Periods of Quadratic Twists of Elliptic Curves

In this paper we prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.


Introduction
One of the central conjectures in Number Theory is the Birch and Swinnerton-Dyer Conjecture, which predicts how one can obtain arithmetic information from the L-function. A simpler question, is to ask: (*) If an elliptic curve satisfies the Birch and Swinnerton-Dyer Conjecture then will its (quadratic) twist also satisfy the Birch and Swinnerton-Dyer Conjecture?
Part two of the Birch and Swinnerton-Dyer Conjecture involves many elliptic curve invariants, namely the order of the Tate-Shafarevich group, the period and the order of the torsion subgroup among other important invariants. In this paper we relate the period of an elliptic curve with the period of its quadratic twists. A relation between the orders of the torsion subgroups has already been proven in [Kwo97]. If a similar result can be drawn for all of the other elliptic curve invariants involved in Part II of the Birch and Swinnerton-Dyer Conjecture then one can prove idea (*). Furthermore, a relation between the arithmetic component group and the regulator, of an elliptic curve and its twist would provide a conjecture for the relation between the order of the Shafarevich-Tate group for an elliptic curve and its twist.
One advantage of idea (*) comes from the fact that quadratic twists of elliptic curves have very different ranks from the original curve. Currently Part two of the Birch and Swinnerton-Dyer Conjecture is only known to be true for families of elliptic curves, usually of low rank; using twists one could possibly extend these results to many different ranks.
In general if F is an elliptic curve, then we denote the invariant differential on F by ω(F ). We will call a global minimal Weierstrass equation of an elliptic curve simply a minimal equation or minimal model, and denote a minimal model of an elliptic curve F by F min .
Let E be an elliptic curve. We use the Birch and Swinnerton-Dyer definition of the period. Recall that this period, denoted by Ω(E), is defined as: Also, recall that the imaginary period, defined up to a sign, is 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 negated by complex conjugation.
Furthermore, recall that the quadratic twist of an elliptic curve E by a nonzero integer d, denoted by E d , is defined as an elliptic curve which is isomorphic to E over Q( √ d) but not over Q. Hence we can assume that d is square-free. We also know that E d is unique up to isomorphism.
The main result of this paper is then and if d < 0, then up to a sign, where c ∞ (E d ) is the number of connected components of E d (R) andũ is a rational number such that 2ũ ∈ Z; it depends on E and d, and is defined explicitly in Proposition 2.5 (the elliptic curve E in Proposition 2.5 should be taken as a minimal model of the E in this theorem).
The main result above is proved as Theorem 3.2 below.
Remark 1.2.ũ is not always 1 and 2ũ can be divisible by an odd prime number. In Section 4, we give an example whereũ is 5 and an example whereũ is 7. Also, in the last paragraph of the appendix (Section 5), there is an example of an optimal elliptic curve for whichũ has positive 3-adic valuation.
A result similar to the second case of the theorem above was derived in [Aga10, Lemma 2.1] for elliptic curves in short Weierstrass form using an assumption on which primes one can twist by. The result here is proved without restrictions.
The main result of this paper allows for a weaker hypothesis for several results in [Aga10]; the details are discussed in the appendix.
We would like to remark that the formulas in the Main Result have been stated incorrectly in the literature. For example, Amod Agashe informed the author that they are quoted withoutũ as formula (12) on p. 463 in the proof of Corollary 3 in [OS98]; he also mentioned that the proof of Corollary 3 in [OS98] still works even after the formula is corrected to includeũ.
The author would like to thank Amod Agashe for suggesting this problem and for his help in revising several drafts of this paper. Furthermore the reference to Connell's book [Con08] was mentioned to the author by Amod Agashe, who in turn heard about it from Randy Heaton.

