Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer

Karl Rubin^{⋆} ^{⋆⋆}

Department of Mathematics, Ohio State University, 231 W. 18 Avenue,

Columbus, Ohio 43210 USA, rubin@math.ohio-state.edu

1. Quick Review of Elliptic Curves  2
2. Elliptic Curves over C  4
3. Elliptic Curves over Local Fields  6
4. Elliptic Curves over Number Fields  12
5. Elliptic Curves with Complex Multiplication  15
6. Descent  22
7. Elliptic Units  27
8. Euler Systems  37
9. Bounding Ideal Class Groups  43
10. The Theorem of Coates and Wiles  47
11. Iwasawa Theory and the “Main Conjecture”  50
12. Computing the Selmer Group  61

The purpose of these notes is to present a reasonably self-contained exposition of recent results concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. The goal is the following theorem.

Theorem. Suppose E is an elliptic curve defined over an imaginary quadratic field K, with complex multiplication by K, and L(E, s) is the L-function of E. If L(E, 1) ≠ 0 then

(i) E(K) is finite, (ii) for every prime p > 7 such that E has good reduction above p, the p-part

of the Tate-Shafarevich group of E has the order predicted by the Birch and Swinnerton-Dyer conjecture.

The first assertion of this theorem was proved by Coates and Wiles in [CW1]. We will prove this in §10 (Theorem 10.1). A stronger version of (ii) (with no assumption that E have good reduction above p) was proved in [Ru2]. The program to prove (ii) was also begun by Coates and Wiles; it can

⋆ Partially supported by the National Science Foundation. The author also grate-

fully acknowledges the CIME for its hospitality. ⋆⋆ current address: Department of Mathematics, Stanford University, Stanford, CA

  94305 USA, rubin@math.stanford.edu

now be completed thanks to the recent Euler system machinery of Kolyvagin [Ko]. This proof will be given in §12, Corollary 12.13 and Theorem 12.19.

The material through §4 is background which was not in the Cetraro lectures but is included here for completeness. In those sections we summarize, generally with references to [Si] instead of proofs, the basic properties of elliptic curves that will be needed later. For more details, including proofs, see Silverman’s book [Si], Chapter 4 of Shimura’s book [Sh], Lang’s book [La], or Cassels’ survey article [Ca].

The content of the lectures was essentially §§5-12.

1 Quick Review of Elliptic Curves

1.1 Notation

Suppose F is a field. An elliptic curve E defined over F is a nonsingular curve defined by a generalized Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}  (1)

with a_{1}, a_{2}, a_{3}, a_{4}, a_{6} ∈ F. The points E(F) have a natural, geometricallydefined group structure, with the point at infinity O as the identity element. The discriminant Δ(E) is a polynomial in the a_{i} and the j-invariant j(E) is a rational function in the a_{i}. (See §III.1 of [Si] for explicit formulas.) The j-invariant of an elliptic curve depends only on the isomorphism class of that curve, but the discriminant Δ depends on the particular Weierstrass model.

Example 1.1. Suppose that E is defined by a Weierstrass equation

y^{2} = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

and d ∈ F^{×}. The twist of E by \sqrt{d} is the elliptic curve E_{d} defined by

y^{2} = x^{3} + a_{2}dx^{2} + a_{4}d^{2}x + a_{6}d^{3}.

Then (exercise:) E_{d} is isomorphic to E over the field F (\sqrt{d}), Δ(E_{d}) = d^{6}Δ(E), and j(E_{d}) = j(E). See also the proof of Corollary 5.22.

1.2 Differentials

See [Si] §II.4 for the definition and basic background on differentials on curves.

Proposition 1.2. Suppose E is an elliptic curve defined by a Weierstrass equation (1). Then the space of holomorphic differentials on E defined over F is a one-dimensional vector space over F with basis

ω_{E} = \frac{dx}{2y + a_{1}x + a_{3}} .

Further, ω_{E} is invariant under translation by points of E(F̄).

Proof. See [Si] Propositions III.1.5 and III.5.1. That ω_{E} is holomorphic is an exercise, using that ω_{E} is also equal to dy/(3x^{2} + 2a_{2}x + a_{4} − a_{1}y). ⊓ ⊔ 1.3 Endomorphisms

Definition 1.3. Suppose E is an elliptic curve. An endomorphism of E is a morphism from E to itself which maps O to O.

An endomorphism of E is also a homomorphism of the abelian group structure on E (see [Si] Theorem III.4.8).

Example 1.4. For every integer m, multiplication by m on E is a endomorphism of E, which we will denote by [m]. If m ≠ 0 then the endomorphism [m] is nonzero; in fact, it has degree m^{2} and, if m is prime to the characteristic of F then the kernel of [m] is isomorphic to (Z/mZ)^{2}. (See [Si] Proposition III.4.2 and Corollary III.6.4.)

Example 1.5. Suppose F is finite, #(F) = q. Then the map φ_{q} : (x, y, z) ↦ (x^{q}, y^{q}, z^{q}) is a (purely inseparable) endomorphism of E, called the q-th power Frobenius morphism.

Definition 1.6. If E is an elliptic curve defined over F, we write End_{F} (E) for the ring (under addition and composition) of endomorphisms of E defined over F. Then End_{F} (E) has no zero divisors, and by Example 1.4 there is an injection Z ↪ End_{F} (E).

Definition 1.7. Write D(E/F) for one-dimensional vector space (see Proposition 1.2) of holomorphic differentials on E defined over F. The map ϕ ↦ ϕ^{∗} defines a homomorphism of abelian groups

ι = ι_{F} : End_{F} (E) → End_{F} (D(E/F)) ≅ F.

The kernel of ι is the ideal of inseparable endomorphisms. In particular if F has characteristic zero, then ι_{F} is injective.

Lemma 1.8. Suppose char(F) = 0, L is a field containing F, and ϕ ∈ End_{L}(E). If ι_{L}(ϕ) ∈ F then ϕ ∈ End_{F} (E).

Proof. If σ ∈ Aut(L̄/F) then

ι_{L}(ϕ^{σ}) = σ(ι_{L}(ϕ)) = ι_{L}(ϕ).

Since L has characteristic zero, ι_{L} is injective so we conclude that ϕ^{σ} = ϕ. ⊓ ⊔

Definition 1.9. If ϕ ∈ End_{F} (E) we will write E[ϕ] ⊂ E(F̄) for the the kernel of ϕ and F (E[ϕ]) for the extension of F generated by the coordinates of the points in E[ϕ]. Note that F (E[ϕ]) is independent of the choice of a Weierstrass model of E over F. By [Si] Theorem III.4.10, #(E[ϕ]) divides deg(ϕ), with equality if and only if ϕ is separable.

Definition 1.10. If ℓ is a rational prime define the ℓ-adic Tate module of E

T_{ℓ}(E) = lim_{\overleftarrow{n}} E[ℓ^{n}],

inverse limit with respect to the maps ℓ : E[ℓ^{n+1}] → E[ℓ^{n}]. If ℓ ≠ char(F) then Example 1.4 shows that

T_{ℓ}(E) ≅ Z_{ℓ}^{2}.

The Galois group G_{F} acts Z_{ℓ}-linearly on T_{ℓ}(E), giving a representation

ρ_{ℓ} : G_{F} → Aut(T_{ℓ}(E)) ≅ GL_{2}(Z_{ℓ})

when ℓ ≠ char(F).

Theorem 1.11. If E is an elliptic curve then End_{F} (E) is one of the following types of rings.

(i) Z, (ii) an order in an imaginary quadratic field, (iii) an order in a division quaternion algebra over Q.

Proof. See [Si] §III.9. ⊓ ⊔

Example 1.12. Suppose char(F) ≠ 2 and E is the curve y^{2} = x^{3} − dx where d ∈ F^{×}. Let ϕ be defined by ϕ(x, y) = (−x, iy) where i = \sqrt{−1} ∈ F̄. Then ϕ ∈ EndF̄(E), and ι(ϕ) = i so ϕ ∈ End_{F} (E) ⇔ i ∈ F. Also, ϕ has order 4 in EndF̄(E)^{×} so we see that Z[ϕ] ≅ Z[i] ⊂ EndF̄(E). (In fact, Z[ϕ] = EndF̄(E) if char(F) = 0 or if char(F) ≡ 1 (mod 4), and EndF̄(E) is an order in a quaternion algebra if char(F) ≡ 3 (mod 4).) The next lemma gives a converse to this example.

Lemma 1.13. Suppose E is given by a Weierstrass equation y^{2} = x^{3}+ax+b. If Aut(E) contains an element of order 4 (resp. 3) then b = 0 (resp. a = 0).

Proof. The only automorphisms of such a Weierstrass elliptic curve are of the form (x, y) ↦ (u^{2}x, u^{3}y) (see [Si] Remark III.1.3). The order of such an automorphism is the order of u in F^{×}, and when u has order 3 or 4 this change of variables preserves the equation if and only if a = 0 (resp. b = 0).

⊓ ⊔

2 Elliptic Curves over C

Remark 2.1. Note that an elliptic curve defined over a field of characteristic zero can be defined over Q[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}], and this field can be embedded in C. In this way many of the results of this section apply to all elliptic curves in characteristic zero.

2.1 Lattices

Definition 2.2. Suppose L is a lattice in C. Define the Weierstrass ℘- function, the Weierstrass σ-function, and the Eisenstein series attached to L

℘(z; L) = \frac{1}{z^{2}} + ∑_{0≠ω∈L} \frac{1}{(z + ω)^{2}} − \frac{1}{ω^{2}}
σ(z; L) = z ∏_{0≠ω∈L} (1 − \frac{z}{ω} ) e^{(z/ω)+(z/ω)^{2}/2}

G_{k}(L) = ∑_{0≠ω∈L} \frac{1}{ω^{k}} for k even, k ≥ 4.

We will suppress the L from the notation in these functions when there is no danger of confusion. See [Si] Theorem VI.3.1, Lemma VI.3.3, and Theorem VI.3.5 for the convergence and periodicity properties of these functions.

Theorem 2.3. (i) If L is a lattice in C then the map

z ↦ (℘(z; L), ℘′(z; L)/2)

is an analytic isomorphism (and a group homomorphism) from C/L to E(C) where E is the elliptic curve y^{2} = x^{3} − 15G_{4}(L)x − 35G_{6}(L).

(ii) Conversely, if E is an elliptic curve defined over C given by an equation y^{2} = x^{3} +ax+b then there is a unique lattice L ⊂ C such that 15G_{4}(L) = −a and 35G_{6}(L) = −b, so (i) gives an isomorphism from C/L to E(C).

(iii) The correspondence above identifies the holomorphic differential ω_{E} with dz.

Proof. The first statement is Proposition VI.3.6 of [Si] and the second is proved in [Sh] §4.2. For (iii), we have that

dx/2y = d(℘(z))/℘′(z) = dz.
⊓ ⊔

Remark 2.4. If E is the elliptic curve defined over C with a Weierstrass model y^{2} = x^{3} + ax + b and ω_{E} is the differential dx/2y of Proposition 1.2, then the lattice L associated to E by Theorem 2.3(ii) is

{∫_{γ} ω_{E} : γ ∈ H_{1}(E, Z)}

and the map

P ↦ ∫_{O}^{P} ω_{E}

is the isomorphism from E(C) to C/L which is the inverse of the map of Theorem 2.3(i).

Definition 2.5. If L ⊂ C is a lattice define

Δ(L) = (60G_{4}(L))^{3} − 27(140G_{6}(L))^{2}
j(L) = −1728(60G_{4}(L))^{3}/Δ(L).

Then Δ(L) is the discriminant and j(L) the j-invariant of the elliptic curve corresponding to L by Theorem 2.3.

Proposition 2.6. Suppose E is an elliptic curve defined over C, corresponding to a lattice L under the bijection of Theorem 2.3. Then the map ι of Definition 1.7 is an isomorphism

End_{C}(E) ⟶^{∼} {α ∈ C : αL ⊂ L}.

Proof. See [Si] Theorem VI.4.1. ⊓ ⊔

Corollary 2.7. If E is an elliptic curve defined over a field F of characteristic zero, then End_{F} (E) is either Z or an order in an imaginary quadratic field.

Proof. If E is defined over a subfield of C then Proposition 2.6 identifies End_{C}(E) with {α ∈ C : αL ⊂ L}. The latter object is a discrete subring of C, and hence is either Z or an order in an imaginary quadratic field.

Using the principle of Remark 2.1 at the beginning of this section, the same holds for all fields F of characteristic zero. ⊓ ⊔

The following table gives a dictionary between elliptic curves over an arbitrary field and elliptic curves over C.

over abitrary field

over C

(E, ω_{E})  (C/L, dz)
x, y  ℘(z; L), ℘′(z; L)/2
isomorphism class of E {αL : α ∈ C^{×}}
End_{C}(E)  {α ∈ C : αL ⊂ L}
Aut_{C}(E)  {α ∈ C^{×} : αL = L}
E[m]  m^{−1}L/L

3 Elliptic Curves over Local Fields

For this section suppose

– p is a rational prime,

– F is a finite extension of Q_{p},

– O is the ring of integers of F,

– 𝔭 is the maximal ideal of F,

– π is a generator of 𝔭

– 𝕜 = O/𝔭 is the residue field of O

– v : F → Z ∪ {∞} is the valuation on F, v(π) = 1.

We fix an elliptic curve E defined over F.

3.1 Reduction

Definition 3.1. A Weierstrass equation (1) for E is minimal if

– a_{1}, a_{2}, a_{3}, a_{4}, a_{6} ∈ O,

– the valuation of the discriminant of this equation is minimal in the set of valuations of all Weierstrass equations for E with coefficients in O.

Every elliptic curve E has a minimal Weierstrass equation, or minimal model, and the minimal discriminant of E is the ideal of O generated by the discriminant of a minimal Weierstrass model of E.

The reduction Ẽ of E is the curve defined over the residue field 𝕜 by the Weierstrass equation

y^{2} + ã_{1}xy + ã_{3}y = x^{3} + ã_{2}x^{2} + ã_{4}x + ã_{6}  (2)

where the a_{i} are the coefficients of a minimal Weierstrass equation for E and ã_{i} denotes the image of a_{i} in 𝕜. The reduction Ẽ is independent (up to isomorphism) of the particular minimal equation chosen for E (see [Si] Proposition VII.1.3(b)).

The curve Ẽ may be singular, but it has at most one singular point ([Si] Proposition III.1.4(a)). In that case the quasi-projective curve

Ẽ_{ns} = Ẽ − {singular point on Ẽ}

has a geometrically-defined group law just as an elliptic curve does (see [Si] Proposition III.2.5).

If Δ is the minimal discriminant of E, then one of the following three possibilities holds (see for example [Si] Proposition III.2.5):

(i) Δ ∈ O^{×} and Ẽ is nonsingular, i.e., Ẽ = Ẽ_{ns} is an elliptic curve, (ii) Δ ∉ O^{×}, Ẽ is singular, and Ẽ_{ns}(𝕜) ≅ 𝕜^{×}, or (iii) Δ ∉ O^{×}, Ẽ is singular, and Ẽ_{ns}(𝕜) ≅ 𝕜.

We say that E has good (resp. multiplicative, resp. additive) reduction if (i) (resp. (ii), resp. (iii)) is satisfied.

We say that E has potentially good reduction if there is a finite extension F′ of F such that E has good reduction over F′.

Lemma 3.2. (i) E has potentially good reduction if and only if j(E) ∈ O. (ii) If E has potentially good reduction then E has either good or additive

reduction.

Proof. See [Si] Propositions VII.5.4 and IV.5.5. ⊓ ⊔

Definition 3.3. There is a natural reduction map

P^{2}(F) → P^{2}(𝕜).

By restriction this defines a reduction map from E(F) to Ẽ(𝕜). We define E_{0}(F) ⊂ E(F) to be the inverse image of Ẽ_{ns}(𝕜) and E_{1}(F) ⊂ E(F) to be the inverse image of Õ ∈ Ẽ_{ns}(𝕜).

Proposition 3.4. There is an exact sequence of abelian groups

0 → E_{1}(F) → E_{0}(F) → Ẽ_{ns}(𝕜) → 0

where the map on the right is the reduction map. If E has good reduction then the reduction map induces an injective homomorphism

End_{F} (E) → End_{𝕜}(Ẽ).

Proof. See [Si] Proposition VII.2.1. ⊓ ⊔

If E has good reduction and ϕ ∈ End_{F} (E), we will write ϕ̃ for the endomorphism of Ẽ which is the reduction of ϕ.

Lemma 3.5. If E is defined by a minimal Weierstrass equation then

E_{1}(F) = {(x, y) ∈ E(F) : v(x) < 0} = {(x, y) ∈ E(F) : v(y) < 0}.

If (x, y) ∈ E_{1}(F) then 3v(x) = 2v(y) < 0.

Proof. It is clear from the definition of the reduction map that (x, y, 1) reduces to (0, 1, 0) if and only if v(y) < 0 and v(y) < v(x). If (x, y) ∈ E(F) then, since x and y satisfy a Weierstrass equation with coefficients in O, it is clear that

v(x) < 0 ⇔ v(y) < 0

and in that case v(y) = (3/2)v(x) < v(x). ⊓ ⊔

Lemma 3.6. Suppose E has good reduction, ϕ ∈ End_{F} (E), and ϕ̃ is purely inseparable. Then

(i) ϕ̃ is injective on Ẽ(𝕜). (ii) ker(ϕ) ⊂ E_{1}(F)

Proof. Clear. ⊓ ⊔

3.2 The Formal Group

Theorem 3.7. Fix a minimal Weierstrass model (1) of E. There is a formal group Ê defined by a power series F_{E} ∈ O[[Z, Z′]], and a power series

w(Z) = Z^{3} + a_{1}Z^{4} + (a_{1}^{2} + a_{2})Z^{5} + ⋯ ∈ O[[Z]],

such that if we define

x(Z) = Z/w(Z) ∈ Z^{−2}O[[Z]], y(Z) = −1/w(Z) ∈ Z^{−3}O[[Z]]

then

(i) (x(Z), y(Z)) ∈ E(O((Z))), (ii) (x(Z), y(Z)) + (x(Z′), y(Z′)) = (x(F_{E}(Z, Z′)), y(F_{E}(Z, Z′)))

as points on E with coordinates in the fraction field of F ((Z, Z′)), (iii) there is a map End_{F} (E) → End(Ê) (which we will denote by ϕ ↦ ϕ(Z) ∈

O[[Z]]) such that for every ϕ ∈ End_{F} (E),

ϕ((x(Z), y(Z))) = (x(ϕ(Z)), y(ϕ(Z)))

in E(F ((Z))).

Proof. See [Ta] or [Si], §IV.1 for an explicit construction of the power series w(Z) and F_{E}(Z, Z). The idea is that Z = −x/y is a uniformizing parameter at the origin of E, and everything (x, y, the group law, endomorphisms) can be expanded as power series in Z. ⊓ ⊔

For every n ≥ 1 write Ê(𝔭^{n}) for the commutative group whose underlying set is 𝔭^{n}, with the operation (z, z′) ↦ F_{E}(z, z′).

Corollary 3.8. With notation as in Theorem 3.7,

Z ↦ (Z/w(Z), −1/w(Z))

is an isomorphism from Ê(𝔭) to E_{1}(F) with inverse given by

(x, y) ↦ −x/y.

Proof. See [Si] Proposition VII.2.2. The first map is a map into E_{1}(F) by Lemma 3.5 and Theorem 3.7(i), and is a homomorphism by Theorem 3.7(ii). It is injective because the only zero of w(Z) in 𝔭 is Z = 0. The second map is clearly a left-inverse of the first, and it maps into 𝔭 by Lemma 3.5. We only need show that the second map is also one-to-one.

If we rewrite our Weierstrass equation for E with variables w = −1/y and z = −x/y we get a new equation

a_{6}w^{3} + (a_{4}z + a_{3})w^{2} + (a_{2}z^{2} + a_{1}z − 1)w + z^{3} = 0.

Fix a value of z ∈ 𝔭 and consider the set S of roots w of this equation. If (z, w) corresponds to a point in E_{1}(F) then by Lemma 3.5, v(w) = v(z^{3}) > 0. It follows easily that S contains at most one root w corresponding to a point of E_{1}(F), and hence the map (x, y) ↦ −x/y is one-to-one on E_{1}(F). ⊓ ⊔

Corollary 3.9. Suppose #(𝕜) = q, E has good reduction, and ϕ ∈ End_{K}(E) reduces to the Frobenius endomorphism φ_{q} ∈ End_{𝕜}(Ẽ). Then

ϕ(Z) ≡ Z^{q} (mod 𝔭O[[Z]]).

Proof. If the reduction of ϕ is φ_{q} then by Theorem 3.7(iii)

(x(ϕ(Z)), y(ϕ(Z))) = ϕ((x(Z), y(Z))) ≡ (x(Z)^{q}, y(Z)^{q})

≡ (x(Z^{q}), y(Z^{q})) (mod 𝔭O((Z))).

Since y(Z) is invertible in O((Z)), we conclude that

ϕ(Z) = −x(ϕ(Z))/y(ϕ(Z)) ≡ −x(Z^{q})/y(Z^{q}) = Z^{q} (mod 𝔭O((Z))).

⊓ ⊔

Definition 3.10. Recall that

ω_{E} = \frac{dx}{2y + a_{1}x + a_{3}} = \frac{dy}{3x^{2} + 2a_{2}x + a_{4} − a_{1}y}

is the holomorphic, translation-invariant differential on E from Proposition 1.2. Define

ω̂(Z) = \frac{\frac{d}{dZ} x(Z)}{2y(Z) + a_{1}x(Z) + a_{3}} ∈ 1 + ZO[[Z]].

Let λÊ(Z) be the unique element of Z +Z^{2}F [[Z]] such that \frac{d}{dZ} λÊ(Z) = ω̂(Z).

Lemma 3.11. (i) The power series λÊ is the logarithm map of Ê, the iso-

morphism from Ê to the additive formal group G_{a} such that λ′_{E}(0) = 1. (ii) The power series λÊ converges on 𝔭. If ord_{𝔭}(p) < p − 1 then λÊ is an

isomorphism from Ê(𝔭) to the additive group 𝔭.

Proof. Let F_{E} ∈ O[[Z, Z′]] be the addition law for Ê. We need to show that

λ_{E}(F_{E}(Z, Z′)) = λ_{E}(Z) + λ_{E}(Z′).

Since ω_{E} is translation invariant (Proposition 1.2),

ω̂(F_{E}(Z, Z′))d(F_{E}(Z, Z′)) = ω̂(Z)dZ

and therefore

\frac{d}{dZ} λ_{E}(F_{E}(Z, Z′)) = \frac{d}{dZ} λ_{E}(Z).

Therefore λ_{E}(F_{E}(Z, Z′)) = λ_{E}(Z) + c(Z′) with c(Z′) ∈ F [[Z′]]. Evaluating at Z = 0 shows c(Z′) = λ_{E}(Z′) as desired.

The uniqueness of the logarithm map and (ii) are standard elementary results in the theory of formal groups. ⊓ ⊔

Definition 3.12. Define λ_{E} : E_{1}(F) → F to be the composition of the inverse of the isomorphism of Corollary 3.8 with λÊ.

Corollary 3.13. If ord_{𝔭}(p) < p−1 then λ_{E} : E_{1}(F) → 𝔭 is an isomorphism.

Proof. This is immediate from Lemma 3.11. ⊓ ⊔

Recall the map ι : End_{F} (E) → F of Definition 1.7 defined by the action of an endomorphism on holomorphic differentials.

Proposition 3.14. For every ϕ ∈ End_{F} (E), ϕ(Z) = ι(ϕ)Z + O(Z^{2}).

Proof. By definition of ι, ω̂(ϕ(Z)) = ι(ϕ)ω̂(Z), i.e.,

\frac{d(x(ϕ(Z)))}{2y(ϕ(Z)) + a_{1}x(ϕ(Z)) + a_{3}} = ι(ϕ)\frac{d(x(Z))}{2y(Z) + a_{1}x(Z) + a_{3}} .

