The Local Symbols

In this chapter, we shall define, for each v ∈ T₀\ {2}, a group homomorphism Φv : E(Q^v)/2E(Q^v) −→ f^v,

where

fv ' E(Qv)/ N_Lv/Qv(E(L_v))

for a particular quadratic extension Lv/Q^v. Then we define the local symbol {ϕ}_υ := Φ_v(P_ϕv),

for each ϕ ∈ Sel₂(E/Q). Also, we denote

N_v = NLv/Qv(E(L_v)). (36) 1. The Archimedean Place

Suppose v = ∞. Then Qv = R and we take Lv = C. Also, since E[2] ⊂ E(Qv), the topological group E(Qv) = E(R) has two components and NC/R(E(C)) is just the identity component of E(R). Therefore, E(Qv)/ NLv/Qv(E(L_v)) ' Z/2Z. Let Φ_v be the composition

E(Qv)/2E(Qv) −→ E(Qv)/ N_Lv/Qv(E(L_v))−→ F^∼ 2.

Suppose an element [P ] ∈ E(Qv)/2E(Qv) is obtained from a point P = (ξ, η) ∈ E(Qv). Then we have

Φ_v([P ]) =

(0 ∈ F2, if ξ ≥ b,

1 ∈ F2, otherwise. (37)

From this, we see that Φ_v([P ]) = 0 if and only if P ∈ 2E(Qv). Therefore, we have the following:

Lemma IV.1. An element ϕ ∈ Sel₂(E/Q) has local symbol {ϕ}^∞ = 0 if and only if ϕ∞ is unramified.

2. The Split Multiplicative Places

The main reference of this as well as next section is [Sil86], section C.14. Sup-pose v is an odd place at which E has split multiplicative reduction. Then over Qv, E is isomorphic to the Tate curve with some period Q_v ∈ Q^∗v so that

E( ¯Qv) ' ¯Q^∗v/Q^Z_v (38) as G_Qv-modules. For an z ∈ ¯Q^∗v, let P_z ∈ E( ¯Qv) denote the point corresponding to z modulo Q^Z_v (the Tate’s uniformization).

29

Let L_v/Qv be the unique unramified quadratic extension. As E[2] ⊂ E(Qv), we have

pQ_v ∈ Q^∗v. In particular, Q_v = (√

Q_v)² ∈ NLv/Qv(L^∗_v), and hence N_Lv/QvE(L_v) ' N_Lv/Qv(L^∗_v)/Q^Z_v.

This shows that E(Q^v)/ NLv/QvE(Lv) ' Q^∗v/ NLv/Qv(L^∗_v) ' Z/2Z. Also, since L_v/Qv is unramified, the local unit group Z^∗v is contained in N_Lv/Qv(L^∗_v). There-fore, an z ∈ Q^∗v is contained in N_Lv/Qv(L^∗_v) if and only if the valuation ord_v(z) is even, or equivalently, the extension Qv(√

z)/Qv is unramified. Since, by the isomor-phism (38), Qv(√

z) is the same as the field generated by the coordinates of one (and hence all) of the 2-division points of P_z, we see that P_z ∈ E(Qv) is in the norm NLv/QvE(L_v) if and only if some of its 2-division points is rational over an unramified extension of Qv

For a point P = (ξ, η) ∈ E(Q^v), we use the formula in Section II.2 to compute a 2-division point Q = (x, y) of P . Then we define Φv so that for the class [P ] ∈ E(Qv)/2E(Qv),

Φ_v([P ]) =

(0 ∈ F2, if Qv(x, y)/Qv is unramified,

1 ∈ F2, otherwise. (39)

In view of this, we have the following lemma.

Lemma IV.2. Suppose v is an odd place at which E has split multiplicative reduction. An element ϕ ∈ Sel₂(E/Q) has local symbol {ϕ}v = 0 if and only if ϕ_v is unramified. In this case the reduction at v of ϕ([v]) ∈ E[2], where [v] is the Frobenius element at v, is a smooth point on ¯E, the reduction curve of E.

Proof. To prove the last statement, we note that ϕ([v]) = Pz for some z ∈ Z^∗v. That the reduction of such point is smooth is well known for Tate’s uniformization

(see [Sil86], theorem C.14.1(c)). 