Quadratic Twists and Minimal Models
First we recall some useful facts. An elliptic curve over Q can be described in the following general Weierstrass form: with a 1 , a 2 , 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 called minimal if its Weierstrass equation is minimal. Let E be an elliptic curve, and let ∆(E), j(E), c 4 (E) and 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 denote the standard p-adic valuation, the p-adic signature of E is the triple v p (c 4 (E)), v p (c 6 (E)), v p (∆(E)).
Remark 2.1. A transformation E → E ′ of elliptic curves over Q preserving the Weierstrass equation and the point at infinity is given by: x = u 2 x ′ + r and y = u 3 y ′ + u 2 sx ′ + t, for some u, r, s, t ∈ Q. We will often abbreviate this transformation as the ordered tuple [u, r, s, t]. Such a transformation has the following useful properties: The above facts can be found in any standard book on elliptic curves, for example see Silverman [Sil92].
Since the period of an elliptic curve depends 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 model, i.e. E = E min . So henceforth, let E be an elliptic curve given by the minimal equation: y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 . (E) Lemma 2.2 (Connell). Let d be a square-free integer. Then a Weierstrass equation for E d is: Proof. See [Con08, Proposition 4.3.2] and the paragraph preceding it.
Remark 2.3. The signature for elliptic curve (E d ) is: Next we recall a proposition from Connell which is displayed below for convenience. It describes v p (∆) for a minimal model of the twist for each prime p.
Proposition 2.4 (Connell). Recall that E is a minimal elliptic curve over Q and E d is its quadratic twist by a square-free integer d. Let ∆ be the discriminant of E, let ∆ ′ be the discriminant of E d min , and for every valuation v on Z let λ v = min{3v(c 4 (E)), 2v(c 6 (E)), v(∆)}. If p is a prime number, then let v p denote the standard p-adic valuation. Then 1. If p is an odd prime divisor of d then: If p is an odd prime not dividing d, then v p (∆ ′ ) = v p (∆).
Proposition 2.5. Recall that E is a minimal elliptic curve over Q and E d is its quadratic twist by a square-free integer d. Let ∆ be the discriminant of E, let ∆ ′ be the discriminant of E d min , and for every valuation v on Z, let λ v = min{3v(c 4 (E)), 2v(c 6 (E)), v(∆)}. If p is a prime number, then let v p denote the standard p-adic valuation. Define u p for all primes p, as follows (the cases correspond exactly to the cases of Proposition 2.4): 1. If p is an odd prime divisor of d, then: (a) If λ vp < 6 or if p = 3 and v p (c 6 (E)) = 5, then u p = 1.
If p is an odd prime not dividing d, then u p = 1.
The transformation which will decrease the valuation of the discriminant by 12 is [p, 0, 0, 0] by Remark 2.1; hence proving the Proposition in this case. Applying a similar process to the other cases will derive 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 send E d to an elliptic curve E ′ with the correct minimal discriminant, but which may not have integer coefficients.
We will now show that we can find r, s, t ∈ R so that the transformation [ũ, r, s, t] applied to E d also gives an integral model for E d , and therefore a minimal model. Since E d min ∼ = E ′ , we know that there is a transformation [u, r, s, t] that sends E ′ to E d min [Sil92, Cor. 7.8.3]. By comparing discriminants 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 desired morphism, [ũ, r, s, t], sending E d to E d min . For the benefit of the reader we remark that often the transformation [ũ, 0, 0, 0] will in fact transform E d to an equation with integral coefficients, hence a minimal model, but for our purposes only the u coefficient of the transformation will play a role later.
Corollary 2.6. We use the notation of Proposition 2.5. Suppose d is coprime to ∆. Thenũ is a power of 2. Moreover if d ≡ 1 mod 4, thenũ = 1.
Proof. Let p be an odd prime. If p does not divide d, then by Proposition 2.5, u p = 1. If p divides d, then v p (∆) = 0 since d is coprime to ∆, and so λ vp < 6, and thus by Proposition 2.5, u p = 1. In both cases, u p = 1 for odd primes, which proves the first claim of the corollary. . If d ≡ 1 mod 4, then by Case 2(a) Proposition 2.5 u 2 = 1. The second claim of the corollary follows, sincẽ u = p u p = 1 Definition 2.7. We define E d min to be the specific minimal model of elliptic curve E d obtained via Proposition 2.5.