Using the definitions of x(Z) and y(Z), the right-hand side is (ι(ϕ)+O(Z))dZ, and the left-hand side is (ϕ′(0) + O(Z))dZ. This completes the proof. ⊓ ⊔

3.3 Applications to Torsion Subgroups

Theorem 3.15. Suppose ϕ ∈ End_{F} (E) and ι(ϕ) ∈ O^{×}.

(i) ϕ is an automorphism of E_{1}(F). (ii) If E has good reduction then the reduction map E[ϕ] ∩ E(F) → Ẽ(𝕜) is

injective.

Proof. By definition of a formal group, F_{E}(X, Y) = X +Y +O(X^{2}, XY, Y^{2}). Using Proposition 3.14, for every n ≥ 1 we have a commutative diagram

Ê(𝔭^{n})/Ê(𝔭^{n+1}) ⟶^{∼} 𝔭^{n}/𝔭^{n+1} ⟶^{∼} 𝕜
ϕ↓  ^{ι(ϕ)}↓  ^{ι(ϕ)}↓
Ê(𝔭^{n})/Ê(𝔭^{n+1}) ⟶^{∼} 𝔭^{n}/𝔭^{n+1} ⟶^{∼} 𝕜

Since ι(ϕ) ∈ O^{×} we see that ϕ is an automorphism of Ê(𝔭^{n})/Ê(𝔭^{n+1}) for every n ≥ 1, and from this it is not difficult to show that ϕ is an automorphism of Ê(𝔭). Therefore by Corollary 3.8, ϕ is an automorphism of E_{1}(F). This proves (i), and (ii) as well since E_{1}(F) is the kernel of the reduction map and (i) shows that E_{1}(F) ∩ E[ϕ] = 0. ⊓ ⊔

Remark 3.16. Theorem 3.15 shows in particular that if E has good reduction and m is prime to p, then the reduction map E[m] → Ẽ[m] is injective.

Corollary 3.17. Suppose E has good reduction, ϕ ∈ End_{F} (E), and ι(ϕ) ∈ O^{×}. If P ∈ E(F̄) and ϕ(P) ∈ E(F), then F (E[ϕ], P)/F is unramified.

Proof. Let F′ = F (E[ϕ], P) and let 𝕜′ be its residue field. Then F′/F is Galois and we let I ⊂ Gal(F′/F) denote the inertia group.

Suppose σ ∈ I. Then the reduction σ̃ of σ is the identity on 𝕜′, so if R ∈ E(F̄) and ϕ(R) ∈ E(F) then σR − R ∈ E[ϕ] and

{ \atop σR − R} = σ̃R̃ − R̃ = 0.

By Theorem 3.15(ii), since ι(ϕ) ∈ O^{×} we conclude that σR = R. In other words σ fixes E[ϕ] and P, so σ fixes F′, i.e., σ = 1. Hence I is trivial and F′/F is unramified. ⊓ ⊔

Corollary 3.18. Suppose ℓ ≠ p, and let I denote the inertia subgroup of G_{F}.

(i) If E has good reduction then I acts trivially on T_{ℓ}(E). (ii) If E has potentially good reduction then I acts on T_{ℓ}(E) through a finite

quotient.

Proof. This is clear by Corollary 3.17. ⊓ ⊔

The converse of Corollary 3.18 is the following.

Theorem 3.19 (Criterion of Néron-Ogg-Shafarevich). Let I ⊂ G_{F} denote the inertia group.

(i) If ℓ ≠ p and I acts trivially on T_{ℓ}(E), then E has good reduction. (ii) If ℓ ≠ p and T_{ℓ}(E)^{I} ≠ 0, then E has good or multiplicative reduction.

Proof. See [Si] Theorem VII.7.1 for (i).. The proof of (ii) is the same except that we use the fact that if E has additive reduction, then over any unramified extension F′ of F with residue field 𝕜′, Ẽ_{ns}(𝕜′) is killed by p and hence has no points of order ℓ. ⊓ ⊔

4 Elliptic Curves over Number Fields

For this section suppose F is a number field and E is an elliptic curve defined over F. Our main interest is in studying the Mordell-Weil group E(F).

If 𝔮 is a prime of F we say that E has good (resp. potentially good, bad, additive, multiplicative) reduction at 𝔮 if E, viewed as an elliptic curve over the local field F_{𝔮} (F completed at 𝔮) does. We will write Δ(E) for the minimal discriminant of E, the ideal of F which is the product over all primes 𝔮 of the minimal discriminant of E over F_{𝔮}. This is well-defined because (every Weierstrass model of) E has good reduction outside of a finite set of primes.

Since F has characteristic zero, the map ι : End_{F} (E) → F of Definition 1.7 (giving the action of End_{F} (E) on differentials) is injective, and from now on we will identify End_{F} (E) with its image O ⊂ F. By Corollary 2.7, O is either Z or an order in an imaginary quadratic field.

If α ∈ O we will also write α for the corresponding endomorphism of E, so E[α] ⊂ E(F̄) is the kernel of α and F (E[α]) is the extension of F generated by the coordinates of the points in E[α].

Definition 4.1. Suppose α ∈ O, α ≠ 0. Multiplication by α is surjective on E(F̄), so there is an exact sequence

0 → E[α] → E(F̄) ⟶^{α} E(F̄) → 0.

Taking G_{F} -cohomology yields a long exact sequence

E(F) ⟶^{α} E(F) → H^{1}(F, E[α]) → H^{1}(F, E) ⟶^{α} H^{1}(F, E)

where H^{1}(F, E) = H^{1}(F, E(F̄)). We can rewrite this as

0 → E(F)/αE(F) → H^{1}(F, E[α]) → H^{1}(F, E)_{α} → 0  (3)

where H^{1}(F, E)_{α} denotes the kernel of α on H^{1}(F, E). Concretely, the connecting map E(F)/αE(F) ↪ H^{1}(F, E[α]) is the “Kummer theory” map defined by

P ↦ (σ ↦ σQ − Q)  (4)

where Q ∈ E(F̄) satisfies αQ = P.

In exactly the same way, if 𝔮 is a prime (finite or infinite) of F we can replace F by the completion F_{𝔮} in (3), and this leads to the diagram

\begin{matrix}0 → & E(F)/αE(F) & ⟶ & H^{1}(F, E[α]) & ⟶ & H^{1}(F, E)_{α} → 0 \\ ↓ & & ↓^{res_{𝔮}} & & ↓^{res_{𝔮}} \\ 0 → & E(F_{𝔮})/αE(F_{𝔮}) & ⟶ & H^{1}(F_{𝔮}, E[α]) & ⟶ & H^{1}(F_{𝔮}, E)_{α} → 0. \end{matrix} (5)

We define the Selmer group (relative to α)

S_{α}(E) = S_{α}(E_{/F}) ⊂ H^{1}(F, E[α])

by

S_{α}(E) = {c ∈ H^{1}(F, E[α]) : res_{𝔮}(c) ∈ image(E(F_{𝔮})/αE(F_{𝔮})) for every 𝔮}
= {c ∈ H^{1}(F, E[α]) : res_{𝔮}(c) = 0 in H^{1}(F_{𝔮}, E) for every 𝔮}.

Proposition 4.2. Suppose α ∈ O, α ≠ 0. Under the Kummer map (3), S_{α}(E) contains the image of E(F)/αE(F).

Proof. Clear. ⊓ ⊔

Remark 4.3. One should think of the Selmer group S_{α}(E) as the smallest subgroup of H^{1}(F, E[α]) defined by natural local conditions which contains the image of E(F)/αE(F).

Proposition 4.4. Suppose α ∈ O, α ≠ 0. Then the Selmer group S_{α}(E) is finite.

Proof. Suppose first that E[α] ⊂ E(F), so H^{1}(F, E[α]) = Hom(G_{F} , E[α]). Let L be the maximal abelian extension of F of exponent deg(α) which is unramified outside of the (finite) set of primes

Σ_{E,α} = {𝔭 of F : 𝔭 divides αΔ(E) or 𝔭 is infinite}.

If c ∈ S_{α}(E) ⊂ Hom(G_{F} , E[α]) then c is trivial on

– commutators,

– deg(α)-th powers,

– inertia groups of primes outside of Σ_{E,α},

the first two because E[α] is abelian and annihilated by deg(α), and the last because of (4) and Corollary 3.17. Therefore c factors through Gal(L/F), so

S_{α}(E) ⊂ Hom(Gal(L/F), E[α]).

Class field theory shows that L/F is finite, so this proves the proposition in this case.

In general, the restriction map

0 → H^{1}(F (E[α])/F, E[α]) → H^{1}(F, E[α]) ⟶^{res} H^{1}(F (E[α]), E[α])

sends S_{α}(E_{/F}) into S_{α}(E_{/F (E[α])}). The case above shows that S_{α}(E_{/F (E[α])}) is finite, and H^{1}(F (E[α])/F, E[α]) is finite, so S_{α}(E_{/F}) is finite. ⊓ ⊔

Corollary 4.5 (Weak Mordell-Weil Theorem). For every nonzero α ∈ O, E(F)/αE(F) is finite.

Proof. This is clear from Propositions 4.2 and 4.4. ⊓ ⊔

Theorem 4.6 (Mordell-Weil). E(F) is finitely generated.

Proof. See [Si] §VIII.6. ⊓ ⊔

Definition 4.7. The Tate-Shafarevich group � (E) of E over F is the subgroup of H^{1}(F, E(F̄)) defined by

� (E) = ker (H^{1}(F, E(F̄)) → ∏_{v of F} H^{1}(F_{v}, E)) .

Proposition 4.8. If α ∈ O, α ≠ 0, then the exact sequence (3) restricts to an exact sequence

0 → E(F)/αE(F) → S_{α}(E) → � (E)_{α} → 0

where � (E)_{α} is the subgroup of elements of � (E) killed by α.

Proof. This is clear from the definitions and the diagram (5). ⊓ ⊔

5 Elliptic Curves with Complex Multiplication

Fix a subfield F of C and an elliptic curve E defined over F.

Definition 5.1. We say E has complex multiplication over F if End_{F} (E) is an order in an imaginary quadratic field, i.e., if End_{F} (E) ≠ Z.

Assume from now on that E has complex multiplication, and let

O = ι(End_{F} (E)) ⊂ F.

As in §4 we will use ι to identify End_{F} (E) with O. Let K = QO ⊂ F be the imaginary quadratic field containing O, and denote the full ring of integers of K by O_{K}. If 𝔞 is an ideal of O we will write E[𝔞] = ∩_{α∈𝔞}E[α].

Fix an embedding of F into C. Viewing E as an elliptic curve over C and using Proposition 2.6 we can write

E(C) ≅ C/L where L ⊂ K ⊂ C and OL = L.  (6)

(A priori L is just a lattice in C, but replacing L by λL where λ^{−1} ∈ L we may assume that L ⊂ K.) Thus if O = O_{K}, then L is a fractional ideal of K.

5.1 Preliminaries

In this section we record the basic consequences of complex multiplication. Put most simply, if E has complex multiplication over F then all torsion points in E(F̄) are defined over abelian extensions of F.

Remark 5.2. It will simplify the exposition to assume that O = O_{K}. The following proposition shows that this restriction is not too severe. Two elliptic curves are isogenous if there is an isogeny (a nonzero morphism sending one origin to the other) from one to the other.

Proposition 5.3. There is an elliptic curve E′, defined over F and isogenous over F to E, such that End_{F} (E) ≅ O_{K}.

Proof. Suppose the conductor of O is c, i.e., O = Z+cO_{K}, and let 𝔠 = cO_{K} ⊂ O. The subgroup E[𝔠] is stable under G_{F}, so by [Si] Proposition III.4.12 and Exercise III.3.13 there is an elliptic curve E′ over F and an isogeny from E to E′ with kernel E[𝔠]. We only need to check that End_{F} (E′) = O_{K}.

With the identification (6), E′(C) ≅ C/L′ where

L′ = {z ∈ C : z𝔠 ⊂ L}.

Suppose α ∈ O_{K}. For every z ∈ L′,

(αz)𝔠 = z(α𝔠) ∈ z𝔠 ⊂ L

so αz ∈ L′. Therefore by Proposition 2.6, α ∈ End_{C}(E′). By Lemma 1.8, since α ∈ K ⊂ F we conclude that α ∈ End_{F} (E′). ⊓ ⊔

From now on we will assume that O is the maximal order O_{K}.

Proposition 5.4. If 𝔞 is a nonzero ideal of O then E[𝔞] ≅ O/𝔞 as Omodules.

Proof. Using the identification (6) we see that E[𝔞] ≅ 𝔞^{−1}L/L where L is a fractional ideal of K, and then 𝔞^{−1}L/L ≅ O/𝔞. ⊓ ⊔

Corollary 5.5. If 𝔞 is a nonzero ideal of O then the action of G_{F} on E[𝔞] induces an injection

Gal(F (E[𝔞])/F) ↪ (O/𝔞)^{×}.

In particular F (E[𝔞])/F is abelian.

Proof. If β ∈ O, σ ∈ G_{F}, and P ∈ E(F̄) then, since the endomorphism β is defined over F, σ(βP) = β(σP). Thus there is a map

Gal(F (E[𝔞])/F) ↪ Aut_{O}(E[𝔞]).

By Proposition 5.4,

Aut_{O}(E[𝔞]) ≅ Aut_{O}(O/𝔞) = (O/𝔞)^{×}.
⊓ ⊔

If 𝔞 is a nonzero ideal of O let E[𝔞^{∞}] = ∪_{n}E[𝔞^{n}].

Corollary 5.6. The action of G_{F} on E[𝔞^{∞}] induces an injection

Gal(F (E[𝔞^{∞}])/F) ↪ (lim_{\overleftarrow{n}} O/𝔞^{n})^{×}.

In particular for every prime p,

Gal(F (E[p^{∞}])/F) ↪ (O ⊗ Z_{p})^{×}.

Proof. Immediate from Corollary 5.5 ⊓ ⊔

Theorem 5.7. Suppose F is a finite extension of Q_{ℓ} for some ℓ.

(i) E has potentially good reduction. (ii) Suppose 𝔭 is a prime of O and n ∈ Z^{+} is such that the multiplicative

group 1 + 𝔭^{n}O_{𝔭} is torsion-free (where O_{𝔭} is the completion of O at 𝔭). If 𝔭 ∤ ℓ then E has good reduction over F (E[𝔭^{n}]) at all primes not dividing 𝔭.

Proof. Suppose p is a rational prime. By Corollary 5.6, the Galois group Gal(F (E[p^{∞}])/F (E[p])) is isomorphic to a subgroup of the multiplicative group 1 + pO ⊗ Z_{p}. If p > 3 then the p-adic logarithm map shows that 1 + pO ⊗ Z_{p} ≅ pO_{𝔭} ≅ Z_{p}^{2}. Thus

Gal(F (E[p^{∞}])/F (E[p])) ≅ Z_{p}^{d}

with d ≤ 2. If p ≠ ℓ, class field theory shows that such an extension is unramified. Thus by the criterion of Néron-Ogg-Shafarevich (Theorem 3.19(i)) E has good reduction over F (E[p]). This proves (i).

The proof of (ii) is similar. Write F_{∞} = F (E[𝔭^{∞}]) and F_{n} = F (E[𝔭^{n}]), and suppose 𝔮 is a prime of F_{n} not dividing 𝔭. By (i) and Corollary 3.17, the inertia group I_{𝔮} of 𝔮 in Gal(F_{∞}/F_{n}) is finite. But Corollary 5.6 shows that

Gal(F_{∞}/F_{n}) ⊂ 1 + 𝔭^{n}O_{𝔭},

which has no finite subgroups, so I_{𝔮} acts trivially on E[𝔭^{∞}]. Therefore by Theorem 3.19(ii), E has good or multiplicative reduction at 𝔮. Since we already know that the reduction is potentially good, Lemma 3.2(ii) allows us to conclude that E has good reduction at 𝔮. ⊓ ⊔

Remark 5.8. The hypothesis of Theorem 5.7(ii) is satisfied with n = 1 if the residue characteristic of 𝔭 is greater than 3.

Proposition 5.9. Suppose 𝔮 is a prime of F where E has good reduction and q = N_{F/Q}𝔮. There is an endomorphism α ∈ O whose reduction modulo 𝔮 is the Frobenius endomorphism φ_{q} of Ẽ.

Proof. If φ_{q} = [m] for some m ∈ Z then the proposition is clear. So suppose now that φ_{q} ∉ Z, and write 𝕜 for the residue field of F at 𝔮. Since φ_{q} commutes with every endomorphism of Ẽ, we see from Theorem 1.11 that the only possibility is that End_{𝕜}(Ẽ) is an order in an imaginary quadratic field. But the reduction map End_{F} (E) → End_{𝕜}(Ẽ) is injective (Proposition 3.4) so its image, the maximal order of K, must be all of End_{𝕜}(Ẽ). This proves the proposition. ⊓ ⊔

5.2 The Main Theorem of Complex Multiplication

In this section we study further the action of G_{F} on torsion points of E. We will see that not only are torsion points abelian over F, in fact they are “almost” abelian over K, so that (using class field theory) we can describe the action of G_{F} on torsion points in terms of an action of the ideles of K.

The reference for this section is [Sh] Chapter 5; see also [ST]. We continue to suppose that E has complex multiplication by the maximal order of K.

Definition 5.10. Let A_{K}^{×} denote the group of ideles of K. There is a natural map from A_{K}^{×} to the group of fractional ideals of K, and if x ∈ A_{K}^{×} and 𝔞 is a fractional ideal of K we will write x𝔞 for the product of 𝔞 and the fractional ideal corresponding to x.

If 𝔭 is a prime of K let O_{𝔭} ⊂ K_{𝔭} denote the completions of O and K at 𝔭. If 𝔞 is a fractional ideal of K, write 𝔞_{𝔭} = 𝔞O_{𝔭} and then

K/𝔞 = 𝔞 ⊗ (K/O) = 𝔞 ⊗ (⊕_{𝔭}(K_{𝔭}/O_{𝔭})) = ⊕_{𝔭}K_{𝔭}/𝔞_{p}  (7)

If x = (x_{𝔭}) ∈ A_{K}^{×} then multiplication by x_{𝔭} gives an isomorphism from K_{𝔭}/𝔞_{𝔭} to K_{𝔭}/x_{𝔭}𝔞_{𝔭} = K_{𝔭}/(x𝔞)_{𝔭}, so putting these maps together in (7) we get an isomorphism

x : K/𝔞 ⟶^{∼} K/x𝔞.

The following theorem is Theorem 5.4 in Shimura’s book [Sh]. Let K^{ab} denote the maximal abelian extension of K and [ · , K^{ab}/K] the Artin map of global class field theory. If σ is an automorphism of C let E^{σ} denote the elliptic curve obtained by applying σ to the coefficients of an equation for E.

Theorem 5.11 (Main theorem of complex multiplication). Fix a fractional ideal 𝔞 of K and an analytic isomorphism

ξ : C/𝔞 → E(C)

as in (6). Suppose σ ∈ Aut(C/K) and x ∈ A_{K}^{×} satisfies [x, K^{ab}/K] = σ |_{K^{ab}}. Then there is a unique isomorphism ξ′ : C/x^{−1}𝔞 → E^{σ}(C) such that the following diagram commutes

K/𝔞 ⟶^{ξ} E_{tors}
x^{−1}↓  ↓^{σ}
K/x^{−1}𝔞 ⟶^{ξ′} E_{tors}^{σ}

where E_{tors} denotes the torsion in E(C) and similarly for E_{tors}^{σ}.

Proof. See [Sh] Theorem 5.4. ⊓ ⊔

Let H denote the Hilbert class field H of K.

Corollary 5.12. (i) K(j(E)) = H ⊂ F, (ii) j(E) is an integer of H.

Proof. Suppose σ ∈ Aut(C/K). With the notation of Theorem 5.11, as in Proposition 2.6 we see that

j(E) = j(E)^{σ} ⇔ E ≅ E^{σ} ⇔ C/𝔞 ≅ C/x𝔞 ⇔ x𝔞 = λ𝔞 for some λ ∈ C

⇔ x ∈ K^{×} ∏_{𝔭∤∞} O_{𝔭}^{×} ∏_{𝔭|∞} K_{𝔭}^{×} ⇔ [x, H/K] = 1 ⇔ σ is the identity on H.

This proves (i), and (ii) follows from Theorem 5.7(i) and Lemma 3.2(i). ⊓ ⊔ Corollary 5.13. There is an elliptic curve defined over H with endomorphism ring O = O_{K}.

Proof. By Theorem 2.3(i) there is an elliptic curve E′ defined over C with E′(C) ≅ C/O, and by Proposition 2.6, End_{C}(E′) ≅ O. Corollary 5.12 shows that j(E′) ∈ H, so (see Proposition III.1.4 of [Si]) there is an elliptic curve E defined over H with j(E) = j(E′). Hence E is isomorphic over C to E′, so End_{C}(E) ≅ O.

The map ι : End_{C}(E) → C of Definition 1.7 is injective, so the image is O ⊂ H. By Lemma 1.8 we conclude that End_{C}(E) = End_{H}(E). Thus E has the desired properties. ⊓ ⊔

Exercise 5.14. Let A be the ideal class group of K. If E ≅ C/𝔞, 𝔟 is an ideal of K, σ_{𝔟} is its image under the isomorphism A_{K} ⟶^{∼} Gal(H/K), and σ ∈ G_{K} restricts to σ_{𝔟} on H, then

E^{σ}(C) ≅ C/𝔟^{−1}𝔞.

For the rest of this section we suppose that F is a number field.

Theorem 5.15. There is a Hecke character

ψ = ψ_{E} : A_{F}^{×}/F^{×} → C^{×}

with the following properties.

(i) If x ∈ A_{F}^{×} and y = N_{F/K}x ∈ A_{K}^{×}, then

ψ(x)O = y_{∞}^{−1}(yO) ⊂ C.

(ii) If x ∈ A_{F}^{×} is a finite idele (i.e., the archimedean component is 1) and 𝔭 is a prime of K, then ψ(x)(N_{F/K}x)_{𝔭}^{−1} ∈ O_{𝔭}^{×} and for every P ∈ E[𝔭^{∞}]

[x, F^{ab}/F]P = ψ(x)(N_{F/K}x)_{𝔭}^{−1}P.

(iii) If 𝔮 is a prime of F and U_{𝔮} denotes the local units in the completion of F at 𝔮, then

ψ(U_{𝔮}) = 1 ⇔ E has good reduction at 𝔮.

Proof. Suppose x ∈ A_{F}^{×}, and let y = N_{F/K}x, σ = [x, F^{ab}/F]. Then σ restricted to K^{ab} is [y, K^{ab}/K] so we can apply Theorem 5.11 with σ and y. Since σ fixes F, E^{σ} = E so Theorem 5.11 gives a diagram with isomorphisms

ξ : C/𝔞 → E(C) and ξ′ : C/y^{−1}𝔞 → E(C). Then ξ^{−1} ∘ ξ′ : C/y^{−1}𝔞 ⟶^{∼} C/𝔞

is an isomorphism, so it must be multiplication by an element ψ_{fin}(x) ∈ K^{×} satisfying ψ_{fin}(x)O = yO. Define

ψ(x) = y_{∞}^{−1}ψ_{fin}(x).

It is clear that ψ : A_{F}^{×}/F^{×} → C^{×} is a homomorphism and that (i) is satisfied. If 𝔭 is a prime of K and k > 0 then Theorem 5.11 gives a diagram

𝔭^{−k}𝔞_{𝔭}/𝔞_{𝔭}  ⟶^{∼}  𝔭^{−k}𝔞/𝔞  ⟶^{ξ} E[𝔭^{k}]
y_{𝔭}^{−1}↓  ^{y^{−1}}↓  ↓^{σ}
𝔭^{−k}y_{𝔭}^{−1}𝔞_{𝔭}/y_{𝔭}^{−1}𝔞_{𝔭} ⟶^{∼} 𝔭^{−k}y^{−1}𝔞/y^{−1}𝔞 ⟶^{ψ_{fin}(x)ξ} E[𝔭^{k}]

(where the left-hand square comes from the definition of the action of y on K/𝔞) which proves (ii).