3. The Non-Split Multiplicative Places

Suppose v is an odd place at which E has non-split multiplicative reduction and let L_v/Qv be the unique unramified quadratic extension. Then L_v = Qv(√

µ) for some non-square local unit µ ∈ Z^∗v \ (Z^∗v)² and the quadratic twist E_µ has split multiplicative reduction at v. Thus, as an abstract group, we have

E(L_v) = E_µ(L_v). (40)

Also, since E_µ is the µ-twist of E and L_v = Qv(√

µ), for the natural action of the Galois group Gal(L_v/Qv) =: {τ, id} on E_µ(L_v), we have

E(Qv) = {P ∈ E_µ(L_v) | P +^τP = 0}, (41) and

NLv/Qv(E(Lv)) = {P ∈ Eµ(Lv) | P =^τQ − Q, for some Q ∈ Eµ(Lv)}. (42)

4. THE POTENTIAL GOOD ADDITIVE PLACES 31

This shows that E(Qv)/ N_Lv/Qv(E(L_v)) = H¹(Gal(L_v/Qv), E_µ(L_v)) which, by Corol-lary II.5, is the dual group of E_µ(Qv)/ N_Lv/Qv(E_µ(L_v)). Therefore, from the discus-sion in the previous section, we can deduce that

E(Qv)/ NLv/Qv(E(L_v)) ' Z/2Z.

We can explicitly describe the group NLv/Qv(E(Lv)) as follow. Let P = (ξ, η) ∈ E(Qv). Then P⁰ = (µξ, µ^3/2η) ∈ E_µ(L_v) is the corresponding point under the identification (40). Here as usual, the defining equation for E_µ is

y² = x(x − µa)(x − µb).

Lemma IV.3. The following conditions are equivalent:

(a) P ∈ NL_v/Qv(E(L_v)).

(b) P⁰ = P_z with z =^τt/t for some t ∈ L^∗_v. (c) P⁰ = P_z with z ∈ O^∗_Lv.

(d) The reduction ¯P⁰ is a smooth point on ¯E_µ (e) The reduction ¯P is a smooth point on ¯E.

Proof. That (a) ⇐⇒ (b) is from (42), and (b) ⇐⇒ (c) is from (41) as well as Hilbert’s Theorem 90. Also, (c) ⇐⇒ (d) is by [Sil86], theorem C.14.1(c)), and (d)

⇐⇒ (e) is obvious. 

Then we define Φ_v so that for the class [P ] ∈ E(Qv)/2E(Qv), Φ_v([P ]) =

(0 ∈ F2, if the reduction ¯P is a smooth point on ¯E,

1 ∈ F², otherwise. (43)

Lemma IV.4. Suppose v is an odd place at which E has non-split multiplicative reduction. An element ϕ ∈ Sel₂(E/Q) has local symbol {ϕ}v = 0 if and only if ϕ_v is unramified and the point ϕ([v]) ∈ E[2], where [v] is the Frobenius element at v, has smooth reduction.

Proof. Write Pϕ⁰ = P_z ∈ E_µ(L_v) for the point corresponding to P_ϕ. The local symbol {ϕ}_v = 0 if and only if z can be chosen in O_L^∗_v, or equivalently, the field extension Qv(√

z)/Qv is unramified and^[v]√ z ·√

z = u × Q^m_v where u is a local unit and m ∈ Z.

 4. The Potential Good Additive Places

Suppose v is an odd place at which E has additive reduction so that its π_v -twist E_πv, for some prime element π_v, has good reduction. Since, for every u ∈ Z^∗v, E_uπv = (E_πv)_u also has good reduction, we see that if L_v/Qv is a ramified quadratic extension with L_v = Qv(√

µ), then E_µ has good reduction. In this case, the equalities (40), (41), (42) also hold, and hence the group E(Qv)/ NL_v/Qv(E(L_v)) is dual to the cohomology group H¹(Gal(L_v/Qv), E_µ(L_v)). We claim that in this case, E(Q^v)/ NLv/Qv(E(Lv)) ' E[2].

