REPRÉSENTATIONS p-ADIQUES SURCONVERGENTES - IMJ-PRG

p
K0
p p
K
K0
p p K
(ϕ, Γ) p
e
E
e
B
(ϕ, ΓK)GK
ψ
B†
F
ΓD†(V)ψ=0
ΓKD†(V)ψ=0
e
AKe A†
K
e
B†e B†
Ke
B†
Kϕ−∞(B† K)
p

pGKKQp
(ϕ, Γ)
V(ϕ, Γ) D(V)
V
V D(V)
pGK
HK
HKHK
GK
HK
ΓK=GK/HK
pGK
Cp
GK
VBcris,Bst,BdR, . . .
DdR(V), Dcris(V). . . D(V)
DdR(V)
D(V)
VLog(h)
V
D(V)
ΓK
GK
p
Qp
p

p
(ϕ, Γ) p
e
Ee A A e B B Frac(R)
W(Frac(R)) Od
Enr W(Frac(R))[1
p]d
Enr
kFpOF=W(kF)
kFF=OF[1
p]OF
F F F
FGFGal(F /F )χ:GF→Z∗
p
K F n ∈NKn=K(µpn)⊂F K∞
K KnGKGal(F /K)
HKχGKHK= Gal(F /K∞) ΓK=GK/HK=
Gal(K∞/K)
e
E
b
F F p e Ex= (x(0), . . . , x(n), . . . )
b
F(x(n+1))p=x(n)e E+·x+y=s
s(n)= lim
m→+∞(x(n+m)+y(n+m))pmx·y=t t(n)=x(n)y(n)e
E
p vEvE(x) = vp(x(0))
e
E+e E
ε= (1, ε(1), . . . , ε(n). . . )e Eε(1) 6= 1 n>1
ε(n)pnvE(ε−1) = p
p−1EF
kF((ε−1)) e E
E EFe E E+
e
EGal(E/EF)
HFK F
EK=EHK,e
EK=e EHK,E+
K= (E+)HKet e
E+
K= (e
E+)HK.
EKEFHF/HK= [K∞:F∞]e EK
e
E+
KE+ K
e
EKEK
e
B
e
A=W(e E)e Ee B=e A[1
p] = Frac( e
A)
e
Be A
e
Ex∈e E[x]e A
xe AP+∞
i=0 pi[xi]e
B
P+∞
i−∞ pi[xi]

x=P+∞
i=0 pi[xi]∈e
Bk∈Zwk(x) = infi6kvE(xi)wk
wk(x) = +∞x∈pk+1 e A
wk(x+y)>inf(wk(x), wk(y)) wk(x)6=wk(y)
wk(xy)>infi+j6k(wi(x) + wj(y))
wk(ϕ(x)) = pwk(x)
r∈Rk∈NUk,r ={x∈e A|wk(x)>r}e A
Uk,r 0
x→(xk)k∈Ne Ae ENe E
vEe B=∪
n∈Np−ne
A
GFe Ee Ae B
ϕ
π= [ε]−1AFe AOF[π, 1
π]e
A
Pn∈ZanπnanOF0n
−∞ AFEF
ϕ(π) = (1 + π)p−1 et g(π) = (1 + π)χ(g)−1 si g∈GF,
AFBF=AF[1
p]ϕGF
B BFe B
A=B∩e A B =A[1
p]A
B E A B
ϕGF
K F
AK=AHK,BK=BHK,e AK=e AHKet e BK=e BHK.
K=FAKBKAKe AK
EKe
EK
BK=AK[1
p]e
BK=e AK[1
p]e
AKe BK
ϕ−∞(AK) = ∪
n∈Nϕ−n(AK)ϕ−∞(BK) = ∪ n∈Nϕ−n(BK)
e
AKAK
L K BLBK[L∞:K∞]
L/K e BL/e BKBL/BK
Gal( e BL/e BK) = Gal(BL/BK) = Gal(EL/EK) =
Gal(L∞/K∞) = HK/HL
K F γ ΓK
Bγ=1
KK∞∩Fnr F
Aγ=1
KOFnr
AKpEγ=1
K⊂kF
x∈Eγ=1
KvE(x)>0Eγ=1 KkF((x))
EK/Eγ=1
Kγ

p
Γ ΓKΓKΓEγ=1
K
ΓKEK
σ(ε) = εχ(σ)6=ε σ 6= 1
(ϕ, ΓK)GK
ZpGKZp
ZpGK
pGKQp
GK
K F (ϕ, ΓK)AK
BKAKBK
ΓKϕΓK
(ϕ, ΓK)DAKBK0D
AKϕ(D)DAK
K F V Zpp
GK
D(V) = (A⊗
Zp
V)HK.
ϕAKGKD(V)ϕ
GK/HK= ΓKD(V) (ϕ, ΓK)ϕ(A)
A AKBKVZp
pGK
D(ϕ, ΓK)AKBK
V(D) = (A⊗
AK
D)ϕ=1.
ZppGK
DAKBKGKD(V)
GKBVZpp
GKV(D(V)) VGK
V(ϕ, ΓK)D(V)V
pGKdimBKD(V) = dimQpV
B⊗
BK
D(V)−→ B⊗
Qp
V
VQpD(V)BKV
ψ
Bp ϕ(B)
ψ:B→Bψ(x) = 1
pϕ−1TrB/ϕ(B)(x)
1,[ε], . . . , [ε]p−1Bϕ(B)ψ(x0+x1[ε]+· · ·+xp−1[ε]p−1) =
ϕ−1(x0)x0, . . . , xp−1∈ϕ(B)ψGKψ(ϕ(x)) = x
x∈Bψ(A)⊂A