Suppose 𝔮 is a prime of F and p is a rational prime not lying below 𝔮. By (ii), if u ∈ U_{𝔮} then [u, F^{ab}/F] acts on T_{p}(E) as multiplication by ψ(u). Since [U_{𝔮}, F^{ab}/F] is the inertia group at 𝔮, (iii) follows from Theorem 3.19 and Corollary 3.18(i).

Thus for almost all 𝔮, ψ(U_{q}) = 1. Even for primes 𝔮 of bad reduction, since the reduction is potentially good (Theorem 5.7(i)) the action of [U_{𝔮}, F^{ab}/F] on T_{p}(E) factors through a finite quotient (Corollary 3.18(ii)) so the argument above shows that ψ vanishes on an open subgroup of U_{𝔮}. Therefore ψ is continuous, and the proof of the theorem is complete. ⊓ ⊔

Let 𝔣 = 𝔣_{E} denote the conductor of the Hecke character ψ of Theorem 5.15. We can view ψ as a character of fractional ideals of F prime to 𝔣 in the usual way.

Corollary 5.16. As a character on ideals, ψ satisfies

(i) if 𝔟 is an ideal of F prime to 𝔣 then ψ(𝔟)O = N_{F/K}𝔟, (ii) if 𝔮 is a prime of F not dividing 𝔣 and 𝔟 is an ideal of O prime to 𝔮, then

[𝔮, F (E[𝔟])/F] acts on E[𝔟] by multiplication by ψ(𝔮). (iii) if 𝔮 is a prime of F where E has good reduction and q = N_{F/Q}𝔮 then

ψ(𝔮) ∈ O reduces modulo 𝔮 to the Frobenius endomorphism φ_{q} of Ẽ.

Proof. The first two assertions are just translations of Theorem 5.15(i) and (ii). If P ∈ E_{tors} has order prime to 𝔮, P̃ denotes its reduction modulo a prime of F̄ above 𝔮, and σ_{𝔮} = [𝔮, F (E[𝔟])/F], then

ψ̃(𝔮)P̃ = σ̃_{𝔮}P = φ_{q}P̃

where the first equality is from (ii) and the second is the definition of the Artin symbol [𝔮, F (E[𝔟])/F]. Since the reduction map is injective on primeto-𝔮 torsion (Theorem 3.15) this proves (iii). ⊓ ⊔

Remark 5.17. Note that Corollary 5.16(iii) gives an explicit version of Proposition 5.9. Proposition 5.9 is one of the key points in the proof of the Main Theorem of Complex Multiplication, of which Corollary 5.16 is a direct consequence.

Corollary 5.18. Suppose F = K and 𝔭 is a prime of K such that the map O^{×} → (O/𝔭)^{×} is not surjective. Then E[𝔭]⊄E(K).

Proof. By Theorem 5.15(ii), [O_{𝔭}^{×}, K^{ab}/K] acts on E[𝔭] via the character ψ(x)x^{−1} of O_{𝔭}^{×}, and by Theorem 5.15(i), ψ(O_{𝔭}^{×}) ⊂ O^{×}. The corollary follows. ⊓ ⊔

Corollary 5.19. Suppose F = K. Then the map O^{×} → (O/𝔣)^{×} is injective. In particular E cannot have good reduction at all primes of K.

Proof. Let u ∈ O^{×}, u ≠ 1 and let x be the idele defined by x_{∞} = 1 and x_{𝔭} = u for all finite 𝔭. Then ψ(x) = ψ(u^{−1}x) = u ≠ 1, so by definition of 𝔣, u ≢ 1 (mod 𝔣). The second assertion now follows from Theorem 5.15(iii). ⊓ ⊔

If 𝔞 is an ideal of K let K(𝔞) denote the ray class field of K modulo 𝔞.

Corollary 5.20. Suppose E is defined over K, 𝔞 is an ideal of K prime to 6𝔣, and 𝔭 is a prime of K not dividing 6𝔣.

(i) E[𝔞𝔣] ⊂ E(K(𝔞𝔣)). (ii) The map Gal(K(E[𝔞])/K) → (O/𝔞)^{×} of Corollary 5.5 is an isomorphism. (iii) If 𝔟 | 𝔞 then the natural map Gal(K(𝔞𝔣)/K(𝔟𝔣)) → Gal(K(E[𝔞])/K(E[𝔟]))

is an isomorphism. (iv) K(E[𝔞𝔭^{n}])/K(E[𝔞]) is totally ramified above 𝔭.

(v) If the map O^{×} → (O/𝔞)^{×} is injective then K(E[𝔞𝔭^{n}])/K(E[𝔞]) is unramified outside of 𝔭.

Proof. Suppose x ∈ A_{K}^{×}, x_{𝔭} ∈ O_{𝔭}^{×} for all finite 𝔭 and x_{∞} = 1. If x ≡ 1(mod^{×}𝔣) then Theorem 5.15(ii) shows that [x, K^{ab}/K] acts on E_{tors} as multiplication by x^{−1}. If x ≡ 1(mod^{×}𝔞) Theorem 5.15 shows that [x, K^{ab}/K] acts on E[𝔞] as multiplication by ψ(x). Thus

– if 𝔭 | 𝔣 then the kernel of O_{𝔭}^{×} → [O_{𝔭}^{×}, K(E[𝔞])/K] is the kernel of the

composition O_{𝔭}^{×} ⟶^{ψ} O^{×} → (O/𝔞)^{×};

– if 𝔭^{n} | 𝔞 and 𝔭^{n+1} ∤ 𝔞 then O_{𝔭}^{×}/(1 + 𝔭^{n}O_{𝔭}) ↪ [O_{𝔭}^{×}, K(E[𝔞])/K] ↪ (O/𝔭^{n})^{×} is an isomorphism.

All assertions of the corollary follow without difficulty from this. ⊓ ⊔

Remark 5.21. In fact, without much more difficulty one can strengthen Corollary 5.20(i) (see [CW1] Lemma 4) to show that E[𝔞𝔣] = E(K(𝔞𝔣)), but we will not need this.

Corollary 5.22. Suppose 𝔮 is a prime of F. There is an elliptic curve E′ defined over F, such that

– E′ is isomorphic to E over F̄,

– E′ has good reduction at 𝔮.

Proof. Let ψ_{E} be the Hecke character attached to E and U_{𝔮} the group of local units at 𝔮, viewed as a subgroup of A_{F}^{×}. By Theorem 5.15(i), ψ_{E}(U_{𝔮}) ⊂ O^{×}. Therefore we can find a continuous map

χ : A_{F}^{×}/F^{×} → O^{×}

such that χ = ψ_{E} on U_{𝔮}. We will take E′ to be the twist of E by χ^{−1} (see [Si] §X.5).

Explicitly, suppose E is given by a Weierstrass equation

y^{2} = x^{3} + ax + b.

and let w = #(O^{×}). By class field theory we can view χ as an element of

Hom(G_{F} , O^{×}) = H^{1}(F, μ_{w}) ≅ F^{×}/(F^{×})^{w}.

In other words, there is a d ∈ F^{×} such that

(d^{1/w})^{σ} = χ(σ)d^{1/w} for every σ ∈ G_{F}.

Define

E′ = \begin{cases}y^{2} = x^{3} + d^{2}ax + d^{3}b & if w = 2 \\ y^{2} = x^{3} + dax & if w = 4 \\ y^{2} = x^{3} + db & if w = 6 \end{cases}

(see Example 1.1). The map

(x, y) ↦ \begin{cases}(dx, d^{3/2}y) & if w = 2 \\ (d^{1/2}x, d^{3/4}y) & if w = 4 \\ (d^{1/3}x, d^{1/2}y) & if w = 6 \end{cases}

defines an isomorphism ϕ : E ⟶^{∼} E′ over F (d^{1/w}) (where we are using Lemma 1.13). If P ∈ E(F̄) and σ ∈ G_{F}, then

σ(ϕ(P)) = ϕ^{σ}(σP) = χ(σ)^{−1}ϕ(σP).

From the definition of the Hecke character ψ_{E′} of E′ we see that ψ_{E′} = χ^{−1}ψ_{E}. By construction this is trivial on U_{𝔮}, so by Theorem 5.15(iii) E′ has good reduction at 𝔮. ⊓ ⊔

6 Descent

In this section we use the results of §5 to compute the Selmer group of an elliptic curve with complex multiplication. After some cohomological lemmas in §6.1, we define an enlarged Selmer group S′(E) in §6.2 which is easier to compute (Lemma 6.4 and Theorem 6.5) than the true Selmer group S(E).

The main result describing the Selmer group S(E) is Theorem 6.9. The methods of this section closely follow the original work of Coates and Wiles [CW1] (see for example [Co]).

We continue to assume that E is an elliptic curve defined over a field F of characteristic 0, with complex multiplication by the maximal order O of an imaginary quadratic field K.

6.1 Preliminaries

Lemma 6.1. Suppose 𝔭 is a prime of K lying above a rational prime p > 3, and n ≥ 0. Let C be a subgroup of (O/𝔭^{n})^{×}, acting on O/𝔭^{n} via multiplication. If either C is not a p-group or C is cyclic, then for every i > 0

H^{i}(C, O/𝔭^{n}) = 0.

Proof. If C is cyclic this is a simple exercise. If C′, the prime-to-p-part of C, is nontrivial, then (O/𝔭^{n})^{C′} = 0 and H^{i}(C′, O/𝔭^{n}) = 0 for every i, so the inflation-restriction exact sequence

0 → H^{i}(C/C′, (O/𝔭^{n})^{C′}) → H^{i}(C, O/𝔭^{n}) → H^{i}(C′, O/𝔭^{n})

shows that H^{i}(C, O/𝔭^{n}) = 0. ⊓ ⊔

Lemma 6.2. Suppose 𝔭 is a prime of K lying above a rational prime p > 3, and n ≥ 0.

(i) If O_{𝔭} = Z_{p} or if E[𝔭] ⊄ E(F) then the restriction map gives an isomor-

phism

H^{1}(F, E[𝔭^{n}]) ≅ H^{1}(F (E[𝔭^{n}]), E[𝔭^{n}])^{Gal(F (E[𝔭^{n}])/F)}.

(ii) Suppose F is a finite extension of Q_{ℓ} for some ℓ ≠ p. Then the restriction map gives an injection

H^{1}(F, E)_{𝔭^{n}} ↪ H^{1}(F (E[𝔭^{n}]), E)_{𝔭^{n}}.

Proof. Use Proposition 5.4 and Corollary 5.5 to identify E[𝔭^{n}] with O/𝔭^{n} and Gal(F (E[𝔭^{n}])/F) with a subgroup C of (O/𝔭^{n})^{×}. Then C is cyclic if O_{𝔭} = Z_{p}, and C is a p-group if and only if E[𝔭] ⊂ E(F) (since Gal(F (E[𝔭])/F) ⊂ (O/p)^{×} has order prime to p). Thus (i) follows from Lemma 6.1 and the inflation-restriction exact sequence.

The kernel of the restriction map in (ii) is H^{1}(F_{n}/F, E(F_{n}))_{𝔭^{n}}, where F_{n} = F (E[𝔭^{n}]). We may as well assume that n ≥ 1, or there is nothing to prove. By Theorem 5.7(ii), E has good reduction over F_{n}, so by Proposition 3.4 there is a reduction exact sequence

0 → E_{1}(F_{n}) → E(F_{n}) → Ẽ(𝕜_{n}) → 0

where 𝕜_{n} is the residue field of F_{n}. Thus E_{1}(F_{n}) is a profinite O-module, of finite index in E(F_{n}), on which (by Theorem 3.15(i)) every α prime to ℓ acts invertibly. It follows that the pro-𝔭 part of E(F_{n}) is finite, say E[𝔭^{m}] for some m ≥ n, and hence

H^{1}(F_{n}/F, E(F_{n}))_{𝔭^{n}} ⊂ H^{1}(F_{n}/F, E[𝔭^{m}]) = H^{1}(F (E[𝔭^{m}])/F, E[𝔭^{m}]).

If E[𝔭] ⊂ E(F) then E has good reduction by Theorem 5.7(ii) (and Remark 5.8) so F_{n}/F is unramified and hence cyclic. Hence exactly as in (i), Lemma 6.1 shows that H^{1}(F (E[𝔭^{m}])/F, E[𝔭^{m}]) = 0, and (ii) follows. ⊓ ⊔

6.2 The Enlarged Selmer Group

Suppose for the rest of this section that F is a number field.

Definition 6.3. If α ∈ O define S′_{α}(E) = S′_{α}(E_{/F}) ⊂ H^{1}(F, E[α]) by

S′_{α}(E) = {c ∈ H^{1}(F, E[α]) : res_{𝔮}(c) ∈ image(E(F_{𝔮})/αE(F_{𝔮})) for every 𝔮 ∤ α}

= {c ∈ H^{1}(F, E[α]) : res_{𝔮}(c) = 0 in H^{1}(F_{𝔮}, E(F̄_{𝔮})) for every 𝔮 ∤ α}

in the diagram (5). Clearly S_{α}(E) ⊂ S′_{α}(E).

Lemma 6.4. Suppose 𝔭 is a prime of K not dividing 6, n ≥ 1, E[𝔭^{n}] ⊂ E(F) and 𝔭^{n} = αO. Then

S′_{α}(E_{/F}) = Hom(Gal(M/F), E[𝔭^{n}])

where M is the maximal abelian p-extension of F unramified outside of primes above 𝔭.

Proof. Since E[𝔭^{n}] ⊂ E(F),

H^{1}(F, E[𝔭^{n}]) = Hom(G_{F} , E[𝔭^{n}]), H^{1}(F_{𝔮}, E[𝔭^{n}]) = Hom(G_{F_{𝔮}}, E[𝔭^{n}]).

Suppose 𝔮 is a prime of F not dividing 𝔭. By Theorem 5.7(ii), E has good reduction at 𝔭 so by (4) and Corollary 3.17, the image of E(F_{𝔮})/αE(F_{𝔮}) under (5) is contained in Hom(G_{F_{𝔮}}/I_{𝔮}, E[𝔭^{n}]), where I_{𝔮} is the inertia group in G_{F_{𝔮}}, and we have O-module isomorphisms

Hom(G_{F_{𝔮}}/I_{𝔮}, E[𝔭^{n}]) ≅ E[𝔭^{n}] ≅ O/𝔭^{n}O.

On the other hand, using Theorem 3.15 and writing 𝕜 for the residue field of F_{𝔮},

E(F_{𝔮})/αE(F_{𝔮}) ≅ Ẽ(𝕜)/αẼ(𝕜) ≅ O/𝔭^{n}O.

Thus the image of E(F_{𝔮})/αE(F_{𝔮}) ↪ H^{1}(F_{𝔮}, E[𝔭^{n}]) under (5) must be equal to Hom(G_{F_{𝔮}}/I_{𝔮}, E[𝔭^{n}]), and the lemma follows from the definition of S′_{α}. ⊓ ⊔ Theorem 6.5. Suppose E is defined over K, 𝔭 is a prime of K not dividing 6, n ≥ 1, and 𝔭^{n} = αO. Let K_{n} = K(E[𝔭^{n}]). Then

S′_{α}(E_{/K}) = Hom(Gal(M_{n}/K_{n}), E[𝔭^{n}])^{Gal(K_{n}/K)}

where M_{n} is the maximal abelian p-extension of K_{n} unramified outside of primes above 𝔭.

Proof. Let G = Gal(K_{n}/K). By Lemma 6.2(ii) and Corollary 5.18, the restriction map gives an isomorphism

H^{1}(K, E[𝔭^{n}]) ≅ H^{1}(K_{n}, E[𝔭^{n}])^{G}.

Clearly the image of S′_{α}(E_{/K}) under this restriction isomorphism is contained in S′_{α}(E_{/K_{n}}). Conversely, every class in H^{1}(K, E[𝔭^{n}]) whose restriction lies in S′_{α}(E_{/K_{n}}) already lies in S′_{α}(E_{/K}), because by Lemma 6.2(iii) the restriction map

H^{1}(K_{𝔮}, E(K̄_{𝔮})) → H^{1}(K_{𝔮}(E[𝔭^{n}]), E(K̄_{𝔮}))

is injective for every prime 𝔮 not dividing 𝔭. This proves that

S′_{α}(E_{/K}) = S′_{α}(E_{/K_{n}})^{G},

and so the theorem follows from Lemma 6.4. ⊓ ⊔

6.3 The True Selmer Group

For the rest of this section we will suppose that E is defined over K, i.e., F = K. Recall that by Corollary 5.12 this implies that K has class number one. Fix a prime 𝔭 of K not dividing 6𝔣 and a generator π of 𝔭. Let λ_{E} : E_{1}(K_{𝔭}) → 𝔭O_{𝔭} be the logarithm map of Definition 3.12.

Lemma 6.6. The map λ_{E} extends uniquely to a surjective map E(K_{𝔭})↠𝔭O_{𝔭} whose kernel is finite and has no 𝔭-torsion.

Proof. By Corollary 3.13, λ_{E} : E_{1}(K_{𝔭}) → 𝔭O_{𝔭} is an isomorphism, and by Lemma 3.6(i) and Corollary 5.16(iii), E(K_{𝔭})/E_{1}(K_{𝔭}) is finite and has no 𝔭torsion. ⊓ ⊔

Definition 6.7. For every n ≥ 1 let K_{n,𝔭} = K_{𝔭}(E[𝔭^{n}]) and define a Kummer pairing

\begin{aligned}⟨ · , · ⟩_{π^{n}} : & E(K_{𝔭}) & × & K_{n,𝔭}^{×} & → & E[𝔭^{n}] \\ & P & , & x & ↦ & [x, K_{n,𝔭}^{ab}/K_{n,𝔭}]Q − Q \end{aligned}

where [ · , K_{n,𝔭}^{ab}/K_{n,𝔭}] is the local Artin map and Q ∈ E(K̄_{𝔭}) satisfies π^{n}Q = P.

Lemma 6.8. For every n there is a unique Galois-equivariant homomorphism δ_{n} : K_{n,𝔭}^{×} → E[𝔭^{n}] such that if P ∈ E(K_{𝔭}) and x ∈ K_{n,𝔭}^{×},

⟨P, x⟩_{π^{n}} = (π^{−1}λ_{E}(P))δ_{n}(x).

Further, if O_{n,𝔭} denotes the ring of integers of K_{n,𝔭} then δ_{n}(O_{n,𝔭}^{×}) = E[𝔭^{n}].

Proof. Define δ_{n}(x) = ⟨R, x⟩_{π^{n}} where λ_{E}(R) = π, and then everything except the surjectivity assertion is clear.

First note that by Theorem 5.15(ii), if x ∈ O_{𝔭}^{×} then [x, K_{n,𝔭}/K_{𝔭}] acts on E[𝔭^{n}] as multiplication by x^{−1}. Therefore E(K_{𝔭}) has no 𝔭-torsion and E[𝔭] has no proper G_{K_{𝔭}}-stable subgroups.

By Lemma 6.6, E(K_{𝔭})/𝔭^{n}E(K_{𝔭}) ⟶^{∼} O/𝔭^{n}. Since

E(K_{𝔭})/𝔭^{n}E(K_{𝔭}) ↪ H^{1}(K_{𝔭}, E[𝔭^{n}]) ↪ Hom(K_{n,𝔭}^{×}, E[𝔭^{n}])

is injective (the first map by (5) and the second by Lemmas 6.2(ii) and 6.6), the image of δ_{n} is not contained in E[𝔭^{n−1}]. Since the image of δ_{n} is stable under G_{K_{𝔭}}, it must be all of E[𝔭^{n}]. But δ_{n}(K_{n,𝔭}^{×})/δ_{n}(O_{𝔭,n}^{×}) is a quotient of E[𝔭^{n}] on which G_{K_{𝔭}} acts trivially, and (as above) such a quotient must be trivial, so δ_{n}(O_{𝔭,n}^{×}) = E[𝔭^{n}] as well. ⊓ ⊔

Theorem 6.9. With notation as above, let K_{n} = K(E[𝔭^{n}]) and O_{n} its ring of integers, and define

W_{n} = K_{n}^{×} ∏_{v|∞} K_{n,v}^{×} ∏_{v∤𝔭∞} O_{n,v}^{×} · ker(δ_{n}) ⊂ A_{K_{n}}^{×}.

Then

S_{π^{n}}(E_{/K}) = Hom(A_{K_{n}}^{×}/W_{n}, E[𝔭^{n}])^{Gal(K_{n}/K)}.

Proof. By definition we have an injective map

E(K_{𝔭})/π^{n}E(K_{𝔭}) ↪ Hom(K_{n,𝔭}^{×}/ker(δ_{n}), E[𝔭^{n}])^{Gal(K_{n,𝔭}/K_{𝔭})}.

By Lemma 6.6, E(K_{𝔭})/π^{n}E(K_{𝔭}) ≅ O/𝔭^{n}. By Lemma 6.8 K_{n,𝔭}^{×}/ker(δ_{n}) ≅ E[𝔭^{n}], and by Theorem 5.15(ii),

Hom(E[𝔭^{n}], E[𝔭^{n}])^{Gal(K_{n,𝔭}/K_{𝔭})} = Hom_{O}(E[𝔭^{n}], E[𝔭^{n}]) ≅ O/𝔭^{n}.

Therefore the injection above is an isomorphism, and the theorem follows from Proposition 6.5 and class field theory. ⊓ ⊔

Let A denote the ideal class group of K(E[𝔭]), and E the group of global units of K(E[𝔭]).

Corollary 6.10. With notation as above,

S_{π}(E) = 0 ⇔ (Hom(A, E[𝔭])^{Gal(K(E[𝔭])/K)} = 0 and δ_{1}(E) ≠ 0) .

Proof. By Corollary 5.20, K(E[𝔭])/K is totally ramified at 𝔭, of degree N𝔭−1. We identify K_{1,𝔭} with the completion of K(E[𝔭]) at the unique prime above 𝔭, and let O_{1,𝔭} denote its ring of integers and Ē the closure of E in O_{1,𝔭}. Let V = ker(δ_{1}) ∩ O_{1,𝔭}^{×} and Δ = Gal(K(E[𝔭])/K). We have an exact sequence

0 → O_{1,𝔭}^{×}/ĒV → A_{K_{1}}^{×}/W_{1} → A′ → 0

where W_{1} is as in Theorem 6.9 and A′ is a quotient of A by some power of the class of the prime P above 𝔭. Since P^{N𝔭−1} = 𝔭 is principal, Hom(A′, E[𝔭]) = Hom(A, E[𝔭]). Using Theorem 6.9 we conclude that

S_{π}(E) = 0 ⇔ (Hom(A, E[𝔭])^{Δ} = 0 and Hom(O_{1,𝔭}^{×}/ĒV, E[𝔭])^{Δ} = 0) .

By Lemma 6.8, δ_{1} : O_{1,𝔭}^{×}/V → E[𝔭] is an isomorphism. Since E[𝔭] has no proper Galois-stable submodules, it follows that

Hom(O_{1,𝔭}^{×}/ĒV, E[𝔭])^{Δ} = 0 ⇔ Ē ⊄ V ⇔ δ_{1}(E) ≠ 0.

This completes the proof of the corollary. ⊓ ⊔

7 Elliptic Units

In this section we define elliptic units and relate them to special values of L-functions. Elliptic units will be defined as certain rational functions of xcoordinates of torsion points on a CM elliptic curve. The results of §5 will allow us determine the action of the Galois group on these numbers, and hence their fields of definition. We follow closely [CW1] §5; see also [dS] Chapter II and Robert’s original memoir [Ro].

Throughout this section we fix an imaginary quadratic field K with ring of integers O, an elliptic curve E over C with complex multiplication by O, and a nontrivial ideal 𝔞 of O prime to 6. For simplicity we will assume that the class number of K is one; see [dS] for the general case.

7.1 Definition and Basic Properties

Definition 7.1. Choose a Weierstrass equation (1) for E with coordinate functions x, y on E. Define a rational function on E

Θ_{E,𝔞} = α^{−12}Δ(E)^{N𝔞−1} ∏_{P ∈E[𝔞]−O}(x − x(P))^{−6}

where α is a generator of 𝔞 and Δ(E) is the discriminant of the chosen model of E. Clearly this is independent of the choice of α.

