FONCTIONS ZÊTA p-ADIQUES EN s = 0 - IMJ-PRG

p s = 0
p s = 0
p
p
L
p
p
p
KOKK S∞
K S K
S∞OK,S S ζK,∞,S

K s = 1
ζK,∞,S∞K
ζK,∞,S (s)∼ −h(OK,S )R∞(O∗
K,S )s|S|−1(1)
s= 0 h(OK,S )OK,S R∞(O∗
K,S )
p
L p
R∞(O∗
K,S )
p
p
L p
L
Q C Cp
Q Q K0K0
Hom(K, Q) Σ
S∞Q C
KΣ Hom(K, Q)
KΣS∞
K0KΣ0K0Σ
Hom(K, Q)K0Σ0Σ
K0(K0,Σ0)K0=QK
K
Q
ΣK p Σ
Hom(K, Q)−ΣSpK p

p s = 0
Q CpK
Hom(K, Q)K p p
K p
K0K K+
0
K0K+
0p
K0
ΣK
pΣ Σ SpΣ∗Hom(K, Q)−Σ
SpΣ∩Σ∗=∅Σ∪Σ∗=Spv
K| |v,∞R+K
v
| |v,∞v
| |v,∞Qv
v K Sp∪S∞x∈K|x|v,Σ=
|x|v,∞∈Q⊂Cpv∈S∞σvΣv|x|v,Σ=
σv(x)∈Q⊂Cpv∈Σ∗x∈K|x|v,Σ=NKv/Qp(x)
| |v,Σv∈Σ
?∈ {∞,Σ}S={v1, . . . , v|S|}K
S∞?=Σ S S =S∞
S∩Sp= Σ∗U= (u1, . . . , u|S|−1) (|S| − 1) O∗
K,S
M?(U)i j
log?|uj|vi,?log∞R+logΣ
CplogΣp= 0
Pv∈Slog?|u|v,?= 0 u∈O∗
K,S
? = ∞
|u|v,∞= 1 u∈O∗
K,S v /∈S?=Σ S∩Sp= Σ∗
Qv∈S|u|v,Σp u ∈O∗
K,S
S=S∞ΣK0K+
0
K0OK+
0
OK0Σ0K0Σu∈O∗
K+
0
Qσ∈Σ0σ(u) = NK+
0/Q(u)∈ {±1}u∈O∗
K
Y
σ∈Σ
σ(u) = Y
σ∈Σ0
σ(NK/K0(u))
U
hUiO∗
K,S UO∗ K,S Mi
?(U)
M?(U)i

1
[O∗
K,S :hUi]sign(det Mi
∞(U)) det Mi ?(U)
U i R?(O∗
K,S )
O∗
K,S RΣ(O∗ K,S )
RΣ(O∗
K,S )
Q Cp
KΣ
KHKHom(K, Q)χ
K A0mχ
χA∗
KK
K∗K
mχT(χ) = Pσ∈Hom(K,Q)kσ(χ)σ HK
χ χ (α)α
χ((α)) = Qσ∈Hom(K,Q)σ(α)−kσ(χ)α≡1[mχ]α
K N
−Pσ∈Hom(K,Q)σ K kσ(χ)
σ k(χ)
χΣK χ
K kσ(χ)≥1σ∈Σ
kσ(χ)≤0σ /∈Σ Σ
K
(k(χ)≥1χ((α)) = Qσ∈Hom(K,Q)σ(α)−k(χ)
k(χ)≤0χ((α)) = Qσ∈Hom(K,Q)sign(σ(α))σ(α)−k(χ)si α≡1 [mχ].
N((α)) = Y
σ∈Hom(K,Q)sign(σ(α))σ(α)
k(χ)
χNk(χ)
χΣkσ(χ) = 0 σ /∈Σ
K
v K p KvK v Ov
Kvχvχ K∗
vγvOv
v(γv) = v(mχdK)x∈Qpexp(2iπx) exp(2iπx) =

p s = 0
exp(2iπ˜x) ˜x x Qp/Zp⊂Q/Ze
v(mχ)
Wv(χ) = 
χv(γv)v(mχ) = 0
χv(γv)N(v)−ePx∈(Ov/πe
v)∗χv(x−1) exp2iπTrKv/Qp(γ−1
vx)v(mχ)≥1
Wv(χ)γve Wv(χ)
pN(v)v(mχ)≥1
L
χΣkσ(χ)≥1
σ∈Σkσ(χ)≤0σ∈ΣK
M(χ)χ M(χ)+
B,τ
τ M(χ)Q
|S∞|
χΣK L(χ, s)L
Λ∞(χ) = 
Qσ∈Σ
Γ(kσ(χ))
(2iπ)kσ(χ)L(χ, 0)
L(χ, 0)
KΣ
KΛ∞(χ)
Λ∞(χ)
K K Σ (K0,Σ0)
A K0Q
Σ0σ∈Σ0ωσQ
H1
DR(A)α∈K0σ(α)u
H1(A(C),Q)K0
Ω∞(K0,Σ0, σ) = Ruωσχ
ΣK
˜
Λ∞(χ) = Λ∞(χ)Y
σ∈Σ0
Ω∞(K0,Σ0, σ)kσ(χ)−kσ(χ)[K:K0]∈Q.
p p
Ωp(K0,Σ0, σ)CpB+
DR,p Ω∞(K0,Σ0, σ)
p