Descent theory over extensions of H

Proof. By Lemma 3.2.4, we have

#(^Y

v∈T

H¹(H_v, E)(p)) =







2^b ·^Q_v∈P1− ^ψE/H_Nv^(v)−1

p if p= 2

Q

v∈P

1− ^ψE/H_Nv^(v)−1

p if p >2.

In addition, by Lemma 3.2.8

#(Sel^(T_p^∞)(E/H)) = #(X(E/H)(p))·#(^Y

v∈T

H¹(H_v, E)(p))/#(E_p^∗∞(H)), and #(Ep^∗) is equal to 2 if p= 2 and 1 if p >2. Hence the result follows.

(ii) If p= 2, the restriction map satisfies res(Sel^(T_p^∞)(E/H))⊂Sel^′_p∞(E/F)^∆, and we have an exact sequence

0→H¹(∆, E_p^∞(F))→Sel^(T_p^∞)(E/H)−→^res Sel^′_p∞(E/F)^∆→0.

Proof. The proof of Theorem 3.3.1 is easy for p > 2, hence we shall omit the details in this case. Now assume that p= 2. Then the following lemma shows that res surjects onto Sel^′_p∞(E/F)^∆. Recall first that the action of ∆ on H¹(Gal(F /F), E_p^∞) is given by inner automorphisms, i.e., given τ ∈ ∆, ξ ∈ H¹(Gal(F /F), E_p^∞), σ ∈ Gal(F /F) and any cocycle f representing ξ, we have

τ ·f(σ) =τ f(τ⁻¹·σ) = τ f(τ⁻¹στ),

whereτ is a lift ofτ in Gal(F /F). The next result a modification of [11, Lemma2.3.5], which was left incomplete, because the fact that the cohomology class is invariant does not mean that one can choose a cocycle which is invariant under the action of ∆. This problem has been fixed in the following proof.

Lemma 3.3.2. Let p= 2. Then we have Sel^′_p∞(E/F)^∆⊂res(H¹(H, E_p^∞)).

Proof. Choose a prime q∈B, and fix a primeQ ofF aboveq. LetI_q⊂Gal(F /H) be the corresponding inertia subgroup. Let τ denote the unique element of H which acts as multiplication by −1 on E_p^∞. Then τ has order 2 and its restriction toH generates

∆. Also qramifies in F/H so τ generates the inertia group ofq in F/H, hence we can find a lifting τ of τ in I_q. Then τ² restricted toF is τ² = id, so τ² ∈Gal(F /F)∩I_q. Furthermore, we know that every ξ_p ∈ Sel^′_p∞(E/F)^∆ is unramified at q since q - p.

Hence ξ_p ∈ ker(H¹(Gal(F /F), E_p^∞) → H¹(Gal(F /F)∩I_q, E_p^∞)), i.e. g(τ²) = 0 for any 1-cocycle g representing ξ_p.

Givenx∈E_p^∞, letd(x) denote the 1-coboundary on Gal(F /H) defined byd(x)(σ) = (σ−1)x. Then we have τ ·d(x) =d(τ(x)) =d(−x) =−d(x) (for any σ ∈Gal(F /H),

we haveτ·d(x)(σ) =τ(τ⁻¹στ−1)x= (σ−1)τ(x)). Let ξ_p∈H¹(F, E_p^∞)^∆ and pick a 1-cocycle g representing ξ_p. Then since ξ_p is ∆-invariant, we have (1−τ)·g = d(x) for some x∈E_p^∞. Now we take y∈E_p^∞ such that 2y=x and define f =g−d(y), then

(1−τ)·f = (g−d(y))−τ ·(g−d(y))

=g−d(y)−(g−d(x)) +τ ·d(y)

=g−d(y)−(g−d(x))−d(y)

since τ ·d(y) = −d(y) from above, and (1−τ)·g = d(x) implies τ ·g = g−d(x).

Therefore, we have (1−τ)·f =d(x)−2d(y) = 0, hence we can pickf as the cocycle representing ξ_p and f is invariant under the action of ∆.

It is clear that every element of Gal(F /H) can be written in the form στⁱ with σ ∈Gal(F /F) and i∈ {0,1}. We now define the map

