k) {0, /q }, f 2χ d(q) χ d (k k ) {0, }, f, /q k (Z/ q Z) k (Z/ q Z) 13

14 This copletes the proof of the lea by noting that T d, = k χ d (k) {0, }, f. q /q k (Z/ q Z) Now we are ready to prove Theore 1.1. When the discriinant of E is negative, we have the following result. Theore 4.2. Let E be an optial elliptic curve over Q with conductor C, and with odd Manin constant. Assue that E has negative discriinant, and satisfies E(Q)[2] 0 and ord 2 (L(E, 1)/Ω + f ) = 1. Let be any integer of the for = q 1q 2 q r(), with r() 1 and q 1,..., q r() arbitrary distinct odd pries congruent to 1 odulo 4, and with (, C) = 1. If ord 2 (N qi ) = 1 for 1 i r(), then L(E (), 1) 0, and we have ord 2 (L(E (), 1)/Ω + E () ) = r() 1. Proof. We will prove the theore by induction on r(), of course we have got the arguent when r() = 1 in [21, Theore 1.5]. We firstly note that T d, = χ d (k) {0, k }, f d d k (Z/Z) = S + χ d (k) {0, k }, f + T. d k (Z/Z) 1<d< By Lea 4.1, it is easy to see that Hence, d 1<d< T d, = q d (a q 2χ d (q)) T d,d = q d χ d (k) {0, k }, f = k (Z/Z) d q d 1<d< (a q 2χ d (q)) T d. (a q 2χ d (q)) T d. We then apply Theore 3.1 and get the following equation S /Ω + f + (a q 2χ d (q)) T d /Ω + f + T /Ω + f = 2r() Ψ, d q d 1<d< where Ψ is an integer with ord 2 (Ψ ) 0. Note that ord 2 (L(E, 1)/Ω + f ) = 1 and ord 2 (N qi ) = 1, we then have ord 2 (S /Ω + f ) = r() 1 14

15 by Lea 2.3. Now assue that r() 2, and that this theore has been proved for all products of less than r() such pries q i, and note that we have assued the Manin constant is odd, so we have that ord 2 (T d /Ω + f ) = r(d) 1, with 1 < d < and d. Moreover, we also have ord 2 (a q 2χ d (q)) = 1. Consequently, we have that ord 2 ( (a q 2χ d (q)) T d /Ω + f ) = r() 1. q d Hence we have that ord 2 ( (a q 2χ d (q)) T d /Ω + f ) = r(), d q d 1<d< by noting that the nuber of the ters in this suation is even. So we ust have that is This copletes the proof of this theore. ord 2 (T /Ω + f ) = r() 1, ord 2 (L(E (), 1)/Ω + E () ) = r() 1. When the discriinant of E is positive, we have the following parallel result. Theore 4.3. Let E be an optial elliptic curve over Q with conductor C, and with odd Manin constant. Assue that E has positive discriinant, and satisfies E(Q)[2] 0 and ord 2 (L(E, 1)/Ω + f ) = 0. Let be any integer of the for = q 1q 2 q r(), with r() 1 and q 1,..., q r() arbitrary distinct odd pries congruent to 1 odulo 4, and with (, C) = 1. If ord 2 (N qi ) = 1 for 1 i r(), then L(E (), 1) 0, and we have ord 2 (L(E (), 1)/Ω + E () ) = r(). Proof. We will also prove the theore by induction on r(), of course we have got the arguent when r() = 1 in [21, Theore 1.7]. Note that ord 2 (L(E, 1)/Ω + f ) = 0 and ord 2 (N qi ) = 1, we then have ord 2 (S /Ω + f ) = r() by Lea 2.3. Now assue that r() 2, and that this theore has been proved for all products of less than r() such pries q i, and note that we have assued the Manin constant is odd, so we have that ord 2 (T d /Ω + f ) = r(d), with 1 < d < and d. Moreover, we also have ord 2 (a q 2χ d (q)) = 1. Consequently, we have that ord 2 ( (a q 2χ d (q)) T d /Ω + f ) = r(). q d 15

