Finally, since (p, h) = 1 by assumption, we can further decompose µ− and µ+ as elements in ⊕χ∈G∗eχΛI(Γ).
For k≡1 mod #(∆), we have χkp(1 +δ) = 0 and χkp(1−δ) = 2. Thus, we obtain µE = 1
2(1−δ)µ−E
interpolating the values ofL(ψkE/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(ψkE/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 ϕkK is (1). LetS denote the set of primes of H dividing f. Then Theorem 4.1.11 gives the imprimitive Hecke L-function LS(ψkE/H, k) =L(ψkE/H, k)·Qv∈S 1−ψ
k E/H(v)
Nvk
!
on the complex side.
The aim of the next section will be to obtain ap-adic L-function which interpolates the values of theL(ψkE/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.
Definition 4.2.1. Gk(L) = P
w∈L\{0}
1
wk for k >3, G2(L) = lim
s→0+
X
w∈L\{0}
w−2|w|−2s,
and G1(L) = 0.
Proposition 4.2.2. Let s be an integral ideal of K prime to f such that σs =σ. Then for any D= (ai, ni)∈I and k >2 an even integer, we have
d
dz logRσD(P) =
r
X
i=1
∞
X
kk=2even
−ni ϕkK(s) Λ(s)kΩk∞
Nai−ϕkK(ai)L(ϕkK, σ, k))zk−1.
In particular, we have d
dz
!k
logRDσ(P)|z=0 =
r
X
i=1
−ni(k−1)! ϕkK(s) Λ(s)kΩk∞
Nai−ϕkK(ai)L(ϕkK, σ, k)).
Proof. We modify the usual σ-function slightly, and define Θ(z, L) = exp
(
−s2(L)z2 2
)
σ(z, L).
Then for any integral ideala= (α) with (a,6f) = 1, we have Θ2(z, Lσ)Na
Θ2(z, α−1Lσ) = Y
w∈α−1Lσ/Lσ
w̸=0
(℘(z, Lσ)−℘(w, Lσ))−1,
so we can write
Rσa(Φ(z, Lσ))2 =cE(a)2 Θ2(z, Lσ)Na Θ2(z, α−1Lσ),
since the product in the definition of Raσ was over the representatives of Ea\{O}
modulo {±1}, and x(⊖P) = x(P) for any P. Hence, RDσ(Φ(z, Lσ))2 =
r
Y
i=1
cE(ai)2 Θ2(z, Lσ)Nai Θ2(z, α−1i Lσ)
!ni
.
Now, (4.1.7) gives d
dz log Θ(z, Lσ) =
∞
X
k=1
(−1)k−1Gk(Lσ)zk−1
and Gk(Lσ) = 0 for k odd. Therefore, d
dz logRσD(Φ(z, Lσ)) =
r
X
i=1
X
k>2 keven
−ni(NaiGk(Lσ)−Gk(α−1i Lσ))zk−1
=
r
X
i=1
X
k>2 keven
−ni(Nai−αki)Gk(Lσ)zk−1
by the homogeneity of Gk.
Let b be an ideal of K. Setting k = s , g = (1) and ρ = Ω∞ in [10, Proposition 5.5], we obtain that the partial Hecke L-function L(ϕkK, σb, k) is identically equal to
Gk(Lσb) = ϕkK(b)
Λ(b)kΩk∞L(ϕkK, σb, k).
Hence, setting b=s, we obtain
(Nai−αki)Gk(Lσ) = Nai
ϕkK(s)
Λ(s)Ωk∞L(ϕkK, σ, k)− ϕkK(sai)
Λ(s)Ωk∞L(ϕkK, σ, k)
= ϕkK(s) Λ(s)kΩk∞
Nai−αkiL(ϕkK, σ, k).
This completes the proof of the proposition.
Define
ΨσD(P) = RσD(P)Np RσσDp(λEσ(p)(P)). Then we have
Y
R∈Epσ
ΨσD(P ⊕R) = 1. (4.2.1) Let AσD(t) be the development as a power series int of the rational function ΨσD(P).