h: Gal(F /H)→Ep^∞

byh(στⁱ) =f(σ), and claim that this is an element of H¹(H, E_p^∞) which maps to ξ_p under res. This map is well-defined, since h(τⁱ) = 0 for all i> 0. Indeed, it is clear that h(τⁱ) =f(id) = 0 for i ∈ {0,1} since H is a 1-cocycle, hence it suffices to show h(τ²) = f(τ²) = 0, which we proved earlier. To see thathis a 1-cocycle, takeρ₁ =σ₁τⁱ¹, ρ₂ =σ₂τⁱ² ∈Gal(F /H). We need to show h(ρ₁ρ₂) =h(ρ₁) +ρ₁h(ρ₂). Since Gal(F /F) is a normal subgroup of Gal(F /H), we can find an element σ₂^′ ∈Gal(F /F) such that τⁱ¹σ₂ = σ^′₂τⁱ¹. Then h(ρ₁ρ₂) = h(σ₁σ₂^′τⁱ¹⁺ⁱ²) = f(σ₁σ₂^′) = f(σ₁) +σ₁f(σ₂^′), where the last equality follows from the fact that f is a 1-cocycle. Also,h(ρ₁) +ρ₁h(ρ₂) = f(σ₁) +σ₁τⁱ¹f(σ₂), and since f is ∆-invariant, we have τ ·f = f, and ∆ acts by inner-automorphism so τ ·f(σ) = τ f(τ⁻¹σ) = τ f(τ⁻¹στ). Therefore, τⁱ¹f(σ₂) = f(τⁱ¹σ₂τ⁻ⁱ¹) =f(σ₂^′τⁱ¹τ⁻ⁱ¹) =f(σ^′₂). Hence f(σ₁) +σ₁τⁱ¹f(σ₂) =f(σ₁) +σ₁f(σ₂^′), as required. Finally, res(h(στⁱ)) = h(σ) = f(σ), so indeed res(h) =f.

We now finish the proof of Theorem 3.3.1. Recall that for a group profinite group G and aG-moduleA, ˆH⁰(G, A) is the modified 0-th cohomology group defined to be equal to A^G/NGA where NG :A→A, a 7→ ^P

σ∈G

σadenotes the norm map. In addition, if G is cyclic andAis finite, then we haveH¹(G, A) = ˆH⁰(G, A) since the Herbrand quotient is equal to 1. Hence, in order to work out the order of H¹(G, A), we will calculate the order of A^G/NGA instead. First, we have H¹(∆, E_p^∞(F)) = H¹(∆, E_p²) = ˆH⁰(∆, E_p²) because ∆ = Gal(F/H) is cyclic of order 2 by Theorem 3.2.1 and F =H(E_p²). We have E_p^∆2 = E_p. Now ∆ = {1, δ} where δ acts as −1 on E_p², so we have N_∆(P) = P + (−1)P =O for all P ∈E_p². Hence|H¹(∆, E_p²)|=|E_p|= 2.

The fact that H¹(∆, E_p²) injects into Sel^(T_p^∞)(E/H) follows from the fact that H¹(Gal(FQ/Hq), E) = 0 (3.3.1) for any prime q /∈ B and any prime Q of F above q. This is because E has good reduction atqandF_Q/H_qis unramified, which implies thatN_FQ/Hq is surjective. Hence it remains to show res(Sel^(T_p^∞)(E/H)) = Sel^′_p∞(E/F)^∆. Given any prime Q of F and

prime q of H lying below Q, we have the commutative diagram H¹(H, E_p^∞) −^λ→^q H¹(H_q, E)_p^∞

↓res ↓res_Q H¹(F, E_p^∞)^∆ −→^λQ H¹(F_Q, E)^∆_p∞.

To show Sel^′_p∞(E/F)^∆ ⊂ res(Sel^(T_p^∞)(E/H)), take any x ∈ Sel^′_p∞(E/F)^∆. Then since H¹(H, E_p^∞) −→^res Sel^′_p∞(E/F)^∆ is surjective by Lemma 3.3.2, there exists a ∈ H¹(H, E_p^∞) with x = res(a). Then for any prime q ∈/ B of H and a prime Q of F above q, we have

res_Q(λ_q(a)) =λ_Q(x) = 0,

since x∈Sel^′_p^∞(E/F)^∆ ⊂kerλ_Q. But res_Q is injective by (4.1.8), so λ_q(a) = 0. Hence a∈Sel^(T_p^∞)(E/H), and x∈res(Sel^(T_p^∞)(E/H)) as required. To show the other inclusion, it suffices to show that for any prime q - p of H and Q of F lying above q we have res(Sel^(T_p^∞)(E/H)) ⊂ kerλ_Q. First let Q be a prime of F lying above q ∈ B. Then we have shown that H¹(F_q, E)_p^∞ = E_p in Lemma 3.2.4, and furthermore, we have H¹(∆_Q, E)_p^∞ ≃H¹(∆_Q, E_p^∞) sinceE(F_Q) =E(F_Q)_p^∞ ⊕A for a p-divisible groupA.

So H¹(∆_Q, E)_p^∞ = E_p since ∆_Q = ∆ for such Q. Hence res_Q is the zero map. But given any a∈Sel^(T_p^∞)(E/H), the above diagram commutes so

λ_Q(res(a)) = res_Q(λ_q(a)) = 0, (3.3.2) giving res(Sel^(T_p^∞)(E/H)) ∈ kerλ_Q as required. Finally, let Q be a prime of F lying above q∈/ B. Then Sel^(T_p^∞)(E/H)⊂kerλ_q, so fora∈Sel^(T_p^∞)(E/H) (4.2.2) holds again, and so res(Sel^(T_p^∞)(E/H))⊂kerλ_Q. This completes the proof of Theorem 3.3.1.

The following is an immediate consequence of Theorem 3.3.1.

