Post on 23-Jun-2020
The p-adic Simpson
correspondence
Ahmed AbbesJoint work with Michel Gros
Padova, February 8, 2018.
K cdvf of char. 0, perfect res. field ofchar. p > 0, K, OK, OK, OC, C.
G = Gal(K/K) ⊃ I ⊃ P , It = I/P .
It∼→ lim
p-nµn(k), σ 7→ (σ(π1/n)/π1/n)p-n,
where π is a unif. of OK.
For prime ` 6= p, we get a hom. t`: I → Z`.
Proposition 1 (Grothendieck) Assumeno finite extension of k contains all rootsof the unity of order a power of `. Then,for any cont. rep. ρ of G on a finite dim.Q`-vector space V , there exists an opensubgroup J of I such that for any σ ∈ J,ρ(σ) is unipotent.
In fact, there exist an open subgroup J ⊂ Iand a nilpotent endomorphism N of V s.t.
ρ(σ) = exp(t`(σ)N), ∀σ ∈ J.
The local structure of p-adic cont. rep. ofG is due to Sen.
χ:GK → Z×p the cyclotomic character,log:Z×p → Zp the usual p-adic logarithm(log(x) =
∑n≥1(−1)n+1(x−1)n/n if x−1 ∈
pZp and log(x) = log(xp−1)/(p−1) in gen-eral).
K∞ the cyclotomic Zp-extension of K con-tained in K, i.e., the unique Zp-extensionof K contained in the sub-field of K gen-erated by the p-power order roots of 1.
H = Gal(K/K∞), Γ = Gal(K∞/K), L theclosure of K∞ in C.
We call C-rep. of G a cont. semi-linearrep. of G on finite dim. C-vector spaces.RepC(G) the cat. of C-rep. of G (idemRepL(Γ)...)
Theorem 2 (Sen) The functors
RepK∞(Γ)→ RepL(Γ), V 7→ V ⊗K∞ L,RepL(Γ)→ RepC(G), W 7→W ⊗L C,
are equivalences of categories.
The functor that associates to a C-rep. Xof G the L-rep. XH of Γ is a quasi-inverseof the second functor. Sen constructed anexplicit quasi-inverse of the first functor.
Theorem 3 (Sen) For any K∞-rep. V ofΓ, ∃ a unique K∞-endo. σ of V such that:∀x ∈ V , ∃ an open subgroup Γx ⊂ Γ s.t.
γ(x) = exp(logχ(γ) · σ)(x), ∀γ ∈ Γx.
Moreover, the characteristic polynomial ofσ has coefficients in K.
Let SK∞ be the category of finite dim.K∞-vector spaces equipped with a K∞-endo. We obtain an exact and faithfulfunctor
RepC(G)→ SK∞, W 7→ (V, σ).
Moreover, (V, σ) determines W .
Inspired by Sen’s theory and by the com-plex Simpson correspondence, Faltings ini-tiated in 2005 a project that aims at de-scribing all continuous p-adic rep. of thegeometric fundamental group of a propersmooth variety over K in termes of Higgsbundles.
I recently published a joint book with M.Gros and T. Tsuji that undertook a sys-tematic development of the p-adic Simp-son correspondence following two new ap-proaches, one by M. Gros and me and an-other by T. Tsuji.
S = Spec(OK), S = Spec(OK), s, η, η.
X a smooth S-scheme of f.t. s.t. Xη isconnected, x a geom. point of Xη.
RepQp(π1(Xη, x)) the cat. of p-adic rep.of π1(Xη, x), i.e., cont. rep. of π1(Xη, x)
on finite dim. Qp-vector spaces.
X the p-adic formal completion of X ×S S,S = Spf(OC).
HB(X/S ) the cat. of Higgs OX[1p]-bundles,
i.e., locally free OX[1p]-modules M with an
OX-linear morphism θ: M → M ⊗OXΩ1
X/Ssuch that θ ∧ θ = 0.
Our goal is to construct a functor
H :RepQp(π1(Xη, x))→ HB(X/S ).
If X is proper over S, we expect the fol-lowing properties:
(i) ∀V1, V2 ∈ RepQp(π1(Xη, x)), we have
HomQp[π1(Xη,x)](V1, V2)⊗Qp C∼→ HomHB(H (V1),H (V2)).
(ii) ∀V ∈ RepQp(π1(Xη, x)) satisfying anadmissibility condition, we have a can. isom.
H∗et(Xη,L(V ))⊗Qp C∼→ H∗(X,K•(H (V ))),
where L(V ) is the étale local system as-sociated to V and K•(H (V )) is the Dol-beault complex of H (V )
H (V )θ→H (V )⊗OX
Ω1X/S
θ∧→H (V )⊗OXΩ2
X/S . . .
(iii) The functor is not essentially surjec-tive in general, but we expect that its im-age consists of semi-stable Higgs bundleswith vanishing Chern classes.
