1 Calculation of dimensions of tangent spaces

Deformations and Rigidity of `-adic Sheaves ^∗

Lei Fu

Yau Mathematical Sciences Center, Tsinghua University, Beijing, China leifu@mail.tsinghua.edu.cn

Abstract

Let X be a smooth connected algebraic curve over an algebraically closed field, letS be a finite closed subset inX, and letF0be a lisse`-torsion sheaf onX−S. We study the deformation ofF0. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisseQ`-sheafFis irreducible and physically rigid, then it is cohomologically rigid, under the extra condition thatF mod `is absolutely irreducible or thatF has finite monodromy.

Key words: deformation of Galois representations, formal scheme, rigid analytic space.

Mathematics Subject Classification: 14D15, 14G22.

Introduction

In this paper, we work over an algebraically closed fieldk of characteristicpeven though our results can be extended to non-algebraically closed fields. LetX be a smooth connected projective curve over k, letS be a finite closed subset ofX, and let`be a prime number distinct from p. For any s∈S, letηsbe the generic point of the strict henselization ofX ats. A lisseQ`-sheafF onX−Sis called physically rigidif for any lisseQ`-sheafGonX−Swith the propertyF |ηs ∼=G|ηs for any closed point sin S, we haveF ∼=G. The lisse Q`-sheafF onX−S corresponds to a Galois representation

ρ: Gal(K(X)/K(X))→GL(Q

r

`)

of the function field K(X) unramified everywhere on X −S. F is physically rigid if and only if for any Galois representation ρ⁰ of Gal(K(X)/K(X)) such that ρ⁰ and ρ induce isomorphic Galois representations of local fields obtained by taking completions of K(X) at places of K(X), we have ρ∼=ρ⁰. To get a good notion of rigidity, we have to assumeX=P¹. Indeed, the abelian-pro-`quotient of the ´etale fundamental groupπ<sub>1</sub>(X) ofX is isomorphic toZ<sup>2g</sup>` , wheregis the genus ofX. Ifg≥1, then there exists a character χ: π1(X) →Q^∗` such that χ<sup>n</sup> are nontrivial for all n. So there exists a lisseQ`-sheafLof rank 1 onX such that L^⊗n are nontrivial. For any lisseQ`-sheaf F onX−S, the lisse sheafG = F ⊗ Lis not isomorphic to F since they have non-isomorphic determinant, but

∗I would like to thank Yongquan Hu for interesting discussion, and Junyi Xie for providing the proof of Lemma 0.7. I am also thankful for H. Esnault who invited me to visit Essen in 2005 and suggested the problem studied in this paper.

This research is supported by the NSFC.

F |_ηs ∼=G|_ηs for alls∈X sinceχ|_{Gal( ¯ηs/ηs)}is unramfied and hence trivial. Thus F is not physically rigid. A lisse irreducibleQ`-sheafF onX−S is calledcohomologically rigidif we have

χ(X, j∗End(F)) = 2,

wherej:X−S ,→X is the canonical open immersion andχ(−,−) is the Euler characteristic. In [13, 5.0.2 and 1.1.2], Katz shows that for an irreducible lisse sheaf, cohomological rigidity implies physical rigidity, and conjectures that the converse is true. He proves the conjecture for complex local systems onX in the case wherek is the complex field. In this paper, we prove the conjecture for any k of characteristicpunder the extra condition thatF mod`is absolutely irreducible or thatF has finite monodromy.

Katz’s proof for the complex field case ([13, 1.1.2]) can be interpreted as a study of the moduli space of representations of the topological fundamental group ofX −S. In [4, Theorem 4.10], Bloch and Esnault study deformations of locally free O_X−S-modules provided with connections while keeping local (formal) data undeformed, and prove that the universal deformation space is algebraizable. Using this fact, they prove that physical rigidity and cohomological rigidity are equivalent for irreducible locally freeO_X−S-modules provided with connections. Our method is similar. We study deformations of lisse`-torsion sheaves. The universal deformation space is a formal scheme, and its generic fiber is a rigid analytic space which can be used to produce families ofQ`-sheaves. By a counting argument on dimensions of rigid analytic spaces, we prove Katz’s conjecture under the extra condition mentioned above.