Corollary 3.3.3. Sel^′_p∞(E/F)^∆ is finite if and only if Sel_p^∞(E/H) is.

Assumingp satisfies the good ordinary hypothesis for E, we have

#(H¹(∆, E_p^∞(F))) = 1 or 2,

depending as p >2 or p= 2. Hence combining Theorem 3.2.9 and Theorem 3.3.1, we obtain:

Theorem 3.3.4. Letp be a prime satisfying the good ordinary hypothesis for E. Then Sel^′_p∞(E/F)^∆ is finite if and only if Sel_p^∞(E/H) is finite. Moreover, if we assume that E(H) and X(E/H)(p) are finite, then we have:

(i) If p > 2, then

#(Sel^′_p∞(E/F)^∆) = ^Y

v∈P

1− ψ_E/H(v) Nv

!

−1

p

·#(X(E/H)(p))

(ii) If p= 2, then

#(Sel^′_p∞(E/F)^∆) = 2^{b−2 Y}

v∈P

1− ψ_E/H(v) Nv

!

−1

p

·#(X(E/H)(p)), where b = #(B).

We define the p^∞-Selmer group Sel_p^∞(E/F∞) by Sel_p^∞(E/F∞) = ker H¹(F∞, E_p^∞)→^Y

w

H¹(F∞,w, E)

!

.

We also define modified Selmer group Sel^′_p∞(E/F∞) = ker



H¹(F∞, E_p^∞)→^Y

w-p

H¹(F∞,w, E)



.

The next result is [2, Lemma 8].

Theorem 3.3.5. We have Sel_p^∞(E/F∞) = Sel^′_p∞(E/F∞).

Proof. It suffices to show that H¹(F∞,P, E)(p) is trivial any placeP of F∞ above p.

Let F_n,Pn denote the completion of F_n at the prime P_n of F_n lying below P. Then we have ∪_n>0F_n,Pn =F∞,P, soH¹(F∞,P, E) = lim−→H¹(F_n,Pn, E) where the limit is taken with respect to the restriction. Recall that π is an element of O satisfying (π) =p^h. Let n be of the form n^′hwith n^′ >0. By Tate’s local duality, the Pontryagin dual of H¹(F_n,Pn, E)(p) is equal to

E(F_n,Pn) = lim←−E(F_n,Pn)/π^∗mE(F_n,Pn),

where the limit is taken with respect to the norm map. Note that P_n lies above p and p ̸= p^∗, so π^∗m is an automorphism of the group E₁(F_n,Pn) of F_n,Pn-rational points in the kernel of reduction modulo P_n. Thus reduction modulo P_n induces an isomorphism E(F_n,Pn)/π^∗mE(F_n,Pn) ∼= E(^e F]_n,Pn)/π^∗mE(^e F]_n,Pn), where F]_n,Pn denotes the residue field of F_n,Pn at P_n. Thus we haveE(F_n,Pn)≃E^e(F]_n,Pn)(p)≃E_p^∗∞(F_n,Pn).

This is a finite group of bounded order, sincePis totally ramified inF∞,P/F_n,Pn, so the

residue fieldF]_n,Pn is finite. So there existsm andn₀ such that E(^e F]_n,Pn) =E_p^∗m for all n >n₀. Take (P_n)∈lim←−E(F_n,Pn). Then P_n0 = [π^∗m]P_n0+m =O, so in fact (P_n) = 0.

Therefore lim←−E(F_n,Pn) is trivial, and it immediately follows that H¹(F∞,P, E)(p) is trivial.

WriteX(F_∞) = Gal(M(F_∞)/F_∞), whereM(F_∞) is the maximal abelianp-extension of F_∞ unramified outside the primes of F_∞ abovep. Note that, by maximality, M(F_∞) is Galois over H, and we can define an action of H= Gal(F_∞/H) on X(F_∞) by

g·x=gxe ge⁻¹,

whereg_e denotes any lifting of g inH to an element of Gal(M(F_∞)/H). Let Λ(Γ) = lim←−Zp[Γ/U]

be the Iwasawa algebra of Γ. The continuous action of Γ on X(F_∞) extends to an action of Λ(Γ). The following classical result is well-known granted that E has good reduction everywhere overF_∞, and is omitted (see Theorem 9, Theorem 12 and Lemma 13 of [2]).

Theorem 3.3.6. We have

Sel_p^∞(E/F∞) = Hom(X(F∞), E_p^∞).

Furthermore, X(F_∞) is a finitely generated torsion Λ(Γ)-module, and the restriction map gives an isomorphism

Sel^′_p∞(E/F)−→^∼ Sel_p^∞(E/F∞)^Γ.

We wish to find a criterion for when Sel^′_p∞(E/F)^∆ is finite, and to compute its order when it is finite. We will study this in more detail in Chapter 7 in order to state the main conjecture for E/H. Before we do this, we will construct in the next chapter the p-adic L-functions which appear in the statement of the main conjectures.