Since v is odd, the formal group bE_µ(OLv) is divisible by 2, and from the exact sequence

0 −→ bE_µ(O_Lv) −→ E_µ(L_v) −→ ¯E_µ(Fv) −→ 0

induced from the reduction map, where Fv is the residue field, we deduce that H¹(Gal(L_v/Qv), E_µ(L_v)) ' H¹(Gal(L_v/Qv), ¯E_µ(Fv)).

The Galois group Gal(Lv/Q^v) acts trivially on F^v, as Lv/Q^v is totally ramified.

Therefore, H¹(Gal(L_v/Qv), ¯E_µ(Fv)) is isomorphic to E[2] and the claim is proved.

Then we make another claim that E(Qv)/2E(Qv) ' E[2], too. To see this, we apply the exact sequence ([Sil86], VII, §2)

0 −→ bE(Zv) −→ E₀(Qv) −→ ¯E_ns(Fv) −→ 0,

where ¯E_ns(Fv) ' Fv is the subgroup consisting of non-singular points. It implies that E₀(Qv) is 2-divisible. On the other hand, the Kodaira and N´eron’s theorem (see [Sil86],p. 183 Theorem 6.1) implies that E(Qv)/E₀(Qv) is a finite group of order at most 4. Since E[2] is a subset of E(Qv), we have E₀(Qv) = 2E(Qv) and the claim is proved.

The above two claims together imply E(Qv)/2E(Qv) ' E(Qv)/ NL_v/Qv(E(L_v)), if L_v/Qv is ramified. Suppose L_v/Qv is the unramified quadratic extension. Then over L_v, our elliptic curve E also has additive reduction, and an argument similar to the above one shows that E₀(L_v) is two divisible and E(L_v)/2E(L_v) ' E[2]. Then we also have E(Qv)/ NLv/Qv(E(L_v)) = E(Qv)/2E(Qv). We summarize as follow.

Lemma IV.5. If L_v/Qv is a quadratic extension, then N_Lv/Qv(E(L_v)) = E₀(Qv) = 2E(Qv) and the composition

E[2] −→ E(Q^v) −→ E(Q^v)/ NLv/Qv(E(Lv)) is an isomorphism.

We let fv = E[2] and define Φ_v so that for the class [P ] ∈ E(Qv)/2E(Qv), Φv([P ]) = A ∈ E[2], if the reduction P − A is a smooth point on ¯E.

Thus, for ϕ ∈ Sel₂(E/Q), the local symbol {ϕ}v = 0 if and only if P_ϕv ∈ 2E(Qv).

Hence, we have the following:

Lemma IV.6. Suppose v is an odd place at which E has potential good additive reduction. An element ϕ ∈ Sel₂(E/Q) has local symbol {ϕ}v = 0 if and only if ϕ_v = 0.

5. The map s For simplicity, denote

T00= T0\ {2}.

6. THE d-TWIST OF E FOR d = ±p OR d = ±pq 33

In the previous sections, we have defined, for each v ∈ T₀₀, the groups N_v ⊂ E(Qv), fv ' E(Qv)/N_v and the local symbol {ϕ}_υ. Now, we set

f = M

v∈T00

fv

and define

s: Sel₂(E/Q) −→ f,

by putting s(ϕ) = ({ϕ}_υ)_v∈T00 ∈ f for each ϕ ∈ Sel2(E/Q).

Lemma IV.1, Lemma IV.2, Lemma IV.4, and Lemma IV.6 together imply the following:

Lemma IV.7. If ϕ ∈ Sel₂(E/Q) with {ϕ}v0 = 0 for some v₀ ∈ T₀₀, then ϕ is unramified at v₀.

Corollary IV.1. The kernel of s is contained in Hom(Gal(Q(√

2)/Q), E[2]).

6. The d-Twist of E for d = ±p or d = ±pq

In this section, we assume that the integers a and b in the defining equation (1) have at most one odd common prime factor. Let q denote such prime, if it exists.

We shall consider the d-twist of E with either d = ±p or d = ±pq for some p 6∈ T₀. Note that if q exists, then E has potential good additive reduction at q and we are in the situation of Section IV.4. In particular, the map that sends each P ∈ E[2]

to the local symbol {ϕ_P}_q is an isomorphism

E[2] ' E(Qq)/2E(Qq).

