The Proof of the Main Theorem

In this chapter, we complete the proof of our Main Theorem I. According to Proposition IV.1, if the local symbols map s is injective but not surjective, then the theorem is proved. But this condition does not always hold, as shown in the case of congruent curve. Our strategy is to find a prime number q 6∈ T₀, so that for the q-twist E_q, the associated local symbols map,

s^(q) : Sel₂(E_q/Q) −→ f^(q),

is injective but not surjective. Then we can complete the proof of the Main Theorem I, by applying Proposition IV.1 again, taking the ±pq twist of Eq, which is the same as E_±p. Thus, it is enough for us to prove the following:

Proposition V.1. Let E/Q be an elliptic curve defined by the Weierstrass equation

y² = x(x − a)(x − b),

with a, b ∈ Z, 0 < a < b and (a, b) = 1 or 2. Then there exists an odd prime number q, relatively prime to ab(b − a), so that the local symbols map s^(q) is injective but not surjective.

For the rest of this Chapter, we assume that E is defined as in the proposition.

1. The Injectivity of s^(q)

By Corollary IV.1, the kernel of s^(q)is contained in Hom(Gal(Q(√

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

On the other hand, since Eq has potential good additive reduction at q, an element ϕ ∈ Sel2(Eq/Q) has local symbol {ϕ}^q= 0 if and only if Pϕq ∈ 2Eq(Q^q) (see Section IV,4), or equivalently, the extension K^(ϕ)/Q splits at q. However, the extension Q(

√2)/Q only splits at odd primes that are congruent to ±1 modulo 8. Therefore, we have proved the following:

Lemma V.1. If q is a prime number relatively prime to ab(b − a) with q ≡ ±3 (mod 8),

then the map s^(q) is injective.

2. The Non-surjectivity of s^(q)

Let q be congruent to ±3 modulo 8 so that by Lemma V.1, the map s^(q) is injective. Write T₀₀^(q) = T₀₀∪ {q} and f^(q) =Q

v∈T₀₀^(q)f^(q)v .

Suppose that s^(q) is surjective and let Λ ⊂ Sel₂(E_q/Q) denote the pre-image of f^(q)q considered as a subspace of f^(q). Then by Lemma IV.5, we have Λ ' Eq[2], and

39

hence |Λ| = 4. Lemma IV.7 implies that each element in Λ is unramified outside field of the decomposition subgroup < [v] > of Gal(Q(√

2,√ q)/Q).

Lemma V.2. Let the notation be as above. Then there exists a proper subgroup D ⊂ E[2] such that for each ϕ ∈ Λ, we have ϕ([v]) ∈ D.

Proof. Since Eq has multiplicative reduction at v, by Lemma IV.2 and Lemma IV.4, for each ϕ ∈ Λ, the point ϕ([v]) has smooth reduction at v. Let D be the subgroup of E_q[2] consisting of points with smooth reduction. Then |D| = 2.  Since |Λ| = 4 and |D| = 2, there exists a non-trivial element ϕ ∈ Λ such that

Since E has good reduction at q, for the reduction curve ¯E, we have

| ¯E(Fq)| = 1 + q − α, for some integer α of absolute value not greater than 2√

q. Let ¯E⁰/Fq denote the quadratic twist ¯E. Then we have

| ¯E⁰(Fq)| = 1 + q + α.

Consider the composition

j_q: Hom(Gal(K/Q), Eq[2]) ,→ H¹(Q, Eq[2]) −→ H¹(Q, Eq) −→ H¹(Qq, E_q), where the first arrow is the natural embedding, the second is induced from the inclusion E_q[2] ⊂ E_q( ¯Q) and the last is the localization map.

Lemma V.3. Let the notation be as above. Then the following are true:

(a) If K = Q(√

2), then j_q is injective.

(b) If K = Q(√

q) and | ¯E(Fq)| ≡ 4 mod 8, then j_q is injective.

(c) If K = Q(√

2q) and | ¯E⁰(Fq)| ≡ 4 mod 8, then j_q is injective.

Proof. For a local field K of residual characteristic q, the topological group E_q(K) = A × B where A is a compact q-group and B is a finite discrete group. We will simply call the Sylow 2-subgroup of B the 2-part of E_q(K).

Suppose K = Q(√

2). Then the extension K/Q is inert at q. Therefore, Eq/K_q has additive reduction and the the 2-part of E_q(K_q) is E_q[2] (Lemma IV.5). In par-ticular, the natural map E_q(Qq)/2E_q(Qq) −→ E_q(K_q)/2E_q(K_q) is an isomorphism.