Supposekis an algebraic closure of the finite fieldFq, and X is obtained from an algebraic curve X0overFq by base change. In [9], Deligne studied the counting of fixed points of the Frobenius map on the set of isomorphic classes of lisseQ`-sheaves onX, a problem which is studied by Drinfeld for the case where the sheaves have rank 2. In [9, 1.3-1.6], Deligne makes some speculation on the moduli space ofQ`-sheaves, which is not algebraizable. The construction in this paper suggests that a piece of the moduli space might have a rigid analytic space structure.

In the following, we take Λ to be either a finite extension E ofQ`, or the integer ring Oof such E, or the residue field of O. Let m be the maximal ideal of Λ, and let κ = Λ/m be the residue field of Λ. Denote by CΛ the category of Artinian local Λ-algebras with same residue field κ as Λ. Morphisms in CΛ are homomorphisms of Λ-algebras. Using the fact that the maximal ideal of an Artinian local ring coincides with its nilpotent radical, one can check that morphisms inCΛ are necessarily local homomorphisms, and they induce the identity homomorphism on the residue field.

IfAis an object inC_Λ, we denote bym_Athe maximal ideal ofA. Letη be the generic point ofX, and letπ1(X−S,η) be the ´¯ etale fundamental group ofX−S. Fix an embedding Gal(¯ηs/ηs),→Gal(¯η/η) for each s ∈ S and let Gal(¯ηs/ηs) → π1(X −S,η) be its composite with the canonical surjection¯ Gal(¯η/η)→ π₁(X−S,η). A homomorphism¯ ρ :π₁(X −S,η)¯ →GL(A^r) is called a representation if it is continuous. Here, if Λ =O, thenA is finite and we put the discrete topology on GL(A^r). If Λ =E, thenAis a finite dimensional vector space overE, and we put the topology induced from the

`-adic topology on GL(A^r).

Suppose we are given a representationρΛ:π1(X−S,η)¯ →GL(Λ^r). Letρ0:π1(X−S,η)¯ →GL(κ^r) be the representation obtained fromρ_Λby passing to residue field. We study deformations ofρ₀. Our

treatment is similar to Mazur’s theory of deformations of Galois representations ([15]) and Kisin’s theory of framed deformations of Galois representations ([14]).

Throughout this paper, we assume thatS is nonempty. Suppose we are givenP0,s∈GL(κ^r) for eachs∈S. In application, we often takeP0,s to be the identity matrix I for alls∈S. In this case, we denote the data (ρ0,(P0,s)_s∈S) by (ρ0,(I)_s∈S). For anyA∈obCΛ, denote the composite

π₁(X−S,η)¯ →^ρΛGL(Λ^r)→GL(A^r)

also byρΛ. DefineF(A) to be the set of deformations (ρ,(Ps)s∈S) of the data (ρ0,(P0,s)s∈S) with the prescribed local monodromyρΛ|_{Gal( ¯ηs/ηs)}. More precisely, we define

F(A) ={(ρ,(P_s)_s∈S) | ρ:π₁(X−S,η)¯ →GL(A^r) is a representation, P_s∈GL(A^r), ρ modm_A=ρ₀, P_s modm_A=P_0,s,

P_s⁻¹ρ|_{Gal( ¯ηs/ηs)}P_s=ρ_Λ|_{Gal( ¯ηs/ηs)} for alls∈S}/∼,

where two tuples (ρ⁽ⁱ⁾,(Ps⁽ⁱ⁾)s∈S) (i= 1,2) are equivalent if there exists P∈GL(A^r) such that (ρ⁽¹⁾,(P_s⁽¹⁾)_s∈S) = (P⁻¹ρ⁽²⁾P,(P⁻¹P_s⁽²⁾)_s∈S).

