aa r X i v : . [ m a t h . N T ] F e b ON A CONJECTURE OF AGASHE
MENTZELOS MELISTAS
Department of Mathematics, University of Georgia, Athens, GA 30602,email: [email protected]
Abstract.
Let E/ Q be an optimal elliptic curve, − D be a negative fundamental discriminantcoprime to the conductor N of E/ Q and let E − D / Q be the twist of E/ Q by − D . A conjectureof Agashe predicts that if E − D / Q has analytic rank , then the square of the order of thetorsion subgroup of E − D / Q divides the product of the order of the Shafarevich-Tate group of E − D / Q and the orders of the arithmetic component groups of E − D / Q , up to a power of . Thisconjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyerconjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more generalstatement without using the optimality hypothesis. Introduction
Let E/ Q be an optimal elliptic curve and let − D be a negative fundamental discriminant, i.e., − D is negative and is the discriminant of a quadratic number field. Assume that D is coprimeto the conductor N of E/ Q and let E − D / Q denote the twist of E/ Q by − D . Let L ( E − D , s ) bethe L -function of E − D / Q and assume that L ( E − D , = 0 , i.e., that E − D / Q has analytic rank . In [1], under some additional mild hypotheses, Agashe proved that L ( E − D , E − D ) ∈ Z [ ] . Thesecond part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of rank predictsthat L ( E − D , E − D ) = | X ( E − D / Q ) | · Q p c p ( E − D ) | E − D ( Q ) | , see Conjecture . of [1]. Here X ( E − D / Q ) denotes the Shafarevich-Tate group of E − D / Q and c p ( E − D ) is the order of the arithmetic component group of E − D / Q at p , also called theTamagawa number of E − D / Q at p . This, along with results of [25], led Agashe, in Section . of the appendix of [25], to propose the following conjecture. Conjecture 1.1.
Let E/ Q be an optimal elliptic curve of conductor N and let − D be anegative fundamental discriminant such that D is coprime to N . Let E − D / Q denote the twistof E/ Q by − D . Suppose that L ( E − D , = 0 . Then | E − D ( Q ) | divides | X ( E − D / Q ) | · Y p | N c p ( E − D ) , up to a power of 2.Theorem 1.2 below, which is the main theorem of this paper, implies Conjecture 1.1 (seeCorollary 3.19 for the proof of this implication). We note that Conjecture 1.1 and, hence, our Date : February 26, 2021.
Mathematics Subject Classification.
Primary 11G05; Secondary 11G07,11G40. heorem can be seen as further evidence for the second part of the Birch and Swinnerton-Dyerconjecture in the analytic rank zero case. Theorem 1.2. (see Corollary 2.6 and Theorem 3.18)
Let E/ Q be an elliptic curve ofconductor N and let d = ± be a square-free integer such that d is coprime to N .(i) If d = ± , then the torsion subgroup of E d ( Q ) contains only points of order dividing .Assume also that L ( E d , = 0 . Then the groups E d ( Q ) and | X ( E d / Q ) | are finite and, up to apower of , the following conditions hold.(ii) If d = 3 , then | E d ( Q ) | divides | X ( E d / Q ) | · Y p | N c p ( E d ) . (iii) If d = 3 , then | E d ( Q ) | divides | X ( E d / Q ) | · Y p | N c p ( E d ) . Remark 1.3.
If moreover d = − , then Part ( i ) of Theorem 1.2 implies Part ( ii ) . We alsonote that the case d = ± is by far the most difficult case to treat in the proof. Remark 1.4.
Clearly, the analogue of Part (i) of Theorem 1.2 with d = 1 does not hold, sincethere exist elliptic curves E/ Q of rank which have Q -rational torsion points of order 3, 5, or7. Consider now the elliptic curve with LMFDB [21] label 176.a2. Then E − / Q , the twist of E/ Q by − , is the elliptic curve with LMFDB label 44.a2 and E − ( Q ) = Z / Z . This provesthat Part ( i ) of Theorem 1.2 is false if d = − .Moreover, | X ( E − / Q ) | = 1 and Q p c p ( E − ) = c ( E − ) = 3 . Therefore, | E − ( Q ) | = 9 while | X ( E − / Q ) | Q p c p ( E − ) = 3 . This proves that the analogues of Theorem 1.2, Parts ( ii ) , ( iii ) are false when d = − .The proof of Part ( i ) of Theorem 1.2 is postponed to Corollary 2.6 and the proof of Parts ( ii ) and ( iii ) is completed in Theorem 3.18. To prove Parts ( ii ) and ( iii ) we prove Theorem 3.1,which has applications to another consequence of the Birch and Swinnerton-Dyer conjecture,discovered by Agashe and Stein in [2] (see Remark 3.17).This paper is organized as follows. First, we prove that if d = ± , ± , then E d ( Q ) onlycontains points of order a power of . This explains some observations made by Agashe, inthe case where d is a negative fundamental discriminant, which can be found on page of [1].Next, we prove that for d = − or , E d ( Q ) cannot contain points of order or . All this isdone in section 2. Finally, in section 3 we consider in detail the case where d = ± and E d ( Q ) contains a point of order . Acknowledgements.
This work is part of the author’s doctoral dissertation at the Universityof Georgia. The author would like to thank his advisor Dino Lorenzini for valuable help duringthe preparation of this work as well as for many useful suggestions on improving the expositionof this manuscript. The author thanks the anonymous referee for many insightful comments.2.
Restrictions on the torsion subgroup
Let E/ Q be an elliptic curve and let d be any non-zero integer. Recall that the twist of E/ Q by d , which is denoted by E d / Q , is an elliptic curve over Q that becomes isomorphicover Q ( √ d ) to E Q ( √ d ) / Q ( √ d ) but which is not in general isomorphic to E/ Q over Q . Moreexplicitly, if E/ Q is given by a Weierstrass equation of the form y = x + ax + b , then E d / Q is given by dy = x + ax + b or, after making a change of variables, by y = x + d ax + d b . n this section, we obtain results that put restrictions on the rational torsion subgroup of E d / Q provided that d and the conductor of E/ Q are coprime. Proposition 2.1.