Lemma 7.2. (i) Θ_{E,𝔞} is independent of the choice of Weierstrass model. (ii) If ϕ : E′ ⟶^{∼} E is an isomorphism of elliptic curves then Θ_{E′,𝔞} = Θ_{E,𝔞}∘ϕ.

(iii) If E is defined over F then the rational function Θ_{E,𝔞} is defined over F.

Proof. Any other Weierstrass model has coordinate functions x′, y′ given by

x′ = u^{2}x + r, y′ = u^{3}y + sx + t

where u ∈ C^{×} ([Si] Remark III.1.3), and then a′_{i} = u^{i}a_{i} and

Δ(E′) = u^{12}Δ(E).

Since #(E[𝔞]) = N𝔞, this proves (i), and (ii) is just a different formulation of (i). For (iii) we need only observe that α ∈ F, Δ(E) ∈ F, and G_{F} permutes the set {x(P) : P ∈ E[𝔞] − O}, so G_{F} fixes Θ_{E,𝔞}. ⊓ ⊔

Lemma 7.3. Suppose E is defined over K and 𝔭 is a prime of K where E has good reduction. Fix a Weierstrass model for E which is minimal at 𝔭. Let 𝔟 and 𝔠 be nontrivial relatively prime ideals of O and P ∈ E[𝔟], Q ∈ E[𝔠] points in E(K̄) of exact orders 𝔟 and 𝔠, respectively. Fix an extension of the 𝔭-adic order ord_{𝔭} to K̄, normalized so ord_{𝔭}(𝔭) = 1.

(i) If n > 0 and 𝔟 = 𝔭^{n} then ord_{𝔭}(x(P)) = −2/(N𝔭^{n−1}(N𝔭 − 1)). (ii) If 𝔟 is not a power of 𝔭 then ord_{𝔭}(x(P)) ≥ 0. (iii) If 𝔭 ∤ 𝔟𝔠 then ord_{𝔭}(x(P) − x(Q)) = 0.

Proof. Suppose that 𝔟 = 𝔭^{n} with n ≥ 1. Let Ê be the formal group over O_{𝔭} associated to E in Theorem 3.7. Let π = ψ_{E}(𝔭), let [π^{m}](X) ∈ O[[X]] be the endomorphism of Ê corresponding to π^{m} for every m, and define

f(X) = [π^{n}](X)/[π^{n−1}](X) ∈ O[[X]].

Since π reduces to the Frobenius endomorphism of the reduction Ẽ of E modulo 𝔭 (Corollary 5.16(iii)), it follows from Corollary 3.9 and Proposition 3.14 that

– f(X) ≡ X^{N𝔭^{n}−N𝔭^{n−1}} (mod 𝔭)

– f(X) ≡ π (mod X).

Thus by the Weierstrass preparation theorem,

f(X) = e(X)u(X)

where e(X) is an Eisenstein polynomial of degree N𝔭^{n−1}(N𝔭−1) and u(X) ∈ O[[X]]^{×}.

Since the reduction of π is a purely inseparable endomorphism of Ê, Lemma 3.6 shows that E[𝔭^{n}] ⊂ E_{1}(K̄_{𝔭}). Thus z = −x(P)/y(P) is a root of f(X), and hence of e(X), so ord_{𝔭}(x(P)/y(P)) = 1/(N𝔭^{n−1}(N𝔭 − 1)). Now (i) follows from Lemma 3.5.

If 𝔟 is not a power of 𝔭 then by Theorem 3.15(i), P ∉ E_{1}(K̄_{𝔭}). Hence by Lemma 3.5, ord_{𝔭}(x(P)) ≥ 0, which is (ii). Further, writing P̃ and Q̃ for the reductions of P and Q, we have

ord_{𝔭}(x(P) − x(Q)) > 0 ⇔ x(P̃) = x(Q̃) ⇔ P̃ = ±Q̃ ⇔

⇔ { \atop P ∓ Q} = Õ ⇔ P ∓ Q ∈ E_{1}(K̄_{𝔭}).

Since 𝔟 and 𝔠 are relatively prime, the order of P ± Q is not a power of 𝔭. So again by Theorem 3.15(i), P ± Q ∉ E_{1}(K̄_{𝔭}), and (iii) follows. ⊓ ⊔

For every ideal 𝔟 of O write K(𝔟) for the ray class field of K modulo 𝔟.

Theorem 7.4. Suppose 𝔟 is a nontrivial ideal of O relatively prime to 𝔞, and Q ∈ E[𝔟] is an O-generator of E[𝔟].

(i) Θ_{E,𝔞}(Q) ∈ K(𝔟). (ii) If 𝔠 is an ideal of O prime to 𝔟, c is a generator of 𝔠, and σ_{𝔠} = [𝔠, K(𝔟)/K],

then

Θ_{E,𝔞}(Q)^{σ_{𝔠}} = Θ_{E,𝔞}(cQ).

(iii) If 𝔟 is not a prime power then Θ_{E,𝔞}(Q) is a global unit. If 𝔟 is a power of a prime 𝔭 then Θ_{E,𝔞}(Q) is a unit at primes not dividing 𝔭.

Proof. Since we assumed that K has class number one, by Corollary 5.13 and Lemma 7.2(i) we may assume that E is defined over K by a Weierstrass model (1). Then by Lemma 7.2(iii) Θ_{E,𝔞} belongs to the function field K(E).

Let ψ be the Hecke character associated to E by Theorem 5.15. Suppose x ∈ ∏_{𝔭} O_{𝔭}^{×} ⊂ A_{K}^{×} and x ≡ 1 mod^{×}𝔟, and let σ_{x} = [x, K^{ab}/K]. By Theorem 5.15, ψ(x) ∈ O^{×} = Aut(E) and σ_{x}Q = ψ(x)Q. Therefore

Θ_{E,𝔞}(Q)^{σ_{x}} = Θ_{E,𝔞}(Q^{σ_{x}}) = Θ_{E,𝔞}(ψ(x)Q) = Θ_{E,𝔞}(Q),

the last equality by Lemma 7.2(ii). Since these σ_{x} generate Gal(K̄/K(𝔟)), this proves (i).

For (ii), let x ∈ A_{K}^{×} be an idele with xO = 𝔠 and x_{𝔭} = 1 for 𝔭 dividing 𝔟. Then Theorem 5.15 shows that ψ(x) ∈ cO^{×} and σ_{𝔠}Q = ψ(x)Q. So again using Lemma 7.2(ii),

Θ_{E,𝔞}(Q)^{σ_{𝔠}} = Θ_{E,𝔞}(ψ(x)Q) = Θ_{E,𝔞}(cQ).

This is (ii).

For (iii), let 𝔭 be a prime of K such that 𝔟 is not a power of 𝔭. By Corollary 5.22 and Lemma 7.2, we may assume that our Weierstrass equation for E has good reduction at 𝔭, so that Δ(E) is prime to 𝔭. Let n = ord_{𝔭}(𝔞). Then

ord_{𝔭}(Θ_{E,𝔞}(Q))/6 = −2n − ∑_{P ∈E[𝔭^{n}]−O} ord_{𝔭}(x(Q) − x(P))

− ∑_{P ∈E[𝔞]−E[𝔭^{n}]} ord_{𝔭}(x(Q) − x(P)).

By Lemma 7.3, since 𝔟 is not a power of 𝔭,

ord_{𝔭}(x(Q) − x(P))

= \begin{cases}−2/(N𝔭^{m} − N𝔭^{m−1}) & if P has order exactly 𝔭^{m}, m > 0 \\ 0 & if the order of P is not a power of 𝔭. \end{cases}

From this one verifies easily that ord_{𝔭}(Θ_{E,𝔞}(Q)) = 0. ⊓ ⊔

7.2 The Distribution Relation

Lemma 7.5. Θ_{E,𝔞} is a rational function on E with divisor

12N𝔞[O] − 12 ∑_{P ∈E[𝔞]}[P].

Proof. The coordinate function x is an even rational function with a double pole at O and no other poles. Thus for every point P, the divisor of x − x(P) is [P] + [−P] − 2[O] and the lemma follows easily. ⊓ ⊔

Theorem 7.6. Suppose 𝔟 is and ideal of O relatively prime to 𝔞, and β is a generator of 𝔟. Then for every P ∈ E(K̄),

∏_{R∈E[𝔟]} Θ_{E,𝔞}(P + R) = Θ_{E,𝔞}(βP).

Proof. Lemmas 7.2(iii) and 7.5 show that both sides of the equation in the theorem are rational functions on E, defined over K, with divisor

12 ∑_{Q∈E[𝔞𝔟]}[Q] − 12N𝔞 ∑_{R∈E[𝔟]}[R].

Thus their ratio is a constant λ ∈ K^{×}, and we need to show that λ = 1.

Let w_{K} = #(O^{×}) and fix a generator α of 𝔞. Evaluating this ratio at P = O one sees that

λ = \frac{Δ(E)^{(N𝔞−1)(N𝔟−1)}}{α^{12(N𝔟−1)}β^{12(N𝔞−1)}} ∏_{{R∈E[𝔟] \atop R≠0}} ∏_{{P ∈E[𝔞] \atop P ≠0}}(x(R) − x(P))^{−6} = μ^{w_{K}}

with

μ = \frac{Δ(E)^{(N𝔞−1)(N𝔟−1)/w_{K}}}{α^{12(N𝔟−1)/w_{K}} β^{12(N𝔞−1)/w_{K}}} ∏(x(R) − x(P))^{−12/w_{K}} ,

where the final product is over R ∈ E[𝔟] − O and P ∈ (E[𝔞] − O)/ ± 1 (recall 𝔞 is prime to 6). Since w_{K} divides 12, all of the exponents in the definition of μ are integers.

Exactly as in the proof of Theorem 7.4(iii), one can show that μ ∈ O^{×}, and therefore λ = 1. ⊓ ⊔ Corollary 7.7. Suppose 𝔟 is an ideal of O prime to 𝔞, Q ∈ E[𝔟] has order exactly 𝔟, 𝔭 is a prime dividing 𝔟, π is a generator of 𝔭, and 𝔟′ = 𝔟/𝔭. If the reduction map O^{×} → (O/𝔟′)^{×} is injective then

N_{K(𝔟)/K(𝔟′)}Θ_{E,𝔞}(Q) = \begin{cases}Θ_{E,𝔞}(πQ) & if 𝔭 | 𝔟′ \\ Θ_{E,𝔞}(πQ)^{1−Frob_{𝔭}^{−1}} & if 𝔭 ∤ 𝔟′ \end{cases}

where in the latter case Frob_{𝔭} is the Frobenius of 𝔭 in Gal(K(𝔟′)/K).

Proof. Let C denote the multiplicative group 1 + 𝔟′(O/𝔟). Because of our hypotheses that O^{×} injects into (O/𝔟′)^{×}, C is isomorphic to the kernel of the map

(O/𝔟)^{×}/O^{×} → (O/𝔟′)^{×}/O^{×}.

Thus class field theory gives an isomorphism

C ⟶^{∼} Gal(K(𝔟)/K(𝔟′))

which we will denote by c ↦ σ_{c}. Therefore

N_{K(𝔟)/K(𝔟′)}Θ_{E,𝔞}(Q) = ∏_{c∈C} Θ_{E,𝔞}(Q)^{σ_{c}} = ∏_{c∈C} Θ_{E,𝔞}(cQ)

by Theorem 7.4(ii).

One sees easily that

{cQ : c ∈ C} = {P ∈ E[𝔟] : πP = πQ and P ∉ E[𝔟′]}

= \begin{cases}\{Q + R : R ∈ E[𝔭]\} & if 𝔭 | 𝔟′ \\ \{Q + R : R ∈ E[𝔭], R ≢ −Q (mod E[𝔟′])\} & if 𝔭 ∤ 𝔟′ \end{cases}

Thus if 𝔭 | 𝔟′

N_{K(𝔟)/K(𝔟′)}Θ_{E,𝔞}(Q) = ∏_{R∈E[𝔭]} Θ_{E,𝔞}(Q + R) = Θ_{E,𝔞}(πQ)

by Theorem 7.6. Similarly, if 𝔭 ∤ 𝔟′

Θ_{E,𝔞}(Q + R_{0})N_{K(𝔟)/K(𝔟′)}Θ_{E,𝔞}(Q) = Θ_{E,𝔞}(πQ)

where R_{0} ∈ E[𝔭] satisfies Q + R_{0} ∈ E[𝔟′]. But then by Theorem 7.4(ii) (note that our assumption on 𝔟′ implies that 𝔟′ ≠ O)

Θ_{E,𝔞}(Q + R_{0})^{Frob^{𝔭}} = Θ_{E,𝔞}(πQ + πR_{0}) = Θ_{E,𝔞}(πQ)

so this completes the proof. ⊓ ⊔ 7.3 Elliptic Curves over K

Since the function Θ_{E,𝔞} depends only on the isomorphism class of E over C, we need to provide it with information that depends on E itself to make it sensitive enough to “see” the value of the L-function of E at 1. Following Coates and Wiles [CW1] we will write down a product of translates of Θ_{E,𝔞} and then show that it has the connections we need with L-values.

From now on suppose that our elliptic curve E is defined over K, ψ is the Hecke character attached to E by Theorem 5.15, 𝔣 is the conductor of ψ, and 𝔞 is prime to 𝔣 as well as to 6. For P ∈ E(K̄) let τ_{P} denote translation P, so τ_{P} is a rational function defined over K(P).

Fix an O-generator S of E[𝔣]. By Corollary 5.20(i) S ∈ E(K(𝔣)), and we define

Λ_{E,𝔞} = Λ_{E,𝔞,S} = ∏_{σ∈Gal(K(f)/K)} Θ_{E,𝔞} ∘ τ_{S^{σ}}.

Proposition 7.8. (i) Λ_{E,𝔞} is a rational function defined over K. (ii) If B is a set of ideals of O, prime to 𝔞𝔣, such that the Artin map 𝔟 ↦

[𝔟, K(𝔣)/K] is a bijection from B to Gal(K(𝔣)/K), then

Λ_{E,𝔞}(P) = ∏_{𝔟∈B} Θ_{E,𝔞}(ψ(𝔟)S + P).

(iii) If 𝔯 is an ideal of O and Q ∈ E[𝔯], Q ∉ E[𝔣], then Λ_{E,𝔞}(Q) is a global unit in K(E[𝔯]).

Proof. The first assertion is clear, (ii) is immediate from Corollary 5.16(ii), and (iii) follows from Theorem 7.4(iii). ⊓ ⊔

7.4 Expansions over C

We continue to suppose that E is defined over K. Fix a Weierstrass model of E (over K) and let L ⊂ C be the corresponding lattice given by Theorem 2.3(ii); then OL = L (Proposition 2.6) so we can choose Ω ∈ C^{×} such that L = ΩO. The map ξ(z) = (℘(z; L), ℘′(z; L)/2) is an isomorphism C/L ⟶^{∼} E(C), and we define Θ_{L,𝔞} = Θ_{E,𝔞} ∘ ξ, i.e.,

Θ_{L,𝔞}(z) = α^{−12}Δ(L)^{N𝔞−1} ∏_{u∈𝔞^{−1}L/L−0}(℘(z; L) − ℘(u; L))^{−6}.

Definition 7.9. Define

A(L) = π^{−1}area(C/L),

s_{2}(L) = lim_{s→0^{+}} ∑_{0≠ω∈L} ω^{−2}|ω|^{−2s},

η(z; L) = A(L)^{−1}z̄ + s_{2}(L)z,

θ(z; L) = Δ(L)e^{−6η(z;L)z}σ(z; L)^{12}.

Lemma 7.10. Θ_{L,𝔞}(z) = θ(z; L)^{N𝔞}/θ(z; 𝔞^{−1}L).

Proof. Write f(z) = θ(z; L)^{N𝔞}/θ(z; 𝔞^{−1}L). Note that although θ(z; L) is not holomorphic (because of the z̄ in the definition of η(z; L)), f(z) is holomorphic. One can check explicitly, using well-known properties of σ(z; L) (see [dS] §II.2.1), that f(z) is periodic with respect to L and its divisor on C/L is 12N𝔞[0] − 12 ∑_{v∈𝔞^{−1}L/L}[v].

Thus by Lemma 7.5, Θ_{L,𝔞} = λf for some λ ∈ C^{×}. At z = 0, both functions have Laurent series beginning α^{−12}Δ(L)^{N𝔞−1}z^{12(N𝔞−1)}, so λ = 1. ⊓ ⊔

Definition 7.11. for k ≥ 1 define the Eisenstein series

E_{k}(z; L) = lim_{s→k} ∑_{ω∈L} \frac{(z̄ + ω̄)^{k}}{|z + ω|^{2s}}

= ∑_{ω∈L} \frac{1}{(z + ω)^{k}} if k ≥ 3

where the limit means evaluation of the analytic continuation at s = k.

Proposition 7.12.

E_{1}(z; L) = log(σ(z; L))′ − s_{2}(L)z − A(L)^{−1}z̄, E_{2}(z; L) = ℘(z; L) + s_{2}(L),

E_{k}(z; L) = \frac{(−1)^{k}}{(k − 1)!} (\frac{d}{dz} )^{(k−2)} ℘(z; L) if k ≥ 3.

Proof. The third equality is immediate from the definition of ℘(z; L). For the first two, see [CW1] pp. 242–243 or [GS] Proposition 1.5. ⊓ ⊔

Theorem 7.13. For every k ≥ 1,

(\frac{d}{dz} )^{k} log Θ_{L,𝔞}(z) = 12(−1)^{k−1}(k − 1)!(N𝔞E_{k}(z; L) − E_{k}(z; 𝔞^{−1}L)).

Proof. By Lemma 7.10

(\frac{d}{dz} )^{k} log Θ_{L,𝔞}(z) = (\frac{d}{dz} )^{k−1} (N𝔞\frac{d}{dz} log(θ(z; L)) − \frac{d}{dz} log(θ(z; 𝔞^{−1}L))) .

The definition of θ shows that

log(θ(z; L)) = log(Δ(L)) − 6s_{2}(L)z^{2} − 6A(L)^{−1}zz̄ + 12 log(σ(z; L)).

Now the theorem follows from Proposition 7.12 ⊓ ⊔ Definition 7.14. Define the Hecke L-functions associated to powers of ψ̄ to be the analytic continuations of the Dirichlet series

L(ψ̄^{k}, s) = ∑ \frac{ψ̄^{k}(𝔟)}{N𝔟^{s}},

summing over ideals 𝔟 of O prime to the conductor of ψ̄^{k}. If 𝔪 is an ideal of O divisible by 𝔣 and 𝔠 is an ideal prime to 𝔪, we define the partial L-function L_{𝔪}(ψ̄^{k}, s, 𝔠) be the same formula, but with the sum restricted to ideals of K prime to 𝔪 such that [𝔟, K(𝔪)/K] = [𝔠, K(𝔪)/K].

Recall that Ω ∈ C^{×} is such that L = ΩO.

Proposition 7.15. Suppose v ∈ KL/L has order 𝔪, where 𝔪 is divisible by 𝔣. Then for every k ≥ 1,

E_{k}(v; L) = v^{−k}ψ(𝔠)^{k}L_{𝔪}(ψ̄^{k}, k, 𝔠)

where 𝔠 = Ω^{−1}v𝔪.

Proof. Let μ be a generator of 𝔪, so that v = αΩ/μ for some α ∈ O prime to 𝔪. For s large,

∑_{ω∈L} \frac{(v̄ + ω̄)^{k}}{|v + ω|^{2s}} = \frac{Nμ^{s}}{μ̄^{k}} \frac{Ω̄^{k}}{|Ω|^{2s}} ∑_{β∈O,β≡α (mod 𝔪)} \frac{β̄^{k}}{|β|^{2s}} .

By Corollary 5.16(i), if we define

ϵ(β) = ψ(βO)/β

then ϵ is a multiplicative map from {β ∈ O : β is prime to 𝔣} to O^{×}. By definition of the conductor, ϵ factors through (O/𝔣)^{×}. Thus if β ≡ α (mod 𝔪),

β̄ = ψ̄(βO)\frac{ψ(αO)}{α}.

Therefore

∑_{β∈O,β≡α (mod 𝔪)} \frac{β̄^{k}}{|β|^{2s}} = \frac{ψ(αO)^{k}}{α^{k}} ∑_{𝔟⊂O,[𝔟,K(𝔪)/K]=[αO,K(𝔪)/K]} \frac{ψ̄(𝔟)^{k}}{N𝔟^{s}}

= \frac{ψ(𝔠)^{k}}{α^{k}} L_{𝔪}(ψ̄^{k}, s, σ_{𝔠})

and the proposition follows. ⊓ ⊔

Definition 7.16. Fix a generator f of 𝔣 and a set B of ideals of O, prime to 𝔞𝔣, such that the Artin map 𝔟 ↦ [𝔟, K(𝔣)/K] is a bijection from B to Gal(K(𝔣)/K). Let u = Ω/f ∈ 𝔣^{−1}L and define

Λ_{L,𝔞}(z) = Λ_{L,𝔞,f}(z) = Λ_{E,𝔞,ξ(u)}(ξ(z)) = ∏_{𝔟∈B} Θ_{L,𝔞}(ψ(𝔟)u + z).

By Proposition 7.8(ii), Λ_{L,𝔞} = Λ_{E,𝔞} ∘ ξ.

Theorem 7.17. For every k ≥ 1,

(\frac{d}{dz} )^{k} log Λ_{L,𝔞}(z) |_{z=0}= 12(−1)^{k−1}(k − 1)!f^{k}(N𝔞 − ψ(𝔞)^{k})Ω^{−k}L_{𝔣}(ψ̄^{k}, k).

Proof. By Theorem 7.13

(\frac{d}{dz} )^{k} log Λ_{L,𝔞}(z) |_{z=0} = ∑_{𝔟∈B} (\frac{d}{dz} )^{k} log Θ_{L,𝔞}(z) |_{z=ψ(𝔟)u}

= 12(−1)^{k−1}(k − 1)! (N𝔞 ∑_{𝔟∈B} E_{k}(ψ(𝔟)u; L) − ∑_{𝔟∈B} E_{k}(ψ(𝔟)u; 𝔞^{−1}L)) .

By Proposition 7.15,

∑_{𝔟∈B} E_{k}(ψ(𝔟)u; L) = ∑_{𝔟∈B}(ψ(𝔟)u)^{−k}ψ(𝔟)^{k}L_{𝔣}(ψ̄^{k}, k, 𝔟) = u^{−k}L_{𝔣}(ψ̄^{k}, k).

By inspection (and Corollary 5.16(i)) E_{k}(z; 𝔞^{−1}L) = ψ(𝔞)^{k}E_{k}(ψ(𝔞)z; L), so

∑_{𝔟∈B} E_{k}(ψ(𝔟)u; 𝔞^{−1}L) = u^{−k}ψ(𝔞)^{k}L_{𝔣}(ψ̄^{k}, k).

⊓ ⊔

Although we will not use it explicitly, the following theorem of Damerell is a corollary of this computation.

Corollary 7.18 (Damerell’s Theorem). For every k ≥ 1,

Ω^{−k}L(ψ̄^{k}, k) ∈ K.

Proof. By Proposition 7.8(i), Λ_{L,𝔞}(z) is a rational function of ℘(z; L) and ℘′(z; L) with coefficients in K. Differentiating the relation (from Theorem 2.3)

℘′(z; L)^{2} = 4℘(z; L)^{3} + 4a℘(z; L) + 4b

shows that all derivatives ℘^{(k)}(z; L) also belong to K(℘(z; L), ℘′(z; L)), and hence Λ_{L,𝔞}^{(k)} does as well. Thus the corollary follows from Theorem 7.17. ⊓ ⊔

7.5 𝔭-adic Expansions

Keep the notation of the previous sections. Fix a prime 𝔭 of K where E has good reduction, 𝔭 ∤ 6. Suppose that our chosen Weierstrass model of E has good reduction at 𝔭 and that the auxiliary ideal 𝔞 is prime to 𝔭 as well as 6𝔣. Let Ê be the formal group attached to E over O_{𝔭} as in §3.2, and x(Z), y(Z) ∈ O_{𝔭}[[Z]] the power series of Theorem 3.7.