Note that the equationPs⁽¹⁾=P⁻¹Ps⁽²⁾ implies thatP ≡I modm_A since we assumeS is nonempty and Ps⁽¹⁾ ≡ Ps⁽²⁾ modmA = P0,s. The column vectors of P0,s can be regarded as a basis, that is, a frame for κ^r, and Ps can be regarded as a lifting of this frame to a frame of A^r. Two tuples (ρ⁽ⁱ⁾,(Ps⁽ⁱ⁾)s∈S) (i= 1,2) are equivalent if any only if there exists an isomorphism of representations P:A^r→A^rfrom ρ⁽¹⁾ to ρ⁽²⁾ which transforms the framePs⁽¹⁾ to the framePs⁽²⁾ for each s∈S. For any morphismA⁰ →A in CΛ, defineF(A⁰)→F(A) to be the map induced by GL(A^0r)→GL(A^r).

We thus get a covariant functorF :CΛ→(Sets).

LetA⁰→A andA⁰⁰→Abe morphisms inC_Λ. Consider the map F(A⁰×AA⁰⁰)→F(A⁰)×F(A)F(A⁰⁰).

Using the fact thatS is nonempty, it is straightforward to verify that this map is bijective ifA⁰⁰→A is surjective. Proposition 0.1 below shows thatF(κ[]) is a finite dimensional vector space. So by the Schlessinger criteria [17, Theorem 2.11], the functorF is pro-representable.

Proposition 0.1. Let κ[] be the ring of dual numbers. The κ-vector space F(κ[]) is finite dimen- sional. Suppose furthermore that X = P¹ and all elements in the set Endπ₁(X−S,¯η)(κ^r) are scalar multiplications, whereκ^r is considered as aπ1(X−S,η)-module through the representation¯ ρ0. (This condition holds ifρ₀ is absolutely irreducible by Schur’s lemma). Then the functor F is smooth, and we have

dimF(κ[]) =−χ(X, j_∗End⁽⁰⁾(F₀)) +X

s∈S

dimH⁰(Gal(¯η_s/η_s),Ad(ρ₀))−1,

where Ad(ρ0) is the κ-vector space of r×r matrices with entries in κ on which π1(X −S,η)¯ and Gal(¯η_s/η_s) act by the composition of ρ₀ with the adjoint representation of GL(κ^r), F0 is the lisse

κ-sheaf on X −S corresponding to the representation ρ₀ : π₁(X −S,η)¯ → GL(κ^r), End⁽⁰⁾(F₀) is the subsheaf of End(F0) formed by sections of trace 0, and j : X −S ,→ X is the canonical open immersion.

Denote by R(ρΛ) the universal deformation ring for the functor F. It is a complete noetherian local Λ-algebra with residue fieldκ. We have a homomorphism

ρuniv:π1(X−S,η)¯ →GL(r, R(ρΛ)) with the property

ρ_univ modm_R(ρΛ)=ρ₀, and we have matricesPuniv,s∈GL(r, R(ρΛ)) with the property

Puniv,s modm_R(ρΛ)=P0,s, P_univ,s⁻¹ ρuniv|_{Gal( ¯ηs/ηs)}Puniv,s=ρΛ|_{Gal( ¯ηs/ηs)} such that the homomorphismπ₁(X−S,η)¯ →GL(r, R(ρ_Λ)/mⁱ_R(ρ

Λ)) induced byρ_univare continuous for all positive integersi, and for any element (ρ,(Ps)s∈S) inF(A), there exists a unique local Λ-algebra homomorphismR(ρΛ)→A which brings (ρuniv,(Puniv,s)_s∈S) to the equivalent class of (ρ,(Ps)_s∈S).

More generally, we have the following.