Let E/ Q be an elliptic curve defined over Q of conductor N , and let d = ± be any square-free integer that is coprime to N . Then E d ( Q ) cannot contain a rational pointof order or .Proof. We prove the proposition by contradiction. Let ℓ be equal to or , assume that E d ( Q ) contains a point of order ℓ , and that d = ± . Since E d / Q has a Q -rational point of order ℓ ,Proposition . in Chapter XV of [6] implies that E d / Q is semi-stable away from ℓ .Assume that there exists an odd prime p with p | d . Then p ∤ N because d and N arecoprime. Therefore, Proposition of [4] implies that E d / Q has reduction type I ∗ modulo p .As a result, since E d / Q is semi-stable away from ℓ , we obtain that p can only be equal to ℓ .However, Proposition 2.2 below implies that E d / Q cannot have reduction of type I ∗ modulo ℓ .Therefore, d = ± since d is square-free.Assume now that d = − or . We will arrive at a contradiction. Since we assume that d and N are coprime, we get that ∤ N . Since E/ Q has good reduction modulo , by Table IIof [4] (see also the hypotheses for this table at the bottom of page 58 of [4]) we obtain that E d / Q has reduction of type I ∗ or II modulo . In [4], good reduction is denoted as usual withthe symbol I , and our twist E d is denoted by E χ . Note that Table II is independent of theexistence of a torsion point. We now use our assumption that E d has a Q -rational point oforder ℓ . Since E d has a Q -rational point of order ℓ , it has semi-stable reduction modulo ,which is a contradiction. This proves that E d ( Q ) cannot contain a point of order ℓ . (cid:3) A proof of the following proposition can be found in the forthcoming paper [24].
Proposition 2.2.
Let E/ Q be an elliptic curve with a Q -rational point of order p , where p isa prime number. Assume that E/ Q has additive reduction modulo p . Then p ≤ , and(i) If p = 5 , then E/ Q can only have reduction of type II or III modulo .(ii) If p = 7 , then E/ Q can only have reduction type II modulo . We now investigate whether E d ( Q ) can contain a point of order . Let E/ Q be an ellipticcurve with a Q -rational point of order . Then, by translating that point to (0 , and performinga change of variables if necessary (see Remark . in Section . of [17]) we obtain a Weierstrassequation for E/ Q of the form y + cxy + dy = x , with c, d ∈ Q . If u ∈ Z , then the transformation ( x, y ) → ( xu , yu ) gives a new Weierstrassequation of the same form with c replaced by uc and with d replaced by u d (see page 185 of[30]). Therefore, by picking u to be a large power of the product of the denominators of c, d (ifany), we can arrange that c, d ∈ Z . Moreover, by applying the transformation ( x, y ) → ( x, − y ) if necessary, we can arrange that d > . We now show that we can find a new Weierstrassequation of the above form with coefficients, which we will call a and b below, that also havethe property that for every prime q either q ∤ a or q ∤ b . If there is no prime q such that q | c and q | d , then set a = c and b = d . Otherwise, let q , ..., q s be the set of primes such that q i | c and q i | d , and let n i = min { ord q i ( c ) , ⌊ ord qi ( d )3 ⌋} , where ⌊−⌋ is the floor function. Let u = Q si =1 q n i i . By applying the transformation ( x, y ) → ( u x, u y ) , we obtain a new Weierstrassequation of the form y + cu xy + du y = x . etting a = cu and b = du we see that a, b ∈ Z , b > , and for every prime q either q ∤ a or q ∤ b .Therefore, we have proved that we can choose a Weierstrass equation for E/ Q of the form y + axy + by = x , ( . )where a, b are integers, b > , and for every prime q either q ∤ a or q ∤ b . Also, we must have a − b = 0 , since the discriminant of Equation (2 . is ∆ = b ( a − b ) and we also have c = a ( a − b ) . Proposition 2.4. (See Proposition . and Lemma . of [20] ) Let E/ Q be an elliptic curvegiven by Equation (2 . . Write D := a − b and let p be any prime (note that either p ∤ a or p ∤ b ). Then the reduction of E/ Q modulo p is determined as follows:(i) If ord p ( a ) ≤ ord p ( b ) : ord p ( a ) < ord p ( b ) → split I ord p ( b ) , c p ( E ) = 3 ord p ( b )3 ord p ( a ) = ord p ( b ) → ( ord p ( D ) > → I ord p ( D ) ord p ( D ) = 0 → Good reduction I (ii) If ord p ( a ) > ord p ( b ) : ord p ( b ) = 0 → ( p = 3 → Go to ( iii ) p = 3 → Good reduction I ord p ( b ) = 1 → IV, c p ( E ) = 3 ord p ( b ) = 2 → IV ∗ , c p ( E ) = 3 . (iii) If p = 3 and ord p ( a ) > ord p ( b ) : ord p ( D ) = → II or III → II → IV n → I ∗ n − , for n ≥ . Proposition 2.5.
Let E/ Q be an elliptic curve over Q of conductor N and let d = ± , ± bea square-free integer coprime to N . Then E d ( Q ) cannot contain a point of order .Proof. The proof is by contradiction. Assume that E d ( Q ) contains a point of order and that d has a prime divisor p = 3 . Assume first that p = 2 . Since p ∤ N , Proposition of [4] implies that E d / Q has reduction of type I ∗ modulo p . However, this is impossible because by Proposition2.4, E d ( Q ) cannot have reduction of type I ∗ modulo p for p = 3 . Assume now that p = 2 . Since ∤ N , we obtain that E/ Q has good reduction modulo and, hence, by Table II of [4] (see alsothe hypotheses for this table at the bottom of page 58 of [4]), we get that E d / Q has reductionof type I ∗ or II modulo . However, E d / Q cannot have reduction of type I ∗ or II modulo byProposition 2.4. This implies that ∤ d . Therefore, d can only be divisible by so d = ± or ± . (cid:3) Corollary 2.6.
Let E/ Q be an elliptic curve over Q of conductor N and let d = ± be asquare-free integer such that d is coprime to N . Then(i) E d ( Q ) tors contains only points of order α β where α, β ≥ .(ii) If d = ± , then E d ( Q ) tors contains only points of order dividing .Proof. By a Theorem of Mazur (see [23] Theorem (8) ) the only primes that can divide | E d ( Q ) tors | are , , and . Therefore, Proposition 2.1 implies Part ( i ) . Moreover, Propositions 2.1 and2.5 imply that if d = ± , ± , then E d ( Q ) tors contains only points of order dividing α , where α ≥ . Finally, Part ( ii ) follows by applying Mazur’s Theorem (see [23] Theorem (8) ). (cid:3) . Elliptic curves over Q with a rational point of order 3 In this section, we prove Parts ( ii ) and ( iii ) of Theorem 1.2 in Theorem 3.18. To achievethis goal, we will use Theorem 3.1 below, which might be of independent interest (see Remark3.17). Theorem 3.1.