Definition 7.19. Let λÊ(Z) ∈ Z + Z^{2}K_{𝔭}[[Z]] be the logarithm map of Ê from Definition 3.10, so that λ′ ; Ê(Z) ∈ O_{𝔭}[[Z]]^{×}, and define an operator D on O_{𝔭}[[Z]] by

D = \frac{1}{λ′ ; Ê(Z)} \frac{d}{dZ} .

Proposition 7.20. Identifying (x, y) both with (℘(z; L), \frac{1}{2}℘′(z; L)) and with (x(Z), y(Z)) leads to a commutative diagram

K(℘(z), ℘′(z)) ⟵^{∼} K(E) ⟶^{∼} K(x(Z), y(Z)) ↪ K_{𝔭}((Z))
\frac{d}{dz} ↓  ↓  ↓^{D}  ↓^{D}
K(℘(z), ℘′(z)) ⟵^{∼} K(E) ⟶^{∼} K(x(Z), y(Z)) ↪ K_{𝔭}((Z)).

Proof. Differentiating the relation ℘′(z)^{2} = 4℘(z)^{3} + 4a℘(z) + 4b shows that

℘′′(z) = 6℘(x)^{2} + 2a ∈ K_{𝔭}(℘(z), ℘′(z)).

Thus, since both vertical maps are derivations, we need only check that D(x(Z)) = 2y(Z) and D(y(Z)) = 3x(Z)^{2} + a. (In fact, it would be enough to check either equality.) Both equalities are immediate from the definition (Definition 3.10) of ω̂ and λÊ. ⊓ ⊔

Definition 7.21. Let Λ_{𝔭,𝔞}(Z) be the image of Λ_{E,𝔞} in K_{𝔭}((Z)) under the map of Proposition 7.20.

Theorem 7.22. (i) Λ_{𝔭,𝔞}(Z) ∈ O_{𝔭}[[Z]]^{×}. (ii) For every k ≥ 1,

D^{k} log(Λ_{𝔭,𝔞}(Z)) |_{Z=0} = 12(−1)^{k−1}(k − 1)!f^{k}(N𝔞 − ψ(𝔞)^{k})Ω^{−k}L(ψ̄^{k}, k).

Proof. Fix an embedding K̄ ↪ K̄_{𝔭} so that we can view x(R) ∈ K̄_{𝔭} when R ∈ E[𝔣]. Let R be the ring of integers of K̄_{𝔭}.

Consider one of the factors x(ψ(𝔟)S + P) − x(Q) of Λ_{E,𝔞}(P), with Q ∈ E[𝔞] − O. The explicit addition law for x(P) ([Si] §III.2.3) shows that

x(ψ(𝔟)S + P) − x(Q) = \frac{(y(P) − y(ψ(𝔟)S))^{2}}{(x(P) − x(ψ(𝔟)S))^{2}} − x(P) − x(ψ(𝔟)S) − x(Q).

By Lemmas 7.3(ii) and 3.5, x(ψ(𝔟)S), y(ψ(𝔟)S), x(Q) ∈ R. Substituting x(Z) for x(P), y(Z) for y(P) and using the expansions in Theorem 3.7 to show that

x(Z) ∈ Z^{−2} + ZO_{𝔭}[[Z]], y(Z) ∈ −Z^{−3} + O_{𝔭}[[Z]]

gives

x(ψ(𝔟)S + P) − x(Q) ↦ g_{𝔟,Q}(Z) ∈ R[[Z]]

under the map of Proposition 7.20, where g_{𝔟,Q} satisfies

g_{𝔟,Q}(0) = x(ψ(𝔟)S) − x(Q) ∈ R^{×}

by Lemma 7.3(iii), so g_{𝔟,Q}(Z) ∈ R[[Z]]^{×}. Also Δ(E), α ∈ O_{𝔭}^{×} since our Weierstrass equation has good reduction at 𝔭 and 𝔭 ∤ 𝔞. Thus

Λ_{𝔭,𝔞}(Z) = Δ(E)^{(N𝔞−1)#(B)}α^{−12#(B)} ∏_{𝔟,Q} g_{𝔟,Q}(Z)^{−6} ∈ R[[Z]]^{×}.

Since we already know Λ_{𝔭,𝔞} ∈ K_{𝔭}((Z)), this proves (i).

The second assertion is immediate from Theorem 7.17 and Proposition 7.20. ⊓ ⊔

8 Euler Systems

In this section we introduce Kolyvagin’s concept of an Euler system (of which the elliptic units of §7 are an example) and we show how to use an Euler system to construct certain principal ideals in abelian extensions of K. In the next section we use these principal ideals (viewed as relations in ideal class groups) to bound the ideal class groups of abelian extensions of K.

As in the previous section, fix an imaginary quadratic field K and an elliptic curve E defined over K with complex multiplication by the ring of integers O of K. Let 𝔣 be the conductor of the Hecke character ψ of E, and fix a generator f of 𝔣.

Fix a prime 𝔭 of K not dividing 6𝔣, and for n ≥ 1 let K_{n} = K(E[𝔭^{n}]). Let p denote the rational prime below 𝔭. Fix a nontrivial ideal 𝔞 of O prime to 6𝔣𝔭. Let R = R(𝔞) denote the set of squarefree ideals of O prime to 6𝔣𝔞𝔭, and if 𝔯 ∈ R let K_{n}(𝔯) = K_{n}(E[𝔯]) = K(E[𝔯𝔭^{n}]). The letter 𝔮 will always denote a prime of R.

Also as in the previous section, fix a Weierstrass model of E which is minimal at 𝔭, let L = ΩO ⊂ C be the corresponding lattice given by Theorem 2.3(ii), and define ξ = (℘( · ; L), ℘′( · ; L)/2) : C/L ⟶^{∼} E(C).

8.1 The Euler System

Definition 8.1. If 𝔯 ∈ R and n ≥ 0 define

η_{n}(𝔯) = η_{n}^{(𝔞)}(𝔯) = Λ_{E,𝔞,ξ(Ω/f)}(ξ(ψ(𝔭^{n}𝔯)^{−1}Ω)) = Λ_{L,𝔞}(ψ(𝔭^{n}𝔯)^{−1}Ω).

where Λ_{L,𝔞} is as in Definition 7.16.

Proposition 8.2. Suppose 𝔯 ∈ R and n ≥ 1.

(i) η_{n}(𝔯) is a global unit in K_{n}(𝔯).

(ii) If 𝔮 is a prime and 𝔯𝔮 ∈ R, then

N_{K_{n}(𝔯𝔮)/K_{n}(𝔯)}η_{n}(𝔮𝔯) = η_{n}(𝔯)^{1−Frob_{𝔮}^{−1}}.

(iii) N_{K_{n+1}(𝔯)/K_{n}(𝔯)}η_{n+1}(𝔯) = η_{n}(𝔯).

Proof. Assertion (i) is just a restatement of Proposition 7.8(iii), and (ii) and (iii) are immediate from Corollary 7.7. ⊓ ⊔

8.2 Kolyvagin’s Derivative Construction

Definition 8.3. Write G_{𝔯} = Gal(K_{n}(𝔯)/K_{n}). By Corollary 5.20(ii), G_{𝔯} is independent of n ≥ 1, and we have natural isomorphisms

G_{𝔯}  =  ∏_{𝔮|𝔯} G_{𝔮}
↓  ↓
(O/𝔯)^{×} = ∏_{𝔮|𝔯}(O/𝔮)^{×}.

If 𝔮 | 𝔯 this allows us to view G_{𝔮} either as a quotient or a subgroup of G_{𝔯}. By Corollary 5.20 if 𝔮𝔯 ∈ R then K_{n}(𝔮𝔯)/K_{n}(𝔯) is cyclic of degree N𝔮 − 1, totally ramified at all primes above 𝔮 and unramified at all other primes.

For every 𝔯 ∈ R define

N_{𝔯} = ∑_{σ∈G_{𝔯}} σ ∈ Z[G_{𝔯}]

so we clearly have

N_{𝔯} = ∏_{𝔮|𝔯} N_{𝔮}.

For every n ≥ 1 and 𝔯 ∈ R, let x_{n,𝔯} be an indeterminate and define X_{n,𝔯} to be the Gal(K_{n}(𝔯)/K)-module Y_{n,𝔯}/Z_{n,𝔯} where

Y_{n,𝔯} = ⨁_{𝔰|𝔯} Z[Gal(K_{n}(𝔰)/K)]x_{n,𝔰},
Z_{n,𝔯} = ∑_{𝔮𝔰|𝔯} Z[Gal(K_{n}(𝔯)/K)] (N_{𝔮}x_{n,𝔮𝔰} − (1 − Frob_{𝔮}^{−1})x_{n,𝔰}) ⊂ Y_{n,𝔯}.

In other words, X_{n,𝔯} is the quotient of the free Z[Gal(K_{n}(𝔯)/K)]-module on {x_{n,𝔰} : 𝔰 | 𝔯} by the relations

– G_{𝔯/𝔰} acts trivially on x_{n,𝔰}, and

– N_{𝔮}x_{n,𝔮𝔰} = (1 − Frob_{𝔮}^{−1})x_{n,𝔰} if 𝔮𝔰 | 𝔯.

For every prime 𝔮 ∈ R fix once and for all a generator σ_{𝔮} of G_{𝔮} and define

D_{𝔮} = ∑_{i=1}^{N𝔮−2} iσ_{𝔮}^{i} ∈ Z[G_{𝔮}]

and for 𝔯 ∈ R

D_{𝔯} = ∏_{𝔮|𝔯} D_{𝔮} ∈ Z[G_{𝔯}].

If M is a power of p and n ≥ 1 define R_{n,M} ⊂ R to be the set of ideals 𝔯 ∈ R such that for every prime 𝔮 dividing 𝔯,

– 𝔮 splits completely in K_{n}/K

– N𝔮 ≡ 1 (mod M).

Proposition 8.4. Suppose M is a power of p, n ≥ 1, and 𝔯 ∈ R_{n,M}.

(i) X_{n,𝔯} has no Z-torsion. (ii) D_{𝔯}x_{n,𝔯} ∈ (X_{n,𝔯}/MX_{n,𝔯})^{G_{𝔯}}.

Proof. For every prime 𝔮 ∈ R and divisor 𝔰 of 𝔯, define

B_{𝔮} = G_{𝔮} − {1},
B_{𝔰} = ∏_{𝔮|𝔰} B_{𝔮} = {∏_{𝔮|𝔰} g_{𝔮} : g_{q} ∈ B_{𝔮}} ⊂ G_{𝔯},
B = ∪_{𝔰|𝔯}B_{𝔰}x_{n,𝔰} ⊂ X_{n,𝔯}

Then one can show by an easy combinatorial argument (see [Ru2] Lemma 2.1) that X_{n,𝔯} is a free Z-module with basis B, which proves (i).

Note that

(σ_{𝔮} − 1)D_{𝔮} = N𝔮 − 1 − N_{𝔮}.

We will prove (ii) by induction on the number of primes dividing 𝔯. Suppose 𝔮 | 𝔯, 𝔯 = 𝔮𝔰. Then

(σ_{𝔮} − 1)D_{𝔯}x_{n,𝔯} = (σ_{𝔮} − 1)D_{𝔮}D_{𝔰}x_{n,𝔯}
= (N𝔮 − 1)D_{𝔰}x_{n,𝔯} − (1 − Frob_{𝔮}^{−1})D_{𝔰}x_{n,𝔰}.

Since 𝔮 ∈ R_{n,M}, M | N𝔮 − 1 and Frob_{𝔮} ∈ G_{𝔰}, so by the induction hypothesis

(σ_{𝔮} − 1)D_{𝔯}x_{n,𝔯} ∈ MX_{n,𝔯}.

Since the σ_{𝔮} generate G_{𝔯}, this proves the proposition. ⊓ ⊔ Definition 8.5. An Euler system is a collection of global units

{η(n, 𝔯) ∈ K_{n}(𝔯)^{×} : n ≥ 1, 𝔯 ∈ R}

satisfying

N_{K_{n}(𝔮𝔯)/K_{n}(𝔯)}η(n, 𝔮𝔯) = η(n, 𝔯)^{1−Frob_{𝔮}^{−1}},  (8)
N_{K_{n+1}(𝔯)/K_{n}(𝔯)}η(n + 1, 𝔯) = η(n, 𝔯).  (9)

Equivalently, an Euler system is a Galois equivariant map

η : lim_{\overrightarrow{n,𝔯}} X_{n,𝔯} → ⋃_{n,𝔯} K_{n}(𝔯)^{×}

such that η(x_{n,𝔯}) is a global unit for every n and 𝔯. We will use these two definitions interchangeably.

For example, by Proposition 8.2 we can define an Euler system by

η(n, 𝔯) = η_{n}(𝔯).

Proposition 8.6. Suppose η is an Euler system and 𝔮 ∈ R is a prime. Write N𝔮−1 = dp^{k} with d prime to p. Then for every n ≥ 1 and every 𝔯 ∈ R prime to 𝔮,

η(n, 𝔮𝔯)^{d} ≡ η(n, 𝔯)^{dFrob_{𝔮}^{−1}}

modulo every prime above 𝔮.

Proof. Suppose m ≥ n, and let G = Gal(K_{m}(𝔮𝔯)/K_{n}(𝔮𝔯)). Fix a prime 𝔔 of K_{m}(𝔮𝔯) above 𝔮, and let H be the decomposition group of 𝔮 in G. Let H′ ⊂ G be a set of coset representatives for G/H, and define

N_{H} = ∑_{γ∈H} γ, N_{H′} = ∑_{γ∈H′} γ

so that N_{H}N_{H′} = ∑_{γ∈G} γ.

Since 𝔮 is totally ramified in K_{m}(𝔮𝔯)/K_{m}(𝔯), the Euler system distribution relation (8) reduces modulo 𝔔 to

η(m, 𝔮𝔯)^{N𝔮−1} ≡ (η(m, 𝔯)^{Frob{−1 \atop 𝔮}})^{N𝔮−1} (mod 𝔔).

On the other hand, since H is generated by the Frobenius of 𝔮, if h denotes the degree of the residue field extension at 𝔮 in K_{n}(𝔯)/K then (9) reduces to

η(n, 𝔯) = η(m, 𝔯)^{N_{H′} N_{H}} ≡ (η(m, 𝔯)^{N_{H′}})^{t} (mod 𝔔)

and similarly η(n, 𝔮𝔯) ≡ (η(m, 𝔮𝔯)^{N_{H′}})^{t} (mod 𝔔), where

t = ∑_{i=0}^{#(H)−1}(N𝔮^{h})^{i} ≡ #(H) (mod N𝔮 − 1).

Recall that p^{k} is the highest power of p dividing N𝔮 − 1. Since the decomposition group of 𝔮 in K_{∞}/K is infinite, for m sufficiently large we will have p^{k} | t, and then combining the congruences above proves the proposition. ⊓ ⊔

For the Euler system of elliptic units, one can prove directly, using Lemma 7.3, that the congruence of Proposition 8.6 holds with d = 1.

Definition 8.7. Suppose η is an Euler system, n ≥ 1 and 𝔯 ∈ R. Using the map X_{n,𝔯} → K_{n}(𝔯)^{×} corresponding to η, we define a 1-cocycle c = c_{η,n,𝔯} : G_{𝔯} → K_{n}(𝔯)^{×} by

c(σ) = η (\frac{(σ − 1)D_{𝔯}x_{n,𝔯}}{M} ) for σ ∈ G_{𝔯}.

This is well defined by Proposition 8.4. Since H^{1}(G_{𝔯}, K_{n}(𝔯)^{×}) = 0, there is a β ∈ K_{n}(𝔯)^{×} such that c(σ) = β^{σ}/β for every σ ∈ G_{𝔯}. Then η(x_{n,𝔯})^{D_{𝔯}}/β^{M} ∈ K_{n}^{×} and we define

κ_{n,M}(𝔯) = η(x_{n,𝔯})^{D_{𝔯}}/β^{M} ∈ K_{n}^{×}/(K_{n}^{×})^{M}.

Since β is uniquely determined modulo K_{n}^{×}, κ_{n,M}(𝔯) is independent of the choice of β.

Remark 8.8. It is quite easy to show for every Euler system η, every n, and every 𝔯 ∈ R_{n,M} that η(n, 𝔯)^{(σ−1)D_{𝔯}} is an M-th power (Proposition 8.4(ii)). The reason for introducing the “universal Euler system” X_{n,𝔯} is to show that η(n, 𝔯)^{(σ−1)D_{𝔯}} has a canonical M-th root, even when K_{n}(𝔯) contains M-th roots of unity (Proposition 8.4(i)). This fact was used to construct the cocycle c above.

We next want to determine the ideal generated by κ_{n,M}(𝔯) (modulo M-th powers).

Definition 8.9. Fix n ≥ 1, a power M of p, and temporarily write F = K_{n}, R_{F,M} = R_{n,M}. Let O_{F} denote the ring of integers of F and

I_{F} = I = ⊕_{𝔔}Z𝔔

the group of fractional ideals of F, written additively. For every prime 𝔮 of K let

I_{F,𝔮} = I_{𝔮} = ⊕_{𝔔|𝔮}Z𝔔,

and if y ∈ F^{×} let (y) ∈ I denote the principal ideal generated by y, and (y)_{𝔮}, [y], and [y]_{𝔮} the projections of (y) to I_{𝔮}, I/MI, and I_{𝔮}/MI_{𝔮}, respectively. Note that [y] and [y]_{𝔮} are well defined for y ∈ F^{×}/(F^{×})^{M}.

Suppose 𝔮 ∈ R_{F,M}, 𝔔 is a prime of F above 𝔮, and 𝔔̄ is a prime of K̄ above 𝔔. Recall that 𝔔 is completely split in F/K, and totally ramified of degree N𝔮 − 1 = N𝔔 − 1 in F (𝔮)/F. Fix a lift σ_{𝔔} of σ_{𝔮} to G_{K} so that σ_{𝔔} belongs to the inertia group of 𝔔̄. Then there is an isomorphism

Z/MZ ⟶^{∼} μ_{M}

given by a ↦ (π^{a/M})^{1−σ_{𝔔}} where π ∈ K is a generator of 𝔮. Let Frob𝔔̄ ∈ G_{F_{𝔔}} denote a Frobenius of 𝔔 and define

ϕ_{𝔔} : F_{𝔔}^{×}/(F_{𝔔}^{×})^{M} → Z/MZ

to be the image of Frob_{𝔔} under the composition

G_{F_{𝔔}} → Hom(F_{𝔔}^{×}, μ_{M}) ⟶^{∼} Hom(F_{𝔔}^{×}, Z/MZ)

where the first map is the Kummer map and the second is induced by the isomorphism above. Concretely, since σ_{𝔔} belongs to the inertia group, we have ϕ_{𝔔}(α) = a where a is characterized by

(α^{1/M})^{Frob_{𝔔}−1} = (π^{a/M})^{1−σ_{𝔔}} ≡ (β^{1/M})^{1−σ_{𝔔}}  (10)

modulo the maximal ideal of F̄_{𝔔}, where β ∈ F̄_{𝔔}^{×} is an element satisfying ord_{𝔔}(β) = a.

Finally, define

ϕ_{𝔮} : F^{×}/(F^{×})^{M} → I_{𝔮}/MI_{𝔮}

by ϕ_{𝔮}(α) = ∑_{𝔔|𝔮} ϕ_{𝔔}(α)𝔔. It is not difficult to check that ϕ_{𝔮} is Gal(F/K)- equivariant, and that ϕ_{𝔮} induces an isomorphism

ϕ_{𝔮} : (O_{F} /𝔮O_{F})^{×}/((O_{F} /𝔮O_{F})^{×})^{M} ⟶^{∼} I_{𝔮}/MI_{𝔮}.

Proposition 8.10. Suppose η is an Euler system, n ≥ 1, 𝔯 ∈ R_{n,M} and 𝔮 is a prime of K.

(i) If 𝔮 ∤ 𝔯 then [κ_{n,M}(𝔯)]_{𝔮} = 0. (ii) If 𝔮 | 𝔯 then [κ_{n,M}(𝔯)]_{𝔮} = ϕ_{𝔮}(κ_{n,M}(𝔯/𝔮)).

Proof. Suppose first that 𝔮 ∤ 𝔯. Then 𝔮 is unramified in K_{n}(𝔯)/K_{n}, and by definition κ_{n,M}(𝔯) is a global unit times an M-th power in K_{n}(𝔯)^{×}, so ord_{𝔔}(κ_{n,M}(𝔯)) ≡ 0 (mod M) for every prime 𝔔 of K_{n} above 𝔮. This proves (i).

Now suppose 𝔮 | 𝔯, say 𝔯 = 𝔮𝔰. By definition

κ_{n,M}(𝔯) = η(x_{n,𝔯})^{D_{𝔯}}/β_{𝔯}^{M}, κ_{n,M}(𝔰) = η(x_{n,𝔰})^{D_{𝔰}}/β_{𝔰}^{M}

where β_{𝔯} ∈ K_{n}(𝔯)^{×}, β_{𝔰} ∈ K_{n}(𝔰)^{×} satisfy

β_{𝔯}^{σ−1} = η((σ − 1)D_{𝔯}x_{n,𝔯}/M), β_{𝔰}^{σ−1} = η((σ − 1)D_{𝔰}x_{n,𝔰}/M)

for every σ ∈ G_{𝔯}.

We will use (10) to evaluate ϕ_{𝔮}(κ(𝔰)). Fix a prime 𝔔 of K_{n} above 𝔮, let σ_{𝔔} be as in Definition 8.9, and let d be the prime-to-p-part of N𝔮 − 1 as in Proposition 8.6. Modulo every prime above 𝔔 we have

(κ_{n,M}(𝔯)^{d/M})^{1−σ_{𝔔}} = ((η(x_{n,𝔯})^{D_{𝔯}})^{1/M}/β_{𝔯})^{d(1−σ_{𝔔})} ≡ β_{𝔯}^{d(σ_{𝔮}−1)}
= η((σ_{𝔮} − 1)D_{𝔯}x_{n,𝔯}/M)^{d}
= η((N𝔮 − 1 − N_{𝔮})D_{𝔰}x_{n,𝔯}/M)^{d}

= η((N𝔮 − 1)D_{𝔰}x_{n,𝔯}/M)^{d}η((Frob_{𝔮}^{−1} − 1)D_{𝔰}x_{n,𝔰}/M)^{d}

= (η(x_{n,𝔯})^{D_{𝔰}})^{d(N𝔮−1)/M}/β_{𝔰}^{d(1−Frob_{𝔮}^{−1})}
= (η(x_{n,𝔯})^{D_{𝔰}})^{Frob_{𝔮}^{−1}d(N𝔮−1)/M}/β_{𝔰}^{d(1−Frob_{𝔮}^{−1})}
≡ ((η(x_{n,𝔰})^{D_{𝔰}}/β_{𝔰}^{M})^{1/M})^{d(1−Frob_{𝔔}^{−1})}
≡ (κ_{n,M}(𝔰)^{d/M})^{Frob_{𝔔}−1}

using Proposition 8.6 for the second-to-last congruence. By (10) it follows that

dϕ_{𝔔}(κ_{n,M}(𝔰)) = dord_{𝔔}(κ_{n,M}(𝔯)),

and since d is prime to p, (ii) follows. ⊓ ⊔

9 Bounding Ideal Class Groups

In this section we describe Kolyvagin’s method of using the Euler system of elliptic units, or rather the principal ideals deduced from elliptic units as in §8.2, to bound the size of certain ideal class groups. For a similar argument in the case of cyclotomic units and real abelian extensions of Q, see [Ru1].

Keep the notation of the previous section. Let F = K_{1} = K(E[𝔭]) and let μ_{F} denote the roots of unity in F. Let Δ = Gal(F/K), so Δ ≅ (O/𝔭)^{×} is cyclic of order p − 1 or p^{2} − 1.