This implies the injectivity of jq. Suppose K = Q(√

q). Then the extension K/Q is totally ramified at q and Eq/Kq has good reduction and as groups Eq(Kq) and E(Kq) are isomorphic. Thus, the 2-parts of E_q(K_q), E(K_q) and ¯E(Fq) are isomorphic. The condition of the lemma implies that the 2-part of Eq(Kq) equals Eq[2]. Then we also deduce that the natural

2. THE NON-SURJECTIVITY OF s^(q) 41

map E_q(Qq)/2E_q(Qq) −→ E_q(K_q)/2E_q(K_q) is an isomorphism and hence the map j_q is injective.

Finally, if K = Q(√

2q), then the reduction of E_q/K_q equals ¯E⁰. A similar

argument also leads to the injectivity of j_q. 

Lemma V.4. Let q be a prime number congruent to ±3 modulo 8. Then the map s^(q) is not surjective, if there is a prime number l | ab(a − b) satisfying one of the following conditions: as in Lemma V.2. Then K/Q splits at v and hence the residue class of l modulo 8 determines K, since K is an intermediate field of the extension Q(√

2,√ q)/Q.

Suppose (a) holds. Then K = Q(√

2) and Lemma V.3 implies that ϕ can not be contained in Sel2(Eq/Q), a contradiction. If either (b) or (c) holds, then K = K_i = Q(Qi). Consider the following three mutual exclusive cases:

(1) There exists some i ∈ {0, a, b} so that √

(3) There exists some K_i(√

−1) that does not contains √

2 and every such K_i(√

−1) contains √

l for all odd prime divisor l of ab(b − a).

Suppose (1) holds. Then there exists an element λ ∈ Gal(K_i(√

l)) is a non-trivial automorphism of the field. By the density theorem, there is a prime number q so that the Frobenius element [q] ∈ Gal(K_i(√

−1,√ 2,√

l)/Q) equals λ. Then q splits completely over K_i(√ has at least an solution in the residue field of each K_i. Therefore, none of these residue field equals Fq, and consequently, | ¯E(Fq)| ≡ 4 (mod 8). We see that the condition (d) of Lemma V.4 holds.

Suppose the condition (3) holds. Note that K_i = Q(√ α,√

β) where α, β ∈ {±a, ±b, ±(a − b)}. Therefore, we must have

{a, b, b − a} = {2ⁿ⁰, lⁿ₁¹, l₂ⁿ²}, n₁ > 0.

Furthermore, we know that n₀ must be odd, for otherwise we shall deduce the contradiction that neither √

First, let us assume that n₂ > 0. We shall show that it is enough to choose q so that

q)/Q) is the just generator, denoted as h, of the group Gal((Q(√ the equality holds, since both groups are of order 4. For simplicity, write

G = Gal(Qq(√ identified with each other. Via this identification, we deduce that for an element ψ ∈ Hom(G, Eq[2]), the local class [ψv] ∈ H¹(G, Eq(Q^q(√

q))) is trivial if and only

3. THE EXAMPLES OF THE CONGRUENT CURVE 43

the reduction ψ(g) (considered as a element of ¯E(Fq)) is contained in N_G( ¯E(Fq)).

Since G acts trivially on ¯E(Fq), we have N_G( ¯E(Fq)) = 2 ¯E(Fq). Also, the condition (49) implies that F^∗q contains no primitive 4’th root of one, and hence

E[4] * ¯¯ E(Fq). (50)

This shows that the natural map

E[2] −→ ¯E[2] −→ ¯E(Fq)/ N_G( ¯E(Fq))

is not the trivial map. In particular, there is an element ψ ∈ Λ = Hom(G, E_q[2]) such that [ψ_v] 6= 0. This contradicts to the fact that Λ ⊂ Sel₂(E_q/Q).

At last, we consider the case where (a, b) = (1, 3), or (a, b) = (2, 3). We shall show that if q ≡ 5 (mod 8), then the map s^(q) is not surjective, by showing that there is no ψ ∈ Sel2(Eq/Q) so that the local symbols {ψ}^∞ 6= 0 and {ψ}v = 0, for v 6= ∞.

If such ψ were to exist, it must be unramified outside 2 and ∞. Therefore, ψ ∈ Hom(Gal(Q(√

−1,√

2)/Q), Eq[2]). But, since ψ_q is trivial (as E_q has potential good additive reduction at q), we have ψ ∈ Hom(Gal(Q(√

−1)/Q), Eq[2]). Denote G = Gal(Q(√

−1)/Q) =< g >. Then we have ψ(g) = P0 ∈ E_q[2] with P₀ = (0, 0).

However, since g is the Frobenius at 3 and the reduction of P₀ at 3 is not a smooth point on ¯Eq, we know that either [ψ]3 6= 0 or {ψ}3 6= 0, a contradiction.

 3. The examples of the Congruent Curve

In this section, we complete the proof of Proposition V.1. We shall deal with the cases (a, b) = (1, 2), (2, 4) by direct computation. Doing so, we also get a proof that primes congruent to 3 modulo 8 are not congruent numbers.