Periods
We first prove a relation between the invariant differentials of E and E d min and then use this relation to prove the desired relation between the periods in our main result.
Lemma 3.1. We have: Proof If d < 0, then up to a sign, where c ∞ (E d ) is the number of connected components of E d (R).
Proof. We first prove the formula for d > 0: As remarked in the proof of Lemma 3.1, the transformation that takes E to E d takes ω(E) to √ dω(E d ). This transformation sends E(R) bijectively to E d (R) because the transformation and its inverse are defined over R (since d > 0). Then: Using a similar argument we see that: Then from equation (3.1) and equation (3.2) we see that: Next we prove the formula for d < 0: We follow the technique used in the proof of [Aga10, Lemma 2.1]. Let P = (x, y) ∈ E(R) and let σ be the complex conjugation map; then σ(P ) = P . The inverse of the map described in Remark 2.3 is given by: Claim: σ(T (P )) = −T (P ). Proof.
Thus T gives a homeomorphism from E d (R) to E(C) − , where E(C) − is the subgroup of points not fixed under complex conjugation. If G is a Lie group, then let G 0 denote the connected component of G containing the identity. Then T also induces a homeomorphism from E d (R) 0 to E(C) − 0 .
In particular, T gives an isomorphism from H 1 (E d (R) 0 , Z) to H 1 (E(C) − 0 , Z). By Lemma 4.4 in [AS05], the natural map from H 1 (E d (R) 0 , Z) to H 1 (E d (C), Z) + is an isomorphism, and by Lemma 5.2 from the appendix (Section 5), the natural map from H 1 (E d (C) − 0 , Z) to H 1 (E d (C), Z) − is an isomorphism. Let γ be a generator of H 1 (E d (C), Z) + . Then from the statements above, one sees that T (γ) is in H 1 (E(C), Z) − and generates it.
Then it follows that where the last equality is up to a sign. Similar to equation (3.2) we have, since the transformation in this integral involves only real numbers it takes E d (R) to E d min (R). Using equation (5.1) from the appendix and equation (3.4), we see that up to a sign, (3.5) The proof of Lemma 5.1 from the appendix shows that (3.6) Putting equation (3.6) in equation (3.5), we see that up to a sign: where the last equality follows from equation (3.3). This finishes the proof for the case d < 0 and proves the theorem.

Real quadratic twist
Using Sage and GP/Pari we were able to find the following example in which theũ in Theorem 3.2 is 5. Let E be the following elliptic curve which is minimal.
By Proposition 2.5, twisting E by d = 5 falls in cases 1(b) and 2(a), and sõ u = 5. Then by Theorem 3.2, Ω(E d )/Ω(E) = 5 √ 5 = √ 5. We now try to verify this in GP/Pari. Using Lemma 2.2, we compute the twist by d = 5 to be Using the command ellminimalmodel in GP/Pari we see that one of the minimal models for E d is then For an elliptic curve E we can compute the periods in GP/Pari using the command E.omega[1].
Remark 4.1. The period computed this way is similar to the period we use, but instead of using a minimal model it is defined as γ ω(E), where γ is a generator of H 1 (E(C), Z) + . Therefore we have to first compute a minimal model for E, use that to compute the period in GP/Pari, and then multiply the result by c ∞ (E), the number of connected components, to get the period we desire.
We can see that both E and E d have only one connected component, by either plotting them or noticing that they both have negative discriminants, Then one finds that .
4.2 How the complex period of GP/Pari relates to the imaginary period defined above.
Recall that the imaginary period is defined up to a sign as where γ is a generator of H 1 (E min , Z) − . It will be a pure imaginary number since, if σ denotes complex conjugation The second equality holds since ω(E) is defined over R (in fact over Q) and because σ(γ) = −γ. The complex period computed by GP/Pari (using the command E.omega) is in general not a pure imaginary number. Using the periods given by GP/Pari we can however approximately recover the imaginary period. This is because the two periods computed by GP/Pari (called the real and complex periods) are generators for a lattice, which is also generated by the two periods used in this paper (called the period and the imaginary period). For an elliptic curve E, let Ω 1 and Ω − 1 be the period and imaginary period, respectively, defined in this paper. Let Ω 2 and Ω − 2 be the real and complex periods, respectively, that are computed in GP/Pari for E using the function E.omega. Since the pairs are generators for the same lattice we have, Ω − 1 = k 1 Ω − 2 − k 2 Ω 2 for some k 1 , k 2 ∈ Z. We also know that Ω − 1 is a pure imaginary number and that Ω 2 is a real number, therefore k 2 /k 1 = Re(Ω − 2 )/Ω 2 where Re(z) denotes the real part of the complex number z. Then k 1 and k 2 can be chosen such that gcd(k 1 , k 2 ) = 1. Finding such a k 1 and k 2 gives a way to compute the imaginary period using GP/Pari; however, we can only compute Re(Ω − 2 )/Ω 2 approximately and hence we can only make a good guess of what k 1 and k 2 are.