Proposition 0.2. Let A⁰ be a local ArtinianΛ-algebra so that its residue fieldκ⁰ =A⁰/m_A⁰ is a finite extension of κ= Λ/m. Let (ρ⁰,(P_s⁰)_s∈S) be an equivalent class of tuples, where ρ⁰ : π1(X−S,η)¯ → GL(A^0r) is a representation andP_s⁰∈GL(A^0r) such that

ρ⁰ modm_A⁰ =ρ₀, P_s⁰ modm_A⁰ =P_0,s, P_s⁻¹ρ|Gal( ¯η_s/η_s)P_s=ρ_Λ|Gal( ¯η_s/η_s) for alls∈S}, and the equivalence relation is defined as before. Then there exists a unique localΛ-algebra homomor- phismR(ρΛ)→A⁰ which brings(ρuniv,(Puniv,s)_s∈S)to the equivalent class of (ρ⁰,(P_s⁰)_s∈S).

Proof. Let A be the inverse image ofκunder the projection A⁰ → A⁰/mA⁰ =κ⁰. ThenA is a local ring with maximal idealm_A⁰ and its residue field is isomorphic to κ. Moreover, A is complete since mA⁰ is nilpotent. The vector space mA⁰/m²_A0 is finite dimensional overκ⁰ =A⁰/mA⁰ and hence finite dimensional over κ ∼= A/m_A. Choose a basis {x₁, . . . , x_n} of m_A⁰/m²_A0 over κ. Then we have an epimorphism

κ[t1, . . . , tn]→ ⊕_i≥0mⁱ_A0/mⁱ⁺¹_A0 , ti7→xi.

It follows that⊕i≥0mⁱ_A0/mⁱ⁺¹_A0 is Noetherian. By [1, Corollary 2.5],Ais Noetherian. Since its maximal idea is nilpotent, A is Artinian. It follows that A is an object in CΛ. Since ρ⁰ modmA⁰ = ρ0, the image of ρ⁰ lands in GL(A) and henceρ⁰ defines a representation ρ : π₁(X −S,η)¯ → GL(A^r).

SinceP_s⁰ modmA⁰ =P0,s, P_s⁰ defines an elementPs∈GL(A^r). The tuple (ρ,(Ps)_s∈S) then defines an element in F(A). So there exists a unique Λ-algebra homomorphism R(ρ_Λ) → A which brings (ρuniv,(Puniv,s)s∈S) to the equivalent class of (ρ,(Ps)s∈S). The composite

R(ρΛ)→A ,→A⁰

brings (ρuniv,(Puniv,s)_s∈S) to the equivalent class of (ρ⁰,(P_s⁰)_s∈S). Since the residue field ofR(ρΛ) is isomorphic toκ, any local Λ-algebra homomorphismR(ρ_Λ)→A⁰ factors through the subringAofA⁰.

So any local Λ-algebra homomorphismR(ρ_Λ)→A⁰which brings (ρ_univ,(P_univ,s)_s∈S) to the equivalent class of (ρ⁰,(P_s⁰)s∈S) gives rise to a homomorphismR(ρΛ)→A which brings (ρuniv,(Puniv,s)s∈S) to the equivalent class of (ρ,(Ps)_s∈S), and hence is unique.

LetEbe a finite extension ofQ`, letObe its integer ring, and letκbe its residue field. SupposeF_E is a lisseE-sheaf onX−Sof rankr. Choose a torsion free lisseO-sheafFO such thatFE∼=FO⊗OE.

LetF0=F_O⊗_Oκ, letρ_O :π1(X−S,η)¯ →GL(O^r) be the representation corresponding to the sheaf FO, and let ρ_E : π₁(X −S,η)¯ → GL(E^r) and ρ₀ : π₁(X−S,η)¯ → GL(κ^r) be the representations obtained fromρ_O by passing to the fraction field and the residue field ofO, respectively. Note thatρE

andρ₀are also the representations corresponding to theE-sheafF_E and theκ-sheafF₀, respectively.