3.1. Consider that (a, b) = (1, 2). Then the elliptic curve is the congruent curve:

E : y² = x(x − 1)(x − 2).

Let q = 3 and denote the 3-twist by E₃. Then

T₀⁽³⁾ = {∞, 2, 3} and E₃[2] = {O, P₀, P₃, P₆},

where P_i = (i, 0) for i = 0, 3 and 6. Let σ_v, τ_v, δ−1 and δ₃ be as in the section IV.7.

As in Section II.4, we choose

A_T0 := {ρ₍^√_pv,P0), ρ₍^√_pv,P3)| v ∈ T₀⁽³⁾}.

Since E₃ has potential good additive reduction at 3, by the Lemma IV.5, we know that B₃ consists of ρ_P0 and ρ_P3. By the method described in the Section II.2, we compute B₂, B₃ and list them in the following tables.

Table V.1 B₃

No. σ3 τ3

ρ_P0 P₀ P₆ ρ_P3 P₃ P₀

Table V.2 B₂

No. σ₂ δ−1 δ₃ 2.1 P₃ P₀ P₀

2.2 O P₀ P₃

2.3 P₃ P₆ P₆

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

be the solutions to the system of linear equations 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(E₃/Q) are listed in V.4 and V.5 Table V.4

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

Since E₃ has potential good additive reduction at 3, by the tables above we get the table below.

Table V.6

No. ∞ 3

1 1 P₀

2 1 P₃

Thus we get that the local symbols map s⁽³⁾ is injective but not surjective. Therefore the Proposition V.1 holds for (a, b) = (1, 2) when we take q = 3.

The computation above implies that a prime p congruent 3 modulo 8 is not a congruent number, and a prime p congruent to 5 modulo 8 is a congruent number by the Propositon IV.1. Since Sel^◦₂(E₃/Q) contains the zero map only and T00^◦ consists of ∞, we may put that Θ₀ consists of ∞. Thus the condition (46) holds for d = 3p with 3p congruent to 1 modulo 8. Thus the 3p-twist of E3 has rank 0 for the primes

3. THE EXAMPLES OF THE CONGRUENT CURVE 45

p satisfying p congruent to 3 modulo 8. Since the 3p-twist of E₃ is equivalent to the p-twist of E, the primes congruent to 3 modulo 8 are not congruent numbers. We take Θ₁ to be the empty set. Then the condition (47) holds for d = −3p with (−3p) congruent to 1 modulo 8. Since the (−3p)-twist of E₃ is equivalent to the p-twist of E, the primes congruent to 5 modulo 8 are congruent numbers.

3.2. Consider that (a, b) = (2, 4). Then the elliptic curve is given by E2 : y² = x(x − 2)(x − 4),

which is the 2-twist of the congruent curve E. Let q = 5 and denote the 5-twist by E₁₀. Then

T₀⁽⁵⁾ = {∞, 2, 5} and E₁₀[2] = {O, P₀, P₁₀, P₂₀},

where Pi = (i, 0) for i = 0, 10 and 20. Let σv, τv, δ−1 and δ3 be defined as in the section IV.7. As in Section II.4, we choose that

A_T0 := {ρ₍^√_pv,P0), ρ₍^√_pv,P10) | v ∈ T₀⁽⁵⁾}.

Similar to the above case, B₅ consists of ρ_P0 and ρ_P10. By the method described in the Section II.2, we compute B₂, B₅ and list them in the following tables.

Table V.7 B₅

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

be the solutions to the system of linear equations 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(E₁₀/Q) are listed in the follow-ing.

Table V.10

No. 1 2

ρ₍^√_−1,P0) 1 1 ρ₍^√_−1,P10) 0 0 ρ₍^√_2,P0) 1 0 ρ₍^√_2,P

10) 1 1

ρ₍^√_5,P0) 1 0 ρ₍^√_5,P

10) 0 1

Table V.11

No. 1 2

σ_∞ P₀ P₀ σ₂ P₂₀ P₁₀ σ₅ P₀ P₁₀

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

Since E₁₀ has potential good additive reduction at 5, by the tables above we get the table below.

Table V.12

No. ∞ 5

1 1 P₀

2 1 P₁₀

Thus we get that the local symbols map s⁽⁵⁾ is injective but not surjective. Therefore the Proposition V.1 holds for (a, b) = (2, 4) when we take q = 5.