Since #(Δ) is prime to p, the group ring Z_{p}[Δ] is semisimple, i.e.,

Z_{p}[Δ] ≅ ⨁_{χ∈Ξ} R_{χ}

where Ξ denotes the set of all irreducible F_{p}-representations of Δ and R_{χ} denotes the corresponding direct summand of Z_{p}[Δ]. (We will also refer to elements of Ξ as irreducible Z_{p}-representations of Δ.) Since #(Δ) divides p^{2} − 1, we have two cases:

– dim(χ) = 1, R_{χ} = Z_{p},

– dim(χ) = 2, R_{χ} is the ring of integers of the unramified quadratic extension of Q_{p}, and χ splits into two one-dimensional pieces over O_{𝔭}.

If χ ∈ Ξ and B is a Z[Δ]-module, we let M^{(p)} denote the p-adic completion of M and

M^{χ} = M^{(p)} ⊗_{Z_{p}[Δ]} R_{χ}.

Then M^{(p)} = ⊕_{χ∈Ξ}M^{χ}, so we can view M^{χ} either as a quotient of M or a submodule of M^{(p)}. If m ∈ M we write m^{χ} for the projection of m into M^{χ}.

Lemma 9.1. For every nontrivial χ ∈ Ξ, (O_{F}^{×}/μ_{F})^{χ} is free of rank one over R_{χ}.

Proof. The Dirichlet unit theorem gives an exact sequence

0 → O_{F}^{×} ⊗ Q → Q[Δ] → Q → 0

and the lemma follows by taking χ-components. ⊓ ⊔

Let A denote the ideal class group of F, and fix a χ ∈ Ξ. We wish to bound the size of A^{χ}. Fix a power M of p, which we will later take to be large, and set F_{M} = F (μ_{M}).

Lemma 9.2. The composition

Hom(A, Z/MZ) → Hom(G_{F} , Z/MZ) → Hom(G_{F_{M}} , Z/MZ),

given by class field theory and restriction to G_{F_{M}}, is injective.

Proof. The first map is clearly injective, and the kernel of the second is equal to Hom(Gal(F_{M}/F), Z/MZ). Thus to prove the lemma it suffices to show that there is no unramified p-extension of F in F_{M}. But the p-part of Gal(F_{M}/F) is Gal(F_{M}/F (μ_{p})), which is totally ramified at all primes above p. This completes the proof. ⊓ ⊔

Lemma 9.3. The map

F^{×}/(F^{×})^{M} → F_{M}^{×}/(F_{M}^{×})^{M}

is injective.

Proof. Kummer theory shows that F^{×}/(F^{×})^{M} ≅ H^{1}(F, μ_{M}) and similarly for F_{M}, so the kernel of the map in the lemma is H^{1}(F_{M}/F, μ_{M}). Since Gal(F_{M}/F) is cyclic and acts faithfully on μ_{M}, and p > 2, it is easy to check that H^{1}(F_{M}/F, μ_{M}) = 0. (See also Lemma 6.1.) ⊓ ⊔

Write R_{F,M} for R_{1,M}, the set of primes of K defined in §8.

Proposition 9.4. Suppose κ ∈ F^{×}/(F^{×})^{M} and α ∈ Hom(A, Z/MZ), α ≠ 0. Then there is a prime 𝔮 ∈ R_{F,M} and a prime 𝔔 of F above 𝔮 such that

(i) α(𝔠) ≠ 0, where 𝔠 denotes the class of 𝔔 in A, (ii) [κ]_{𝔮} = 0 and for every d ∈ Z, dϕ_{𝔮}(κ) = 0 ⇔ κ^{d} ∈ (F^{×})^{M}.

Proof. Let t be the order of κ in F^{×}/(F^{×})^{M}, and let ρ ∈ Hom(G_{F_{M}} , μ_{M}) be the image of κ under the Kummer map. We view α as a map on G_{F_{M}} via the map of Lemma 9.2. Define two subgroups of G_{F_{M}}

H_{α} = {γ ∈ G_{F_{M}} : α(γ) = 0},
H_{κ} = {γ ∈ G_{F_{M}} : ρ(γ) has order less than t in μ_{M}}.

Since α ≠ 0, Lemma 9.2 shows that H_{α} ≠ G_{F_{M}}. Similarly it follows from Lemma 9.3 that H_{κ} ≠ G_{F_{M}}. Since a group cannot be a union of two proper subgroups, we can choose a γ ∈ G_{F_{M}}, γ ∉ H_{α} ∪ H_{κ}. Let L be a finite Galois extension of F containing F_{M} such that both ρ and α are trivial on G_{L}. By the Cebotarev theorem we can choose a prime 𝔔̃ of L, not dividing 6𝔞𝔣𝔭 and such that [κ]_{𝔮} = 0, whose Frobenius in L/K is γ. Let 𝔔 and 𝔮 denote the primes of F and K, respectively, below 𝔔̃. We will show that these primes satisfy the conditions of the proposition.

First, the fact that γ fixes F (μ_{M}) means that 𝔮 splits completely in F (μ_{M}) and thus 𝔮 ∈ R_{F,M}.

The class field theory inclusion Hom(A, Z/MZ) ↪ Hom(G_{F} , Z/MZ) identifies α(𝔠) with α(Frob_{𝔔}) = α(γ), so (i) follows from the fact that γ ∉ H_{α}.

Since γ ∉ H_{κ}, (κ^{1/M})^{Frob_{𝔔}−1} is a primitive t-th root of unity. Therefore κ has order t(N𝔮 − 1)/M modulo 𝔔, and hence has order at least t (and hence exactly t) in (O_{F} /𝔮O_{F})^{×}/((O_{F} /𝔮O_{F})^{×})^{M}. Since ϕ_{𝔮} is an isomorphism on (O_{F} /𝔮O_{F})^{×}/((O_{F} /𝔮O_{F})^{×})^{M}, this proves (ii). ⊓ ⊔

Suppose η is an Euler system as defined in Definition 8.5. Define C = C_{η} ⊂ O_{F}^{×} to be the group generated over Z[Δ] by μ_{F} and η(1, O).

Theorem 9.5. With notation as above, if η is an Euler system and χ is an irreducible Z_{p}-representation of Δ then

#(A^{χ}) ≤ #((O_{F}^{×}/C_{η})^{χ}).

Proof. If χ is the trivial character then A^{χ} is the p-part of the ideal class group of K, which is zero. Hence we may assume that χ ≠ 1.

By Lemma 9.1

(O_{F}^{×}/C)^{χ} ≅ R_{χ}/mR_{χ}

for some m ∈ R_{χ}. If m = 0 then there is nothing to prove, so we may assume m ≠ 0. Choose M large enough so that M/m annihilates A. For 𝔯 ∈ R_{F,M} we will write κ(𝔯) for the element κ_{1,M}(𝔯) ∈ F^{×}/(F^{×})^{M} constructed in Definition 8.7.

Number the elements of Hom(A^{χ}, Z/MZ) ⊂ Hom(A, Z/MZ) so that

Hom(A^{χ}, Z/MZ) = {α_{1}, …, α_{k}}.

Using Proposition 9.4 we choose inductively a sequence of primes 𝔮_{1}, …, 𝔮_{k} ∈ R_{F,M} and 𝔔_{i} of F above 𝔮_{i} such that, if 𝔠_{i} denotes the class of 𝔔_{i} in A and 𝔯_{i} = ∏_{j≤i} 𝔮_{j} for 0 ≤ i ≤ k,

α_{i}(𝔠_{i}) ≠ 0,  (11)
dϕ_{𝔮_{i}}(κ(𝔯_{i−1})^{χ}) = 0 ⇔ (κ(𝔯_{i−1})^{χ})^{d} = 0 ∈ F^{×}/(F^{×})^{M}  (12)

(just apply Proposition 9.4 with κ = κ(𝔯_{i−1})^{χ} and α = α_{i} to produce 𝔮_{i} and 𝔔_{i}).

First we claim that the classes {𝔠_{i}^{χ}} generate A^{χ}. For if not, then there is an α ∈ Hom(A^{χ}, Z/MZ) such that α(𝔠_{j}) = 0 for every j. But α = α_{i} for some i, so (11) shows this is not the case.

If 1 ≤ i ≤ k let s_{i} denote the order of 𝔠_{i}^{χ} in A^{χ}/⟨𝔠_{1}^{χ}, …, 𝔠_{i−1}^{χ}⟩. Since the 𝔠_{i}^{χ} generate A^{χ} we have

#(A^{χ}) = ∏_{i=1}^{k}[R_{χ} : s_{i}R_{χ}].

If 0 ≤ i ≤ k − 1 let t_{i} denote the order of κ(𝔯_{i})^{χ} in F^{×}/(F^{×})^{M}. By (12) and Proposition 8.10(ii), for i ≥ 1 the order of [κ(𝔯_{i})^{χ}]_{𝔮_{i}} is t_{i−1}. In particular it follows that t_{i−1} | t_{i}. Since κ(𝔯_{0}) is the image of η(1, O) in O_{F}^{×}/(O_{F}^{×})^{M}, the exact sequence

0 → R_{χ}κ(𝔯_{0})/μ_{F} ∩ R_{χ}κ(𝔯_{0}) → (O_{F}^{×}/μ_{F} (O_{F}^{×})^{M})^{χ} → (O_{F}^{×}/C)^{×} → 0

shows that M | t_{0}m.

For each i we can choose ν_{i} ∈ F^{×}/(F^{×})^{M} such that ν_{i}^{M/t_{i}} = κ(𝔯_{i})^{χ}ζ with ζ ∈ μ_{F}. In particular

(M/t_{i})[ν_{i}]_{𝔮_{i}} = [κ(𝔯_{i})^{χ}]_{𝔮_{i}}

so [ν_{i}]_{𝔮_{i}} has order t_{i−1}M/t_{i} in (I_{𝔮}/MI_{𝔮})^{χ} ≅ R_{χ}/MR_{χ}. Thus, using Proposition 8.10(i), there is a unit u ∈ R_{χ}^{×} such that

(ν_{i}) ≡ u(t_{i}/t_{i−1})𝔮_{i}^{χ} (mod I_{𝔮_{1}}, …, I_{𝔮_{i−1}}, t_{i}I).

We know that t_{0} | t_{i} and (M/m) | t_{0}. Thus by our choice of M, t_{i} annihilates A and we conclude that

(t_{i}/t_{i−1})𝔠_{i}^{χ} = 0 in A^{χ}/⟨𝔠_{1}^{χ}, …, 𝔠_{i−1}^{χ}⟩.

Therefore s_{i} | (t_{i}/t_{i−1}) for every i ≥ 1, so

#(A^{χ}) = ∏_{i=1}^{k}[R_{χ} : s_{i}R_{χ}] divides [t_{0}R_{χ} : t_{k}R_{χ}].

Since t_{k} | M and M | t_{0}m, this index divides [R_{χ} : mR_{χ}]. This proves the theorem. ⊓ ⊔ Corollary 9.6. Let C_{𝔞} denote the group of (elliptic) units of F generated over Z_{p}[Δ] by μ_{F} and by η_{1}^{(𝔞)}. If χ is an irreducible Z_{p}-representation of Δ then

#(A^{χ}) divides #((O_{F}^{×}/C_{𝔞})^{χ}).

Proof. Apply Theorem 9.5 with the Euler system η(n, 𝔯) = η_{n}^{(𝔞)}(𝔯). ⊓ ⊔

Remark 9.7. If C_{F} denotes the full group of elliptic units of F (see for example [Ru2] §1), then one can combine Theorem 9.5 with a well-known argument using the analytic class number formula to prove that for every χ,

#(A^{χ}) = #((O_{F}^{×}/C_{F})^{χ}).

See Theorem 3.3 of [Ru2].

Corollary 9.8. With notation as above, if (η_{1}^{(𝔞)})^{χ} ∉ μ_{F}^{χ} ((O_{F}^{×})^{χ})^{p} then A^{χ} = 0.

Proof. Immediate from Corollary 9.6 and Lemma 9.1. ⊓ ⊔

10 The Theorem of Coates and Wiles

Keep the notation of the previous sections. In this section we will prove the following theorem.

Theorem 10.1 (Coates-Wiles [CW1]). If L(ψ̄, 1) ≠ 0 then E(K) is finite.

Suppose for the rest of this section that 𝔭 is a prime of K not dividing 𝔣, of residue characteristic p > 7 (see remark 10.3 below). As in §9 we let F = K(E[𝔭]), Δ = Gal(F/K) and A is the ideal class group of F.

Lemma 10.2. There is an ideal 𝔞 of O, prime to 6𝔭𝔣, such that N𝔞 ≢ ψ(𝔞) (mod 𝔭).

Proof. By Corollary 5.18, E[𝔭̄]⊄E(K). Choose a prime 𝔮 of K, not dividing 6p𝔣, such that [𝔮, K(E[𝔭̄])/K] ≠ 1. By Corollary 5.16(ii) we deduce that ψ(𝔮) ≢ 1 (mod 𝔭̄), and so ψ̄(𝔮) ≢ 1 (mod 𝔭). Since ψ(𝔮)ψ̄(𝔮) = N𝔮, the lemma is satisfied with 𝔞 = 𝔮. ⊓ ⊔

Remark 10.3. Lemma 10.2 is not in general true without the assumption p > 7, since for small p it may happen that E[𝔭̄] ⊂ E(K).

By Corollary 5.20(iv), F/K is totally ramified at 𝔭. Let P denote the prime of F above 𝔭. By Lemma 3.6 and Corollary 5.16 E[𝔭] ⊂ E_{1}(F_{P}), so the isomorphism of Corollary 3.8 restricts to an isomorphism

E[𝔭] ⟶^{∼} Ê[𝔭] ⊂ Ê(F_{P})

where Ê is the formal group attached to E. Let O_{F,P} denote the completion of O_{F} at P.

Lemma 10.4. The map

E[𝔭] ⟶^{∼} Ê[𝔭] ⟶^{1+·} (1 + PO_{F,P})/(1 + P^{2}O_{F,P})

is a Δ-equivariant isomorphism.

Proof. The map in question is a well-defined homomorphism, and by Lemma 7.3 it is injective. Both groups have order N𝔭, so it is an isomorphism. The Δ-equivariance is clear. ⊓ ⊔

Now fix an ideal 𝔞 satisfying Lemma 10.2, a generator Ω of the period lattice of E as in §7.4, and a generator f of the conductor 𝔣. With these choices define the elliptic units η_{n}(𝔯) as in §8.1. Let η = η_{1}(O), a global (elliptic) unit of F which depends on the choice of 𝔞.

Definition 10.5. Define

δ : O_{F,P}^{×} → E[𝔭]

to be the composition of the natural projection

O_{F,P}^{×}↠(1 + PO_{F,P})/(1 + P^{2}O_{F,P})

with the inverse of the isomorphism of Lemma 10.4.

Recall that by Corollary 7.18, L(ψ̄, 1)/Ω ∈ K.

Proposition 10.6. L(ψ̄, 1)/Ω is integral at 𝔭, and

L(ψ̄, 1)/Ω ≡ 0 (mod 𝔭) ⇔ δ(η) = 0.

Proof. Let P = (℘(Ω/ψ(𝔭); ΩO), ℘′(Ω/ψ(𝔭); ΩO)/2) ∈ E[𝔭] and

z = −x(P)/y(P) ∈ P,

the image of P in Ê[𝔭]. Then η = Λ_{𝔭,𝔞}(z), where Λ_{𝔭,𝔞} is the power series of Definition 7.21.

By Theorem 7.22, Λ_{𝔭,𝔞}(0) ∈ O_{𝔭}^{×}, 12f(N𝔞 − ψ(𝔞))(L(ψ̄, 1)/Ω) ∈ O_{𝔭}, and

η ≡ Λ_{𝔭,𝔞}(0)(1 + 12f(N𝔞 − ψ(𝔞))(L(ψ̄, 1)/Ω)z) (mod P^{2}).

Thus

δ(η) = 12f(N𝔞 − ψ(𝔞))(L(ψ̄, 1)/Ω)P

and with our choice of 𝔞, 12f(N𝔞 − ψ(𝔞)) ∈ O_{𝔭}^{×}. This proves the proposition. ⊓ ⊔

Definition 10.7. Let χ_{E} denote the representation of Δ on E[𝔭]; by Corollary 5.20 χ_{E} is F_{p}-irreducible. Then in the notation of §9 we have E[𝔭] ≅ R_{χ_{E}}/𝔭R_{χ_{E}} as Δ-modules.

Theorem 10.8. Suppose L(ψ̄, 1)/Ω is a unit at 𝔭. Then

A^{χ_{E}} = 0.

Proof. Since the map δ is Δ-equivariant,

δ(η^{χ_{E}}) = δ(η)^{χ_{E}} = δ(η) ≠ 0

by Proposition 10.6. Hence

η^{χ_{E}} ∉ ((O_{F}^{×})^{χ_{E}})^{p}.

The Weil pairing (see [Si] Proposition III.8.1) gives a Galois-equivariant isomorphism

E[p] ≅ Hom(E[p], μ_{p}).

If μ_{F}^{χ_{E}} were nontrivial, then E[p]^{G_{K}} would be nontrivial, and this is impossible by Corollary 5.18. Now the theorem follows from Corollary 9.8. ⊓ ⊔

Lemma 10.9. Suppose p splits into two primes in K and Tr_{K/Q}ψ(𝔭) ≠ 1. Then

(i) μ_{p} ⊄ F_{P}, (ii) (O_{F,P}^{×})^{χ_{E}} is free of rank one over R_{χ_{E}}.

Proof. By Theorem 5.15(ii), [ψ(𝔭), F_{P}/K_{𝔭}] = 1. On the other hand, class field theory over Q shows that [p, Q_{p}(μ_{p})/Q_{p}] = 1. Thus we have (again using Theorem 5.15(ii))

μ_{p} ⊂ F_{P} ⇒ F_{P} = K_{𝔭}(μ_{p}) ⇒ [p/ψ(𝔭), F_{P}/K_{𝔭}] = 1
⇒ p/ψ(𝔭) ≡ 1 (mod 𝔭)
⇒ Tr_{K/Q}ψ(𝔭) ≡ 1 (mod 𝔭)
⇒ Tr_{K/Q}ψ(𝔭) = 1,

the last implication because |Tr_{K/Q}ψ(𝔭)| ≤ 2\sqrt{p} < p − 1. This proves (i). We have isomorphisms

O_{F,P}^{×} ⊗ Q_{p} ⟶^{∼} O_{F,P} ⊗ Q_{p} ≅ K_{𝔭}[Δ],

the first one given by the 𝔭-adic logarithm map. Together with (i) this proves (ii). ⊓ ⊔

Theorem 10.10. Suppose L(ψ̄, 1)/Ω is a unit at 𝔭, p splits into two primes in K, and Tr_{K/Q}ψ(𝔭) ≠ 1. Then the natural (injective) map

(O_{F}^{×})^{χ_{E}} → (O_{F,P}^{×})^{χ_{E}}

is surjective.

Proof. As in the proof of Theorem 10.8, δ(η^{χ_{E}}) = δ(η). Thus by Proposition 10.6 (O_{F}^{×})^{χ_{E}} ⊄ ((O_{F,P}^{×})^{χ_{E}})^{p}, so the Theorem follows from Lemma 10.9. ⊓ ⊔

Proof (of the Coates-Wiles Theorem 10.1). Using the Cebotarev theorem we can find infinitely many primes 𝔭 which split in K and such that Tr_{K/Q}ψ(𝔭) ≠ 1. Choose one which does not divide 6𝔣 or L(ψ̄, 1)/Ω. Then by Theorems 10.8 and 10.10 and Corollary 6.10, the Selmer group S_{ψ(𝔭)}(E) = 0. In particular E(K)/𝔭E(K) = 0, so (using the Mordell-Weil Theorem 4.6), E(K) is finite.

⊓ ⊔

Remark 10.11. This proof also shows that for primes 𝔭 satisfying the hypotheses of Theorem 10.10, the 𝔭-part of the Tate-Shafarevich group � (E) is trivial.

Using the Explicit Reciprocity Law of Wiles ([Wil] or [dS] §I.4) one can show that δ = −δ_{1} where δ_{1} is the map of Lemma 6.8. Together with Proposition 10.6, Theorem 10.8 and Corollary 6.10, this shows that S_{ψ(𝔭)}(E) = 0 for every 𝔭 not dividing 2 · 3 · 5 · 7 · 𝔣 · (L(ψ̄, 1)/Ω). We will prove a stronger version of this (Corollary 12.13 and Theorem 12.19) in §12.

11 Iwasawa Theory and the “Main Conjecture”

In order to study the Selmer group under more general conditions than in §10, we need to prove Iwasawa-theoretic versions (Theorem 11.7 and Corollary 11.8 below) of Theorem 9.5 and Remark 9.7. As in the previous sections, we fix an elliptic curve E defined over an imaginary quadratic field K, with End_{K}(E) = O, the ring of integers of K. We fix a prime 𝔭 of K where E has good reduction, and for simplicity we still assume that p > 7 (in order to apply Lemma 10.2).

Write K_{n} = K(E[𝔭^{n}]), n = 0, 1, 2, …, ∞, and let G_{∞} = Gal(K_{∞}/K). By Corollary 5.20(ii), we have

G_{∞} ≅ O_{𝔭}^{×} ≅ Δ × Γ

where

Δ ≅ Gal(K_{1}/K) ≅ (O/𝔭)^{×}

is the prime-to-p part of G_{∞} and

Γ = Gal(K_{∞}/K_{1}) ≅ 1 + 𝔭O_{𝔭} ≅ Z_{p}^{[K_{𝔭}:Q_{p}]}

is the p-part.

11.1 The Iwasawa Algebra

Define the Iwasawa algebra

Λ = Z_{p}[[G_{∞}]] = lim_{\overleftarrow{n}} Z_{p}[Gal(K_{n}/K)] = lim_{\overleftarrow{n}} Z_{p}[Δ][Gal(K_{n}/K_{1})].

Then

Λ = ⨁_{χ∈Ξ} Λ_{χ}

where Ξ is the set of irreducible Z_{p}-representations of Δ as in §9 and

Λ_{χ} = Λ ⊗_{Z_{p}[Δ]} R_{χ} ≅ R_{χ}[[Γ]].

The following algebraic properties of the Iwasawa algebra and its modules are well-known. For proofs, see for example [Iw] and [Se].

For every irreducible Z_{p}-representation χ of Δ, Λ_{χ} is a complete local noetherian ring, noncanonically isomorphic to a power series ring in [K_{𝔭} : Q_{p}] variables over R_{χ}. In particular Λ is not an integral domain, but rather is a direct sum of local integral domains. Let M denote the (finite) intersection of all maximal ideals of Λ, i.e., M is the kernel of the natural map Λ↠F_{p}[Δ].

A Λ-module M will be called a torsion Λ-module if it is annihilated by a non-zero-divisor in Λ. A Λ-module will be called pseudo-null if it is annihilated by an ideal of height at least two in Λ. If Γ ≅ Z_{p} then a module is pseudo-null if and only if it is finite.

If M is a finitely generated torsion Λ-module, then there is an injective Λ-module homomorphism

⨁_{i=1}^{r} Λ/f_{i}Λ ↪ M

with pseudo-null cokernel, where the elements f_{i} ∈ Λ can be chosen to satisfy f_{i+1} | f_{i} for 1 ≤ i ≤ r. The elements f_{i} are not uniquely determined, but the ideal ∏_{i} f_{i}Λ is. We call the ideal ∏_{i} f_{i}Λ the characteristic ideal char(M) of the torsion Λ-module M. The characteristic ideal is multiplicative in exact sequences: if 0 → M′ → M → M′′ → 0 is an exact sequence of torsion Λ-modules then

char(M) = char(M′)char(M′′).

