Periods of quadratic twists of elliptic curves

Transcription

1 Perios of quaratic twists of elliptic curves Vivek Pal with an appenix by Amo Agashe Abstract In this paper we prove a relation between the perio of an elliptic curve an the perio of its real an imaginary quaratic twists. This relation is often misstate in the literature. 1 Introuction One of the central conjectures in Number Theory is the Birch an Swinnerton- Dyer Conjecture, which preicts how one can obtain arithmetic information from the L-function. A simpler question, is to ask: (*) If an elliptic curve satisfies the Birch an Swinnerton-Dyer Conjecture then will its (quaratic) twist also satisfy the Birch an Swinnerton-Dyer Conjecture? Part two of the Birch an Swinnerton-Dyer Conjecture involves many elliptic curve invariants, namely the orer of the Tate-Shafarevich group, the perio an the orer of the torsion subgroup among other important invariants. In this paper we relate the perio of an elliptic curve with the perio of its quaratic twists. A relation between the orers of the torsion subgroups has alreay been proven in [Kwo97]. If a similar result can be rawn for all of the other elliptic curve invariants involve in Part II of the Birch an Swinnerton-Dyer Conjecture then one can prove iea (*). Furthermore, a relation between the arithmetic component group an the regulator, of an elliptic curve an its twist woul provie a conjecture for the relation between the orer of the Shafarevich-Tate group for an elliptic curve an its twist. One avantage of iea (*) comes from the fact that quaratic twists of elliptic curves have very ifferent ranks from the original curve. Currently Part two of the Birch an Swinnerton-Dyer Conjecture is only known to be true for families of elliptic curves, usually of low rank; using twists one coul possibly exten these results to many ifferent ranks. Floria State University, the author was fune by the FSU Office of National Fellowships Amo Agashe was supporte by the National Security Agency Grant number Hg

2 In general if F is an elliptic curve, then we enote the invariant ifferential on F by ω(f ). We will call a global minimal Weierstrass equation of an elliptic curve simply a minimal equation or minimal moel, an enote a minimal moel of an elliptic curve F by F min. Let E be an elliptic curve. We use the Birch an Swinnerton-Dyer efinition of the perio. Recall that this perio, enote by Ω(E), is efine as: Ω(E) := ω(e min ). E min(r) Also, recall that the imaginary perio, efine up to a sign, is Ω (E) := ω(e min ), γ where γ is a generator of H 1 (E min, Z), which is the subgroup of elements in H 1 (E min, Z) which are negate by complex conjugation. Furthermore, recall that the quaratic twist of an elliptic curve E by a nonzero integer, enote by E, is efine as an elliptic curve which is isomorphic to E over Q( ) but not over Q. Hence we can assume that is square-free. We also know that E is unique up to isomorphism. The main result of this paper is then Main Result 1.1. Let E be an elliptic curve an let E enote its quaratic twist by a square-free integer. Then the perios of E an E are relate as follows: If > 0, then Ω(E ) = ũ Ω(E), an if < 0, then up to a sign, Ω(E ) = ũ c (E )Ω (E), where c (E ) is the number of connecte components of E (R) an ũ is a rational number such that ũ Z; it epens on E an, an is efine explicitly in Proposition.5 (the elliptic curve E in Proposition.5 shoul be taken as a minimal moel of the E in this theorem). The main result above is prove as Theorem 3. below. Remark 1.. ũ is not always 1 an ũ can be ivisible by an o prime number. In Section 4, we give an example where ũ is 5 an an example where ũ is 7. Also, in the last paragraph of the appenix (Section 5), there is an example of an optimal elliptic curve for which ũ has positive 3-aic valuation. A result similar to the secon case of the theorem above was erive in [Aga10, Lemma.1] for elliptic curves in short Weierstrass form using an assumption on which primes one can twist by. The result here is prove without restrictions.