Let E/ Q be an elliptic curve with a Q -rational point of order . Assume thatthe analytic rank of E/ Q is and that E/ Q has reduction of type I ∗ n modulo , for some n ≥ .(a) If E/ Q has semi-stable reduction away from , then | | X ( E/ Q ) | · Q p c p ( E ) .(b) If E/ Q has more than two places of additive reduction, then | Q p c p ( E ) .(c) If E/ Q has exactly two places of additive reduction, then | | X ( E/ Q ) | · Q p c p ( E ) .Proof of Theorem 3.1. Let j E be the j -invariant of E/ Q . Since some of our arguments belowdo not work for j E = 0 or , we first handle these cases with the following claim. Claim 3.2.
Let E/ Q be an elliptic curve with j -invariant or that has a Q -rational pointof order . Then E/ Q cannot have reduction of type I ∗ n modulo , for any n ≥ . Proof.
Let E/ Q be an elliptic curve with j -invariant j E = 0 . Since, j E = c ∆ we obtain that c = 0 . Tableau II of [26] implies that E/ Q can only have reduction type II, II ∗ , III, III ∗ , IV,or IV ∗ modulo . Therefore, E/ Q cannot have reduction type I ∗ n modulo , for any n ≥ , asneeded.Let E/ Q be an elliptic curve with j -invariant j E = 1728 and with a Q -rational point of order . Since E/ Q has a Q -rational point of order , it has a Weierstrass equation of the form (2 . . Assume that E/ Q has reduction type I ∗ n modulo , for some n ≥ , and we will find acontradiction. Proposition 2.4, Parts ( i ) and ( ii ) imply that ∤ b and | a . Since j E = c ∆ and j E = 1728 , we obtain that c = 1728∆ . Writing c and ∆ in terms of a and b , we get that ( a ( a − b )) = 1728 b ( a − b ) and, hence, ( a ) + 3ord ( a − b ) = ord (1728) + 3ord ( b ) + ord ( a − b ) . Since | a and ∤ b , we get that ord ( a − b ) = 1 . Therefore, ( a ) + 3 = 3 + 0 + ord ( a − b ) , which yields ( a ) = ord ( a − b ) . If ord ( a ) = 1 , then ord ( a − b ) = 3 , which is a contradiction because Proposition 2.4, Part ( iii ) implies that E/ Q has reduction type II or III. If ord ( a ) ≥ , then ord ( a − b ) = 3 because ord ( b ) = 0 , and this is again a contradiction. This proves our claim. (cid:3) Claim 3.2 implies that to prove our theorem, it is enough to assume that j E = 0 , . Weassume from now on that j E = 0 , .Since E/ Q has a Q -rational point of order , it can be given by an equation of the form (2 . where (0 , is a Q -rational point of order . Let b E := E/ < (0 , > and let φ : E → b E be theassociated -isogeny. We denote by ˆ φ the dual isogeny.We recall now some important facts in preparation for the proof of Lemma 3.6 below. Thefollowing formula is due to Cassels (see Theorem of [18] or Theorem . of [20]) | Sel ( φ ) ( E/ Q ) || Sel ( ˆ φ ) ( b E/ Q ) | = | E [ φ ]( Q ) | Ω( b E ) Q p c p ( b E ) | b E [ ˆ φ ]( Q ) | Ω( E ) Q p c p ( E ) . (3.3) ere Sel ( φ ) ( E/ Q ) and Sel ( ˆ φ ) ( b E/ Q ) are the φ - and ˆ φ -Selmer groups, respectively. Moreover, Ω( E ) := Z E ( R ) | ω min | and Ω( b E ) := Z b E ( R ) | b ω min | , where ω min and b ω min are two minimal invariantdifferentials on E/ Q and b E/ Q respectively (see Section III.1 of [30] and page 451 of [30]). Theinterested reader can consult section X. of [30] concerning the definitions of Selmer groups ofelliptic curves as well as relevant background.Since E [ φ ]( Q ) ∼ = Z / Z , we get that b E [ ˆ φ ]( Q ) is trivial by the Weil pairing (see [30] exerciseIII.3.15 on page ). Hence, | E [ φ ]( Q ) || b E [ ˆ φ ]( Q ) | = 3 . (3.4)We know that φ ∗ b ω min = λω min , for some λ ∈ Z (see the second paragraph of page of[32]). Moreover, the fact that ˆ φ ◦ φ = [3] implies that | λ | = 1 or . Therefore, | ω min φ ∗ b ω min | is equalto or − . By Lemma . of [11], Ω( E )Ω( b E ) = | ker( φ : E ( R ) −→ b E ( R )) || coker( φ : E ( R ) −→ b E ( R )) | · | ω min φ ∗ b ω min | Since ker( φ ) ⊂ E ( R ) and φ has degree , Proposition . of [11] implies that | ker φ : E ( R ) −→ b E ( R ) || coker( φ : E ( R ) −→ b E ( R )) | =3 . Therefore, since | ω min φ ∗ b ω min | is equal to or − , Ω( E )Ω( b E ) = 1 or . (3.5)The following lemma will be used repeatedly in what follows. In pursuing this idea, we wereinspired by work of Byeon, Kim, and Yhee in [3]. Lemma 3.6.