11.2 The Iwasawa Modules

Define

A_{n} = the p-part of the ideal class group of K_{n}, U_{n} = the p-adic completion of the local units of K_{n} ⊗ K_{𝔭} (equivalently, the

1-units of K_{n} ⊗ K_{𝔭}), E_{n} = the global units of K_{n}, Ē_{n} = the p-adic completion of E_{n} (equivalently, since Leopoldt’s conjecture

holds for K_{n}, the closure of the image of E_{n} in U_{n}), C_{n} = the elliptic units of K_{n}, the subgroup of E_{n} generated over the group

ring Z[Gal(K_{n}/K)] by the η_{n}^{(𝔞)} = η_{n}^{(𝔞)}(O) (see Definition 8.1) for all choices of ideal 𝔞 prime to 6𝔭𝔣, and the roots of unity in K_{n}, C̄_{n} = the p-adic completion of C_{n} (equivalently, the closure of the image of

C_{n} in U_{n}),

and

A_{∞} = lim_{\overleftarrow{n}} A_{n}, U_{∞} = lim_{\overleftarrow{n}} U_{n}, E_{∞} = lim_{\overleftarrow{n}} Ē_{n}, C_{∞} = lim_{\overleftarrow{n}} C̄_{n},

inverse limits with respect to norm maps. Also define X_{∞} = Gal(M_{∞}/K_{∞}) where M_{∞} is the maximal abelian p-extension of K_{∞} unramified outside of the prime above 𝔭.

Class field theory identifies A_{∞} with Gal(L_{∞}/K_{∞}), where L_{∞} is the maximal everywhere-unramified abelian p-extension of K_{∞}, and identifies the inertia group in X_{∞} of the unique prime above 𝔭 with U_{∞}/E_{∞}. Thus there is an exact sequence of Λ-modules

0 → E_{∞}/C_{∞} → U_{∞}/C_{∞} → X_{∞} → A_{∞} → 0.  (13)

For every n ≥ 0, let Λ_{n} = Z_{p}[Gal(K_{n}/K)] and let J_{n} ⊂ Λ denote the kernel of the restriction map Λ → Λ_{n}. In particular J_{0} is the augmentation ideal of Λ.

Lemma 11.1. For every n ≥ 1, the natural map

A_{∞}/J_{n}A_{∞} → A_{n}

is an isomorphism.

Proof. When Γ = Z_{p} this is a standard argument going back to Iwasawa [Iw], using the fact that only one prime of K ramifies in K_{∞} and it is totally ramified.

For the general case, consider the diagram of fields below, where L_{n} is the maximal unramified abelian p-extension of K_{n}, and L′_{n} is the fixed field of J_{n}A_{∞} in L_{∞}. Since K_{∞}/K_{n} is totally ramified above 𝔭, K_{∞} ∩ L_{n} = K_{n}, and so Gal(K_{∞}L_{n}/K_{∞}) = A_{n} and the map A_{∞}/J_{n}A_{∞} → A_{n} is just the restriction map. We will show that K_{∞}L_{n} = L′_{n}, and the lemma will follow.

Since Gal(K_{∞}/K_{n}) acts on Gal(L_{∞}/K_{∞}) by conjugation, J_{n}A_{∞} is generated by commutators

Gal(L_{∞}/L′_{n}) = [Gal(L_{∞}/K_{n}), Gal(L_{∞}/K_{∞})].

Such commutators are trivial on L_{n}, so K_{∞}L_{n} ⊂ L′_{n}.

On the other hand, only the unique prime above 𝔭 ramifies in the abelian extension L′_{n}/K_{n}, and it is totally ramified in K_{∞}/K_{n}. If we write I for the inertia group of this prime in Gal(L′_{n}/K_{n}), the inverse of the projection isomorphism I ⟶^{∼} Gal(K_{∞}/K_{n}) gives a splitting of the exact sequence

0 → A_{∞}/J_{n}A_{∞} → Gal(L′_{n}/K_{n}) → Gal(K_{∞}/K_{n}) → 0.

It follows that L′_{n}^{I} is an abelian, everywhere-unramified p-extension of K_{n}, and hence L′_{n}^{I} ⊂ L_{n} and so L′_{n} = K_{∞}L_{n}. ⊓ ⊔

[figure: K; K_{n}; K_{∞}; L_{n}; K_{∞}L_{n}; L′_{n}; L_{∞}; A_{n}; A_{n}; J_{n}A_{∞}]

Proposition 11.2. A_{∞} is a finitely-generated torsion Λ-module. Proof. By Lemma 11.1, A_{∞}/J_{n}A_{∞} is finite for every n, and the proposition follows. ⊓ ⊔ Proposition 11.3. (i) X_{∞} is a finitely-generated Λ-module and for every

χ

rank_{Λ_{χ}}X_{∞}^{χ} = [K_{𝔭} : Q_{p}] − 1.

(In particular if K_{𝔭} = Q_{p} then X_{∞} is a finitely-generated torsion Λmodule.)

(ii) X_{∞} has no nonzero pseudo-null submodules. Proof. See [Gr]. ⊓ ⊔ Proposition 11.4. U_{∞} is a finitely-generated, torsion-free Λ-module, and for every χ

rank_{Λ_{χ}}(U_{∞}^{χ}) = [K_{𝔭} : Q_{p}].

Further, if [K_{𝔭} : Q_{p}] = 2 then U_{∞}^{χ_{E}} is free of rank 2 over Λ_{χ}. Proof. See [Iw] §12 or [Win]. ⊓ ⊔ Proposition 11.5. E_{∞} is a finitely-generated Λ-module, and for every χ

rank_{Λ_{χ}}(E_{∞}^{χ}) = 1.

Proof. The natural map E_{∞} → U_{∞} is injective, so the proposition follows from (13) and Propositions 11.2, 11.3 and 11.4 ⊓ ⊔ Proposition 11.6. C_{∞}^{χ_{E}} is free of rank one over Λ_{χ_{E}}.

Proof. Choose an ideal 𝔞 of O such that ψ(𝔞) ≢ N𝔞 (mod 𝔭) (Lemma 10.2). We will show that C_{∞}^{χ_{E}} is generated over Λ_{χ_{E}} by {(η_{n}^{(𝔞)})^{χ_{E}}}_{n} with this choice of 𝔞. Suppose 𝔟 is some other ideal of O prime to 6𝔭𝔣. It follows from Theorem 7.4(ii) and Lemma 7.10 that for every n

(η_{n}^{(𝔞)})^{σ_{𝔟}−N𝔟} = (η_{n}^{(𝔟)})^{σ_{𝔞}−N𝔞}

where σ_{𝔞} = [𝔞, K_{n}/K], σ_{𝔟} = [𝔟, K_{n}/K]. Since ψ(𝔞) ≢ N𝔞 (mod 𝔭), and σ_{𝔞} acts as ψ(𝔞) on E[𝔭] (Corollary 5.16(ii)), we see that σ_{𝔞} − N𝔞 acts bijectively on E[𝔭]. But E[𝔭] ≅ Λ_{χ_{E}}/M_{χ_{E}} where M_{χ_{E}} denotes the maximal ideal of the local ring Λ_{χ_{E}}. Therefore σ_{𝔞} − N𝔞 is invertible in Λ_{χ_{E}}, so

{(η_{n}^{(𝔟)})^{χ_{E}}}_{n} ∈ Λ_{χ_{E}}{(η_{n}^{(𝔞)})^{χ_{E}}}_{n}

as claimed. Since U_{∞} is torsion-free (Proposition 11.4), C_{∞}^{χ_{E}} must free of rank 1. ⊓ ⊔

11.3 Application of the Euler System of Elliptic Units

Theorem 11.7. char(A_{∞}) divides char(E_{∞}/C_{∞}).

The rest of this section will be devoted to a proof of this theorem. The techniques are similar to those of the proof of Theorem 9.5, but messier and more technically complicated because one needs to study modules over Z_{p}[Gal(K_{n}/K)] rather than Z_{p}[Δ]. See [Ru2] for the details which are not included below, and see [Ru1] for the analogous result for cyclotomic fields.

We also record, but will not prove, the following corollary. With a better definition of elliptic units, it would hold for more representations χ of Δ. See [Ru2] Theorem 4.1 for a precise statement and [Ru2] §10 (see also [dS] §III.2) for the proof, which is an application of the analytic class number formula.

Corollary 11.8. char(A_{∞}^{χ_{E}}) = char(E_{∞}^{χ_{E}}/C_{∞}^{χ_{E}}).

Definition 11.9. Since A_{∞} is a torsion Λ-module, we can fix once and for all an injective Λ-module map with pseudo-null cokernel

⨁_{i=1}^{r} Λ/f_{i}Λ ↪ A_{∞}

with f_{i} ∈ Λ, f_{i+1} | f_{i} for 1 ≤ i ≤ r. Let A_{∞}^{0} denote the image of this map, so

A_{∞}^{0} = ⊕_{i=1}^{r}Λy_{i} ⊂ A_{∞}

where y_{i} ∈ A_{∞} is the image of 1 ∈ Λ/f_{i}Λ. Then A_{∞}^{0} is an “elementary” submodule of A_{∞} and A_{∞}/A_{∞}^{0} is pseudo-null.

Let Ω = K_{∞}(μ_{p^{∞}}). If σ ∈ G_{Ω} we write [σ] ∈ A_{∞} for the restriction of σ to L_{∞}. Note that if K_{𝔭} ≠ Q_{p} then the Weil pairing (see [Si] Proposition III.8.1) shows that Ω = K_{∞}, and if K_{𝔭} = Q_{p} then Ω/K_{∞} is totally ramified at the prime p/𝔭. Thus in either case the map G_{Ω} → A_{∞} is surjective.

If 0 ≤ k ≤ r, a Frobenius sequence σ of length k is a k-tuple (σ_{1}, …, σ_{k}) of elements of G_{Ω} satisfying

[σ_{i}] − y_{i} ∈ MA_{∞}^{0}

for 1 ≤ i ≤ k, where M is as defined in §11.1, the intersection of all maximal ideals of Λ.

Suppose n ≥ 1 and M is a power of p. Recall the subset R_{n,M} of R defined in §8.2. For 0 ≤ k ≤ r we call a k-tuple (π̃_{1}, …, π̃_{k}) of primes of K_{n} a Kolyvagin sequence (for n and M) if

– the π̃_{i} lie above distinct primes of K belonging to R_{n,M}, and

– there is a Frobenius sequence σ = (σ_{1}, …, σ_{k}) such that for 1 ≤ i ≤ k,

Frobπ̃_{i} = σ_{i} on L_{n}

where L_{n} is the maximal unramified abelian p-extension of K_{n}.

If π is a Kolyvagin sequence of length k we will write π_{i} for the prime of K below π̃_{i} and we define

𝔯(π) = ∏_{i=1}^{k} π_{i} ∈ R_{n,M}.

Let Π(k, n, M) be the set of all Kolyvagin sequences of length k for n and M.

Fix an ideal 𝔞 so that {(η_{n}^{(𝔞)})^{χ_{E}}}_{n} generates C_{∞}^{χ_{E}}, as in the proof of Proposition 11.6. Using the Euler system of elliptic units η_{n}^{(𝔞)}(𝔯), for 𝔯 ∈ R_{n,M} we obtain the Kolyvagin derivative classes

κ_{n,M}(𝔯) ∈ K_{n}^{×}/(K_{n}^{×})^{M}

as in Definition 8.7. For every n recall that Λ_{n} = Z_{p}[Gal(K_{n}/K)] and let Λ_{n,M} = (Z/MZ)[Gal(K_{n}/K)] = Λ_{n}/MΛ_{n}. If 0 ≤ k ≤ r define Ψ(k, n, M) to be the ideal of Λ_{n,M} generated by

{ψ(κ_{n,M}(𝔯(π))) : π ∈ Π(k, n, M), ψ ∈ Hom_{Λ_{n}}(Λ_{n,M}κ_{n,M}(𝔯(π)), Λ_{n,M})}

When k = 0, Π(k, n, M) has a single element (the empty sequence) and

Ψ(0, n, M) ⊃ {ψ(η_{n}^{(𝔞)}) (mod M) : ψ ∈ Hom_{Λ_{n}}(E_{n}, Λ_{n})}  (14)

It follows from Lemma 11.1 that A_{∞}/J_{n}A_{∞} is finite for every n. From this it is not difficult to show that Λ_{n}/char(A_{∞})Λ_{n} is also finite for every n. For every n define N_{n} to be the product of #(A_{n}) and the smallest power of p which annihilates Λ_{n}/char(A_{∞})Λ_{n}.

The following proposition is the key to the proof of Theorem 11.7.

Proposition 11.10. There is an ideal B of height at least two in Λ such that for every n ≥ 1, power M of p, and 0 ≤ k < r,

BΨ(k, n, MN_{n})Λ_{n,M} ⊂ f_{k+1}Ψ(k + 1, n, M).

We will first show how to complete the proof of Theorem 11.7 assuming Proposition 11.10, and then we will prove Proposition 11.10.

Lemma 11.11. Suppose G is a finite abelian group and B is finitely generated Z_{p}[G]-module with no p-torsion. If f ∈ Z_{p}[G] is not a zero-divisor, b ∈ B, and

{ψ(b) : ψ ∈ Hom_{Z_{p}[G]}(B, Z_{p}[G])} ⊂ fZ_{p}[G],

then b ∈ fB.

Proof. Let B′ = Z_{p}[G]b + fB. Since f is not a zero-divisor, we have a commutative diagram

Hom_{Z_{p}[G]}(B′, fZ_{p}[G]) ⟵^{f} Hom_{Z_{p}[G]}(B′, Z_{p}[G]) ⟶^{∼} Hom_{Z_{p}}(B′, Z_{p})
↓  ↓  ↓
Hom_{Z_{p}[G]}(fB, fZ_{p}[G]) ⟵^{f} Hom_{Z_{p}[G]}(fB, Z_{p}[G]) ⟶^{∼} Hom_{Z_{p}}(fB, Z_{p})

in which the horizontal maps are all isomorphisms.

Choose φ̄ ∈ Hom_{Z_{p}[G]}(fB, fZ_{p}[G]). Since B has no p-torsion and f is not a zero-divisor, φ̄ extends uniquely to a map φ : B → Z_{p}[G], and by our assumption, φ ∈ Hom_{Z_{p}[G]}(B′, fZ_{p}[G]). Thus all the vertical maps in the diagram above are isomorphisms. Since B′ and fB are free Z_{p}-modules, the surjectivity of the right-hand vertical map shows that B′ = fB, which proves the lemma. ⊓ ⊔

Proof (of Theorem 11.7, assuming Proposition 11.10). Fix n ≥ 1 and ψ ∈ Hom_{Λ_{n}}(Ē_{n}, Λ_{n}), and let B ⊂ Λ be an ideal of height at least two satisfying Proposition 11.10. We will show that, for every choice of 𝔞,

B^{r}ψ(η_{n}^{(𝔞)}) ⊂ char(A_{∞})Λ_{n}.  (15)

Assuming this, Lemma 11.11 applied with B = Ē_{n}/(Ē_{n})_{tors} shows that

B^{r}η_{n}^{(𝔞)} ⊂ char(A_{∞})Ē_{n} + (Ē_{n})_{tors}.

Since E_{∞} has no Λ-torsion (Lemma 11.5), it follows that

B^{r}{η_{n}^{(𝔞)}}_{n} ⊂ char(A_{∞})E_{∞}.

Thus B^{r}C_{∞} ⊂ char(A_{∞})E_{∞}, and since B^{r} is an ideal of height at least two the theorem follows.

It remains to prove (15). Suppose 0 ≤ k < r and M is a power of p. Proposition 11.10 shows that

BΨ(k, n, MN_{n}^{r−k})Λ_{n,M} ⊂ f_{k+1}Ψ(k + 1, n, MN_{n}^{r−k−1})Λ_{n,M},

so by induction we conclude that

B^{r}Ψ(0, n, MN_{n}^{r})Λ_{n,M} ⊂ (∏_{i=1}^{r} f_{i}) Ψ(r, n, M) ⊂ char(A_{∞})Λ_{n,M}.  (16)

Using (14) it follows that

B^{r}ψ(η_{n}^{(𝔞)})Λ_{n,M} ⊂ char(A_{∞})Λ_{n,M},

and since this holds for every M, it proves (15). This completes the proof of Theorem 11.7. ⊓ ⊔

The rest of this section is devoted to proving Proposition 11.10. If σ = (σ_{1}, …, σ_{k}) is a Frobenius sequence define

A_{σ} = ∑_{i=1}^{k} Λ[σ_{i}] ⊂ A_{∞}^{0}.

Lemma 11.12. If σ is a Frobenius sequence of length k then A_{σ} is a direct summand of A_{∞}^{0} and A_{σ} = ⊕_{i=1}^{k}Λ/f_{i}Λ.

Proof. Recall that A_{∞}^{0} = ⊕_{i=1}^{r}Λy_{i}. Define Y_{k} = ∑_{i=k+1}^{r} Λy_{i}. The image of A_{σ} + Y_{k} in A_{∞}^{0}/MA_{∞}^{0} contains all the y_{i}, so by Nakayama’s Lemma, A_{σ} + Y_{k} = A_{∞}^{0}. We will show that A_{σ} ∩ Y_{k} = 0, and thus A_{∞}^{0} = A_{σ} ⊕ Y_{k} and

A_{σ} ≅ A_{∞}^{0}/Y_{k} ≅ ⊕_{i=1}^{k}Λ/f_{i}Λ.

For 1 ≤ i ≤ k write

[σ_{i}] = y_{i} + v_{i} + w_{i}

where v_{i} ∈ M(⊕_{i≤k}Λy_{i}) and w_{i} ∈ MY_{k}. Suppose

∑_{i=1}^{k} a_{i}[σ_{i}] ∈ Y_{k}

with a_{i} ∈ Λ. Then we must have ∑_{i=1}^{k} a_{i}(y_{i} + v_{i}) = 0. We can write this in matrix form, using the basis y_{1}, …, y_{k} of ⊕_{i≤k}Λy_{i}, as

(a_{1}, …, a_{k})B ∈ (f_{1}Λ, …, f_{k}Λ)

where B is a k × k matrix with entries in Λ, congruent to the identity matrix modulo M. Therefore B is invertible, and, since f_{k} | f_{i} for every i ≤ k, we conclude that each a_{i} is divisible by f_{k}. But f_{k} annihilates Y_{k}, so we deduce that

∑_{i=1}^{k} a_{i}[σ_{i}] = ∑_{i=1}^{k} a_{i}w_{i} = 0.

This completes the proof of the lemma. ⊓ ⊔

Recall Ω = K_{∞}(μ_{p^{∞}}).

Proposition 11.13. Suppose W is a finite subgroup of K_{n}^{×}/(K_{n}^{×})^{M} for some n and M. Then G_{K} acts trivially on the cokernel of the natural Kummer map

G_{Ω} → Hom(W, μ_{M}).

Proof. Let W̄ denote the image of W in Ω^{×}/(Ω^{×})^{M}. The map in question factors

G_{Ω} → Hom(W̄ , μ_{M}) → Hom(W, μ_{M}).

where the first (Kummer) map is surjective and the cokernel of the second is Hom(V, μ_{M}) with

V = ker(W → W̄) ⊂ ker(H^{1}(K_{n}, μ_{M}) → H^{1}(Ω, μ_{M})) = H^{1}(Ω/K_{n}, μ_{M}).

Since Ω is abelian over K, G_{K} acts on V via the cyclotomic character, and hence G_{K} acts trivially on Hom(V, μ_{M}). The proposition follows. ⊓ ⊔

Let A denote the annihilator in Λ of A_{∞}/A_{∞}^{0}, so A is an ideal of height at least two.

Lemma 11.14. Suppose n ≥ 0, M is a power of p, k < r, and π = {π̃_{1}, …, π̃_{k+1}} ∈ Π(k + 1, n, MN_{n}). Let 𝔔 = π̃_{k+1}, 𝔮 = π_{k+1} and 𝔯 = 𝔮^{−1}𝔯(π). If ρ ∈ A then there is a Galois-equivariant homomorphism

ψ : Λ_{n,M}κ_{n,M}(𝔯𝔮) → Λ_{n,M}

such that

ρϕ_{𝔮}(κ_{n,M′}(𝔯)) ≡ f_{k+1}ψ(κ_{n,M}(𝔯𝔮))𝔔 (mod M)

where ϕ_{𝔮} : Λ_{n,MN_{n}}κ_{n,MN_{n}}(𝔯) → Λ_{n,MN_{n}}𝔔 is the map of Definition 8.9.

Proof. Write M′ = MN_{n}, and let σ be a Frobenius sequence corresponding to π. Let Ā_{n} denote the quotient of A_{n} by the Λ_{n}-submodule generated by the classes of π̃_{1}, …, π̃_{k}, and let [𝔔] denote the class of 𝔔 in Ā_{n}. Since the Frobenius of 𝔔 on the Hilbert class field of K_{n} is σ, [𝔔] is the projection of [σ] to Ā_{n}. By Lemma 11.12 the annihilator of [σ] in A_{∞}^{0}/A_{σ} is f_{k+1}Λ and A_{σ} is a direct summand of A_{∞}^{0}, so the annihilator of [σ] in (A_{∞}^{0}/A_{σ}) ⊗ Λ_{n} is f_{k+1}Λ_{n}. By Lemma 11.1, Ā_{n} = A_{∞}/(A_{σ} + J_{n}A_{∞}) so the kernel of the natural map (A_{∞}^{0}/A_{σ}) ⊗ Λ_{n} → Ā_{n} is annihilated by A. Therefore if A′ ⊂ Λ_{n} is the annihilator of [𝔔] in Ā_{n}, then

AA′ ⊂ f_{k+1}Λ_{n}.

Since #(A_{n}) divides N_{n}, Proposition 8.10(i) shows that [κ_{n,M′}(𝔯𝔮)]_{𝔮} is 0 in Ā_{n}. Therefore [κ_{n,M′}(𝔯𝔮)]_{𝔮} ∈ A′Λ_{n,M′}𝔔, so if ρ ∈ A then

ρ[κ_{n,M′}(𝔯𝔮)]_{𝔮} ∈ f_{k+1}Λ_{n,M′}𝔔.

Since f_{k+1} divides f_{1} and f_{1} divides N_{n} in Λ_{n}, the map

f_{k+1}^{−1} : Λ_{n,M′} → Λ_{n,M}

is well-defined, and we will define ψ : Λ_{n,M}κ_{n,M}(𝔯𝔮) → Λ_{n,M}𝔔 by

ψ(κ_{n,M}(𝔯𝔮))𝔔 = f_{k+1}^{−1}ρ[κ_{n,M′}(𝔯𝔮)]_{𝔮}.

If we can show that ψ is well-defined, then by Proposition 8.10(ii) we will have

ρϕ_{𝔮}(κ_{n,M′}(𝔯)) = ρ[κ_{n,M′}(𝔯𝔮)]_{𝔮} ≡ f_{k+1}ψ(κ_{n,M}(𝔯𝔮))𝔔 (mod M)

as desired.

We need to show that ψ is well-defined, i.e., if η ∈ Λ_{n} and κ_{n,M}(𝔯𝔮)^{η} ∈ (K_{n}^{×})^{M} then ηρ[κ_{n,M′}(𝔯𝔮)]_{𝔮} ∈ f_{k+1}MΛ_{n,M′}𝔔. But this is essentially the same argument as above. If η annihilates κ_{n,M}(𝔯𝔮) then κ_{n,M′}(𝔯𝔮)^{η} = α^{M} for some α ∈ K_{n}^{×}. Again using Proposition 8.10(i), [α]_{𝔮} is 0 in Ā_{n}, so ρ[α]_{𝔮} ∈ f_{k+1}Λ_{n,N_{n}}𝔔 and the desired inclusion follows. ⊓ ⊔

Proof (of Proposition 11.10). Let Λ_{Ω} = Z_{p}[[Gal(Ω/K)]] and denote by ϵ both the cyclotomic character Gal(Ω/K) → Z_{p}^{×} and the induced map Λ_{Ω} → Z_{p}. Define

tw_{ϵ} : Λ_{Ω} → Λ_{Ω}

to be the homomorphism induced by γ ↦ ϵ(γ)γ^{−1} for γ ∈ Gal(Ω/K). Recall that A is the annihilator of A_{∞}/A_{∞}^{0}, and define

B = \begin{cases}A & if K_{𝔭} = Q_{p} \\ A tw_{ϵ}(MAJ_{0}) & if K_{𝔭} ≠ Q_{p} \end{cases}

(recall that Ω = K_{∞} if K_{𝔭} ≠ Q_{p}). Then B is an ideal of height at least two, and we will show that Proposition 11.10 holds with this choice of B.