Take Λ =O. Consider the universal deformation ring R(ρO) of the functorF :CO →(Sets) for the data (ρ0,(I)_s∈S), whereP0,s=Ifor alls∈S. As a localO-algebra, it is isomorphic to a quotient of O[[y1, . . . , y_n]] for somen.

Let’s recall a construction of Berthelot which associates a rigid analytic space to any noetherian adic formal schemeXover SpfOwhose reductionX_redis a scheme of finite type overκ. First consider the case whereX = SpfR is affine, whereR is a complete adic noetherianO-algebra such that the largest ideal of definition J of R contains m_OR, and R/J is a finitely generated κ-algebra. One can show ([2, Lemma 1.2]) thatR is a quotient of the ring O{x1, . . . , xn}[[y1, . . . , ym]]. Recall that O{x1, . . . , xn} is the ring of power seriesP

i1,...,inai₁...i_nxⁱ₁¹· · ·xⁱ_nⁿ with the propertyai₁...i_n∈ Oand a_i1...in →0 as i₁+· · ·+i_n → ∞. We define the rigid analytic spaceX^rig to beE(0,1)ⁿ×D(0,1)^m for R = O{x1, . . . , xn}[[y1, . . . , ym]], where E(0,1) = SpE{x} is the closed unit disc over E, and D(0,1) = S∞

i=1SpE{r_i⁻¹x} is the open unit disc. Here E{x} = E⊗_O O{x}, {r_i} is an increasing sequence of positive real numbers with limit 1 such that for eachri, an integral power of ri is equal to the norm of an element inE, andE{r⁻¹x}is the ring of power seriesP

i≥0aixⁱwith the property a_i ∈E and a_irⁱ →0 as i→ ∞. In general, ifR is the quotient ofO{x1, . . . , x_n}[[y1, . . . , y_m]] by an ideal generated byg1, . . . , gk, we defineX^rig to be the closed analytic subvarietyg1=. . .=gk= 0 of E(0,1)ⁿ×D(0,1)^m. One extends this construction to a formal schemeXoverO by gluing the rigid analytic spaces constructed from an affine open covering ofX. The rigid analytic spaceX^rig can be thought as the generic fiber of the formal schemeXover SpfO. The constructionX→X^rig defines a functor from the category of noetherian adic formal schemesXover SpfO whose reductionXred are schemes locally of finite type over Specκto the category of rigid analytic spaces overE. This functor commutes with fiber products. We refer the reader to [2, §1] and [12, §7] for details of Berthelot’s construction.

Let F= SpfR(ρ_O) be the formal scheme associated to the universal deformation ring R(ρ_O) of the functorF : CO → (Sets) for the data (ρ0,(I)s∈S), and let F^rig be the associated rigid analytic space. By [12, 7.1.10], there is a one-to-one correspondence between the set of point inF^rig and the set of equivalent classes of local homomorphismsR(ρ_O)→ O⁰ ofO-algebras, whereO⁰ is the integer ring of a finite extensionE⁰ of E, and two such homomorphismsR(ρ_O)→ O⁰ andR(ρ_O)→ O⁰⁰ are equivalent if there exists a commutative diagram

R(ρO) → O⁰

↓ ↓

O⁰⁰ → O⁰⁰⁰

such thatO⁰⁰⁰ is the integer ring of a finite extensionE⁰⁰⁰ ofE containing both the fraction fieldsE⁰ andE⁰⁰ofO⁰andO⁰⁰respectively. Applying the universal property ofR(ρO) to the tuples (ρO,(I)s∈S) modmⁱ_O for alli, we get a uniqueO-algebra homomorphism

ϕ₀:R(ρ_O)→ O

which brings the universal representation ρuniv : π1(X −S,η)¯ → GL(r, R(ρ_O)) to ρ_O, and brings P_univ,s to I for all s∈S. The homomorphismϕ₀ defines a point t₀ in F^rig. Let tbe a point in F^rig corresponding to a local homomorphism