Let E/ Q be an elliptic curve given by a Weierstrass equation of the form (2 . and assume that E/ Q has analytic rank . Let b E := E/ < (0 , > and let φ : E → b E be theassociated -isogeny.(i) If ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ or dim F (Sel ( φ ) (E / Q )) ≥ , then divides | X ( E/ Q ) | .(ii) If | Sel ( φ ) ( E/ Q ) || Sel (ˆ φ ) ( b E/ Q ) | = 1 , then divides | X ( E/ Q ) | .Proof. Since the analytic rank of E/ Q is zero, work of Gross and Zagier, on heights of Heegnerpoints [14], as well as work of Kolyvagin, on Euler systems [19], imply that E/ Q has (algebraic)rank and that X ( E/ Q ) is finite (see Theorem . of [8] for a sketch of the proof). Thus thestatements of the lemma make sense. Proof of ( i ) : First we prove that if dim F (Sel ( φ ) ( E/ Q )) ≥ , then divides | X ( E/ Q ) | . Thereis a short exact sequence −→ E ( Q ) / E ( Q ) −→ Sel ( E/ Q ) −→ X ( E/ Q )[3] −→ (see Theorem X. . of [30]). Therefore, using the fact that E ( Q ) / E ( Q ) ∼ = Z / Z , since E/ Q has rank and E [3]( Q ) ∼ = Z / Z , we see that dim F (Sel ( E/ Q )) ≥ implies that X ( E/ Q )[3] has a nontrivial element. Since the order of X ( E/ Q ) is a square (see [30] Corollary . . ) if X ( E/ Q )[3] has a nontrivial element, then divides | X ( E/ Q ) | . By Corollary 1, section 2 of [18]and the fact that b E [ ˆ φ ]( Q ) is trivial, we obtain that the natural map Sel ( φ ) ( E/ Q ) −→ Sel ( E/ Q ) is injective. Therefore, if dim F (Sel ( φ ) ( E/ Q )) ≥ , then divides | X ( E/ Q ) | . inally, if ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ , then Equation ( ) , combined with ( ) and ( ) , impliesthat dim F (Sel ( φ ) ( E/ Q )) ≥ so by the first part of the proof, we find that divides | X ( E/ Q ) | .This concludes the proof of Part ( i ) . Proof of ( ii ) : By Corollary 1 of section of [18] and the fact that b E [ ˆ φ ]( Q ) is trivial there isan exact sequence: → Sel ( φ ) ( E/ Q ) → Sel ( E/ Q ) → Sel ( ˆ φ ) ( b E/ Q ) → X ( b E/ Q )[ ˆ φ ] /φ ( X ( E/ Q )[3]) → . (3.7)Moreover, by Theorem . of [28], the Cassels-Tate pairing on X ( b E/ Q ) restricts to a non-degenerate alternating pairing on X ( b E/ Q )[ ˆ φ ] /φ ( X ( E/ Q )[3]) (see also Theorem of [13]).Therefore, it follows that dim F X ( b E/ Q )[ ˆ φ ] /φ ( X ( E/ Q )[3]) is even.Let α := dim F X ( b E/ Q )[ ˆ φ ] /φ ( X ( E/ Q )[3]) . Since | Sel ( φ ) ( E/ Q ) || Sel (ˆ φ ) ( b E/ Q ) | = 1 by hypothesis, we get that β := dim F Sel ( φ ) ( E/ Q ) = dim F Sel ( ˆ φ ) ( b E/ Q ) . If γ := dim F Sel ( E/ Q ) , then by the exact sequence ( ) we obtain that β − γ + β − α = 0 and, therefore, γ must be even. There is a short exact sequence −→ E ( Q ) / E ( Q ) −→ Sel ( E/ Q ) −→ X ( E/ Q )[3] −→ (see Theorem X. . of [30]). Since E/ Q has a Q -rational point of order three and rank , weobtain that E ( Q ) / E ( Q ) ∼ = Z / Z and, hence, γ ≥ . Since γ is even, we obtain that γ ≥ .Since γ = dim F Sel ( E/ Q ) ≥ and E ( Q ) / E ( Q ) ∼ = Z / Z , X ( E/ Q )[3] is not trivial. Finally,since the order of X ( E/ Q ) is a square (see [30] Corollary . . ) if X ( E/ Q )[3] has a nontrivialelement, then divides | X ( E/ Q ) | . This concludes the proof of Part ( ii ) . (cid:3) We now collect some facts concerning root numbers. Let w ( E ) := Y p ∈ M Q w p ( E ) where w p ( E ) is the local root number of E/ Q at p and M Q is the set of places of Q . The number w ( E ) iscalled the global root number of E/ Q . By Theorem . of [8], X ( E/ Q ) is finite and the rankof E/ Q is . Consequently, by Theorem . of [10] we get that − rk ( E/ Q ) = w ( E ) .Recall that we assume that j E = 0 , . The local root number of E/ Q at p is as follows(see [5] page and [27] page for j E = 0 , and p ≥ , and [16] page 1051 for p = 2 or ) w ∞ ( E ) = − w p ( E ) = 1 if E/ Q has modulo p good or nonsplit multiplicative reduction .w p ( E ) = − E/ Q has modulo p split multiplicative reduction .w ( E ) = − E/ Q has modulo 3 reduction of type I ∗ n , n ≥ . Lemma 3.9.
Let E/ Q be an elliptic curve given by a Weierstrass equation of the form (2 . such that E/ Q has modulo reduction of type I ∗ n , for some n ≥ . If ∤ Q p c p ( E ) , then either b = 1 or b = r m for some prime r = 3 . roof. Since E/ Q has modulo reduction of type I ∗ n , for some n ≥ , Proposition 2.4 impliesthat ∤ b , as we now explain. First, if | b and ∤ a , then Proposition 2.4, Part ( i ) , impliesthat E/ Q has modulo multiplicative reduction, which is a contradiction. Moreover, if | b and | a , then, ∤ b so ord ( a ) > ord ( b ) . Therefore, using Proposition 2.4, Part ( ii ) , weobtain that E/ Q has modulo reduction of type IV or IV ∗ , which is again a contradiction.If a prime r divides b , then either r ∤ b or r ∤ a . Therefore, since r = 3 , by Proposition 2.4,Parts ( i ) and ( ii ) , E/ Q has reduction of type IV, IV ∗ or (split) I ord r ( b ) , and in any of thesecases | c r ( E ) . Hence, if ∤ Q p c p ( E ) , then b can have at most one prime divisor. (cid:3) Proof of Theorem 3.1 Part ( a ) . Assume that E/ Q is semi-stable away from . If divides Q p c p ( E ) , then Theorem 3.1, Part ( a ) is true. Therefore, we can assume from now on that ∤ Q p c p ( E ) and, hence, by Lemma 3.9 that either b = 1 or b = r m , for some prime r with r = 3 . We now split the proof into two cases depending on whether b = 1 or b = r m .Case 1: b = 1 . By a proposition of Hadano (see Theorem . of [15]) since b = 1 , b E/ Q isgiven by y + ( a + 6) xy + ( a + 3 a + 9) y = x . The discriminant of this equation is b ∆ = ( a − and b c = a ( a + 216) .We first show that b E/ Q has a prime q of split multiplicative reduction. By Proposition 2.4,Part ( iii ) , we obtain that | a because E/ Q has reduction of type I ∗ n modulo . Therefore, wecan write a = 3 a ′ for some integer a ′ . Then a + 3 a + 9 = 9(( a ′ ) + a ′ + 1) and ( a ′ ) + a ′ + 1 cannot be a power of unless a ′ = 1 , − because it is always non zero modulo . If ( a ′ ) + a ′ + 1 = 3 , then a ′ = 1 or − . Moreover, a ′ = 1 gives a = 3 , and together with b = 1 gives an equation of the form (2 . which has discriminant and, hence, is not an ellipticcurve. Therefore, a ′ = 1 is not allowed. If a ′ = − , which implies a = − , we obtain, combinedwith b = 1 , an elliptic curve E/ Q of the form (2 . which does not have reduction of type I ∗ n at 3 because a −
27 = ( − −
27 = 243 is not divisible by , see Proposition 2.4, Part ( iii ) .Moreover, if ( a ′ ) + a ′ + 1 = ± , then a ′ = − . If a ′ = − , which implies a = − , we obtain,combined with b = 1 , an elliptic curve E/ Q of the form (2 . which does not have reductionof type I ∗ n at 3 because a −
27 = ( − −
27 = − is not divisible by , see Proposition 2.4,Part ( iii ) . What this proves is that a + 3 a + 9 has a prime divisor not equal to , say q . Bylooking at the Weierstrass equation for b E/ Q we get that b E/ Q has split multiplicative reductionmodulo q . Indeed, if q | a + 3 a + 9 and q | a + 6 , then q | a + 3 a + 9 − ( a + 6) = a + 3 a + 9 − a − a −
36 = − a + 3) and, hence, q | a + 3 . Since q | a + 6 , we obtain that q = 3 , which is a contradiction. Thisproves that q ∤ a + 6 and, thus, b E/ Q has split multiplicative reduction modulo q .Since b E/ Q has split multiplicative modulo q , we get that c q ( b E ) = ord q (( a − ) =3ord q ( a − . Since E/ Q and b E/ Q are isogenous over Q , they have the same L -function(see Korollar 1 of [12]) and, hence, a q ( E ) = a q ( b E ) (see page of [9]). Moreover, a q ( b E ) = 1 because b E/ Q has split multiplicative reduction modulo q (see page of [9]), which implies that a q ( E ) = 1 . Therefore, b E/ Q has split multiplicative reduction modulo q and c q ( E ) = ord q (∆) .However, ∆ = a − so c q ( E ) = ord q ( a − , which implies that ord ( c q ( b E ) c q ( E ) ) > . e will now show that divides | X ( E/ Q ) |. By Part ( i ) of Lemma 3.6, it is enough to showthat ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ . We achieve this in the following claim.