3 The main result of this paper allows for a weaker hypothesis for several results in [Aga10]; the etails are iscusse in the appenix. We woul like to remark that the formulas in the Main Result have been state incorrectly in the literature. For example, Amo Agashe informe the author that they are quote without ũ as formula (1) on p. 463 in the proof of Corollary 3 in [OS98]; he also mentione that the proof of Corollary 3 in [OS98] still works even after the formula is correcte to inclue ũ. The author woul like to thank Amo Agashe for suggesting this problem an for his help in revising several rafts of this paper. Furthermore the reference to Connell s book [Con08] was mentione to the author by Amo Agashe, who in turn hear about it from Rany Heaton. Quaratic Twists an Minimal Moels First we recall some useful facts. An elliptic curve over Q can be escribe in the following general Weierstrass form: y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6, with a 1, a, a 3, a 4, a 6 Q. In this paper, by an elliptic curve, we mean a curve given by a Weierstrass equation. An elliptic curve will be calle minimal if its Weierstrass equation is minimal. Let E be an elliptic curve, an let (E), j(e), c 4 (E) an c 6 (E) be the usual Weierstrass invariants of the elliptic curve E. Then the signature of the elliptic curve E is the triple c 4 (E), c 6 (E), (E). If p is a prime, then letting v p enote the stanar p-aic valuation, the p-aic signature of E is the triple v p (c 4 (E)), v p (c 6 (E)), v p ( (E)). Remark.1. A transformation E E of elliptic curves over Q preserving the Weierstrass equation an the point at infinity is given by: x = u x + r an y = u 3 y + u sx + t, for some u, r, s, t Q. We will often abbreviate this transformation as the orere tuple [u, r, s, t]. Such a transformation has the following useful properties: 1. u 4 c 4 (E ) = c 4 (E). u 6 c 6 (E ) = c 6 (E) 3. u 1 (E ) = (E) 4. j(e ) = j(e) 5. ω(e ) = uω(e) 3

4 The above facts can be foun in any stanar book on elliptic curves, for example see Silverman [Sil9]. Since the perio of an elliptic curve epens only on the isomorphism class, for the purpose of proving Main Result 1.1 or for computing Ω(E), we can assume that E is a minimal moel, i.e. E = E min. So henceforth, let E be an elliptic curve given by the minimal equation: y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6. (E) Lemma. (Connell). Let be a square-free integer. equation for E is: Then a Weierstrass y + a 1 xy + a 3 y = (E ) ( ) ( = x 3 + a + a 1 1 x + a 4 ) ( 1 + a 1 a 3 x + a a 3 ) Proof. See [Con08, Proposition 4.3.] an the paragraph preceing it. Remark.3. The signature for elliptic curve (E ) is: c 4 (E ) = c 4 (E), c 6 (E ) = c 6 (E) 3 an (E ) = (E) 6. Let α = 1/. Then the transformation from E to E is: x = α x y = α 3 y + a 1α (α 1) x + a 3(α 3 1). Next we recall a proposition from Connell which is isplaye below for convenience. It escribes v p ( ) for a minimal moel of the twist for each prime p. Proposition.4 (Connell). Recall that E is a minimal elliptic curve over Q an E is its quaratic twist by a square-free integer. Let be the iscriminant of E, let be the iscriminant of Emin, an for every valuation v on Z let λ v = min{3v(c 4 (E)), v(c 6 (E)), v( )}. If p is a prime number, then let v p enote the stanar p-aic valuation. Then 1. If p is an o prime ivisor of then: (a) If λ vp < 6 or if p = 3 an v p (c 6 (E)) = 5, then v p ( ) = v p ( ) + 6. (b) Otherwise v p ( ) = v p ( ) 6. If p is an o prime not iviing, then v p ( ) = v p ( ).. If p = then: (a) If 1 mo 4, then v ( ) = v ( ). (b) If 3 mo 4, then 4