Write

Sel2(E/Q)^◦ = {ϕ ∈ Sel2(E/Q) | {ϕ}^q = 0}

and

T₀₀^◦ = T₀₀\ {q}.

If q does not exist, we denote Sel2(E/Q)^◦ = Sel2(E/Q) and T00^◦ = T00. In either case, let s^◦ denote the composition

Sel₂(E/Q)^◦ −→ f −→ f^{s ◦} := Y

v∈T₀₀^◦

fv

where the last arrow is the projection.

Assume that s is injective. Then so is s^◦. For each subset Θ ⊂ T₀₀^◦, set fΘ :=Y

v∈Θ

fv ⊂ f^◦.

By the Gaussian elimination, we can find a subset Θ₀ ⊂ T₀₀^◦ so that fΘ0, considered as a subspace of f^◦, forms a direct summand of the image Im(s^◦). Namely, we have Im(s^◦) ∩ fΘ0 = {0}, and Im(s^◦) + fΘ0 = f^◦. (44) Let L = Q(√

d). In this case, we have T = T₀∪ {p}. Suppose d is chosen so that Θ0∪ {2} = {v ∈ T | L/Q splits at v}. (45)

For each v ∈ T₀\ (Θ₀∪ {2}), we recall the notation in the previous sections and put L_v = L_v. Then N_Lv/Qv(E(L_v)) = N_v. For v ∈ Θ₀∪ {2}, we have N_Lv/Qv(E(L_v)) = E(Qv). Thus, the condition (44) actually implies that the equality (19) holds so that

Im(R_T0) + (N_L/Q(E_T0(L))/2E_T0(Q)) = ET0(Q)/2ET0(Q).

By Lemma I.1, the extension L/Q is independent and tL = 2, and Theorem III.1 says that E_dhas rank zero over Q. For v ∈ T00, x ∈ Z, let (^x_v) denote the Lengendre symbol in the sense that if v = ∞, then (^x_v) = 1 for x > 0 and (^x_v) = −1 for x < 0.

Then the condition (45) is equivalent to d ≡ 1 (mod 8), d In the above condition, the value of the symbol (_∞^d) determines the sign of d, and others give congruent conditions for p. Obviously, the density of positive prime numbers p satisfying these congruent conditions is (¹₂)^m(E)+1.

Assume that s is injective but not surjective. Then our Θ₀ is non-empty, and we choose a subset Θ1 ⊂ Θ0 so that |Θ1| = |Θ0| − 1. An argument similar to the above shows that if L = Q(√

d) with d satisfying the condition d ≡ 1 (mod 8), d then L/Q is an independent extension and we have tL= 3. Hence, by Theorem III.1, if the Parity Conjecture holds for E_d over Q, then then rk(Ed(Q)) = 1. Again, the condition (47) not only determines the sign of d but also defines a set of congruent conditions for positive prime numbers of density (¹₂)^m(E)+1. We summarize the above discussion as follows.

Proposition IV.1. Suppose s is injective and Θ₀ is chosen so that (44) holds.

Then for d = ±p or d = ±pq satisfying the condition (46), the d-twist E_d has zero rank over Q. Furthermore, if s is not surjective and Θ1 is a subset of Θ₀ with cardinality less by one, then for d = ±p or d = ±pq satisfying the condition (47), the d-twist Edshould have rank one over Q unless the Parity Conjecture fails to hold for E_d over Q. In both cases, the set of prime number p satisfying the corresponding conditions has density (¹₂)^m(E)+1.

7. An Example

Let the elliptic curve E/Q given by the Weierstrass equation E : y² = x(x − 15)(x − 77).

Then

T₀ = {∞, 2, 3, 5, 7, 11, 31}

E[2] = {O, P₀ = (0, 0), P₁₅ = (15, 0), P₇₇= (77, 0)}.

For finite place v, let p_v be the corresponding prime number. For infinite place

∞, let p∞ = −1. The field Q⁽²⁾T0 defined in the Section II.4 is generated by √ p_v

7. AN EXAMPLE 35

where v ∈ T₀. Let G⁽²⁾

Q,T0 be the Galois group of Q⁽²⁾T0 over Q. For v ∈ T0 we choose σ_v ∈ G⁽²⁾

Q,T0 such that

σ_v(√

p) =  −√

p , if p = p_v

√p , if p 6= pv.