16 Hence we have that ord 2 ( d (a q 2χ d (q)) T d /Ω + f ) = r() + 1, q d 1<d< by noting that the nuber of the ters in this suation is even. So we ust have by the following equation S /Ω + f + d q d 1<d< ord 2 (T /Ω + f ) = r(), (a q 2χ d (q)) T d /Ω + f + T /Ω + f = 2r()+1 Ψ, which is deduced fro Theore 3.2. Hence we have This copletes the proof of this theore. ord 2 (L(E (), 1)/Ω + E () ) = r(). This copletes the proof of Theore 1.1 by cobining the above two theores and the celebrated theores of Gross Zagier and Kolyvagin part of the Birch Swinnerton-Dyer conjecture In this section, we will prove that the 2-part of the Birch Swinnerton-Dyer conjecture holds for soe certain failies of the quadratic twists of elliptic curves in the previous section. In particular, we will prove the following result, cobining with the nonvanishing result in Theore 1.1, to give a proof of Theore 1.5. Proposition 5.1. Let M = q 1 q r be a square free product of r pries in S. (1) Then E (M) (Q)[2] = Z/2Z. (2) Let l 0 C be the prie such that ord 2 (c l0 (E)) = 1. Assue l 0 splits in Q( M) and ord 2 ( l c l(e)) = 1. Then { ord 2 (c l (E (M) 0, if l l 0, and l M, )) = 1, if l = l 0, or l M. In particular, ord 2 ( l c l(e (M) )) = r + 1. (3) Assue further that Sel 2 (E)[2] = Z/2Z and X(E )[2] = 0. If all pries l 2C split in Q( M), then 1 di Sel 2 (E (M) ) 2. In particular, if X(E (M) ) is finite, then X(E (M) )[2] = 0 and Sel 2 (E (M) ) = Z/2Z. Proof. (1) It follows fro the facts that E[2] = E (M) [2] as G Q -odules and E(Q)[2] = Z/2Z. (2) First consider l l 0 and l M. Let E and E (M) be the Néron odel over Z l of E and E (M) respectively. Notice that E (M) /Q l is the unraified quadratic twist 16

17 of E (M). Since Néron odels coute with unraified base change, we know that the coponent groups Φ E and Φ E (M) are quadratic twists of each other as Gal(F l /F l )- odules. In particular, Φ E [2] = Φ E (M)[2] as Gal(F l /F l )-odules and thus Φ E (F l )[2] = Φ E (M)(F l )[2]. It follows that c l (E) and c l (E (M) ) have the sae parity, and hence c l (E (M) ) is odd. Next consider l M. Since E (M) has additive reduction at l and l is odd, we know that Φ E (M)(F l )[2] = E (M) (Q l )[2]. On the other hand, E (M) (Q l )[2] = E(Q l )[2] = E(F l )[2], which is Z/2Z since l S. Thus ord 2 (c l (E (M) )) = 1 for any l M. Finally consider l = l 0. By our extra assuption that l 0 is split in Q( M), we know that E (M) /Q l and E/Q l are isoorphic, hence c l (E (M) ) = c l (E), which has 2-adic valuation 1. (3) We use the following well-known exact sequence relating the 2-Seler group and φ, ˆφ-Seler groups (see [17, Lea 6.1]):- 0 A (K)[ ˆφ]/φ(A(K)[2]) S (φ) (A/K) Sel (2) (A/K) S ( ˆφ) (A/K ) X(A/K )[ ˆφ]/φ(X(A/K)[2]) 0. By our assuption Sel 2 (E) = Z/2Z and X(E )[2] = 0, it follows fro the above exact sequence that Sel φ (E) = E (Q)[φ]/φ(E(Q)[2]) = Z/2Z, Sel ˆφ(E ) = Sel 2 (E) = Z/2Z. By abuse of notation we denote the 2-isogeny E (M) E (M) again by φ (note that E (M) = E (M) ). We first clai that the isoorphis of G Q -representations E (M) [φ] = E[φ] induces an isoorphis of φ-seler groups Sel φ (E (M) ) = Sel φ (E). For v a place of Q, we denote the local condition defining the φ-seler group Sel φ (E) to be L v (E) := i(e (Q v )/φ(e(q v ))) H 1 (Q v, E[φ]). To show the clai it suffices to prove for any v, L v (E (M) ) = L v (E). We now prove the clai by the following 4 cases. (1) For v 2CM, then both E and E have good reduction at v 2 and hence L v (E (M) ) = L v (E) = H 1 ur(q v, E[φ]) is the unraified condition. (2) For v M, the desired equality of local condition at v follows fro [12, Lea 6.8]. 17