5 i. If the -aic signature of E is 0, 0, c (c 0) or a, 3, 0 (4 a ), then v ( ) = v ( ) + 1. ii. If the -aic signature of E is 4, 6, c (c 1 an 6 c 6 (E) 1 mo 4) or a, 9, 1 (a 8 an 9 c 6 (E) 1 mo 4), then v ( ) = v ( ) 1. iii. Otherwise v ( ) = v ( ). (c) If mo 4, let w = / then i. If the -aic signature of E is 0, 0, c (c 0), then v ( ) = v ( ) ii. If the -aic signature of E is 6, 9, c with (c 18) an 9 c 6 (E)w 1 mo 4, then v ( ) = v ( ) 18. iii. If v (c 4 (E)) = 4, 5 or v (c 6 (E)) = 3, 5, 7 or the -aic signature of E is a, 6, 6 with (a 6) an 6 c 6 (E)w 1 mo 4, then v ( ) = v ( ) + 6. iv. Otherwise v ( ) = v ( ) 6. Proof. This proposition is the correcte form of [Con08, 5.7.3], which is misstate in Connell s book. The proof given by Connell in [Con08, 5.7.1] is however correct. This was pointe out to the author by the referee. Proposition.5. Recall that E is a minimal elliptic curve over Q an E is its quaratic twist by a square-free integer. Let be the iscriminant of E, let be the iscriminant of Emin, an for every valuation v on Z, let λ v = min{3v(c 4 (E)), v(c 6 (E)), v( )}. If p is a prime number, then let v p enote the stanar p-aic valuation. Define u p for all primes p, as follows (the cases correspon exactly to the cases of Proposition.4): 1. If p is an o prime ivisor of, then: (a) If λ vp < 6 or if p = 3 an v p (c 6 (E)) = 5, then u p = 1. (b) Otherwise u p = p. If p is an o prime not iviing, then u p = 1.. If p = then: (a) If 1 mo 4, then u = 1. (b) If 3 mo 4, then i. If the -aic signature of E is 0, 0, c (c 0) or a, 3, 0 (4 a ), then u = 1/. ii. If the -aic signature of E is 4, 6, c (c 1 an 6 c 6 (E) 1 mo 4) or a, 9, 1 (a 8 an 9 c 6 (E) 1 mo 4), then u =. iii. Otherwise u = 1. (c) If mo 4, let w = /, then 5

6 i. If the -aic signature of E is 0, 0, c (c 0), then u = 1/. ii. If the -aic signature of E is 6, 9, c with (c 18) an 9 c 6 (E)w 1 mo 4, then u = 4. iii. If v (c 4 (E)) = 4, 5 or v (c 6 (E)) = 3, 5, 7 or the -aic signature of E is a, 6, 6 with (a 6) an 6 c 6 (E)w 1 mo 4, then u = 1. iv. Otherwise u =. Let ũ = p u p. Then there exist r, s, t Q such that the transformation [ũ, r, s, t] will transform equation E to a minimal moel. Proof. The iea of the proof is to apply Proposition.4 to the elliptic curve E an then to fin a transformation sening E to a minimal moel. We claim that [ũ, 0, 0, 0] transforms E to a curve with the correct minimal iscriminant. This follows on a case by case basis using Proposition.4, Remark.3, an Remark.1. Take for example the case 1(b): this is the case where, by Proposition.4, v p ( (Emin )) = v p( (E)) 6. By Remark.3, we know that v p ( (E )) = v p ( 6 (E)) = v p ( (E)) + 6, since in this case, p ivies (an is square-free). Therefore v p ( (Emin )) = v p( (E )) 1. The transformation which will ecrease the valuation of the iscriminant by 1 is [p, 0, 0, 0] by Remark.1; hence proving the Proposition in this case. Applying a similar process to the other cases will erive the respective u p. Since the u p s are coprime to each other, composing the transformations [u p, 0, 0, 0] will give the transformation [ũ, 0, 0, 0]. Thus the transformation, [ũ, 0, 0, 0], will sen E to an elliptic curve E with the correct minimal iscriminant, but which may not have integer coefficients. We will now show that we can fin r, s, t R so that the transformation [ũ, r, s, t] applie to E also gives an integral moel for E, an therefore a minimal moel. Since Emin = E, we know that there is a transformation [u, r, s, t] that sens E to Emin [Sil9, Cor ]. By comparing iscriminants we see that u = ±1; we can assume u = 1 since we can compose this morphism with [ 1, 0, 0, 0] to change the sign of u. Composing the morphism [ũ, 0, 0, 0] with [1, r, s, t] gives the esire morphism, [ũ, r, s, t], sening E to Emin. For the benefit of the reaer we remark that often the transformation [ũ, 0, 0, 0] will in fact transform E to an equation with integral coefficients, hence a minimal moel, but for our purposes only the u coefficient of the transformation will play a role later. Corollary.6. We use the notation of Proposition.5. Suppose is coprime to. Then ũ is a power of. Moreover if 1 mo 4, then ũ = 1. Proof. Let p be an o prime. If p oes not ivie, then by Proposition.5, u p = 1. If p ivies, then v p ( ) = 0 since is coprime to, an so λ vp < 6, an thus by Proposition.5, u p = 1. In both cases, u p = 1 for o primes, which proves the first claim of the corollary.. If 1 mo 4, then by Case (a) Proposition.5 u = 1. The secon claim of the corollary follows, since ũ = p u p = 1 6