Then as before, AσD(t) ∈ 1 +POP[[t]], and so CDσ(t) = Np1 logAσD(t) ∈ OP[[t]]. Let JDσ(W) =CDσ◦δσ,v(W)∈I[[W]]. LetµD,σ be theI-valued measure onZp determined by JDσ(W). Then µD,σ is supported on Z×p, and writing also µD,σ for the corresponding measure on ΛI(H), we have
Z
H
χkpdµD,σ =
Z
Zp
xkdµD,σ =DkJDσ(W)|W=0, (4.2.2)
where D = (1 +W)dWd . We have an isomorphism ˆGm
−→∼ Gˆa given by W 7→ ez−1, hence we see immediately that D= dzd. Moreover, we have δσ,v(W) = Ωσ,vW +· · ·, so
DkJDσ(W)|W=0 = d dz
!k
(JDσ(ez−1))|z=0 = 1
NpΩkσ,v d dz
!k
log ΨσD(Φ(z, Lσ))|z=0. (4.2.3) Lemma 4.2.3. For an even integer k >2, we have
Λ(s)−kΩv−k
Z
H
χkpdµD,σ =
r
X
i=1
−ni(k−1)!ϕkK(s)Λ(s)−kΩ−k∞ck(ai) L(ϕkK, σ, k)− ϕkK(p)
Np L(ϕkK, σσp, k)
!
,
where we recall that ck(ai) = Nai−αki.
Proof. We have λE(p)Φ(z, Lσ) = Φ(Λ(p)σz, Lσσp) and Λ(sp) = Λ(s)Λ(p)σ, so d
dz
!k
logRσσDp(λE(p)Φ(z, Lσ))|z=0 =
r
X
i=1
−ni(k−1)! ϕkK(sp)
Λ(s)kΩk∞ck(ai)L(ϕkK, σσp, k).
Therefore, d dz
!k
log ΨσD(Φ(z, Lσ))|z=0 = (4.2.4)
r
X
i=1
−ni(k−1)!ϕkK(s)Np
Λ(s)kΩk∞ck(ai) L(ϕkK, σ, k)−ϕkK(p)
Np L(ϕkK, σσp, k)
!
.
Combining (4.2.2), (4.2.3) and (4.2.4), the proof of the proposition is complete.
Let
DD,σ(k) =ϕK(s)−k
Z
H
χkpdµD,σ,
and put for each χ∈G∗,
DD(χ, k) = X
σ∈G
χ(σ)DD,σ(k).
Defining
M(χ, k) = X
σ∈G
χ(σ)L(ϕkK, σ, k)Λ(s)kΩkv Λ(s)kΩk∞,
we conclude immediately that
DD(χ, k) =ck(D)(k−1)!M(χ, k) 1− ϕkK(p)χ−1(σp) Np
!
,
whereck(D) =Pri=1−nick(ai). LetC denotes a set of integral ideals representing of the ideal class group ofK with (c,pf) = 1 for anyc∈ C, and set Ω∞(E/H) =Qc∈CΛ(c)Ω∞
and Ωp(E/H) = Qc∈CΛ(c)Ωv. Taking the product over χ∈G∗, we obtain Lemma 4.2.4. For any even integer k >2, we have
Y
χ∈G∗
DD(χ, k) =ck(D)h((k−1)!)hΩp(E/H)kΩ∞(E/H)−kL(ψkE/H, k)·Y
w|p
1− ψE/Hk (w) Nw
!
.
Lemma 4.2.5. There exists a measure νD in ΛI(G) such that for all k > 1, k ≡ 0 mod #(∆), we have
Ωp(E/H)−k
Z
G
χkpdνD =ck(D)h((k−1)!)hΩ∞(E/H)−kL(ψkE/H, k)·Y
w|p
1− ψE/Hk (w) Nw
!
.
Note that, since k ≡ 0 mod #(∆), we have χkp(τ) = 1 for any τ ∈ ∆. Hence, we can naturally considerνD as an element of ΛI(G). Again, the only dependence ofνD on D occurs in the factor ck(D)h. We claim that we can remove this factor and obtain a pseudo-measure which is independent of D.
Lemma 4.2.6. There exists D ∈ I and θD ∈ ΛI(G) such that θD|Γ generates the augmentation ideal of ΛI(Γ)⊂ΛI(G) and
Z
G χkpdθD =ck(D)h. for all k>1.