18 (3) For v 2C, by assuption we have v splits in Q( M), hence E (M) and E are isoorphic over Q v, and E (M) and E are isoorphic over Q v. The desired equality of local condition at v follows. (4) For v =, since M > 0, we know that E (M) and E are isoorphic over R, and E (M) and E are isoorphic over R. The desired equality of local condition at v again follows. This copletes the proof of the clai. Now by [12, Theore 6.4], we have Sel φ (E/Q) Sel ˆφ(E /Q) = v L v (E), 2 Sel φ (E (M) /Q) Sel ˆφ(E (M) /Q) = v L v (E (M) ). 2 Since we have shown that L v (E) = L v (E (M) ) for every place v of Q, we obtain Sel φ (E/Q) Sel ˆφ(E /Q) = Sel φ(e (M) /Q) Sel ˆφ(E (M) /Q). Hence Sel ˆφ(E (M) ) = Z/2Z. Now the well-known exact sequence for E (M) iplies that di Sel 2 (E (M) ) di Sel φ (E (M) ) + di Sel ˆφ(E (M) ) = = 2. On the other hand, E (M) (Q)[2] = Z/2Z, so di Sel 2 (E (M) ) 1. If X(E (M) )[2] is finite, then by the Cassels Tate pairing X(E (M) )[2] has square order, hence by the previous bounds it ust be 0, as desired. We are now ready to give the proof of Theore 1.5. Theore 5.2. (Theore 1.5) Let E and M be as in Theore 1.1. Assue further that (1) X(E )[2] = 0; (2) all pries l which divide 2C split in Q( M); (3) the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E. Then the 2-priary coponent of X(E (M) /Q) is zero, and the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E (M). Proof. If the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E, then ( l ord c ) l(e) X(E) 2 = 1. E(Q) tor 2 Since E(Q)[2] = Z/2Z and X(E)[2] has square order, we know that X(E)[2] = 0, Sel 2 (E) = Z/2Z and ord 2 ( l c l(e)) = 1. By Theore 1.1, we have ord 2 (L (alg) E (M), 1) = r 1, and X(E (M) ) is finite. The assuptions of Proposition 5.1 are all satisfied, and hence E (M) (Q)[2] = Z/2Z, ord 2 ( l (c l (E (M) )) = r + 1, X(E (M) )[2] = 0. 18

19 We then have ( l ord c ) l(e M ) X(E (M) ) 2 = r 1. E (M) (Q) tor 2 Therefore, the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E (M). 6. Applications In this section we will apply Theore 1.1 and Theore 1.5 to give soe failies of quadratic twists of elliptic curves which satisfy the 2-part of the exact Birch Swinnerton-Dyer forula. In particular, we give a full discussion of quadratic twists of X 0 (14), and soe analogous exaples on the quadratic twists of 34A1, 56B1, and 99C1 (in Creona s label), for which we will not give the proofs in details since they are siilar to the case of X 0 (14), and all the nuerical exaples are verified by Maga. Moreover, we also include a faily of elliptic curves satisfying the full Birch Swinnerton-Dyer conjecture. More exaples have been included in Wan s paper [20]. In the following, we always denote A to be the 2-isogenous curve of a given elliptic curve A. For each square free integer M, prie to the conductor of A, with M 1 od 4, as usual, we define L (alg) (A (M), 1) = L(A (M), 1)/Ω A (M) Quadratic twists of X 0 (14). Let A be the odular curve X 0 (14), which has genus 1, and which we view as an elliptic curve by taking [ ] to be the origin of the group law. It has a inial Weierstrass equation given by A : y 2 + xy + y = x 3 + 4x 6, which has non-split ultiplicative reduction at 2. Moreover, A(Q) = Z/6Z. The discriinant of A is Also, a siple coputation shows that Q(A[2]) = Q( 7). Writing L(A, s) for the coplex L-series of A, we have L(A, 1)/Ω + A = 1/6. Let q 1,..., q r be r 0 distinct pries, which are all 1 od 4. Recall that the L-function of an elliptic curve E over Q is defined as an infinite Euler product L(E, s) = q C (1 a q q s + q 1 2s ) 1 q C (1 a q q s ) 1 =: a n n s, 19