7 Definition.7. We efine Emin to be the specific minimal moel of elliptic curve E obtaine via Proposition.5. 3 Perios We first prove a relation between the invariant ifferentials of E an E min an then use this relation to prove the esire relation between the perios in our main result. Lemma 3.1. We have: an ω(e ) = ω(e) ω(e min) = ũ ω(e). Proof. Using the properties liste in Remarks.1 an.3 regaring transformations, the transformation taking E to E has u = α = 1/. Then by Remark.1, ω(e ) = ω(e). By Proposition.5, the transformation taking E to E min has u = ũ. Then ω(e min ) = ũ ω(e ) = ũ ω(e). We now prove the main result relating the perios. Theorem 3.. Recall that E is a minimal elliptic curve an E is its quaratic twist by. Then the perios of E an E are relate as follows If > 0, then Ω(E ) = ũ Ω(E). If < 0, then up to a sign, Ω(E ) = ũ c (E )Ω (E), where c (E ) is the number of connecte components of E (R). Proof. We first prove the formula for > 0: As remarke in the proof of Lemma 3.1, the transformation that takes E to E takes ω(e) to ω(e ). This transformation sens E(R) bijectively to E (R) because the transformation an its inverse are efine over R (since > 0). Then: ω(e) = ω(e ). (3.1) E (R) E(R) Using a similar argument we see that: 7

8 E (R) ω(e ) = 1 ũ E min (R) ω(e min). (3.) Then from equation (3.1) an equation (3.) we see that: Ω(E) = ω(e) = ũ Ω(E ). E(R) Next we prove the formula for < 0: We follow the technique use in the proof of [Aga10, Lemma.1]. Let P = (x, y) E(R) an let σ be the complex conjugation map; then σ(p ) = P. The inverse of the map escribe in Remark.3 is given by: x = 1 α x y = 1 α 3 y a 1 ( 1 α 1 ) α 3 x a 3 (1 1α ) 3 where α = 1/. Let T be this map, T : E E. Claim: σ(t (P )) = T (P ). Proof. (( 1 σ(t (P )) = σ α x, 1 α 3 y a ( 1 1 α 1 ) α 3 x a 3 (1 1α ))) 3 = ( 1 1 = x, α α 3 y a ( 1 1 α + 1 ) α 3 x a 3 (1 + 1α )) 3. Using the efinition of the negative of a point on an elliptic curve, given in [Sil9, III..3]: ( 1 T (P ) = α x, 1 α 3 y a ( 1 1 α 1 ) α 3 x a 3 (1 1α )) 3 = ( ( 1 1 = α x, α 3 y a ( 1 1 α 1 ) α 3 x a 3 (1 1α )) ( ) ) 1α 3 a 1 x a 3 ( 1 1 = x, α α 3 y a ( 1 1 α + 1 ) α 3 x a 3 (1 + 1α )) 3. Then we see that σ(t (P )) = T (P ). Thus T gives a homeomorphism from E (R) to E(C), where E(C) is the subgroup of points not fixe uner complex conjugation. If G is a Lie group, then let G 0 enote the connecte component of G containing the ientity. Then T also inuces a homeomorphism from E (R) 0 to E(C) 0. 8