Fix n and M, and write M′ = MN_{n}. Fix a Kolyvagin sequence π ∈ Π(k, n, M′), let 𝔯 = 𝔯(π), and suppose ψ : Λ_{n,M′}κ_{n,M′}(𝔯) → Λ_{n,M′}. We need to show that

Bψ(κ_{n,M′}(𝔯))Λ_{n,M} ⊂ f_{k+1}Ψ(k + 1, n, M).

We will do this by constructing suitable Kolyvagin sequences of length k + 1 extending π.

There is a Z_{p}-module isomorphism

ι : Hom_{Λ_{n}}(Λ_{n,M′}κ_{n,M′}(𝔯), Λ_{n,M′}) ⊗ μ_{M′} ⟶^{∼} Hom_{Z_{p}}(Λ_{n,M′}κ_{n,M′}(𝔯), μ_{M′})

induced by

(∑_{γ} a_{γ}γ) ⊗ ζ ↦ ζ^{a_{1}}.

One can check that if ϕ ∈ Hom_{Λ_{n}}(Λ_{n,M′}κ_{n,M′}(𝔯), Λ_{n,M′}), ζ ∈ μ_{M′}, and ρ ∈ Λ_{Ω} then

ρι(ϕ ⊗ ζ) = ι((tw_{ϵ}(ρ)ϕ) ⊗ ζ).

Suppose τ ∈ G_{K}, fix a primitive M′-th root of unity ζ_{M′}, and let Kum_{M′} denote the Kummer map G_{Ω} → Hom(K_{n}^{×}, μ_{M′}). By Proposition 11.13 there is a γ_{0} ∈ G_{Ω} such that

Kum_{M′}(γ_{0}) = (τ − 1)ι(ψ ⊗ ζ_{M′}) on Λ_{n,M′}κ_{n,M′}(𝔯).

Choose ρ ∈ Λ_{Ω} such that the projection of ρ to Λ lies in MA and let γ ∈ G_{Ω} be such that γ = γ_{0}^{ρ} on Ω^{ab} (we view Gal(Ω/K) as acting on Gal(Ω^{ab}/Ω) in the usual way).

Let σ be a Frobenius sequence corresponding to π. We define two Frobenius sequences σ′ and σ′′ of length k + 1 extending σ as follows. Let σ′_{k+1} be an element of G_{Ω} such that [σ′_{k+1}] = y_{k+1}, and let σ′′_{k+1} = σ′_{k+1}γ with γ as above.

Since [γ] = ρ[γ_{0}] ∈ MA_{∞}^{0}, both σ′ and σ′′ are Frobenius sequences.

Let 𝔮′ and 𝔮′′ be primes of K whose Frobenius elements (for some choice of primes “upstairs”) in H_{n}(μ_{M′}, (Λ_{n}κ_{n,M′}(𝔯))^{1/M′})/K are the restrictions of σ′ and σ′′, respectively, where H_{n} is the Hilbert class field of K_{n}. Let 𝔔′ and 𝔔′′ be primes of K_{n} above 𝔮′ and 𝔮′′ with these Frobenius elements. It follows from Definition 8.9 that there are integers a′ and a′′ such that

ι(ϕ̄_{𝔮′} ⊗ ζ_{M′}^{a′}) = Kum_{M′}(σ′), ι(ϕ̄_{𝔮′′} ⊗ ζ_{M′}^{a′′}) = Kum_{M′}(σ′′)

where ϕ_{𝔮′} : Λ_{n,M′}κ_{n,M′}(𝔯) → Λ_{n,M′}𝔔′ is the map of Definition 8.9, ϕ̄_{𝔮′} ∈ Hom(Λ_{n,M′}κ_{n,M′}(𝔯), Λ_{n,M′}) is defined by ϕ_{𝔮′} = ϕ̄_{𝔮′}𝔔′, and similarly for 𝔮′′. Now

ι(tw_{ϵ}(ρ(τ − 1))ψ) ⊗ ζ_{M′} = ρKum_{M′}(γ_{0})
= Kum_{M′}(σ′′) − Kum_{M′}(σ′)
= ι(a′′ϕ̄_{𝔮′′} ⊗ ζ_{M′} − a′ϕ̄_{𝔮′} ⊗ ζ_{M′})

and so finally

tw_{ϵ}(ρ(τ − 1))ψ = a′′ϕ̄_{𝔮′′} − a′ϕ̄_{𝔮′}.  (17)

If K_{𝔭} ≠ Q_{p}, then the tw_{ϵ}(ρ)(ϵ(τ)τ − 1), with our choices of τ and ρ, generate tw_{ϵ}(MAJ_{0}). If K_{𝔭} = Q_{p}, then μ_{p} ⊄ K_{∞} (since K_{∞}/K is unramified at p/𝔭) and so we can choose τ and ρ so that tw_{ϵ}(ρ(τ − 1)) projects to a unit in Λ. Now Lemma 11.14 completes the proof of Proposition 11.10. ⊓ ⊔

12 Computing the Selmer Group

In this section we compute the order of the p-power Selmer group for primes p > 7 of dood reduction, and thereby prove assertion (ii) of the theorem of the introduction. The computation divides naturally into two cases depending on whether p splits in K or not.

Keep the notation of the previous section. In particular E is an elliptic curve defined over an imaginary quadratic field K, with complex multiplication by the full ring of integers of K, and 𝔭 is a prime of K of residue characteristic greater than 7 where E has good reduction.

Definition 12.1. Let π = ψ(𝔭) and recall that the π-adic Tate module of E is defined by

T_{π}(E) = lim_{\overleftarrow{n}} E[𝔭^{n}],

inverse limit with respect to multiplication by π. For every n let δ_{n} : U_{n} → E[𝔭^{n}] be the map of Lemma 6.8. It is clear from the definition that we have commutative diagrams

U_{n+1} ⟶^{δ_{n+1}} E[𝔭^{n+1}]
N_{K_{n+1}/Kn} ↓  ↓^{π}
U_{n} ⟶^{δ_{n}} E[𝔭^{n}],

and we define

δ_{∞} = lim_{\overleftarrow{n}} δ_{n} : U_{∞} → T_{π}(E).

Recall the Selmer group S_{π^{n}}(E) of Definition 4.1 and the extended Selmer group S′_{π^{n}}(E) of Definition 6.3. Define

S_{𝔭^{∞}} = lim_{\overrightarrow{n}} S_{π^{n}}(E), S′_{𝔭^{∞}} = lim_{\overrightarrow{n}} S′_{π^{n}}(E).

Thus there is an exact sequence

0 → E(K) ⊗ Q_{p}/Z_{p} → S_{𝔭^{∞}} → � (E)_{𝔭^{∞}} → 0.  (18)

Proposition 12.2. (i) S′_{𝔭^{∞}} = Hom(X_{∞}, E[𝔭^{∞}])^{G_{∞}} = Hom(X_{∞}^{χ_{E}}, E[𝔭^{∞}])^{Γ} . (ii) S_{𝔭^{∞}} is the kernel of the composition

S′_{𝔭^{∞}} ⟶^{∼} Hom(X_{∞}^{χ_{E}}, E[𝔭^{∞}])^{Γ} → Hom(ker(δ_{∞}), E[𝔭^{∞}])

induced by (i) and local class field theory.

Proof. The first assertion is just a restatement of Proposition 6.5, and the second follows from Theorem 6.9. ⊓ ⊔

Theorem 12.3 (Wiles’ explicit reciprocity law [Wil]). Suppose x is an O_{𝔭}-generator of T_{π}(E), z = (z_{n}) is the corresponding generator of T_{π}(Ê), u = (u_{n}) ∈ U_{∞}, and f(Z) ∈ O_{𝔭}[[Z]] is such that f(z_{n}) = u_{n} for every n. Then

δ_{∞}(u) = (ψ(𝔭) − 1)\frac{f′(0)}{f(0)}x.

See [Wil] or [dS] Theorem I.4.2 for the proof.

Corollary 12.4. δ_{∞}(C_{∞}) = \frac{L(ψ̄, 1)}{Ω}T_{π}(E)

Proof. Using Theorem 12.3, we see that δ_{∞}(C_{∞}) is the ideal of O_{𝔭} generated by the values (Λ′_{𝔭,𝔞}(0)/Λ_{𝔭,𝔞}(0)) where Λ_{𝔭,𝔞} is defined in Definition 7.21, and we allow the ideal 𝔞 to vary. The corollary now follows from Theorem 7.22(ii) and Lemma 10.2. ⊓ ⊔

Remark 12.5. In fact, for every u ∈ U_{∞} there is a power series f_{u} ∈ O_{𝔭}[[X]] such that f_{u}(z_{n}) = u_{n} for every n as in Theorem 12.3. See [Col] or [dS] §I.2.

Definition 12.6. Let ρ_{E} : G_{∞} → O_{𝔭}^{×} be the character giving the action of G_{∞} on E[𝔭^{∞}]. We can also view ρ_{E} as a homomorphism from Λ to O_{𝔭}, and we define A_{E} ⊂ Λ to be the kernel of this homomorphism.

If a, b ∈ K_{𝔭} we will write a ∼ b to mean that a/b ∈ O_{𝔭}^{×}.

12.1 Determination of the Selmer Group when K_{𝔭} = Q_{p}

For this subsection we suppose (in addition to our other assumptions) that K_{𝔭} = Q_{p}.

If M is a Λ-module we will write

M^{A_{E}=0} = {m ∈ M : A_{E}m = 0}.

Proposition 12.7. Suppose that M is a finitely-generated torsion Λ module.

(i) Hom(M, E[𝔭^{∞}])^{G_{∞}} is finite ⇔ ρ_{E}(char(M)) ≠ 0 ⇔ M^{A_{E}=0} is finite. (ii) #(Hom(M, E[𝔭^{∞}])^{G_{∞}}) ∼ ρ_{E}(char(M))#(M^{A_{E}=0}).

Proof. Fix an exact sequence of Λ-modules

0 → ⨁_{i=1}^{k} Λ/f_{i}Λ → M → Z → 0

with pseudo-null (in this case, finite) cokernel Z. Fix a topological generator γ of G_{∞} = Δ × Γ. Then A_{E} = (γ − ρ_{E}(γ))Λ, so multiplication by γ − ρ_{E}(γ) leads to a snake lemma exact sequence of kernels and cokernels

0 → ⨁_{i=1}^{k}(Λ/f_{i}Λ)^{A_{E}=0} → M^{A_{E}=0} → Z^{A_{E}=0}

→ ⨁_{i=1}^{k} Λ/(f_{i}Λ + A_{E}) → M/A_{E}M → Z/A_{E} → 0.

Also

Hom(M, E[𝔭^{∞}])^{G_{∞}} = Hom(M/A_{E}M, E[𝔭^{∞}]).

The map ρ_{E} induces an isomorphism Λ/(f_{i}Λ + A_{E}) ⟶^{∼} Z_{p}/ρ_{E}(f_{i})Z_{p}, and

(Λ/f_{i}Λ)^{A_{E}=0} = {g ∈ Λ, gA_{E} ⊂ f_{i}Λ}/f_{i}Λ

so since A_{E} is a prime ideal,

(Λ/f_{i}Λ)^{A_{E}=0} ≠ 0 ⇔ f_{i} ∈ A_{E} ⇔ (Λ/f_{i}Λ)^{A_{E}=0} is infinite.

Since Z is finite, the exact sequence

0 → Z^{A_{E}=0} → Z ⟶^{γ−ρ_{E}(γ)} Z → Z/A_{E}Z → 0

shows that #(Z^{A_{E}=0}) = #(Z/A_{E}Z). Since char(M) = ∏_{i} f_{i}Λ, the lemma follows. ⊓ ⊔

Theorem 12.8. #(S′_{𝔭^{∞}}) = [Z_{p} : ρ_{E}(char(X_{∞}))].

Proof. This is immediate from Propositions 12.2(i), 11.3, and 12.7 (note that if ρ_{E}(char(X_{∞})) ≠ 0 then X_{∞}^{A_{E}=0} is finite by Proposition 12.7 and hence zero by Proposition 11.3). ⊓ ⊔

Theorem 12.9. char(X_{∞}^{χ_{E}}) = char(U_{∞}^{χ_{E}}/C_{∞}^{χ_{E}}).

Proof. Immediate from Corollary 11.8 and (13). ⊓ ⊔

Theorem 12.10 (Coates and Wiles). Let D denote the ring of integers of the completion of the maximal unramified extension of Q_{p}. Then there is a 𝔭-adic period Ω_{𝔭} ∈ D^{×} such that char(U_{∞}/C_{∞})D[[G_{∞}]] has a generator L_{E} satisfying

ρ_{E}^{k}(L_{E}) = Ω_{𝔭}^{k}(1 − ψ(𝔭^{k})/p)\frac{L(ψ̄^{k}, k)}{Ω^{k}}

for every k ≥ 1.

Proof. See [CW2] or [dS] Corollary III.1.5. ⊓ ⊔ Corollary 12.11. #(S′_{𝔭^{∞}}) ∼ (1 − ψ(𝔭)/p)\frac{L(ψ̄, 1)}{Ω}.

Proof. Immediate from Theorem 12.8, Theorem 12.9, and Theorem 12.10.

⊓ ⊔

Proposition 12.12. [S′_{𝔭^{∞}} : S_{𝔭^{∞}}] ∼ (1 − ψ(𝔭)/p).

For a proof see [PR] Proposition II.8 or [Co] Proposition 2 and Lemma 3.

Corollary 12.13. Suppose 𝔭 ∤ 𝔣, p > 7, and K_{𝔭} = Q_{p}.

(i) If L(ψ̄, 1) = 0 then S_{𝔭^{∞}} is infinite. (ii) If L(ψ̄, 1) ≠ 0 then

#(� (E)_{𝔭^{∞}}) ∼ \frac{L(ψ̄, 1)}{Ω}.

Proof. This is immediate from Corollary 12.11 and Proposition 12.12. (For (ii), we also use (18).) ⊓ ⊔

12.2 Determination of the Selmer Group when [K_{𝔭} : Q_{p}] = 2

For this subsection we suppose that [K_{𝔭} : Q_{p}] = 2, so Γ ≅ Z_{p}^{2} and E[𝔭^{∞}] ≅ K_{𝔭}/O_{𝔭} has Z_{p}-corank 2.

Lemma 12.14. There is a decomposition

U_{∞}^{χ_{E}} = V_{1} ⊕ V_{2}

where V_{1} and V_{2} are free of rank one over Λ_{χ_{E}}, δ_{∞}(V_{2}) = 0, and E_{∞}^{χ_{E}} ⊄ V_{2}.

Proof. By Proposition 11.4, U_{∞}^{χ_{E}} is free of rank two over Λ_{χ_{E}}. Fix a splitting U_{∞}^{χ_{E}} = Λ_{χ_{E}}v_{1} ⊕ Λ_{χ_{E}}v_{2}. By Corollary 5.20(ii), ρ_{E} is surjective, and it follows that δ_{∞}(Λ_{χ_{E}}v_{1}) and δ_{∞}(Λ_{χ_{E}}v_{2}) are O_{𝔭}-submodules of T_{π}(E). Since δ_{∞} is surjective (Lemma 6.8) and δ_{∞}(U_{∞}) = δ_{∞}(U_{∞}^{χ_{E}}), it follows that either δ_{∞}(Λ_{χ_{E}}v_{1}) = T_{π}(E) or δ_{∞}(Λ_{χ_{E}}v_{2}) = T_{π}(E).

Thus, by renumbering if necessary, we may assume that δ_{∞}(Λ_{χ_{E}}v_{1}) = T_{π}(E). In particular we can choose g ∈ Λ_{χ_{E}} so that δ_{∞}(v_{2}) = δ_{∞}(gv_{1}), and (by adjusting g if necessary by an element of the kernel of ρ_{E}) we may assume that E_{∞} ⊄ Λ_{χ_{E}}(v_{2} − gv_{1}). Now the lemma is satisfied with

V_{1} = Λ_{χ_{E}}v_{1}, V_{2} = Λ_{χ_{E}}(v_{2} − gv_{1}).
⊓ ⊔

Definition 12.15. Fix a decomposition of U_{∞}^{χ_{E}} as in Lemma 12.14 and define

Ũ = U_{∞}^{χ_{E}}/V_{2}, X̃ = X_{∞}^{χ_{E}}/image(V_{2})

where image(V_{2}) denotes the image of V_{2} in X_{∞} under the Artin map of local class field theory.

Lemma 12.16. (i) X̃ is a torsion Λ_{χ_{E}}-module with no nonzero pseudo-null

submodules. (ii) char(X̃) = char(Ũ/image(C_{∞}^{χ_{E}}))

Proof. Since E_{∞} ⊄ V_{2} and V_{2} is free, (i) follows from Proposition 11.3. Also, the exact sequence (13) induces an exact sequence

0 → E_{∞}^{χ_{E}}/C_{∞}^{χ_{E}} → Ũ/image(C_{∞}^{χ_{E}}) → X̃ → A_{∞}^{χ_{E}} → 0.

so (ii) follows from Corollary 11.8. ⊓ ⊔

Proposition 12.17. S_{𝔭^{∞}} = Hom(X̃, E[𝔭^{∞}])^{Γ}

Proof. By our choice of V_{1} and V_{2} (Proposition 12.14), we see that ker(δ_{∞}) = A_{E}V_{1} + V_{2}. Thus

X̃/A_{E}X̃ = X_{∞}/(A_{E}X_{∞} + image(V_{2})) = X_{∞}/(A_{E}X_{∞} + image(ker(δ_{∞})))

and so by Proposition 12.2(ii)

S_{𝔭^{∞}} = Hom(X_{∞}/(A_{E}X_{∞} + image(ker(δ_{∞}))), E[𝔭^{∞}]) = Hom(X̃, E[𝔭^{∞}])^{Γ} .

⊓ ⊔

Proposition 12.18. Suppose M is a finitely-generated torsion Λ_{χ_{E}}-module and F is a Z_{p}-extension of K_{1} in K_{∞} satisfying

(i) M has no nonzero pseudo-null submodules, (ii) If γ generates Gal(K_{∞}/F) then char(M) ⊄ (γ − 1)Λ_{χ_{E}}, char(M) ⊄

(γ − ρ_{E}(γ))Λ_{χ_{E}}, and M/(γ − 1)M has no nonzero finite submodules.

Then

#(Hom(M, E[𝔭^{∞}])^{Γ}) = [O_{𝔭} : ρ_{E}(char(M))].

Proof (sketch). For a complete proof see [Ru2], Lemmas 6.2 and 11.15.

Let T̂_{π} = Hom(T_{π}, O_{𝔭}), let Λ_{F} = Z_{p}[[Gal(F/K)]], and let M̄ denote the Λ_{F}^{χ_{E}}-module (M ⊗ T̂_{π})/(γ − 1)(M ⊗ T̂_{π}). Using the hypotheses on M and F it is not difficult to show (see [Ru2] Lemma 11.15) that M̄ has no nonzero finite submodules. Therefore exactly as in Proposition 12.7,

#(Hom(M, E[𝔭^{∞}])^{Γ}) = #(Hom(M ⊗ T̂_{π}, O_{𝔭})^{Γ})
= #(Hom(M̄, O_{𝔭})^{Gal(F/K)})
= [O_{𝔭} : 1l(char_{F} (M̄))]

where 1l denotes the trivial character and char_{F} (M̄) is the characteristic ideal of M̄ as a Λ_{F}^{χ_{E}}-module.

By an argument similar to the proof of Proposition 12.7, one can show that char_{F} (M̄) = char(M ⊗ T̂_{π})Λ_{F}^{χ_{E}}. Therefore

#(Hom(M, E[𝔭^{∞}])^{Γ}) = [O_{𝔭} : 1l(char(M ⊗ T̂_{π}))] = [O_{𝔭} : ρ_{E}(char(M))].

⊓ ⊔

Theorem 12.19. Suppose 𝔭 ∤ 𝔣, p > 7, and K_{𝔭} ≠ Q_{p}.

(i) If L(ψ̄, 1) = 0 then S_{𝔭^{∞}} is infinite. (ii) If L(ψ̄, 1) ≠ 0 then

#(� (E)_{𝔭^{∞}}) = [O_{𝔭} : (L(ψ̄, 1)/Ω)O_{𝔭}].

Proof. Lemma 12.16(i) shows that X̃ satisfies the first hypothesis of Proposition 12.18, and the same argument with K_{∞} replaced by F verifies the second hypothesis for all but finitely many choices of F. Also, Ũ/image(C_{∞}^{χ_{E}}) satisfies the hypotheses of Proposition 12.18 since it is a quotient of one free Λ_{χ_{E}}-module by another (Proposition 11.6). Therefore by Proposition 12.18 and Lemma 12.16(ii),

#(Hom(X̃, E[𝔭^{∞}])^{Γ}) = #(Hom(Ũ/image(C_{∞}^{χ_{E}}), E[𝔭^{∞}])^{Γ}).

The left-hand side of this equality is #(S_{𝔭^{∞}}) by Proposition 12.17. On the other hand,

Hom(Ũ/image(C_{∞}^{χ_{E}}), E[𝔭^{∞}])^{Γ} = Hom(Ũ/(image(C_{∞}^{χ_{E}} + A_{E}Ũ)), E[𝔭^{∞}]),

and δ_{∞} : Ũ/A_{E}Ũ → T_{π}(E) is an isomorphism (Lemmas 6.8 and 12.14). Therefore

#(Hom(Ũ/image(C_{∞}^{χ_{E}}), E[𝔭^{∞}])^{Γ}) = #(T_{π}/δ_{∞}(C_{∞}))

and the Theorem follows from Wiles’ explicit reciprocity law (Corollary 12.4) and (18). ⊓ ⊔

12.3 Example

We conclude with one example. Let E be the elliptic curve y^{2} = x^{3} − x. The map (x, y) ↦ (−x, iy) is an automorphism of order 4 defined over K = Q(i), so End_{K}(E) = Z[i]. Let 𝔭_{2} denote the prime (1 + i) above 2.

Clearly E(Q)_{tors} ⊃ E[2] = {O, (0, 0), (1, 0), (−1, 0)}. With a bit more effort one checks that E(K) contains the point (−i, 1 + i) of order 𝔭_{2}^{3}, and using the Theorem of Nagell and Lutz ([Si] Corollary VIII.7.2) or Corollary 5.18 one can show that in fact E(K)_{tors} = E[𝔭_{2}^{3}].

The discriminant of E is 64, so E has good reduction at all primes of K different from 𝔭_{2}. Since E[𝔭_{2}^{3}] ⊂ E(K), if we write ψ_{E} for the Hecke character of K attached to E, Corollary 5.16 shows that ψ_{E}(𝔞) ≡ 1 (mod 𝔭_{2}^{3}) for every ideal 𝔞 prime to 𝔭_{2}. But every such ideal has a unique generator congruent to 1 modulo 𝔭_{2}^{3}, so this characterizes ψ_{E} and shows that its conductor is 𝔭_{2}^{3}.

Standard computational techniques now show that

L(ψ̄, 1) = .6555143885...
Ω = 2.622057554...

Therefore by the Coates-Wiles theorem (Theorem 10.1), E(K) = E[𝔭_{2}^{3}] and E(Q) = E[2]. Further, L(ψ̄, 1)/Ω is approximately 1/4. By Proposition 10.6 L(ψ̄, 1)/Ω is integral at all primes 𝔭 of residue characteristic greater than 7. In fact the same techniques show that L(ψ̄, 1)/Ω is integral at all primes 𝔭 ≠ 𝔭_{2}, and give a bound on the denominator at 𝔭_{2} from which we can conclude that L(ψ̄, 1)/Ω = 1/4.

Therefore by Corollary 12.13 and Theorem 12.19, S_{𝔭^{∞}} = � (E_{/K})_{𝔭^{∞}} = 0 for all primes 𝔭 of residue characteristic greater than 7, and again the same proof works for all 𝔭 ≠ 𝔭_{2}. It follows easily from this that � (E_{/Q})_{p} = 0 for all odd rational primes. Fermat did the 2-descent necessary to show that � (E_{/Q})_{2} = 0 (see [We] Chap. II), so in fact � (E_{/Q}) = 0. Together with the fact that the Tamagawa factor at 2 is equal to 4, this shows that the full Birch and Swinnerton-Dyer conjecture holds for E over Q.

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