ϕt:R(ρ_O)→ O⁰

of O-algebras. Let (ρ_t,(P_t,s)_s∈S) be the tuple obtained by pushing forward the universal tuple (ρuniv,(Puniv,s)_s∈S) through the homomorphism ϕt. Note that ρt : π1(X −S,η)¯ → GL(O^0r) is a representation,P_t,s∈GL(O^0r), and

ρt0 =ρO, Pt0,s modmO =I,

ρ_t modmO⁰=ρ₀, P_t,s modmO⁰ =I, P_t,s⁻¹ρ_t|Gal( ¯η_s/η_s)P_t,s=ρ_O|Gal( ¯η_s/η_s),

SupposeρEis physically rigid. Then the third line in the above equations implies thatρtis isomorphic toρE as Q`-representations, that is, after enlarging the fieldE⁰, there exists P ∈GL(E^0r) such that P⁻¹ρtP =ρE. We conjecture that for thosetclose tot0, we can chooseP so thatP ∈GL(O^r

Q`

) and P ≡I modmO

Q`, whereO_Q

` is the integer ring ofQ`. More precisely, we should have the following conjecture.

Conjecture 0.3. Notation as above. Suppose that X = P¹, that End(F_E) consists of scalar mul- tiplications, and that FE is physically rigid. Then there exists an admissible neighborhood V of the point t₀ inF^rig such that for any t∈V, there exists P ∈GL(O^r

Q`

) such thatP ≡I modm_OQ

` and P⁻¹ρtP =ρO.

Remark0.4. Note that under the assumption of Conjecture 0.3,Pis uniquely determined up to scalar.

Indeed, providedE^r and E^0r with the π₁(X−S,η)-module structure via the representation¯ ρ_E. By [11, Lemma 1.1], we have

End_{π1(X−S,¯η)}(E^0r)∼= Endπ₁(X−S,¯η)(E^r)⊗EE⁰.

As End_{π1(X−S,¯η)}(E^r)∼= End(FE) consists of scalar multiplications, the same is true for End_{π1(X−S,¯η)}(E^0r).

IfP andP⁰ are two matrices in GL(E^0r) such thatP⁻¹ρtP =P⁰⁻¹ρtP⁰ =ρE, then we have ρ_EP⁻¹P⁰=P⁻¹P⁰ρ_E.

SoP⁻¹P⁰ lies in Endπ₁(X−S,¯η)(E^0r),and hence is a scalar matrix.

In§3, we prove the following proposition:

Proposition 0.5. Conjecture 0.3 holds under either one of the following conditions:

(i)End(F0)consists of scalar multiplications.

(ii) FE has finite monodromy, that is, im(ρE)is finite.

LetG:C_O→(Sets) be the functor defined by

G(A) ={(P_s)_s∈S|P_s∈Aut_{Gal( ¯ηs/ηs)}(A^r), P_s≡I modm_A}/∼,

where A^r is provided with the Gal(¯η_s/η_s)-action via ρ_O, and two tuples (Ps⁽ⁱ⁾)_s∈S (i = 1,2) are equivalent if there exists an invertible scalar (r×r)-matrix P = uI for some unit u in A such that Ps⁽¹⁾ = P⁻¹Ps⁽²⁾ for all s ∈ S. Using Schlessinger’s criteria, one can verify that functor G is pro-representable. One can describe the universal deformation ring ofG as follows: Let H_s be the wild inertia subgroup of Gal(¯ηs/ηs). Then the (tame) quotient group Gal(¯ηs/ηs)/Hsis topologically generated by one element, say by the image ofg_s∈Gal(¯η_s/η_s). SinceH_s is a pro-p-group, its image underρO is finite. Choosehs,1, . . . , hs,n_s ∈Hs so that

imρO ={ρO(hs,1), . . . , ρO(hs,n_s)}.

Set

As=ρ_O(gs), As,k=ρ_O(hs,k) (k= 1, . . . , ns).