9 In particular, T gives an isomorphism from H 1 (E (R) 0, Z) to H 1 (E(C) 0, Z). By Lemma 4.4 in [AS05], the natural map from H 1 (E (R) 0, Z) to H 1 (E (C), Z) + is an isomorphism, an by Lemma 5. from the appenix (Section 5), the natural map from H 1 (E (C) 0, Z) to H 1(E (C), Z) is an isomorphism. Let γ be a generator of H 1 (E (C), Z) +. Then from the statements above, one sees that T (γ) is in H 1 (E(C), Z) an generates it. Then it follows that γ ω(e ) = T (γ) T (ω(e )) = 1 where the last equality is up to a sign. Similar to equation (3.) we have, E (R) ω(e ) = 1 ũ T (γ) ω(e) = 1 Ω (E), (3.3) E min (R) ω(e min), (3.4) since the transformation in this integral involves only real numbers it takes E (R) to Emin (R). Using equation (5.1) from the appenix an equation (3.4), we see that up to a sign, Ω(E ) = ω(emin) = ũ ω(e ). (3.5) Emin (R) E (R) The proof of Lemma 5.1 from the appenix shows that ω(e ) = c (E ) ω(e ). (3.6) E (R) Putting equation (3.6) in equation (3.5), we see that up to a sign: Ω(E ) = ũ c (E ) ω(e ) = ũ c (E ) Ω (E), γ where the last equality follows from equation (3.3). This finishes the proof for the case < 0 an proves the theorem. 4 Examples 4.1 Real quaratic twist Using Sage an GP/Pari we were able to fin the following example in which the ũ in Theorem 3. is 5. Let E be the following elliptic curve which is minimal. E : y = x 3 x 6883x + 137, γ 9

10 By Proposition.5, twisting E by = 5 falls in cases 1(b) an (a), an so ũ = 5. Then by Theorem 3., Ω(E )/Ω(E) = 5 5 = 5. We now try to verify this in GP/Pari. Using Lemma., we compute the twist by = 5 to be E : y = x 3 5x 17075x Using the comman ellminimalmoel in GP/Pari we see that one of the minimal moels for E is then y = x 3 + x 75x For an elliptic curve E we can compute the perios in GP/Pari using the comman E.omega[1]. Remark 4.1. The perio compute this way is similar to the perio we use, but instea of using a minimal moel it is efine as ω(e), where γ is a generator γ of H 1 (E(C), Z) +. Therefore we have to first compute a minimal moel for E, use that to compute the perio in GP/Pari, an then multiply the result by c (E), the number of connecte components, to get the perio we esire. We can see that both E an E have only one connecte component, by either plotting them or noticing that they both have negative iscriminants, thus c (E) = c (E ) = 1. Then one fins that Ω(E ) = Ω(E min) So Ω(E )/Ω(E) 5, as expecte. Ω(E) How the complex perio of GP/Pari relates to the imaginary perio efine above. Recall that the imaginary perio is efine up to a sign as Ω (E) = ω(e), where γ is a generator of H 1 (E min, Z). It will be a pure imaginary number since, if σ enotes complex conjugation σ(ω (E)) = σ(ω(e)) = ω(e) = Ω (E). σ(γ) The secon equality hols since ω(e) is efine over R (in fact over Q) an because σ(γ) = γ. The complex perio compute by GP/Pari (using the comman E.omega) is in general not a pure imaginary number. Using the perios given by GP/Pari γ γ 10