20 where q + 1 #E(F q ) if E has good reduction at q, 1 if E has split ultiplicative reduction at q, a q = 1 if E has non-split ultiplicative reduction at q, 0 if E has additive reduction at q. Here we give a result of the behavior of the coefficients a q of the L-function of elliptic curve A. Theore 6.1. Let q be an odd prie with (q, 14) = 1. Then we have that and a q a 2 = 1, a 7 = 1, 2 od 4 if q 1 od 8, 2 od 4 if q 3 od 8 and q is inert in Q( 7), 2 od 4 if q 5 od 8 and q splits in Q( 7), 0 od 4 if q 7 od 8, 0 od 4 if q 3 od 8 and q splits in Q( 7), 0 od 4 if q 5 od 8 and q is inert in Q( 7). Proof. The assertions for a 2 and a 7 are clear, since A has non-split ultiplicative reduction at 2 and split ultiplicative reduction at 7. Let A denote the 2-isogenous curve of A, which has a inial Weierstrass equation given by A : y 2 + xy + y = x 3 36x 70. It is easy to get that Q(A [2]) = Q( 2). For a q, first note that the 2-division field Q(A[2]) = Q( 7) and Q(A [2]) = Q( 2), and we have the sae L-function of A and A. So we have that A(F q )[2] = Z/2Z Z/2Z when q splits in Q( 7), and A (F q )[2] = Z/2Z Z/2Z when q splits in Q( 2). Since A(F q )[2] and A (F q )[2] are subgroups of A(F q ) and A (F q ), respectively. We have that 4 #A(F q ) and 4 #A (F q ). While for q is both inert in Q( 2) and Q( 7), we have that A(F q )[2] = Z/2Z. It is easy to copute that Q( 2) is a subfield of Q(A[4] ), where A[4] eans any one of the 4-division points which is deduced fro the non-trivial rational 2-torsion point of A(Q). But q is inert in Q( 2), that eans A(F q )[4] = A(F q )[2] = Z/2Z. Hence 2 #A(F q ), but 4 #A(F q ). Hence { 2 od 4 if q is both inert in Q( 2) and Q( 7), N q = #A(F q ) 0 od 4 if q splits in Q( 2) or Q( 7). Then all the assertions follow by applying a q = q + 1 N q. This copletes our proof. We then can apply Theore 1.1 to get the following result. 20

21 Theore 6.2. Let M be any integer of the for M = q 1 q 2 q r, r 1, with q 1,..., q r arbitrary distinct odd pries all congruent to 5 odulo 8, and inert in Q( 7). We then have ord 2 (L (alg) (A (M), 1)) = r 1. In particular, we have L(A (M), 1) 0. Proof. According to Theore 6.1, when q i 3, 5 od 8 and q i is inert in Q( 7), we have ord 2 (N qi ) = 1 for 1 i r. The theore then follows iediately by Theore 4.2. We next prove the 2-part of the Birch and Swinnerton-Dyer conjecture for all the twists E (M) in Theore 1.5. Note that A (M) has bad additive reduction at all pries dividing M. Write c q (A (M) ) for the Taagawa factor of A (M) at a finite odd prie q M. We then have that (6.1) ord 2 (c q (A (M) )) = ord 2 (#A(Q q )[2]). We apply the results in [5, 7] on the Taagawa factors of A (M), we then get the following result. Proposition 6.3. For all odd square-free integers M with (M, 14) = 1, we have (i) A (M) (R) has one connected coponent, (ii) ord 2 (c 2 (A (M) )) = 1, ord 2 (c 7 (A (M) )) = 0, (iii) ord 2 (c q (A (M) )) = 1 if q does not split in Q( 7), and (iv) ord 2 (c q (A (M) )) = 2 if q splits in Q( 7). Proof. Assertion (i) follows iediately fro the fact that Q(A[2]) = Q( 7). Assertion (ii) follows easily fro Tate s algorith. The reaining assertions involving odd pries q of bad reduction follow iediately fro (6.1), on noting that A(Q q )[2] is of order 2 or 4, according as q does not or does split in Q( 7), respectively. To obtain the 2-part of the Birch Swinnerton-Dyer forula, we also have to investigate the 2-part of X(A (M) /Q). If we just apply Theore 1.5, of course we will get that the 2-part of the Birch Swinnerton-Dyer forula holds for a faily of quadratic twists, provided both 2 and 7 split in Q( M), whence M has to have an even nuber of prie factors. However, a classical 2-descent of quadratic twists of X 0 (14) has been carried out earlier by Junhwa Choi, which yields that X(A (M) /Q)[2] is trivial, provided that all the prie factors of M are distinct pries congruent to 3, 5 odulo 8 and inert in Q( 7). We then can get the following theore. Theore 6.4. Let M be any integer of the for M = q 1 q 2 q r, r 1, with q 1,..., q r arbitrary distinct odd pries all congruent to 5 odulo 8, and inert in Q( 7). Then the 2-part of Birch and Swinnerton-Dyer conjecture is valid for A (M). Proof. Under the assuptions of the theore, X(A (M) /Q)[2] is trivial. Then cobining the results of Proposition 6.3, we have that ord 2 ( q M c q(a (M) )) = r. Note also 21