Then we haveP_s∈Aut_{Gal( ¯ηs/ηs)}(A^r) if and only if

P_sA_s=A_sP_s, P_sA_s,k=A_s,kP_s.

LetI be the homogeneous ideal ofO[t_s,ij]s∈S,1≤i,j≤r generated by the entries of the matrices (t_s,ij)A_s−A_s(t_s,ij), (t_s,ij)A_s,k−A_s,k(t_s,ij) (s∈S, 1≤k≤n_s).

It defines a closed subschemeX of the projective spaceP^|S|r

2−1 = ProjO[t_s,ij]s∈S,1≤i,j≤r. Letp be the kernel of the homomorphism

T = (O[ts,ij]s∈S,1≤i,j≤r)/I→κ[t], ts,ij 7→δijt.

Thenpis a homogeneous prime ideal ofT (resp. a Zariski closed pointxofX = ProjT). The universal deformation ring R(G) for the functor G is isomorphic to the completion T_(p)^∧ (resp. OˆX,x) of the homogeneous localizationT_(p) (resp. OX,x). Note that for any localO-algebra Aand any equivalent class of tuples (P_s⁰)_s∈S, where P_s⁰ ∈ Aut_{Gal( ¯ηs/ηs)}(A^0r) and Ps ≡ I modmA⁰ and the equivalence relation is defined as before, there exists a unique local Λ-algebra homomorphismR(G)→A⁰ which brings the universal tuple ((ts,ij)s∈S) to (P_s⁰).

The rigid analytic spaceG^rig associated to the formal scheme SpfR(G) is a group object, and its points can be identified with the set

{(Ps)s∈S|Ps∈AutGal( ¯η_s/η_s)(O^r

Q`), Ps≡I modmO

Q`}/∼, where O^r

Q`

is provided with the Gal(¯ηs/ηs)-action via ρO, and two tuples (Ps⁽ⁱ⁾)s∈S (i = 1,2) are equivalent ifPs⁽¹⁾=u⁻¹Ps⁽²⁾ for alls∈S for some unitsuin O

Q`. Note that dimG^rig=X

s∈S

dim End_{Gal( ¯ηs/ηs)}(E^r)−1,

whereE^ris provided with the Gal(¯η_s/η_s)-action viaρ_E.

Let F : C_O → (Sets) be the functor introduced before for the data (ρ_O,(I)_s∈S). We have a morphism of functorsG→F defined by

G(A)→F(A), (Ps)_s∈S 7→(ρ_O,(Ps)_s∈S).

Lemma 0.6. LetF(resp. G) be the formal scheme associated to the universal deformation ring for the functorF (resp. G), letF^rig (resp. G^rig) be the associated rigid analytic space, and letf :G^rig→F^rig be the morphism on rigid analytic spaces induced by the morphism of functorsG→F. Suppose that X=P¹, thatEnd(FE)consists of scalar multiplications, and thatFEis physically rigid. If Conjecture 0.3 is true, thenf :f⁻¹(V)→ V is surjective in the sense that every point in V is the image of a point inG^rig.

Proof. Lett be a point inV, and let ϕt:R(ρO)→ O⁰ be the corresponding local homomorphism of O-algebras ([12, 7.1.10]), whereO⁰ is the integer ring of a finite extensionE⁰ ofE. Let (ρt,(Pt,s)_s∈S) be the tuple obtained by pushing forward the universal tuple (ρuniv,(Puniv,s)s∈S) through the ho- momorphism ϕt. By our assumption, there exists P ∈ GL(O^r

Q`

) such that P ≡ I modm_O

Q`

and P⁻¹ρ_tP = ρ_O. By enlarging E⁰, we may assume P ∈ GL(O^0r). Then for each i, the tuple (ρt,(Pt,s)_s∈S) modmⁱ_O0 is equivalent to the tuple (ρ_O,(P⁻¹Pt,s)_s∈S) modmⁱ_O0. The family of tu- ples (P⁻¹P_t,s)_s∈S modmⁱ_O0 defines a family of local O-algebra homomorphisms R(G) → O⁰/mⁱ_O0, where R(G) is the universal deformation ring for G. This family is compatible and defines a local homomorphismR(G)→ O⁰ ofO-algebras. It corresponds to a point in G^rig that is mapped to the pointtofF^rig.