Construction of H
Rep∗Qp(π1(Xη, x)) //
_
GR∗(2)
// _
HB∗(X/S ) _
RepQp(π1(Xη, x))(1)
//GR(3)
//HB(X/S )
GR the cat. of gen. rep. of π1(Xη, x), i.e.p-adic cont. semi-linear rep. of π1(Xη, x)on projective modules of f.t. over a certainring (analogue of C in Sen’s theory).
∗ either small rep. (i.e. rep. that are closefrom the trivial rep.) [Faltings]or Dolbeault rep. (an admissibility condi-tion à la Fontaine) [A.-Gros].
Tsuji proved recently that these two no-tions are equivalent. In fact, Faltings’ con-struction was initially incomplete for ratio-nal p-adic rep. Tsuji’s result completes hisconstruction and proves that it coincideswith ours.
Rep∗Qp(π1(Xη, x)) //
_
GR∗(2)
// _
HB∗(X/S ) _
RepQp(π1(Xη, x))(1)
//GR(3)
//HB(X/S )
(1) is fully faithful if X is proper over S(Faltings’ main p-adic comparison thm).
(2) is an equivalence of categories.
(3) is defined by descent (achieved nowonly for curves).
Small generalized representations foraffine small schemes [Faltings]
X = Spec(R) is affine, ∃ an étale map
X → Gdm,S = Spec(OK[T±11 , . . . , T±1
d ]),
Xη is connected and Xs 6= ∅.
x a geom. gen. point of Xη, ∆ = π1(Xη, x).
ti the image of Ti in R. Choosing p∞-rootsof the ti’s in κ(x), we obtain a surjectivehomomorphism
∆→∆∞∼→ Zp(1)d,
⇒ χ1, . . . , χd: ∆ → Zp(1) that describe theaction of ∆ on the p∞-roots of the ti’s.
R1 = R⊗OKOK, R1 its p-adic completion.
M a free R1-module of f.t. with a cont. p-adic rep. ϕ of ∆∞ which is trivial modulopαM for a rational number α > 1
p−1.
logϕ =d∑
i=1
θi ⊗ χi ⊗ ξ−1,
where ξ a Zp-basis of Zp(1), θ1, . . . , θd areR1-endomorphisms of M that commute toeach other.
⇒ the R1-linear morphism
θ =d∑
i=1
θi⊗d log ti⊗ξ−1:M →M⊗RΩ1R/OK
(−1)
is a Higgs field, i.e., θ ∧ θ = 0.
(Vi)i∈I a universal cover of Xη at x, Xi =
Spec(Ri) the norm. of X = X ×S S in Vi,
R = colimi∈IRi
equipped with the natural action of ∆.
GR the cat. RepR
(∆) of cont. p-adic
semi-linear rep. of ∆ on R-modules.
Repsmall
R(∆) the full subcat. of small rep.
i.e., rep. (M,ϕ) such that M is a free R-module of f.t. having a basis of elements∆-invariant modulo p2α for a rational num-ber α > 1
p−1. Idem for RepsmallR1
(∆∞).
Proposition 4 (Faltings) The functor
RepsmallR1
(∆∞)→ Repsmall
R(∆), M 7→M⊗
R1R
is an equivalence of categories.
HBsmall(R1) the cat. of pairs (M, θ) whereM is a free R1-module of finite type andθ:M →M⊗RΩ1
R/OK(−1) is a R1-linear map
s.t. θ ∧ θ = 0 and θ ≡ 0 mod pαM for arational number α > 1
p−1.
RepsmallR1
(∆∞)∼ //
o
HBsmall(R1)
Repsmall
R(∆)
H
55
Two problems:
- Faltings claimed that he can extend thisconstruction for rational coefficients. Thisis not obvious as it requires to descendsmall representations with rational coeffi-cients, which was proved only recently byTsuji (a similar descent result is due toAndreatta-Brinon).- It depends on the choice of the t′is.
In my work with Michel, we solve thesetwo problems by using another approachbased on deformations.
Fontaine’s p-adic inf’mal thickening
OK[ = lim OK/pOK
It is a complete non-discrete val. ring ofht. 1, perfect of char. p.∃ a can. isom. of multiplicative monoids
OK[∼→ (x(n)) ∈ ON
C | (x(n+1))p = x(n).
Choose p = (pn)n≥0 ∈ OK[ s.t. p0 = p and
set
ξ = [p]− p ∈W(OK[).
[Fontaine] we have an exact sequence
0→W(OK[)·ξ→W(O
K[)
θ→ OC → 0,
where
θ(x0, x1, ...) =∑n≥0
pnx(n)n .
A2(OK) = W(OK[)/ker(θ)2,
0 −→ ξOC −→ A2(OK)θ−→ OC −→ 0.
In fact, it is the universal p-adic inf’malthickening of OC of order ≤ 1.
S = Spec(OC), S = Spec(A2(OK)) ⇒ ∃can. closed immersion S → S.
∃ can. hom. Zp(1)→ (OK[)×. ∀ζ ∈ Zp(1),
θ([ζ]− 1) = 0 ⇒ log. hom.
Zp(1)→ A2(OK), ζ 7→ [ζ]− 1.