Construction of the p-adic L-functions

4.1 Construction of the p-adic L-function for F

_∞

/F

We now construct thep-adic L-functions attached toE/H, which we shall subsequently need to formulate the main conjectures. We will follow the ideas in [4], however, we will also deal with the casep= 2, which cannot be found in literature.

Fix once and for all an embedding ofH into C. Write x, y for the coordinates of E/H. We fix a generalised global minimal Weierstrass equation for E over H, which exists by [14, Proposition 3.2], to be

y²+a₁xy+a₃y=x³+a₂x²+a₄x+a₆. (4.1.1) Recall that G denotes the Galois group ofH overK. Then applying σ∈G to (4.1.1) gives a generalised global minimal Weierstrass equation for E^σ/H. Let

ω^σ = dx 2y+a^σ₁x+a^σ₃

be the Néron differentials on E^σ, and note that the discriminant of this equation ∆(E^σ) is equal to (∆(E))^σ = ∆(E). Let g denote the conductor of ϕ_K, so that (g,2p) = 1, and let f = g^h, so that f is principal. Let L (resp. L_σ) be the period lattice of the Neron differential on our global minimal Weierstrass equation for E (resp. E^σ). Then there exists Ω_∞ ∈C^× such that L = Ω_∞O. The uniformisation Φ : C/L−→^∼ E(C) is

accomplished through

Φ(z, L_σ) = ℘(z, L_σ)−((a^σ₁)²+ 4a^σ₂)

12 ,1

2 ℘^′(z, L_σ)−a^σ₁ ℘(z, L_σ)−((a^σ₁)²+ 4a^σ₂) 12

!

−a^σ₃

!!

.

Given a principal ideal a= (α) withα ∈ O and (a,6f) = 1, define Ra(P) :=cE(a)^Y

U

(x(P)−x(U))⁻¹,

where U runs over any set of representatives of E_a\{O} modulo {±1}, and cE(a) is an element of H whose 12-th power is equal to ∆(E)^Na−1/Λ(a)¹², where Λ(a)∈H^× satisfies

λE(a)^∗(ω^σa) = Λ(a)ω.

Thus R_a(P) is a rational function on E with coefficients in H. Let us write P for the generic point on E^σ with coordinates (x, y). Applying σ ∈ G to the coefficients of R_a(P), we obtain a rational functionR^σ_a(P) on the curve E^σ/H.

Proposition 4.1.1. Let b be an integral ideal of K with (b,a) = 1. Then we have R^σσ_a ^b(λ_E^σ(b)(P)) = ^Y

R∈E^σ

b

R_a^σ(P ⊕R).

Proof. Recall that the kernel of λ_E^σ(b) is E_b^σ, and λ_E^σ(b) is injective on E_a^σ since (b,a) = 1. Hence, the left hand side and the right hand side of the above equation

have the same divisor, and

R^σσ_a ^b(λ_E^σ(b)(P))

Q

R∈E^σ

b

R^σ_a(P ⊕R)

is a non-zero element of H. It can be shown, thanks to the unique scaling factor c_E(a) in our definition of the rational functions, that this constant is equal to 1. See [3, Appendix, Theorem 4] for details.

We fix a generator f of the ideal f, and define Q = Φ(Ω∞/f, L) so that Q is a primitive f-division point on E. We then define

R_a(P) = ^Y

τ∈Gal(H(E_f)/H)

R_a(P ⊕Q^τ),

where ⊕denotes the addition on E. ThusR_a(P) is also a rational function on E over H. Similarly, we define

R^σ_a(P) = ^Y

τ∈Gal(H(E_f)/H)

R^σ_a(P ⊕(Q^σ)^τ),

a rational function on E^σ overH. Hence, defining

Ψ^σ_a(P) = R^σ_a(P)^Np/R^σσ_a ^p(λ_E^σ(p)(P)), it follows that

Y

R∈E^σ_p

Ψ^σ_a(P ⊕R) = 1. (4.1.2) Now, we fix an embeddingi_v :K →K_p, and we let v denote the prime ofH above p determined by i_v. We write O_v for the ring of integers of H_v.

Recall that t=−^x_y is a parameter for this formal group.

Lemma 4.1.2. Let B_a^σ(t) denote the t-expansions of R^σ_a(P). Then B_a^σ(t) is a unit in O_v[[t]].

Proof. We claim that, if V denotes any of the points (Q^σ)^τ, then x(P ⊕V)−x(U),

where U is any non-zero element of E_a^σ, has a t-series expansion which is a unit in O_B[[t]], whereB denotes any prime ofH(U, V) above v, andO_B is the ring of integers of the completion of H(U, V) at this prime. Indeed, we have explicitly

x(P ⊕V)−x(U) =D(P, V)²+a^σ₁D(P, V)−a^σ₂ −x(P)−x(V)−x(U), where

D(P, V) = y(P)−y(V) x(P)−x(V).