Imaginary quadratic twist
Using Sage and GP/Pari we were able to find the following example in which theũ in Theorem 3.2 is 7.
Using Lemma 2.2 we compute the twist by d = −7 to be Using the command ellminimalmodel in GP/Pari we see that one of the minimal models for E d is then We can see that both E and E d have only one connected component, by either plotting them or noticing that they both have negative discriminants, thus c ∞ (E) = c ∞ (E d ) = 1.
Using Remark 4.1 one finds that Following the procedure to compute the imaginary period from Section 4.2 we find that k 2 /k 1 ≈ −.50000000000 . . .. Assuming that this is actually −1/2, we get and Ω(E d )/Ω − (E) ≈ √ −7, as expected.

Appendix on periods by Amod Agashe
In Section 5.1, we state and prove some facts about periods that are well known, but whose proofs do not seem to be documented in the literature; some of these results are used in Section 3. In Section 5.2, we give a lemma that is used in Section 3. In Section 5.3, we point out the implications of the results of this article to [Aga10], and in particular, we make a conjecture that strengthens a conjecture made in [Aga10].

Some facts about periods
Let E be an elliptic curve over Q, and let E min denote an elliptic curve given by a global minimal Weierstrass equation for E. Let ω(E min ) denote the invariant differential on E min . Then recall that the period of E is defined as Note that if we take a different global minimal Weierstrass equation for E, call it E ′ min , then E min and E ′ min are isomorphic to each other over Q by a transformation of the type [u, r, s, t] (notation as in Remark 2.1) with u = ±1 (since they have the same discriminant, and the transformation changes the discriminant by a factor of u 12 , by Remark 2.1). Then the invariant differential of E ′ min differs from that of E min by a factor of u (again, see Remark 2.1), i.e., by ±1, and so the definition of Ω(E) given above is independent 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 model, and hence they have the same period.
The Néron model of E is the open subscheme of E min consisting of the regular points (see § III.6 of [Lan91]), and so the period defined above agrees with the period used in the more general version of the Birch and Swinnerton-Dyer conjecture for abelian varieties (as described for example in § III.5 of loc. cit.), which uses Néron differentials. Now as a Lie group, E min (R) is isomorphic to one or two copies of R/Z (see, e.g., [Sil94, Cor. V.2.3.1]). Since the invariant differential has no zeros or poles (see Prop. III.1.5 in [Sil92]), it does not change its sign on any copy of R/Z, and 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) = Emin(R) ω(E min ). Now suppose E min (R) consists of two copies; call them C 1 and C 2 . Without loss of generality, assume that C 1 contains the identity, and choose a point P on C 2 . Then the translation by P map induces a map from C 1 to C 2 (by continuity arguments) and similarly, translation by −P maps C 2 to C 1 . These two maps are inverses to each other, and moreover, ω(E min ) is invariant under translation. Thus we see that the integral of |ω(E min )| over C 1 is the same as that over C 2 and up to a sign is the integral of ω(E min ) over either component. Thus in this case, up to a sign, Ω(E) = 2 · C1 ω(E min ). In either case, we see that up to a sign, 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) and one sees (by integration by substitution) that where ω(E) as usual is the invariant differential on E and φ * denotes the pullback by φ map on differentials. Thus up to a sign, This definition was used in [ARS06], for example.
Considering that C 1 is homeomorphic to the circle, and 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 discussion 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) + and let c ∞ (E min ) denote the number of connected components in E min (R). Then up to a sign, Note that since E and E min are isomorphic over Q, and hence over R, we have c ∞ (E min ) = c ∞ (E), and so we also have that up to a sign, Lemma 5.1 above is well known, and 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 quoted lemma, and uses the result of Lemma 5.1 above as an input without proof.