Claim 3.10.
We have ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ . Proof.
First, we claim that because the analytic rank of E/ Q is zero, E/ Q has an even numberof places of split multiplicative reduction. Indeed, by 3.8 w ( E ) = 1 , w ( E ) = − , and w ∞ ( E ) = − . Moreover, E/ Q is semi-stable away from and by 3.8 for p = 3 we obtain that w p ( E ) = − if and only if E/ Q has split multiplicative reduction modulo p . This proves that E/ Q has aneven number of places of split multiplicative reduction.Let p = 3 be any prime such that E/ Q has multiplicative reduction modulo p . If E/ Q hasnonsplit multiplicative reduction modulo p , then by lines , , and of Theorem . of [11] weobtain that ord ( c p ( b E ) c p ( E ) ) = 0 . Note that in Theorem . of [11], b E is denoted by E ′ and δ, δ ′ are the valuations of the two discriminants. Assume now that E/ Q has split multiplicativereduction modulo p . Recall that we assume that b = 1 so ∆ = a − and b ∆ = ( a − . Iford p (∆) = γ , then ord p ( b ∆) = 3 γ . Therefore, by line of Theorem . of [11] we obtain thatord ( c p ( b E ) c p ( E ) ) = 1 . Since E/ Q is semi-stable away from , the above arguments prove that if p = 3 is a prime, then E/ Q has split multiplicative reduction modulo p if and only if ord ( c p ( b E ) c p ( E ) ) = 1 ,and ord ( c p ( b E ) c p ( E ) ) = 0 otherwise.We now claim that ord ( Q p c p ( b E ) Q p c p ( E ) ) is even. By line of Theorem . of [11], since E/ Q hasmodulo 3 reduction of type I ∗ n for some n ≥ , we obtain that ord ( c ( b E ) c ( E ) ) = 0 . Combining allthe above with the fact that E/ Q has an even number of places of split multiplicative reduction,we obtain that ord ( Q p c p ( b E ) Q p c p ( E ) ) is even.Finally, since ord ( c q ( b E ) c q ( E ) ) > and ord ( Q p c p ( b E ) Q p c p ( E ) ) is even, we get that ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ . Thisproves our claim. (cid:3) Case 2: b = r m for some prime r , and m > . Recall that we assume that E/ Q has semi-stable reduction outside of and reduction of type I ∗ n modulo , for some n ≥ . Lemma 3.9implies that r = 3 .We claim that E/ Q has split multiplicative reduction modulo r . Indeed, by our assumption E/ Q is semi-stable away from and r = 3 . Since ord r ( b ) > and we know that either r ∤ a or r ∤ b , we obtain that either ord r ( a ) > ord r ( b ) or ord r ( a ) = 0 . The case ord r ( a ) > ord r ( b ) gives that E/ Q has reduction of type IV or IV ∗ modulo r , by Part ( ii ) of Proposition 2.4, whichcontradicts our assumption that E/ Q has semi-stable reduction outside of . If ord r ( a ) = 0 ,then we obtain that ord r ( a ) < ord r ( b ) and, hence, by Proposition 2.4, Part ( i ) we get that E/ Q has split multiplicative reduction modulo r .If | m , then by Proposition 2.4, Part ( i ) we obtain that m | Q p c p ( E ) and, hence, | Q p c p ( E ) . Thus, if | m , then Part ( a ) of Theorem 3.1 is satisfied. Therefore, we can assumethat ∤ m in what follows. Finally, since ord r ( a ) < ord r ( b ) , by Theorem . of [20] we obtainord ( c r ( b E ) c r ( E ) ) = − . laim 3.11. The number ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and non-negative. Proof.
