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.