Note that x(V), y(V) are integral at B because (f, p) = 1 by assumption. Similarly, the expansions of x(P) and y(P) as power series in t begin

x(P) = 1 t² − a^σ₁

t −a^σ₂ +· · · , y(P) = −1 t³ +a^σ₁

t² +a^σ₂

t +· · · ,

where the coefficients of all the higher order terms lie inO_B. Note thatx(U) is integral at primes of H(U, V) above B because (a, p) = 1. Thus we see that the coefficients of

thet-series expansion ofx(P⊕V)−x(U) all belong to O_B. Moreover,x(P⊕V)−x(U) is holomorphic att = 0, and so there are no negative powers oftin itst-series expansion.

Moreover, we have

D(P, V) =−1

t −x(V)t+· · · ,

so the constant term of the t-series expansion of is x(P ⊕V)−x(U) is x(V)−x(U).

We claim this must be a unit at B. If not, we would have x(V^e) = x(U^e), where ^e denotes the reduction modulo B. But this would imply that V^e = ±U^e, whence we would have one ofV ±U must lie on the formal group ofE^σ at v. But this is impossible because (pa,f) = 1. The assertion of the lemma now follows immediately, on noting that c_E(a) is a unit at v.

From this we obtain

Corollary 4.1.3. Let A^σ_a(t)denote the t-expansions of Ψ^σ_a(P). Then A^σ_a(t) belongs to 1 +m_v[[t]], where m_v denotes the maximal ideal of O_v.

Proof. Write B_a^σ(t) =

∞

P

n=0

a_nt. Thus, by the previous lemma,a_n∈ O_p for alln >0 and a₀ ∈ O_p^×. Now,A^σ_a(t) = ^Baσ(t)p

B^σσv_a (b^λE(v)(t)) and recall that λb_E(v)(t)≡t^p modv.

Hence, we see that

B_a^σσvλ^b_E(v)(t)=

∞

X

n=0

a^σ_n^v(λ^b_E(v)(t))ⁿ ≡

∞

X

n=0

a^p_nt^pn modv.

On the other hand,

B_a^σ(t)^p ≡

∞

X

n=0

a^p_nt^pn ≡

∞

X

n=0

a_nt^pn modv, so A^σ_a(t)≡1 modv, as required.

Lemma 4.1.4. Define C_a^σ(t) by

C_a^σ(t) := 1

plogA^σ_a(t).

Then C_a^σ(t)∈ O_p[[t]], and

X

ω∈Dσ,p

C_a^σ(t[+]ω) = 0, (4.1.3)

where D_σ,p denotes the group of p-division points on the formal group E^bσ at a place v of H lying abovep and [+] denotes the group law on E^bσ. This group can be identified with E_p^σ.

Proof. We have

C_a^σ(t) = 1 p

X

n>1

(−1)ⁿ⁻¹(A^σ_a(t)−1)ⁿ

n .

The first claim is now clear from the previous lemma as n > ord_p(n) + 1 for n >1.

The final equation then follows from (4.1.2).

LetI be the ring of integers of the completion of the maximal unramified extension K_p^ur of K_p. By [8, Proposition 1.6], we have an isomorphism

δσ,v:G^bm −→^∼ E^bσ

defined over I, where G^bm denotes the formal multiplicative group and E^bσ denotes the formal group of E^σ atv. Define J_a^σ(W) =C_a^σ ◦δ_σ,v(W)∈I[[W]].

Definition 4.1.5. Letµ_a,σ be the I-valued measure onZ^p determined by J_a^σ(W), i.e.

J_a^σ(W) =

Z

Zp

(1 +W)^xdµa,σ(x). (4.1.4) We claim that the measure µ_a,σ is supported on Z^×p. Indeed, let Λ_I(Z^p) (resp.

Λ_I(Z^×p) be the ring ofI-valued measures onZ^p (resp. Z^×p). Then we have an inclusion ι : Λ_I(Z^×p),→Λ_I(Z^p) given by extending the measures on Z^p to Z^×p by zero. Let µ be a measure in Λ_I(Z^p), and let f_µ(W)∈I[[W]] be the corresponding power series given by the isomorphism Λ_I(Z^p)∼=I[[W]]. Then it is well-known (see [8, I.3.3] for more details) that µbelongs to ιΛ_I(Z^×p) if and only iff_µ satisfies the equation

X

ζ∈µ_p

f_µ(ζ(1 +W)−1) = 0.

It follows from (4.1.3) that this is satisfied by J_a^σ.

We know that, writing alsoµ_a,σ for the corresponding measure in Λ_I(H), we have

Z

H

χ^k_pdµ_a,σ =

Z

Zp

x^kdµ_a,σ =D^kJ_a^σ(W)|_W=0, (4.1.5)

where D = (1 +W)_dW^d . We have an isomorphism ˆGm

−→∼ Gˆa given by W 7→ e^z −1.