There exists at least one prime q different from r such that E/ Q has split multiplicativereduction modulo q . Indeed, by 3.8 w ( E ) = 1 , w r ( E ) = − , w ( E ) = − , and w ∞ ( E ) = − .Moreover, since E/ Q is semi-stable away from , by 3.8 for p = 3 , we get that w p ( E ) = − if and only if E/ Q has split multiplicative reduction modulo p . This proves that E/ Q hasan even number of places of split multiplicative reduction. Therefore, since E/ Q has splitmultiplicative reduction modulo r , we obtain that there is a prime q = r , such that E/ Q hassplit multiplicative reduction modulo q .Let p = 3 be any prime such that E/ Q has multiplicative reduction modulo p . If E/ Q hasnonsplit multiplicative reduction modulo p , then by lines , , or of Theorem . of [11] weobtain that ord ( c p ( b E ) c p ( E ) ) = 0 .The elliptic curve b E/ Q is given by a Weierstrass equation of the form y + axy − by = x − ( a + 27 b ) b, where a, b are as in (2 . (see [20] equation (3 . ). The discriminant of this Weierstrass equationis b ∆ = ( a − b ) b. We now prove that for any prime p = 3 , r, if E/ Q has split multiplicative reduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 1 . Assume that p = r is a prime such that E/ Q has split multiplicativereduction modulo p . Note that since b = r m and p = r , p ∤ b . Recall that ∆ = b ( a − b ) and b ∆ = b ( a − b ) . If ord p (∆) = γ , then ord p ( b ∆) = 3 γ . Therefore, by line of Theorem . of[11] we obtain that ord ( c p ( b E ) c p ( E ) ) = 1 .Recall that there exists a prime q different from r such that E/ Q has split multiplicativereduction modulo q . We have proved so far that ord ( c r ( b E ) c r ( E ) ) = − , ord ( c q ( b E ) c q ( E ) ) = 1 , and that forany prime p = 3 , r , if E/ Q has split multiplicative reduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 1 .Moreover, we have proved above that if E/ Q has nonsplit multiplicative reduction modulo p ,then ord ( c p ( b E ) c p ( E ) ) = 0 . Therefore, since E/ Q is semistable away from and has an even numberof places of split multiplicative reduction, we obtain that ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and non-negative,as claimed. (cid:3) If ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ , then Part ( i ) of Lemma 3.6 implies that divides | X ( E/ Q ) | . Other-wise, ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 . Since the degree of φ is , by Proposition of [18] we obtain thatord s ( Q p c p ( b E ) Q p c p ( E ) ) = 0 for every prime s = 3 . Therefore, if Ω( E )Ω( b E ) = 3 , then Equation ( ) impliesthat | Sel ( φ ) ( E/ Q ) || Sel (ˆ φ ) ( b E/ Q ) | = 1 and, hence, by Part ( ii ) of Lemma 3.6 we obtain that divides | X ( E/ Q ) | .If Ω( E )Ω( b E ) = 1 , then, since b = r m with ∤ m , the following lemma shows that divides | X ( E/ Q ) | .This concludes the proof of Part ( a ) of Theorem 3.1. (cid:3) emma 3.12. Let E/ Q be as in Theorem 3.1, and given by a Weierstrass equation as in (2 . .Assume moreover that there exists a prime r = 3 such that b = r m , for some integer m coprimeto . If ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 and Ω( E )Ω( b E ) = 1 , then divides | X ( E/ Q ) | .Proof. Since Ω( E )Ω( b E ) = 1 , by Equation ( ) , combined with ( ) , we get that | Sel ( φ ) ( E/ Q ) | = | Sel ( ˆ φ ) ( b E/ Q ) | · · Q p c p ( b E ) Q p c p ( E ) . Since ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 , we obtain that dim F Sel ( φ ) ( E/ Q ) = dim F Sel ( ˆ φ ) ( b E/ Q ) + 1 . Therefore,if we can show that dim F Sel ( ˆ φ ) ( b E/ Q ) ≥ , then we get that dim F Sel ( φ ) ( E/ Q ) ≥ and, hence,by Part ( i ) of Lemma 3.6 we obtain that divides | X ( E/ Q ) | .We now show that dim F Sel ( ˆ φ ) ( b E/ Q ) ≥ . There is a short exact sequence −→ E ( Q ) / ˆ φ ( b E ( Q )) −→ Sel ( ˆ φ ) ( b E/ Q ) −→ X ( b E/ Q )[ ˆ φ ] −→ (see Theorem X. . of [30] applied to ˆ φ : b E → E ). Recall that E/ Q has rank and that E ( Q ) contains a point of order . The rank of b E/ Q is because it is isogenous to E/ Q . Moreover,since b is not a cube, Theorem . of [15] implies that b E ( Q ) does not contain a point of order . Therefore, E ( Q ) / ˆ φ ( b E ( Q )) contains a point of order which injects into Sel ( ˆ φ ) ( b E/ Q ) . Thisproves that dim F (Sel ( ˆ φ ) ( b E/ Q )) ≥ . (cid:3) Proof of Theorem 3.1 Part ( b ) . We know that E/ Q has a Weierstrass equation as in (2 . . ByProposition 2.4 if p = 3 is any prime such that E/ Q has additive reduction, then p | b , thereduction type modulo p is IV or IV ∗ , and | c p ( E ) . Thus, if E/ Q has more than two placesof additive reduction, then | Q p | N c p ( E ) . (cid:3) Proof of Theorem 3.1 Part ( c ) . Our strategy in this proof is to show that if does not divide Q p | N c p ( E ) , then 9 divides | X ( E/ Q ) | . We know that E/ Q has a Weierstrass equation as in (2 . . By assumption E/ Q has exactly two places of additive reduction, say at and r .Assume that ∤ Q p | N c p ( E ) otherwise the theorem is proved. Our assumptions force b = r or b = r , and a = 3 rm where m is an integer, as we now explain. Indeed, if b = 1 , then Parts ( i ) and ( ii ) of Proposition 2.4 imply that E/ Q is semi-stable away from and that E/ Q hasonly one place of additive reduction. If b had two or more prime divisors, Parts ( i ) and ( ii ) of Proposition 2.4 imply that divides Q p | N c p ( E ) . If r ∤ a , then by Proposition 2.4, Part ( i ) ,we obtain that E/ Q has multiplicative reduction modulo r , contrary to our assumption. Thisproves that r | a . Since r | a and E/ Q has additive reduction modulo r , Proposition 2.4 impliesthat r | b . Moreover, since we must have that r ∤ a or r ∤ b in Equation (2 . , we obtain that b is equal to either r or r . Finally, since E/ Q has reduction I ∗ n modulo , Proposition 2.4, Part ( iii ) implies that | a .By Proposition 2.4, Part ( ii ) , E/ Q has reduction of type IV or IV ∗ modulo r . By the table onpage of [31] for r = 2 and by Tableau IV of [26] for r = 2 , we obtain then that ord r (∆) = 4 or . Therefore, Proposition of [27] implies that w r ( E ) = (cid:0) − r (cid:1) . If (cid:0) − r (cid:1) = 1 , then r ≡ mod and if (cid:0) − r (cid:1) = − , then r ≡ mod . We split the proof into two cases, when w r ( E ) = 1 and r ≡ mod , and when w r ( E ) = − and r ≡ mod .Let us first prove the following claim. laim 3.13. If p is any prime such that E/ Q has split multiplicative reduction modulo p , thenord ( c p ( b E ) c p ( E ) ) = 1 . Proof.