A lemma
In this section, let E be an elliptic curve over R. Recall that E(C) − denotes the subgroup of E(C) on which complex conjugation acts as multiplication by −1 and E(C) − 0 is the component of E(C) − containing the identity. The following lemma is an adaptation of Lemma 4.4 in [AS05], and is used in Section 3.
where the two vertical arrows on the right are the obvious natural maps, the upper horizontal sequence the exact sequence obtained by viewing the real torus E(C) − 0 as the quotient of the tangent space at the identity by the first integral homology, and the lower horizontal sequence is the exact sequence obtained from the exact sequence of complex analytic parametrization of E by taking anti-invariants under complex conjugation. The middle vertical map is an isomorphism of real vector spaces because if it were not, then its kernel would be an uncountable set that maps to 0 in E(C) − 0 (using the rightmost square in the commutative diagram above), and hence would be contained in H 1 (E(C) − 0 , Z), which is countable. The snake lemma then yields an exact sequence 0 → ker(ψ) → 0 → 0 → coker(ψ) → 0, which implies that ψ is an isomorphism, as was to be shown.

Some implications
In this section, we point out the implications of the results of this article to [Aga10].
By Corollary 2.6, if d is coprime to the conductor E (or the discriminant of E), then theũ in Theorem 3.2 is a power of 2. Note that the D in [Aga10] is −d, with d < 0. Thus if one replaces the hypothesis (**) in §2 of loc. cit., with the hypothesis that D is coprime to the conductor N of E, then the conclusions of Lemma 2.1, Proposition 2.2, and Corollary 2.4 are valid up to a power of 2. As a consequence (see the discussion after Corollary 2.4 in loc. cit.), we would like to weaken the hypothesis (**) in Conjecture 2.5 of loc. cit. to the hypothesis that D is coprime to N , and thus make the following conjecture: Conjecture 5.3. Let E be an optimal elliptic curve over Q of conductor N and let −D be a negative fundamental discriminant such that D is coprime to N . As mentioned in loc. cit., using the mathematical software sage, with its inbuilt Cremona's database for all elliptic curves of conductor up to 130000, we verified the conjecture above for all triples (N, E, D) such that N and D are positive integers with N D 2 < 130000, and E is an optimal elliptic curve of conductor N .
Finally, we remark that Proposition 2.5 explains why the concluding statement of Conjecture 2.5 of [Aga10] does not hold in the example of (E, D) = (27a1, 3) in Table 1 of loc. cit. (this example does not satisfy the hypotheses of the conjecture): using SAGE, we find that ∆(E min ) = −3 9 and c 6 (E min ) = 2 3 ·3 6 , and so by part 1(b) of Proposition 2.5, v 3 (ũ) > 0. In particular, this is an example of an optimal elliptic curve for whichũ is not a power of 2. Anyhow, the concluding statement of Corollary 2.4 in loc. cit. does not hold, and so for this pair (E, D), assuming the second part of the Birch and Swinnerton-Dyer conjecture, one does not expect that |E −D (Q)| 2 divides |X(E −D )| · p|N c p (E −D ), even up to a power of 2 (see the discussion just before Corollary 2.5 in loc. cit.); rather one expects that |E −D (Q)| 2 dividesũ · |X(E −D )| · p|N c p (E −D ), and so it is not surprising that |E −D (Q)| 2 divides 3 · |X(E −D )| · p|N c p (E −D ), up to a power of 2.