Hence we see immediately that D= _dz^d. Moreover, we have δ_σ,v(W) = Ω_σ,vW +· · ·, so for any integer k >1, we have

D^kJ_a^σ(W)|_W=0 = d dz

!k

J_a^σ(e^z−1)|_z=0 = 1

pΩ^k_σ,v d dz

!k

log (Ψ^σ_a(Φ(z, L_σ)))|_z=0. (4.1.6) Lemma 4.1.6. We have Ω_σ,v= Λ(s)Ω_v, where Ω_v ∈I^× is the coefficient of W in the formal power series t =δ_v(W), with δ_v :G^bm

−→∼ E^b is an isomorphism defined over I. Proof. We haveλE(s)^∗(ω^σ) = Λ(s)ωby definition, so thatλE(s) (Φ(z, L)) = Φ(Λ(s)z, Lσ).

Hence, writing exp(z, Lσ) for the formal power series in z obtained by express- ing t = −x/y in terms of z using the isomorphism Φ(z, Lσ) for E^σ, we also have λE(exp(z, L)) = exp(Λ(s)z, Lσ)). Now, regarding z as the parameter of the formal ad- ditive group, exp(z, Lσ) is the exponential map of E^bσ. It then follows by the uniqueness of the exponential maps for the formal groups that

δ_σ,v(e^z/Ωσ,v −1) = exp(z, L_σ).

On the other hand, we have δσ,v = λ^bE(s)◦δv(W), where λ^bE(s) : E^b → E^bσ is the isomorphism over Hv of formal groups induced byλE(s). Hence we have

δ_σ,v(e^z−1) = exp(Λ(s)Ω_vz, L_σ).

The assertion follows by comparing the coefficients of z in the above equations.

Proposition 4.1.7. Let s be an integral ideal of K prime to f such that σ_s =σ. Then for any integer k>1, k≡1 mod #(∆), we have

d

dz logR^σ_a(Φ(z, L_σ)) =

∞

X

k=1

(−1)^kϕ^k_K(s)f^k Λ(s)^kΩ^k_∞

Na−ϕ^k_K(a)L(ϕ^k_K, σ, k)z^k−1.

In particular, we have d

dz

!k

logR^σ_a(Φ(z, L_σ))|z=0 = (−1)^k(k−1)!ϕ^k_K(s)f^k Λ(s)^kΩ^k_∞

Na−ϕ^k_K(a)L(ϕ^k_K, σ, k).

Proof. Let L be a complex lattice. We will modify the Weierstrass σ-function slightly, and define

Θ(z,L) = exp

(

−s₂(L)z² 2

)

σ(z,L).

Recall that for any integer k>1, we can define the Kronecker–Eisenstein series Hk(z, s,L) := ^X

w∈L

(z+w)^k

|z+w|^2s,

where the sum in taken over allw∈L, except −z if z∈L. This series converges for Re(s)> ^k₂ + 1, and it has an analytic continuation to the whole complex s-plane. The non-holomorphic Eisenstein series E_k^∗(z,L) is defined by

E_k^∗(z, L) :=H_k(z, k,L).

Furthermore, it is well-known that (see [10, Corollary 1.7] for a proof and the definition of A(L)) for any z₀ ∈C\L, we have

d

dzlog Θ(z+z₀) = z₀A(L)⁻¹+

∞

X

k=1

(−1)^k−1E_k^∗(z₀,L)z^k−1. (4.1.7) By [10, Theorem 1.9], for any principal integral ideala= (α) with (a,6f) = 1, we have

Θ²(z, Lσ)^Na

Θ²(z, α⁻¹L_σ) = ^Y

w∈α⁻¹Lσ/Lσ

w̸=0

(℘(z, L_σ)−℘(w, L_σ))⁻¹,

so we can write

R^σ_a(Φ(z, L_σ))² =c_E(a)² Θ²(z, L_σ)^Na Θ²(z, α⁻¹L_σ),

since the product in the definition ofR^σ_a was over the representatives ofE_a\{O}modulo {±1}, and x(⊖P) = x(P) for any P. Let ρ= Ω∞/f so that our choice of Q^σ is given by Φ(Λ(s)ρ, L_σ). Moreover, letB be a set of integral ideals of K prime to f such that

Gal(H(E_f)/K) = {(b, H(E_f)/K),b ∈ B},

where (b, H(E_f)/K) denotes the Artin symbol of b for H(E_f)/K. Hence, we have R^σ_a(Φ(z, L_σ)) = ^Y

b∈B

R^σ_a (Φ(z+ϕ_K(b)Λ(s)ρ, L_σ)).

Noting thatA(α⁻¹L_σ) = Nα⁻¹A(L_σ) and thatE_k^∗ is homogeneous of degree−k, we obtain

d

dzlogR^σ_a(Φ(z, L_σ)) =

∞

X

k=1

(−1)^kX

b∈B

NaE_k^∗(ϕ_K(b)Λ(s)ρ, L_σ)−α^kE_k^∗(ϕ_K(b)αΛ(s)ρ, L_σ)z^k−1.

By [10, Proposition 5.5], we have ϕ^k_K(s)

Ns^k−s

(Λ(s)ρ)^k

|Λ(s)ρ|^2sL(ϕ^k_K, σ, s) = ^X

b∈B

H_k(ϕ_K(b)Λ(s)ρ,0, s, L_σ), and similarly,

ϕ^k_K(sa) N(sa)^k−s

(Λ(s)αρ)^k

|Λ(s)αρ|^2sL(ϕ^k_K, σσ_a, s) = ^X

b∈B

H_k(ϕ_K(b)Λ(s)αρ,0, s, L_σ).