22 that #(A(Q)[2]) = 2. So we have ord 2 (#(X(A (M) )))+ord 2 ( p c p (A (M) ))+ord 2 (c (A (M) )) 2ord 2 (#(A (M) (Q))) = r 1. Hence, the 2-part of Birch and Swinnerton-Dyer conjecture holds for A (M). Here is the beginning of an infinite set of pries q satisfying the conditions in the above theore:- S = {5, 13, 61, 101, 157, 173, 181, 229, 269, 293, 349, 397,...} More nuerical exaples. For the following three exaples, the analogous ethods of quadratic twists of X 0 (14) would apply, so we will not give the detailed proofs here Quadratic twists of 34A1. Let A be the elliptic curve 34A1 with the inial Weierstrass equation given by A : y 2 + xy = x 3 3x + 1, which has split ultiplicative reduction at 2 and a 2 = 1. Moreover, A(Q) = Z/6Z and L (alg) (A, 1) = 1/6. The discriinant of A is Also, a siple coputation shows that Q(A[2]) = Q( 17) and Q(A [2]) = Q( 2). Here is the beginning of an infinite set of pries q which are congruent to 1 odulo 4 and inert in both the fields Q( 17) and Q( 2):- S = {5, 29, 37, 61, 109, 173, 181, 197, 269, 277, 317, 397,...}. Let M = q 1 q 2 q r, be a product of r distinct pries in S. We then have ord 2 (L (alg) (A (M), 1)) = r 1, and the 2-part of Birch and Swinnerton-Dyer conjecture is valid for all these twists Quadratic twists of 56B1. Let A be the elliptic curve 56B1 with the inial Weierstrass equation given by A : y 2 = x 3 x 2 4, which has potentially supersingular reduction at 2 and a 2 = 0. Moreover, A(Q) = Z/2Z and L (alg) (A, 1) = 1/6. The discriinant of A is Also, a siple coputation shows that Q(A[2]) = Q( 7) and Q(A [2]) = Q( 2). Here is the beginning of an infinite set of pries q which are congruent to 1 odulo 4 and inert in both the fields Q( 7) and Q( 2):- S = {5, 13, 61, 101, 157, 173, 181, 229, 269, 293, 349, 397,...}. Let M = q 1 q 2 q r, be a product of r distinct pries in S. We then have ord 2 (L (alg) (A (M), 1)) = r 1, 22

23 and the 2-part of Birch and Swinnerton-Dyer conjecture is valid for all these twists Quadratic twists of 99C1. Let A be the elliptic curve 99C1 with the inial Weierstrass equation given by A : y 2 + xy = x 3 x 2 15x + 8, which has good reduction at 2 and a 2 = 1. Moreover, A(Q) = Z/2Z and L (alg) (A, 1) = 1/2. The discriinant of A is Also, a siple coputation shows that Q(A[2]) = Q( 33) and Q(A [2]) = Q( 3). Here is the beginning of an infinite set of pries q which are congruent to 1 odulo 4 and inert in both the fields Q( 33) and Q( 3):- S = {5, 53, 89, 113, 137, 257, 269, 317, 353, 389,...}. Let M = q 1 q 2 q r, be a product of r distinct pries in S. We then have ord 2 (L (alg) (A (M), 1)) = r 1, and the 2-part of Birch and Swinnerton-Dyer conjecture is valid for all these twists Exaples satisfying the full Birch Swinnerton-Dyer conjecture. Let A be the elliptic curve 46A1 with the inial Weierstrass equation given by A : y 2 + xy = x 3 x 2 10x 12, which has non-split ultiplicative reduction at 2 and a 2 = 1, a 3 = 0. Moreover, A(Q) = Z/2Z and L (alg) (A, 1) = 1/2. The discriinant of A is The Taagawa factors c 2 = 2, c 23 = 1. Also, a siple coputation shows that Q(A[2]) = Q( 23) and Q(A [2]) = Q( 2). Here is the beginning of an infinite set of pries q which are congruent to 1 odulo 4 and inert in both the fields Q( 23) and Q( 2), and satisfy a q 0:- S = {5, 37, 53, 61, 149, 157, 181, 229, 293, 373,...}. Let M = q 1 q 2 q r be a product of r distinct pries in S. By Theore 1.1, we have L(A (M), 1) 0, and ord 2 (L (alg) (A (M), 1)) = r 1. If we carry out a classical 2-descent on A (M), one shows easily that the 2-priary coponent of X(E (M) /Q) is zero and ord 2 (c qi ) = 1 for 1 i r, and therefore the 2- part of the Birch and Swinnerton-Dyer conjecture holds for E (M). Alternatively, we can just apply Theore 1.5, and take the nuber of prie factors of M, say r(m), to be even, then the assuption that both 2 and 23 split in Q( M) will hold, whence we can also verify the 2-part of the Birch and Swinnerton-Dyer conjecture. Then cobining with the result in [20, Theore 9.3], the full Birch and Swinnerton-Dyer conjecture is valid for A (M). Hence the full Birch and Swinnerton-Dyer conjecture is verified for infinitely any elliptic curves. 23