11 we can however approximately recover the imaginary perio. This is because the two perios compute by GP/Pari (calle the real an complex perios) are generators for a lattice, which is also generate by the two perios use in this paper (calle the perio an the imaginary perio). For an elliptic curve E, let Ω 1 an Ω 1 be the perio an imaginary perio, respectively, efine in this paper. Let Ω an Ω be the real an complex perios, respectively, that are compute in GP/Pari for E using the function E.omega. Since the pairs are generators for the same lattice we have, Ω 1 = k 1Ω k Ω for some k 1, k Z. We also know that Ω 1 is a pure imaginary number an that Ω is a real number, therefore k /k 1 = Re(Ω )/Ω where Re(z) enotes the real part of the complex number z. Then k 1 an k can be chosen such that gc(k 1, k ) = 1. Fining such a k 1 an k gives a way to compute the imaginary perio using GP/Pari; however, we can only compute Re(Ω )/Ω approximately an hence we can only make a goo guess of what k 1 an k are. 4.3 Imaginary quaratic twist Using Sage an GP/Pari we were able to fin the following example in which the ũ in Theorem 3. is 7. Let E be the following elliptic curve E : y + xy + y = x 3 173x + 879, which is minimal. By Proposition.5, twisting E by = 7 falls in cases 1(b) an (a); hence ũ = 7. Then by Theorem 3., Ω(E )/Ω (E) = 7 7 = 7, up to a sign. We now try to verify this in GP/Pari. Using Lemma. we compute the twist by = 7 to be E : y + xy + y = x 3 x 8453x Using the comman ellminimalmoel in GP/Pari we see that one of the minimal moels for E is then y + xy = x 3 + x 3x 4. We can see that both E an E have only one connecte component, by either plotting them or noticing that they both have negative iscriminants, thus c (E) = c (E ) = 1. Using Remark 4.1 one fins that Ω(E ) = Ω(E min) Following the proceure to compute the imaginary perio from Section 4. we fin that k /k Assuming that this is actually 1/, we get Ω (E) ( ) 1 an Ω(E )/Ω (E) 7, as expecte. 11

12 5 Appenix on perios by Amo Agashe In Section 5.1, we state an prove some facts about perios that are well known, but whose proofs o not seem to be ocumente in the literature; some of these results are use in Section 3. In Section 5., we give a lemma that is use in Section 3. In Section 5.3, we point out the implications of the results of this article to [Aga10], an in particular, we make a conjecture that strengthens a conjecture mae in [Aga10]. 5.1 Some facts about perios Let E be an elliptic curve over Q, an let E min enote an elliptic curve given by a global minimal Weierstrass equation for E. Let ω(e min ) enote the invariant ifferential on E min. Then recall that the perio of E is efine as Ω(E) = ω(e min ). E min(r) Note that if we take a ifferent global minimal Weierstrass equation for E, call it E min, then E min an E min are isomorphic to each other over Q by a transformation of the type [u, r, s, t] (notation as in Remark.1) with u = ±1 (since they have the same iscriminant, an the transformation changes the iscriminant by a factor of u 1, by Remark.1). Then the invariant ifferential of E min iffers from that of E min by a factor of u (again, see Remark.1), i.e., by ±1, an so the efinition of Ω(E) given above is inepenent of the choice of a global minimal Weierstrass equation for E. If two elliptic curves are isomorphic over Q, then they have a common minimal moel, an hence they have the same perio. The Néron moel of E is the open subscheme of E min consisting of the regular points (see III.6 of [Lan91]), an so the perio efine above agrees with the perio use in the more general version of the Birch an Swinnerton- Dyer conjecture for abelian varieties (as escribe for example in III.5 of loc. cit.), which uses Néron ifferentials. Now as a Lie group, E min (R) is isomorphic to one or two copies of R/Z (see, e.g., [Sil94, Cor. V..3.1]). Since the invariant ifferential has no zeros or poles (see Prop. III.1.5 in [Sil9]), it oes not change its sign on any copy of R/Z, an so on any copy, we have ω(e min ) = ±ω(e min ). If E min (R) consists of one copy, then we see that up to a sign, Ω(E) = E ω(e min(r) min). Now suppose E min (R) consists of two copies; call them C 1 an C. Without loss of generality, assume that C 1 contains the ientity, an choose a point P on C. Then the translation by P map inuces a map from C 1 to C (by continuity arguments) an similarly, translation by P maps C to C 1. These two maps are inverses to each other, an moreover, ω(e min ) is invariant uner translation. Thus we see that the integral of ω(e min ) over C 1 is the same as that over C an up to a sign is the integral of ω(e min ) over either component. Thus in this case, up to 1