It induces an OC-linear isomorphism
OC(1)∼→ p
1p−1ξOC.
X a smooth scheme over S, X = X ×S S.
The p-adic Simpson correspondence de-pends of a smooth deformation
X //
X
S // S
- Such a deformation may not exist.- We can define a relative version of A2(OK)
over OK. Over this thickening, there ex-ists always a can. deformation. However,it is not possible to build the whole theoryfrom this deformation.
For simplicity, I will explain the construc-tion only in the small affine case. So weassume that X = Spec(R) is affine andsmall, that Xs 6= ∅ and that Xη is con-nected.
x a geom. point of Xη, (Vi)i∈I a universalcover of Xη at x, Xi = Spec(Ri) the norm.of X = X ×S S in Vi, ∆ = π1(Xη, x) and
R = colimi∈IRi.
R[ = lim R/pR.
It is a perfect OK[-algebra and the sequence
0 −→W(R[)·ξ−→W(R[)
θ−→ R −→ 0
is exact. We set
A2(R) = W(R[)/ker(θ)2,
so we have the exact sequence
0 −→ ξR −→ A2(R)θ−→ R −→ 0.
X = Spec(R) and X = Spec(A2(R)), so wehave a com. diagram
X //
X
X //
X
S // S
The set of dotted arrows that completethe diagram so it remains commutative isa torsor L over X under the group
HomR
(Ω1R/OK
⊗R R, ξR).
Let F be the R-module of affine functionson L . It is naturally an R-rep. of ∆ thatfits into an exact sequence
0→ R→ F → ξ−1Ω1R/OK
⊗R R→ 0.
C = colimn≥0Symn
R(F),
where the transition maps are defined by
x1 ⊗ . . .⊗ xn 7→ 1⊗ x1 ⊗ . . .⊗ xn.
It is naturally an R-algebra and Spec(C )
represents L .
dC : C → ξ−1Ω1R/OK
⊗R C
the universal R-derivation of C /R. It is aHiggs R-field with coefficients in ξ−1Ω1
R/OK.
In fact, F is isomorphic to Faltings’ exten-sion up to a p-power, so C is an integralmodel of Hyodo’s ring.
The R-algebra C is equipped with two struc-tures, namely a p-adic cont. action of ∆
and a R-Higgs field. We can use it as aperiod ring for the p-adic Simpson corre-spondence, à la Fontaine.
This correspondence will work only for ad-missible objects. The admissibility condi-tion associated to C is rather strong. Wewill see that it corresponds to p-adic vari-ations of Hodge structures (rep. whoseassociated Higgs field is nilpotent).
C † the p-adic weak completion of C . It isequipped with a p-adic cont. action of ∆
and R-derivation
dC †: C † → ξ−1Ω1R/OK
⊗R C †,
which is a Higgs R-field.
If M is a R-rep. of ∆, we set
H(M) = (M ⊗R
C †)∆,
which is a R1-module with the Higgs fieldinduced by dC †.
If (N, θ) is a R1-Higgs module with coeffi-cients in ξ−1Ω1
R/OK, we set
V(N, θ) = (N ⊗R1
C †)θ⊗id+id⊗dC †=0
,
which is a R-representation of ∆.
Definition 5 We say that a p-adic cont.R[1
p]-rep. M of ∆ is Dolbeault if the fol-lowing conditions are satisfied:(i) M is a projective R[1
p]-module of f.t.with the p-adic topology;(ii) H(M) is projective of finite type overR1[1
p];
(iii) H(M)⊗R1
C † ∼→M ⊗R
C †.
We define in a symmetric way the notionof solvable Higgs R1[1
p]-module.
Proposition 6 The functors H and V in-duce equiv. of cat. inverse to each otherbetween:- the cat. of Dolbeault R[1
p]-rep. of ∆;
- the cat. of solvable Higgs R1[1p]-modules
with coefficients in ξ−1Ω1R/OK
.
Proposition 7 (Faltings, Tsuji, A.-Gros)For any Dolbeault R[1
p]-rep. M of ∆, ∃ afun. can. quasi-isom.
C•cont(∆,M)∼→ K•(H(M), θ).
Proposition 8 LetM be a projective R[1p]-
module of f.t. with a semi-linear rep. of∆. Then the following properties are equiv-alent:(i) M is Dolbeault and H(M) is nilpotent,i.e., it admits a decreasing filtration (Ni)0≤i≤nby sub-R1[1
p]-modules such that N0 = H(M),Nn = 0 and
θ(Ni) ⊂ ξ−1Ω1R/OK
⊗R Ni+1.
(ii) There exists a projective R1[1p]-module
N , a Higgs R1[1p]-field θ on N with coef-
ficients in ξ−1Ω1R/OK
and a ∆-equivariant
C -isom. of Higgs R[1p]-modules
N ⊗R1
C∼→M ⊗
RC .
Moreover, under these conditions, we haveH(M) = (N, θ).
When these conditions are satisfied, we saythat the R[1
p]-rep. M is Hodge-Tate.