Note that p ∤ b because E/ Q has additive reduction modulo and r . Recall that b E/ Q is given by a Weierstrass equation of the form y + axy − by = x − ( a + 27 b ) b, where a, b are as in (2 . (see [20] equation (3 . ). The discriminant of this Weierstrass equationis b ∆ = ( a − b ) b. Since ∆ = b ( a − b ) and b ∆ = b ( a − b ) , if ord p (∆) = γ , then ord p ( b ∆) = 3 γ . Therefore,by line of Theorem . of [11] we obtain that ord ( c p ( b E ) c p ( E ) ) = 1 . This proves that if p be anyprime such that E/ Q has split multiplicative reduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 1 . (cid:3) Case 1: w r ( E ) = 1 and r ≡ mod . Claim 3.14.
The number ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and non-negative. Proof.
First, 3.8 implies that w ∞ ( E ) = − and that w ( E ) = − . Recall that E/ Q is semi-stable away from and r . For p = 3 we have that w p ( E ) = − if and only if E/ Q has splitmultiplicative reduction modulo p . Since w ( E ) = 1 , we obtain that E/ Q has an even numberof places of split multiplicative reduction.Since r ≡ mod , we obtain that ζ ∈ Q r and since we also have that E/ Q has reductionof type IV or IV ∗ modulo r , by line of Theorem . of [11] we obtain that ord ( c r ( b E ) c r ( E ) ) = 0 . If E/ Q has nonsplit multiplicative reduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 0 by line of Theorem . of [11]. Also, If E/ Q has split multiplicative reduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 1 byClaim 3.13. Finally, line of Theorem . of [11] implies that ord ( c ( b E ) c ( E ) ) = 0 . Putting allthose together we obtain that ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and greater than or equal to zero. Thisconcludes the proof of the claim. (cid:3) We are now ready to conclude the proof of Case . Assume first that ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 . Since the degree of φ is , we can use Proposition of [18] and obtain that ord p ( Q p c p ( b E ) Q p c p ( E ) ) = 0 for every prime p = 3 . Hence, if Ω( E )Ω( b E ) = 3 , then Equation ( ) implies that | Sel ( φ ) ( E/ Q ) || Sel (ˆ φ ) ( b E/ Q ) | = 1 . Therefore, Part ( ii ) of Lemma 3.6 implies that divides | X ( E/ Q ) |. If Ω( E )Ω( b E ) = 1 , then, since b is equal to either r or r , and we assume that ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 , Lemma 3.12 implies that divides | X ( E/ Q ) | . Assume now that ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ . Then divides | X ( E/ Q ) | by Part ( i ) of Lemma 3.6. This proves Theorem 3.1, Part ( c ) , in the case where w r ( E ) = 1 .Case 2: w r ( E ) = − and r ≡ mod . Claim 3.15.
The number ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and non-negative. roof. By 3.8 w ( E ) = 1 , w ( E ) = − , and w ∞ ( E ) = − . If p = 3 , r is a prime such that E/ Q has split multiplicative reduction modulo p , then w p ( E ) = − . If p = 3 , r is a prime suchthat E/ Q has nonsplit multiplicative or good reduction modulo p , then w p ( E ) = 1 . Therefore,since E/ Q is semi-stable away from and r , we obtain that E/ Q has an odd number of primesof split multiplicative reduction. Note that in particular E/ Q has at least one prime of splitmultiplicative reduction.If p is any prime such that E/ Q has split multiplicative reduction modulo p , then Claim 3.13implies that ord ( c p ( b E ) c p ( E ) ) = 1 . By line of Theorem . of [11], if E/ Q has nonsplit multiplicativereduction modulo p , then ord ( c p ( b E ) c p ( E ) ) = 0 . Moreover, by line of Theorem . of [11] we obtainthat ord ( c r ( b E ) c r ( E ) ) = − because r ≡ mod implies ζ / ∈ Q r and E/ Q has reduction of typeIV or IV ∗ modulo r . Finally, line Theorem . of [11] implies that ord ( c ( b E ) c ( E ) ) = 0 . Sincethere is an odd number of primes of split multiplicative reduction, and in particular at least oneprime of split multiplicative reduction, we obtain that ord ( Q p c p ( b E ) Q p c p ( E ) ) is even and non-negative.This proves our claim. (cid:3) We are now ready to conclude the proof of Case . Assume first that ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 . Since the degree of φ is , by line Theorem . of [11] (or Proposition of [18]) we obtain thatord q ( Q p c p ( b E ) Q p c p ( E ) ) = 0 for every prime q = 3 . Therefore, if Ω( E )Ω( b E ) = 3 , Equation ( ) implies that | Sel ( φ ) ( E/ Q ) || Sel (ˆ φ ) ( b E/ Q ) | = 1 and, hence, by Part ( ii ) of Lemma 3.6 we obtain that divides | X ( E/ Q ) | . If Ω( E )Ω( b E ) = 1 , since b is equal to r or r , and we assume that ord ( Q p c p ( b E ) Q p c p ( E ) ) = 0 , Lemma 3.12 impliesthat divides | X ( E/ Q ) | . Assume now that ord ( Q p c p ( b E ) Q p c p ( E ) ) ≥ . Then divides | X ( E/ Q ) | byPart ( i ) of Lemma 3.6. This proves Theorem 3.1, Part ( c ) , in the case w r ( E ) = − . Thiscompletes the proof of Theorem 3.1. (cid:3) Remark 3.16.
The condition that E/ Q has modulo reduction I ∗ n for some n ≥ in Theorem3.1 is necessary. Indeed, the conclusion of Theorem 3.1 for the elliptic curve E/ Q with Cremona[7] label 324d1 (or LMFDB [21] label 324.b1) is not true. This curve has reduction II modulo , reduction IV ∗ modulo , Q p c p ( E ) = 3 , X ( E/ Q ) is trivial, and E ( Q ) ∼ = Z / Z .Moreover, the condition that E/ Q has analytic rank in Theorem 3.1 is necessary. Indeed,the conclusion of Theorem 3.1 for the elliptic curve E/ Q with Cremona [7] label 171b2 (orLMFDB [21] label 171.b2) is not true. This curve has reduction I ∗ modulo , rank , Q p c p ( E ) =6 , X ( E/ Q ) is trivial, and E ( Q ) ∼ = Z × Z / Z . Remark 3.17.