13 a sign, Ω(E) = C 1 ω(e min ). In either case, we see that up to a sign, Ω(E) = E min(r) ω(e min ). (5.1) If φ : E E min is an isomorphism (such an isomorphism exists, of course), then φ maps E(R) bijectively to E min (R) an one sees (by integration by substitution) that ω(e min ) = φ ω(e min ), E min(r) E(R) where ω(e) as usual is the invariant ifferential on E an φ enotes the pullback by φ map on ifferentials. Thus up to a sign, Ω(E) = φ ω(e min ). E(R) This efinition was use in [ARS06], for example. Consiering that C 1 is homeomorphic to the circle, an the natural map from the first homology group of C 1 to H 1 (E min (C), Z) + is an isomorphism (e.g., see Lemma 4.4 in [AS05]), from the iscussion two paragraphs above, we get the following lemma: Lemma 5.1. Let γ be a generator of the cyclic free abelian group H 1 (E min (C), Z) + an let c (E min ) enote the number of connecte components in E min (R). Then up to a sign, Ω(E) = c (E min ) ω(e min ). γ Note that since E an E min are isomorphic over Q, an hence over R, we have c (E min ) = c (E), an so we also have that up to a sign, Ω(E) = c (E) ω(e min ). Lemma 5.1 above is well known, an in fact a more general result for abelian varieties is given as Lemma 8.8 in [Man71]. However, in loc. cit., the author only gives a sketch of the proof of the quote lemma, an uses the result of Lemma 5.1 above as an input without proof. 5. A lemma In this section, let E be an elliptic curve over R. Recall that E(C) enotes the subgroup of E(C) on which complex conjugation acts as multiplication by 1 an E(C) 0 is the component of E(C) containing the ientity. The following lemma is an aaptation of Lemma 4.4 in [AS05], an is use in Section 3. Lemma 5.. The natural map from H 1 (E(C) 0, Z) to H 1(E(C), Z) is an isomorphism. γ 13