The computation above implies that a number 2p with the prime p congruent 5 modulo 8 is not a congruent number, and a number 2p with the prime p congruent to 3 modulo 8 is a congruent number by the Propositon IV.1. Since Sel^◦₂(E₁₀/Q) contains the zero map only and T₀₀^◦ consists of ∞, we may put that Θ₀ consists of

∞. Thus the condition (46) holds for d = 5p with 5p congruent to 1 modulo 8. Thus the 5p-twist of E10 has rank 0 for the primes p satisfying p congruent to 5 modulo 8. Since the 5p-twist of E₁₀ is equivalent to the 2p-twist of E, a number 2p with the prime p congruent to 5 modulo 8 is not a congruent number. We take Θ₁ to be the empty set. Then the condition (47) holds for d = −5p with (−5p) congruent to 1 modulo 8. Since the (−5p)-twist of E₁₀ is equivalent to the 2p-twist of E, a number 2p with the prime p congruent to 3 modulo 8 is a congruent number.

Bibliography

[Bir69] B. J. Birch, Diophantine analysis and modular functions, in Algebraic Geometry (Internat.

Colloq., Tata Inst. Fund. Res., Bombay, 1968), 35-42, Oxford Univ. Press, London, 1969.

[Bir70] -, Elliptic curves and modular functions, In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), 27-32 Academic Press, London, 1970.

[CaE56] H. Cartan, S. Eilenberg Homological Algebra, Princenton University Press, Princeton, New Jersey, 1956.

[CLT05] W.-C. Chi, K.F. Lai and K.-S. Tan, Integral points on elliptic curves, Pac. J. Math. 222, No. 2 (2005), 237-252

[CKRS] J.B. Conrey, J.P. Keating, M.O. Rubinstein, N.C. Snaith On the frequency of vanishing of quadratic twists of modular L-functions, Number theory for the millennium, I (Urbana, IL, 2000), 301V315, A K Peters, Natick, MA, 2002.

[Elk94] N. Elkies, Heegner point computations, Algorithmic Number Theory (ANTS-1), Lect. Notes in Comp. Sci. 877, Springer-Verlag, Berlin, 1994, 122-133.

[ElkWeb] -, http://www.math.harvard.edu/ elkies/compnt.html

[Gol79] D. Goldfeld, Conjectures on elliptic curves over quadratic fields, Lecture Notes in Math.

751, Springer-Verlag, 1979, 108-119.

[GouM91] F. Gouvˆea, B. Mazur, The square-free sieve and the rank of elliptic curves J. Amer.

Math. Soc. 4, 1991, no.1, 1-23.

[Hea94] D.R. Heath-Brown, The size of Selmer groups for the congruent number problem. II Invent.

Math. 118, 1994, no.2, 331-370.

[Hee52] K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56,1952, 227-253.

[Mil86] J.S. Milne, Arithmetic Duality Theorems, Academic Press, New York, 1986. , 1952, 227-253.

[M-M94 ] Mai, L. and Murty, R., A note on quadratic twists of an elliptic curve, CRM Proceedings and Lecture Notes, 4, 1994, 121-124.

[Mon90] Paul Monsky. Mock heegner points and congruent numbers Mathematische Zeitschrift, 1990, 204:45-67.

[Kob84] Neal Koblitz, Introduction to elliptic curves and modular forms, Graudate Texts in Math.

97, Springer, New York, 1984.

[Ono97] K. Ono, Twists of Elliptic Curves, Compositio Mathematica 106, 1997, 349-360.

[OS98] K. Ono, C. Skinner, Non-vanishing of quadratic twists of modular L-functions Invent. Math.

134, 1998, 651-660.

[Rei75] Reiner, I. Maximal Orders, Academic Press, New York, 1975.

[RubS01] K. Rubin, A. Silverberg, Rank frequencies for quadratic twists of elliptic curves Exper.

Math. 10, 2001, no.4, 559-569.

[RubS02] -, Ranks of elliptic curves Bull. Amer. Math Soc. 39. 2002, 455-474.

[Ser79] J.-P. Serre, Local Fields, Spronger-Verlag, New York, 1979.

[Sht72] S. Shatz, Profinite groups, Arithmetic, and Geometry, Annals of Math. Studies 67, Prince-ton University Press, 1972.

[Sil86] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math. 106, Springer, New York, 1986.

[Sil94] -, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math. 151, Springer, New York, 1994.

47

[SteT95] C.L. Stewart, J. Top, On ranks of twists of elliptic curves and power-free values of binary forms, J. Amer. Math. Soc. 8, 1967, no.4, 943-973.

[Tat57/58] J. Tate, WC-groups over p-adic fields, S´eminaire Bourbaki, Expos´e 156, 13pp.

[Tat62] J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Intern. Congress Math. Stockholm, 234-241.

[Tat67] J. Tate, Global class field theory, in Algebraic Number Theory, J.W.S. Cassels and A.

Fr¨ohlich, eds., Acdemic Press, 1967, 162-203. Proc. Intern. Congress Math. Stockholm, 234-241.

[Tun83] J.B. Tunnell. A classical diophantine problem and modular forms of weight 3/2 Inventiones Mathematicae, 1983, 72:323-334.

在文檔中 橢圓曲線的二次扭變 (頁 39-48)