24 References [1] L. Cai, Y. Chen, Y. Liu, Heegner points on odular curves, Trans. Aer. Math. Soc. 370 (2018), no. 5, [2] L. Cai, J. Shu, Y. Tian, Explicit Gross-Zagier and Waldspurger forulae, Algebra Nuber Theory 8 (2014), no. 10, [3] L. Cai, C. Li, S. Zhai, On the 2-part of the Birch and Swinnerton-Dyer conjecture for quadratic twists of elliptic curves II, in preparation. [4] K. Česnavičius, The Manin constant in the seistable case, arxiv: (2017). [5] J. Coates, Lectures on the Birch Swinnerton-Dyer conjecture, Notices of the ICCM, [6] J. Coates, Y. Li, Y. Tian, S. Zhai, Quadratic twists of elliptic curves, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, [7] J. Coates, The conjecture of Birch and Swinnerton-Dyer, Open probles in atheatics, , Springer, [Cha], [8] J. Creona, Algoriths for odular elliptic curves, Cabridge University Press, [9] J. Creona, Manin constants and optial curves: conductors , (2016). [10] B. H. Gross, D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, [11] B. H. Gross, Kolyvagin s work on odular elliptic curves, L-functions and arithetic (Durha, 1989), , London Math. Soc. Lecture Note Ser., 153, Cabridge Univ. Press, Cabridge, [12] Z. Klagsbrun, Seler ranks of quadratic twists of elliptic curves with partial rational twotorsion, Trans. Aer. Math. Soc. 369 (2017), no. 5, [13] V. Kolyvagin, Finiteness of E(Q) and X(E, Q) for a subclass of Weil curves, Math. USSR- Izv. 32 (1989), [14] D. Kriz, C. Li, Congruences between Heegner points and quadratic twists of elliptic curves, arxiv: (2016). [15] D. Kriz, C. Li, Prie twists of elliptic curves, arxiv: (2017). [16] Ju. I. Manin, Parabolic points and zeta-functions of odular curves, Math. USSR-Izv. 6 (1972), [17] E. F. Schaefer, M. Stoll, How to do a p-descent on an elliptic curve, Trans. Aer. Math. Soc. 356 (2004), no. 3, [18] A. Sith, 2 -Seler groups, 2 -class groups, and Goldfeld s conjecture, arxiv: v2 (2017). [19] J. T. Tate, The arithetic of elliptic curves, Invent. Math. 23 (1974), [20] X. Wan, Iwasawa ain conjecture for supersingular elliptic curves, arxiv: v5 (2018). [21] S. Zhai, Non-vanishing theores for quadratic twists of elliptic curves, Asian J. Math. 20 (2016), no. 3, Li Cai Yau Matheatical Sciences Center Tsinghua University Beijing China lcai@ath.tsinghua.edu.cn 24

25 Chao Li Departent of Matheatics Colubia University 2990 Broadway, New York, NY U.S.A. Shuai Zhai Institute for Advanced Research & School of Matheatics Shandong University Jinan, Shandong China 25