Then G⁽²⁾

Q,T0 is generated by {σ_v | v ∈ T₀}. As in (26), we choose the set A_T0 := {ρ₍^√_pv,P0), ρ₍^√_pv,P15)| v ∈ T₀}.

Then

ρ₍^√_v,P0)(σ_p) =  P0 , if p = p_v O , if p 6= p_v.

In order to compute Sel₂(E/Q), we need to find B in (25). For each v ∈ T0, let Q^(2)v be the maximal abelian extension of Qv with exponent 2 and G⁽²⁾v the Galois group of Q^(2)v /Q^v. If v = ∞, then G⁽²⁾∞ is generated by σ∞. When v is odd, let τv

be the non-trivial element in G⁽²⁾v which fixes Qv(√

v). Then G⁽²⁾v is generated by σ_v and τ_v. When v is even, let δ−1 (resp. δ₃) be the non-trivila element in G⁽²⁾₂ fixed

√2 and √

3 (resp. √

2 and √

−1). Then G⁽²⁾₂ is generated by σ₂, δ−1 and δ₃. Since B_v is contained in Hom(G⁽²⁾v , E[2]), we use the method described in the Section II.2 and list in the following tabel.

Table IV.1 B_v

No. σ_v τ_v

3.1 O P₇₇

3.2 P₇₇ O

5.1 O P₇₇

5.2 P₇₇ P₀

7.1 O P₁₅

7.2 P₁₅ P₀

11.1 O P₁₅

11.2 P₁₅ P₀

31.1 O P0

31.2 P0 P15

Table IV.2 B₂

No. σ2 δ−1 δ3

2.1 P0 P15 P77

2.2 O O P₁₅

2.3 P₀ P₇₇ O

Now, we use the Proposition II.2 to compute the values of the pairings and list it in the following table.

Table IV.3

be the solutions to the system of linear equations over F2

X

ϕ∈A_T0

< ϕ_v, ψ_v >_Qv ·x_ϕ = 0, ψ_v ∈ B_v, v ∈ T₀.

A basis for V and the corresponding element in Sel₂(E/Q) are listed in following.

Table IV.4

7. AN EXAMPLE 37

Now, we need to determine the local symbols for the generators listed above.

For an odd place v in T₀, the Frobenius element [v] at v is given by [v] =X

p σ_p,

where p runs through T₀ with Legendre symbol ^p_v
= −1. Thus we may use the Lemma IV.1, IV.2 and IV.4 in the above sections to determine the local symbols.

Table IV.6

From the above table, we see that map s is injective but not surjective and Im(s^◦) is a subspace of f with dimension 4. Thus the set Θ0satisfying condition (44) contains two elements. The above table tell us that the choice of Θ₀ is not unique, but an arbitrary choice of two elements of T00 might not work. If we choose Θ0 consisting of ∞ and 31, then the condition (46) tell us that S0 in Main Theorem I consists of the primes p satisfying the condition (47) tell us that S₁ consists of the primes p satisfing

p

3
= ^p₅
= ^p₇
= ₁₁^p
= ₃₁^p

= −1

p ≡ 1 mod 8, (48)

and ε1 = 1. The Main Theorem I implies that if p is an element in S1 so that the Parity Conjecture holds for Ep/Q, then E^p has rank 1 over Q. For example, we find three primes 17, 1097 and 4457 satisfying the condition (48) and list the curves and points with infinite order in the following.

• p = 17

– E₁₇: y² = x(x − 255)(x − 1309) – (17493, 2209116)

• p = 1097

– E1097 : y² = x(x − 16455)(x − 84469)

– (19874174293991588311797

70557598472000809 ,2274823994119540883954733398081412 18741989309475259908083627 )

• p = 4457

– E₄₄₅₇ : y² = x(x − 66855)(x − 343189) – (50492119267754085453

146019164977921 ,27953176195024774075593726564 1764472080507635918719 )

CHAPTER V

在文檔中 橢圓曲線的二次扭變 (頁 29-39)