Let E/ Q be an optimal elliptic curve of analytic rank . It follows from work ofAgashe and Stein that the Birch and Swinnerton-Dyer conjecture, combined with the conjecturethat the Manin constant is , imply that the odd part of | E ( Q ) | divides | X ( E/ Q ) | · Q p c p ( E ) (see the end of Section 4.3 of [2]). Lorenzini has proved the above statement up to a power of (see Proposition 4.2 of [22]).Without the assumption that E/ Q is optimal, if E/ Q has reduction of type I ∗ n modulo ,then Theorem 3.1 proves that the odd part of | E ( Q ) | divides | X ( E/ Q ) | · Q p c p ( E ) . Notehowever that the curve E/ Q with Cremona [7] label 14a4 has a Q -rational point of order , Q p c p ( E ) = 2 , and | X ( E/ Q ) | = 1 . Thus the assumption that the curve E/ Q is optimal is ecessary for the statement that the odd part of | E ( Q ) | divides | X ( E/ Q ) | · Q p c p ( E ) to be truein general. Theorem 3.18.
Let E/ Q be an elliptic curve of conductor N with ∤ N . Let d = − or andassume that L ( E d , = 0 . Then(i) If d = − , then | E d ( Q ) | divides | X ( E d / Q ) | · Y p | N c p ( E d ) , up to a power of . (ii) If d = 3 , then | E d ( Q ) | divides | X ( E d / Q ) | · Y p | N c p ( E d ) , up to a power of . Proof.
We first claim that | E d ( Q ) | divides | X ( E d / Q ) | · Q p c p ( E d ) , up to a power of 2. ByCorollary 2.6, the only primes that can divide | E d ( Q ) | are and . Therefore, if E d / Q doesnot have a point of order , our claim is proved since E d ( Q ) has order a power of . So assumethat E d ( Q ) contains a Q -rational point of order . A theorem of Lorenzini (see [22] Proposition . ) implies that if E/ Q is an elliptic curve with a Q -rational point of order , then divides Q p c p ( E ) except for the curve that has Cremona label [7] 54b3 with Q p c p ( E ) = 27 . If E/ Q isan elliptic curve with ∤ N , then E/ Q has good reduction I modulo . Therefore, if d = − or , we obtain, using Proposition of [4], that E d / Q has reduction of type I ∗ modulo . However,the curve with Cremona label [7] 54b3 does not have reduction of type I ∗ modulo and, hence,it cannot be a twist by d of an elliptic curve with good reduction modulo . Therefore, we canassume that E d ( Q ) does not contain a point of order .Since we assume that E d ( Q ) does not contain a point of order and the only primes thatcan divide | E d ( Q ) | are and , to prove that | E d ( Q ) | divides | X ( E d / Q ) | · Q p c p ( E d ) , up toa power of 2, it is enough to show that divides | X ( E d / Q ) | · Q p c p ( E d ) . Since ∤ N and,hence, E/ Q has good reduction I , by Proposition of [4], we obtain that E d / Q has reductionof type I ∗ modulo . If E d / Q is semi-stable away from , then applying Part ( a ) of Theorem3.1 to E d / Q proves that divides | X ( E d / Q ) | · Q p c p ( E d ) . If E d / Q has more than two placesof additive reduction, then by Part ( b ) of Theorem 3.1 we obtain that | Q p c p ( E d ) . Finally, if E d / Q has exactly two places of additive reduction, then Part ( c ) of Theorem 3.1 implies that divides | X ( E d / Q ) | · Q p c p ( E d ) . Proof of ( i ) : We just need to show that the product can be taken over all the primes of badreduction for E/ Q . If d = − , then the only prime that ramifies in Q ( √ d ) is . Thereforethe primes of bad reduction of E d / Q consist of the primes of bad reduction of E/ Q as well as3. Hence, it is enough to show that ∤ c ( E d ) . By Proposition of [4] we see that E d / Q hasmodulo reduction of type I ∗ and c ( E d ) = 1 , or by page of [29]. Proof of ( ii ) : If d = 3 , then and are the only primes that ramify in Q ( √ d ) . Thereforethe primes of bad reduction of E d / Q consist of all the odd primes of bad reduction of E/ Q aswell as and possibly . By Proposition of [4] we see that E d / Q has modulo reduction oftype I ∗ and c ( E d ) = 1 , or by page of [29]. This completes the proof. (cid:3) We are now ready to prove a slightly stronger form of Agashe’s Conjecture 1.1.
Corollary 3.19.
Let E/ Q be an elliptic curve of conductor N and let − D be a negative fun-damental discriminant such that D is coprime to N . Suppose that L ( E − D , = 0 . Then | E − D ( Q ) | divides | X ( E − D / Q ) | · Y p | N c p ( E − D ) , up to a power of 2. roof. Recall that an integer is called a fundamental discriminant if it is the discriminant of aquadratic number field. If n is a square-free integer, then the discriminant of Q ( √ n ) is n inthe case where n ≡ mod and n otherwise. If − D is square-free, which happens in thecase where − D ≡ mod , then Part ( ii ) of Corollary 2.6 implies that the only prime thatcan divide | E − D ( Q ) tors | is , except possibly for the case where D = 3 . We note that since − is not a fundamental discriminant, we must have that D = 1 . If D = 3 , then Theorem 3.18implies the desired result.Assume now that − D is not square-free. Then we know that − D = 4 n , where n is a square-free integer. Since E − D = E n and n is square-free, by applying Part ( i ) of Corollary 2.6 weobtain that the only prime that can divide | E − D ( Q ) tors | is , except possibly for the cases where n = − or − . If n = − , then since E − = E − , Part ( i ) of Theorem 3.18 implies the desiredresult. Therefore, in order to prove our corollary it remains to provide a proof for the casewhere n = − , i.e., where − D = − .Assume that D = 4 . By our assumption ∤ N . We will show that E − ( Q ) = E − ( Q ) canonly contain points of order a power of . Recall that since L ( E − , = 0 , Theorem . of [8]implies that the rank of E − / Q is zero. Since ∤ N , by Tables I and II of [4] we obtain that E − / Q has reduction of type I ∗ , I ∗ , II, or II ∗ modulo . If E − / Q had a point of order or ,then E − / Q would have had multiplicative reduction modulo which is impossible. Therefore,the only primes that can divide | E − ( Q ) | are and . Assume that E − / Q has a point oforder , and we will arrive at a contradiction. Since E − / Q has a point of order , E − / Q hasan equation of the form (2 . . By Proposition 2.4, E − / Q has either semi-stable reduction orreduction of type IV or IV ∗ modulo . However, this is a contradiction because E − / Q hasreduction of type I ∗ , I ∗ , II or II ∗ modulo . This proves that E − ( Q ) only contains points oforder a power of . This concludes our proof. (cid:3) References [1] A. Agashe. Squareness in the special L -value and special L -values of twists. Int. J. Number Theory ,6(5):1091–1111, 2010. 1, 2[2] Amod Agashe and William Stein. Visible evidence for the Birch and Swinnerton-Dyer conjecture for mod-ular abelian varieties of analytic rank zero.
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