Proof. Choose α∈ OK so that α≡1 modpm+1, α≡1 +pm modp∗m+1 where m= 1 or 2 according as p > 2 or p = 2, and define a = (α). Take a1 = a, a2 = a, n1 = 1, n2 =−1. Then ({a1,a2},{n1, n2})∈I. Writeσa for the Artin symbol (a, H∞/K) of a for the extensionH∞/K. Note that (a, H/K) = 1 since a is principal, so that we can considerσa as an element of Γ.
We will show that the measure
θD=−(Na−σa−(Na−σa))
=σa−σa,
has the desired property. Indeed, we have χkp(θD) = ck(D)h, so it remains to show that θD|Γ generates the augmentation ideal of I[[Γ]]. In order to do this, let us fix a topological generator of γ of Γ, and write σa|Γ = γa, σa = γb where a, b ∈ Zp. It suffices to show that θD|Γ = (1−γ)·u for u ∈ Zp[[Γ]]×. Now, we have Γ ≃ Zp and
1
pm log : 1 +pmZp →Zp sending 1 +pmx7→ P∞
i=1
(−1)i−1xni is an isomorphism. Hence pm+1 |α−1 implies a≡0 modp, and α∗ generates 1 +pmOp so b̸≡0 mod p. Now,
σa|Γ−σa|Γ =γa−γb
=γa(1−γb−a),
where clearlyγa is a unit, and alsob−a̸≡0 modp so 1−γb−a is a product of (1−γ) and a unit, as required
We define
νp =νD/θD. (4.2.5)
This is a pseudo-measure, since (1−γ)· θ1
D = (1−γ)· (1−γ)u1 whereu is a unit by the proof of Lemma 4.2.6.
The following is an immediate consequence of Lemma 4.2.5 and Lemma 4.2.6.
Theorem 4.2.7. There exists a unique elementνpbelonging to the quotient fieldΛI(G) such that, for all integers k >1 with k ≡0 mod #(∆), we have
Ωp(E/H)−k
Z
G χkpdνp= ((k−1)!)hΩ∞(E/H)−kL(ψkE/H, k)Y
v∈P
1− ψE/Hk (v) Nv
!
.
Furthermore, the denominator of νp is given by γ−1, so that(γ−1)νp∈ΛI(G).
Recall from Section 4.1 that we can decompose νp as a sum of elements ineχΛI(Γ) if we in addition assume that (p, h) = 1. Given χ ∈ G∗, let νpχ ∈ΛI(Γ) denote the χ-part of νp in the decomposition. Then we have shown that νpχ ∈ ΛI(Γ) for every χ̸= 1, and (γ−1)νpχ ∈ΛI(Γ) forχ= 1. Thus, identifying ΛI(Γ) with I[[T]] via the map sendingγ to 1 +T, we haveνpχ ∈I[[T]]/T whenχis trivial. The pseudo-measure νp will be used for the main conjecture of H∞/H.
Finally, we define the p-adicL-function attached toE/H. In Section 4.1 we showed that thep-adic L-functionµE ∈ΛI(G) interpolates that values of L(ψkE/H, k) when k is odd, and ofLS(ψkE/H, k) when k is even, where S is the set of primes of H dividing f. Furthermore, our methods readily give an analogue of Theorem 4.2.7 to obtainνp which takes the value 0 at χkp fork odd and interpolates the values ofL(ψkE/H, k) for k even.
Define Ψp =µE +νp, where we consider νp as an element of ΛI(G). Explicitely, for p= 2 we can write
Ψp = 1 2
(1−δ)µ−E + (1 +δ)νp∈ΛI(G).
Then Ψp interpolates the values ofL(ψkE/H, k) fork ranging over all the residue classes modulo #(∆) in the following way.
Theorem 4.2.8. Given a positive integer k, we have
Ωp(E/H)−k
Z
G
χkpdΨp =
L(ψkE/H, k) 1 +fkhQv∈S 1− ψ
k E/H(v)
Nvk
!!
A(k) if k is even L(ψkE/H, k)fkhA(k) if k is odd,
where A(k) = ((k−1)!)hΩ∞(E/H)−kQw|p
1− ψ
k E/H(w)
Nw
.