Sur la fidélité de certaines représentations de GL2(F) sous une

GL2(F)
∗ † ‡
FQpOFE
FpO×
FOF
E[[OF]]
E[[OF]] O×
F
GL2(OF)
GL2(F)
δ
p F QpE
FpG= GL2(F)F=Qp
E
G F 6=Qp
∗
†
‡

U
G
OFF π G E
U π
E[[U]] π
E[[U]] π
π π
E[[U]] π
π E[[U]]
F=QpE[[U]] 'E[[X]]
E[[U]]
[F:Qp]
π G
E[[U]]
Γ = 1 + pOF0
0 1⊂G U
E[[U]]
I⊂E[[U]]
ΓI0 [F:Qp]I= 0 I
E[[U]]
FQp
p E p
FQpnOFFq=Fpf
e F Qpn=ef
G= GL2(F)G
U=1OF
0 1 et Γ = 1 + pOF0 0 1.
UΓp≥3p= 2 Γ2Γ
U G U
E[[U]] U E
E[[U]] := lim
←−
U0
E[U/U0]
U0U
u−1u∈U
7.4

U/Up'Fq[X]/(Xe)$∈ OF
λFqFp
E[[Xi,k; 0 ≤i≤e−1,0≤k≤f−1]] ∼ −→ E[[U]]
Xi,k 7−→ 1$i[λk]
0 1 −1
0≤i≤e−1 0 ≤k≤f−1 [λk]∈ OF
λk
a b A(a:b)
x∈A xb⊂a
VFpP(V)V
1V S ⊂P(V)Ql∈Sl
Ql∈Swl∈Sym(V)wll l ∈S
P(Sym(V))
WFpW∗
FpWFp
δ
A=E[[X1, ..., Xn]] n
EmAm
deg m
x∈Adeg(x)k x ∈mk
deg(0) = +∞deg Adeg(x+
y)≥min{deg(x),deg(y)}deg(xy) = deg(x) + deg(y)x, y ∈A r ∈[0,+∞[
mrx∈Adeg(x)≥r
pA x ∈A ν(x)x
A/p m A/pk x ∈p+mk
ν(x) = +∞x∈Tk≥1(p+mk)A
mAp=Tk≥1(p+mk)
ν(x)=+∞x∈pν(x)≥deg(x)ν
ν(xy)≥ν(x)+ν(y)
x, y ∈A
x, P ∈A
δ(x, P ) := sup
k≥0ν(xkP)−deg(xkP)∈N∪ {+∞}
δ(x, P )=+∞x∈pP∈pδ(x)δ(x, 1)

ν(xk+1P)−deg(xk+1P)≥ν(xkP)−deg(xkP) + ν(x)−deg(x),
k7→ ν(xkP)−deg(xkP)
δ(x, P ) = lim
k→∞ν(xkP)−deg(xkP)
x, P ∈A
gr(A) = Lk≥0mk/mk+1 A
E[X1, ..., Xn]x∈Agr(x) :=
xmod mdeg(x)+1 ∈gr(A)xgr(p)p
gr(p) = {gr(x), x ∈p}.
gr(A)
gr(p) := M
k≥0
(p∩mk)/(p∩mk+1)⊆gr(A)
δ
x∈A
δ0 +∞δ(x)=+∞ ⇔ gr(x)∈pgr(p)
δ(x) = 0 = +∞P /∈pδ(x, P )<+∞
= +∞
x6= 0 δ(x)=+∞ > 0k0≥0k≥k0
ν(xk)≥(1 + ) deg(xk).
gr(x)∈pgr(p)k0≥1
ν(xk0)≥deg(xk0)+1.
gr(x)∈pgr(p)
ν(xmk0)≥mν(xk0)≥m+ deg(xmk0),
δ(x)=+∞gr(x)/∈pgr(p)ν(xk) = deg(xk)
k≥1ν(x)=0
δ(x) = +∞δ(x, P ) = +∞
P∈A P = 1
ν(xkP)≥ν(xk) + deg(P)δ(x, P )=+∞δ(x)=+∞
δ(x)=0
P /∈pk0≥0
((p+mk) : P)⊆(p:P) + mk−k0=p+mk−k0,∀k≥k0

P /∈pk
ν(xkP)≥k0
ν(xk)≥ν(xkP)−k0,
ν(xkP)−deg(xkP)≤(ν(xk)−deg(xk)) + (k0−deg(P)) = k0−deg(P).
P= 1
k0≥1ν(xk0)≥1 + deg(xk0)
δ(x) = +∞deg(x)6= 0 k≥k0m=bk/k0c
k/k0
ν(xk)≥m+ deg(xk)≥deg(xk)1 + m
kdeg(x)≥deg(xk)1 + 1 2k0deg(x).
=1
2k0deg(x)
U=1OF
0 1 E[[U]] U I
E[[U]] V U I V
V I I E[[V]]
I= (I∩E[[V]]) ·E[[U]].
I $U
I $U I
∂
∂X0,k 0≤k≤f−1
∂
∂X0,k E[[U]] Γ
VFp
Sym(V) Sym(V)
HomSym(V)(Sym(V)⊗FpV, Sym(V)) FpV
Sym(V)Fpφ
VSym(V)φpnFpV
Sym(V)φpn(v) = φ(v)pnv∈V1.4
g V g
Sym(V)gSym(V)
φpnn0≤n≤n0+ dim(V)−1