14 Proof. Let ψ enote the natural map from H 1 (E(C) 0, Z) to H 1(E(C), Z). We have the commutative iagram 0 H 1 (E(C) 0, Z) H 1 (E(C) 0, R) ψ E(C) 0 0 H 1 (E(C), Z) H 1 (E(C), R) E(C) 0, where the two vertical arrows on the right are the obvious natural maps, the upper horizontal sequence the exact sequence obtaine by viewing the real torus E(C) 0 as the quotient of the tangent space at the ientity by the first integral homology, an the lower horizontal sequence is the exact sequence obtaine from the exact sequence 0 H 1 (E(C), Z) H 1 (E(C), R) E(C) 0 of complex analytic parametrization of E by taking anti-invariants uner complex conjugation. The mile vertical map is an isomorphism of real vector spaces because if it were not, then its kernel woul be an uncountable set that maps to 0 in E(C) 0 (using the rightmost square in the commutative iagram above), an hence woul be containe in H 1 (E(C) 0, Z), which is countable. The snake lemma then yiels an exact sequence 0 ker(ψ) 0 0 coker(ψ) 0, which implies that ψ is an isomorphism, as was to be shown. 5.3 Some implications In this section, we point out the implications of the results of this article to [Aga10]. By Corollary.6, if is coprime to the conuctor E (or the iscriminant of E), then the ũ in Theorem 3. is a power of. Note that the D in [Aga10] is, with < 0. Thus if one replaces the hypothesis (**) in of loc. cit., with the hypothesis that D is coprime to the conuctor N of E, then the conclusions of Lemma.1, Proposition., an Corollary.4 are vali up to a power of. As a consequence (see the iscussion after Corollary.4 in loc. cit.), we woul like to weaken the hypothesis (**) in Conjecture.5 of loc. cit. to the hypothesis that D is coprime to N, an thus make the following conjecture: Conjecture 5.3. Let E be an optimal elliptic curve over Q of conuctor N an let D be a negative funamental iscriminant such that D is coprime to N. Recall that E D enotes the twist of E by D. Suppose L(E D, 1) 0, so that E D (Q) is finite. Then E D (Q) ivies X(E D ) p N c p(e D ), up to a power of, where X(E D ) enotes the Shafarevich-Tate group of E D an c p (E D ) enotes the orer of the arithmetic component group of E D at p. 14

15 As mentione in loc. cit., using the mathematical software sage, with its inbuilt Cremona s atabase for all elliptic curves of conuctor up to , we verifie the conjecture above for all triples (N, E, D) such that N an D are positive integers with ND < , an E is an optimal elliptic curve of conuctor N. Finally, we remark that Proposition.5 explains why the concluing statement of Conjecture.5 of [Aga10] oes not hol in the example of (E, D) = (7a1, 3) in Table 1 of loc. cit. (this example oes not satisfy the hypotheses of the conjecture): using SAGE, we fin that (E min ) = 3 9 an c 6 (E min ) = 3 3 6, an so by part 1(b) of Proposition.5, v 3 (ũ) > 0. In particular, this is an example of an optimal elliptic curve for which ũ is not a power of. Anyhow, the concluing statement of Corollary.4 in loc. cit. oes not hol, an so for this pair (E, D), assuming the secon part of the Birch an Swinnerton-Dyer conjecture, one oes not expect that E D (Q) ivies X(E D ) p N c p(e D ), even up to a power of (see the iscussion just before Corollary.5 in loc. cit.); rather one expects that E D (Q) ivies ũ X(E D ) p N c p(e D ), an so it is not surprising that E D (Q) ivies 3 X(E D ) p N c p(e D ), up to a power of. References [Aga10] Amo Agashe, Squareness in the special L-value an special L-values of twists, Int. J. Number Theory 6 (010), no. 5, [ARS06] Amo Agashe, Kenneth Ribet, an William A. Stein, The Manin constant, Pure Appl. Math. Q. (006), no., part, [AS05] [Con08] A. Agashe an W. A. Stein, Visible evience for the Birch an Swinnerton-Dyer conjecture for moular abelian varieties of analytic rank zero, Math. Comp. 74 (005), no. 49, Ian Connell, Elliptic curve hanbook, preprint, available at [Kwo97] Soonhak Kwon, Torsion subgroups of elliptic curves over quaratic extensions, J. Number Theory 6 (1997), no. 1, MR (98e:11068) [Lan91] S. Lang, Number theory. III, Springer-Verlag, Berlin, 1991, Diophantine geometry. [Man71] J. I. Manin, Cyclotomic fiels an moular curves, Russian Math. Surveys 6 (1971), no. 6, [OS98] Ken Ono an Christopher Skinner, Fourier coefficients of half-integral weight moular forms moulo l, Ann. of Math. () 147 (1998), no.,

16 [Sil9] [Sil94] J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 199. J. H. Silverman, Avance topics in the arithmetic of elliptic curves, Springer-Verlag, New York,