Putting s=k, we obtain d

dzlogR^σ_a(Φ(z, L_σ)) =

∞

X

k=1

(−1)^kϕ^k_K(s)Λ(s)^−kf^kΩ^−k_∞ NaL(ϕ^k_K, σ, k)−ϕ^k_K(a)L(ϕ^k_K, σσ_a, k)z^k−1.

The result now follows on noting that σ_a = 1 because a= (α) is principal.

Let us remark that Proposition 4.1.7 is true for all integers k > 1. However, the Hecke L-function will no longer be primitive when k is even, for example, if k ≡ 0 mod #(∆), because in this case the conductor of ϕ^k_K is (1). Thus we shall first concentrate on the case k ≡1 mod #(∆) but the arguments extend readily to k ranging over any fixed residue class modulo #(∆).

Lemma 4.1.8. For any positive integer k ≡1 mod #(∆), we have Λ(s)^−kΩ^−k_v

Z

H

χ^k_pdµa,σ=−(k−1)!ϕ^k_K(s)f^k Λ(s)^kΩ^k_∞

Na−ϕ^k_K(a) L(ϕ^k_K, σ, k)−ϕ^k_K(p)

Np L(ϕ^k_K, σσp, k)

! .

Proof. We have λ_E(p)Φ(z, L_σ) = Φ(Λ(p)^σz, L_σσp) and Λ(sp) = Λ(s)Λ(p)^σ, so

d dz

k

logR^σσ_a ^p(λ_E(p)Φ(z, L_σ))|_z=0 =−(k−1)!ϕ^k_K(sp)f^k Λ(s)^kΩ^k_∞

Na−ϕ^k_K(a)L(ϕ^k_K, σσ_p, k)).

Therefore,

d dz

k

log Ψ^σ_a(Φ(z, L_σ))|_z=0=−c_k(a)(k−1)!ϕ^k_K(s)f^kNp

Λ(s)^kΩ^k_∞ L(ϕ^k_K, σ, k)−ϕ^k_K(p)

Np L(ϕ^k_K, σσ_p, k)

! , (4.1.8)

where c_k(a) = Na −ϕ^k_K(a). Combining (4.1.5), (4.1.6), (4.1.8) and the fact that Ω_σ,v = Λ(s)Ω_v by Lemma 4.1.6, the proof of Proposition 4.1.8 is complete.

Let

Da,σ(k) =ϕK(s)^−k

Z

H

χ^k_pdµa,σ, and define

G^∗ = Hom(G,C^×p),

whereG= Gal(H/K) as before. Then for each χ∈G^∗, define D_a(χ, k) = ^X

σ∈G

χ(σ)D_a,σ(k).

It is easy to see that

Da(χ, k) = ck(a)(k−1)!f^k Λ(s)^kΩ^k_v Λ(s)^kΩ^k_∞

X

σ∈G

χ(σ)L(ϕ^k_K, σ, k) 1−ϕ^k_K(p)χ⁻¹(σ_p) Np

!

.

We let C denotes a set of integral ideals representing of the ideal class group of K with (c,pf) = 1 for any c∈ C, and set Ω∞(E/H) =^Q_c∈CΛ(c)Ω∞ and Ω_p(E/H) =

Q

c∈CΛ(c)Ω_v. Recalling

L(ψ^k_E/H, k) = ^Y

χ∈G^∗

X

σ∈G

χ(σ)L(ϕ^k_K, σ, k)

and the factorisation of primes of K in H given by class field theory, we immediately obtain the following.

Lemma 4.1.9. For any positive integer k ≡1 mod #(∆), we have

Y

χ∈G^∗

D_a(χ, k) =c_k(a)^h((k−1)!)^hf^khΩ_p(E/H)^kΩ_∞(E/H)^−kL(ψ^k_E/H, k)·^Y

w|p

1−ψ^k_E/H(w) Nw

! .

DefineG= Gal(F_∞/K). Write µa for the measure on G satisfying

Z

G

χ^k_pdµ_a := ^Y

χ∈G^∗

Z

G^χ

χχ^k_pd(µ_a)^χ= ^Y

χ∈G^∗

X

σ∈G

χ(σ)ϕ_K(s)^−k

Z

H

χ^k_pdµ_a,σ.

Thus

Lemma 4.1.10. There exists a measure µ_a in Λ_I(G) such that for all k > 1, k ≡ 1 mod #(∆), we have

Ω_p(E/H)^−k Z

G

χ^k_pdµ_a=c_k(a)^h((k−1)!)^hf^khΩ_∞(E/H)^−kL(ψ^k_E/H, k)·^Y

w|p

1−ψ_E/H^k (w) Nw

! .

Note that, on the right hand side the equation in Lemma 4.1.10, the only dependence ona occurs in the factor c_k(a)^h. We remove this factor in the next theorem.

Theorem 4.1.11. There exists a unique measure µ_E ∈ Λ_I(G) such that, for all integers k >1 with k ≡1 mod #(∆), we have

Ω_p(E/H)^−k

Z

G

χ^k_pdµ_E = ((k−1)!)^hf^khΩ∞(E/H)^−kL(ψ^k_E/H, k)·^Y

w|p

1−ψ^k_E/H(w) Nw

!

.

Proof. Given an integral ideal a with (a,6pf) = 1, let θ_a be the measure satisfying

Z

G

χ^k_pdθ_a =c_k(a)^h.

Then µ_E =µ_a/θ_a is independent ofa. In order to show that this is an integral measure, it suffices to show that for some a, c_k(a)^h is a unit in I. Pick a= (α) with (a,p) = 1 such that α ≡ 1 modf and α ̸≡ 1 modp^∗. This is possible since we have p^∗ - f by hypothesis. Then ϕ_K(a) =α, σ_a= 1, and c₁(a)^h =α^h(α^∗−1)^h is a unit in O_p. This shows that there is a unit in I[[G]] whose values atk with k ≡1 mod #(∆) is equal toc_k(a)^h.

Theorem 4.1.11 asserts the existence of a good p-adic L-function. Now, let us assume (p, h) = 1. Define Σ = Gal(F∞/K∞) where K∞ is the maximal Zp-extension of K inside F∞, so that Σ≃∆×G. Furthermore, identify Γ with Gal(K∞/K). Let H∞=HK∞ andK∞ is the uniqueZp-extension ofK unramified outside p, and define G = Gal(H∞/K).

F∞

F H∞

H K∞

K

Γ

∆

Σ

G

H

Γ G G

Givenθ ∈Σ^∗, let Λ_I(G)^θ denote the largest submodule of Λ_I(G) on which Σ acts viaθ. If p >2, then #(Σ) is prime to p, so the idempotent e_θ = _#(Σ)^{1 P}_τ∈Σθ⁻¹(τ)τ lie inside Λ_I(G). Thus we can decompose Λ_I(G) = I[[G]] =I[Σ][[Γ]] in the form

Λ_I(G) = ⊕_θ∈Σ^∗e_θΛ_I(Γ).

Therefore, we can write anyµ∈Λ_I(G) as a sum of the form µ=⊕_θ∈Σ^∗e_θµ^θ,

whereµ^θ is an element of Λ_I(Γ).

Ifp = 2, #(∆) = 2 is not coprime top. In this case, let δ denote the non-trivial element of ∆. We have Λ_I(G) = I[∆][[G]]. We claim (1−δ)I[∆] = (1−δ)I. Indeed, let a₁+a_δδ ∈ I[∆]. Then (1−δ)(a₁ +a_δδ) = (1−δ)(a₁ −a_δ)∈ (1−δ)I. Hence (1− δ)I[∆] ⊂ (1−δ)I, and the other inclusion is clear. It follows that (1−δ)Λ_I(G) = (1−δ)Λ_I(G). Hence, givenµ∈Λ_I(G), there exists uniqueµ⁻ ∈Λ_I(G) satisfying (1−δ)µ= (1−δ)µ⁻. Similarly, we have (1 +δ)Λ_I(G) = (1 +δ)Λ_I(G), so givenµ∈Λ_I(G) there existsµ⁺ ∈Λ_I(G) such that (1+δ)µ= (1+δ)µ⁺. Furthermore, we have

µ= 1 2

(1−δ)µ⁻+ (1 +δ)µ⁺.

Finally, since (p, h) = 1 by assumption, we can further decompose µ⁻ and µ⁺ as elements in ⊕χ∈G^∗e_χΛ_I(Γ).

For k≡1 mod #(∆), we have χ^k_p(1 +δ) = 0 and χ^k_p(1−δ) = 2. Thus, we obtain µ_E = 1

2(1−δ)µ⁻_E

interpolating the values ofL(ψ^k_E/H, k) fork ≡1 mod #(∆) from the above construction.

Finally, we remark that our methods readily give an analogue of Theorem 4.1.11 for the p-adic interpolation ofL(ψ^k_E/H, k) when k ranges over any fixed residue class modulo #(∆). However, the HeckeL-function will no longer be primitive when k is even, and in particular, when k ≡0 mod #(∆), because in this case the conductor of ϕ^k_K is (1). LetS denote the set of primes of H dividing f. Then Theorem 4.1.11 gives the imprimitive Hecke L-function L_S(ψ^k_E/H, k) =L(ψ^k_E/H, k)·^Q_v∈S 1−^ψ

k E/H(v)

Nv^k

!

on the complex side.

The aim of the next section will be to obtain ap-adic L-function which interpolates the values of theL(ψ^k_E/H, k) for k even. This will give rise to the p-adic L-function for H∞/H for all p, and as we shall see in Chapter 7, will be an essential ingredient for the main conjecture for E/H for p= 2.

In document On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication (página 56-74)