Lemma 0.7. Let f :X →Y be a separated morphism of rigid analytic spaces over a nonarchimedean fieldk with non-trivial valuation. Suppose X can be covered by countably many k-affinoid subspaces X_n (n= 1,2, ...). Iff is surjective on the underlying sets of points, thendimY ≤dimX.

Proof. The following proof is due to Junyi Xie. Making a base change to the completion of the algebraic closure ofk, we may assumek is a complete algebraically closed field. We may reduced to the case whereY = SpAfor a strictly k-affinoid algebraA(in the language of Berkovich [3]). Then by definition, dimY is the Krull dimension ofA. By the Noether normalization theory ([5, Corollary 6.1.2/2]), there exists a finite monomorphismTd→Afor some Tate algebra Td =k{t1, . . . , td} with d= dimY. Note that the induced morphismY →SpTd is surjective on the underlying set of points.

So we can reduce to the case whereA=T_d andY =E(0,1)^d is the unit polydisc of dimensiond.

LetX (resp. Xn, resp. Y) be the Berkovich space associated toX(resp. Xn, resp. Y). Denote the morphismX → Y corresponding tof :X →Y also byf. For any real multipler= (r₁, . . . , r_d) with 0< ri<1 and ri ∈ |k^∗|, and any rigid pointa= (a1, . . . , ad) inE(0,1)^d, whereai∈k and|ai| ≤1, consider the polydisc

E(a, r) ={(x1, . . . , xd)∈k^d:|xi−ai| ≤ri}.

We haveE(a, r)⊂E(0,1)^d. We define the associated Gauss norm| · |_E(a,r)onT_d by

|f|E(a,r)= max{|f(x)|:x∈E(a, r)}

for anyf ∈T_d. If

f = X

i₁,...,i_d

ai₁...i_d(t1−a1)ⁱ¹· · ·(td−ad)^id is the Taylor expansion off ata, then we have

|f|E(a,r)= maxi1,...,i_d|ai1...i_d|rⁱ₁¹· · ·r_d^id.

The Gauss norms| · |_E(a,r) are points in Y. Let S be the subset of Y consisting of all Gauss norms associated to all polydiscsE(a, r). Note thatS is dense inY. Indeed, as the radiusr= (r1, . . . , rd) approaches to 0, the Gauss norm|·|_E(a,r)approaches to the rigid pointa, and it is known that the set of all rigid points is dense inY([3, 2.1.15]). Moreover, for anyy∈ S, we haves(H(y)/k) =d, whereH(y) is the field defined in [3, 1.2.2 (i)], ands(H(y)/k) = tr.deg(H(y)/] ˜k) is defined in [3, 9.1]. We claim thatf(X)∩ S in nonempty. Otherwise, f(X_n) is disjoint fromS for eachn, that is,S ⊂ Y −f(X_n).

HenceY −f(Xn) is dense inY. SinceXn is affiniod, it is compact ([3, 1.2.1]). Sof(Xn) is a compact subset in the Hausdorff spaceY, and hence it is a closed subset. It follows thatY −f(Xn) is a dense open subset ofY. By [3, 2.1.15], the subset of rigid points (Y −f(Xn))∩Y is open dense inY. Here we provideY with the topology induced from the Berkovich spaceY. But this topology onY is induced by a complete metric. In fact, it is unit polydisc inkⁿprovided with the metric given by the valuation ofk. By the Baire category theorem ([16, 9.1]), the set ∩^∞_n=1(Y −f(Xn))∩Y = (Y −f(X))∩Y is dense in Y. In particular, it is nonempty. This contradicts to the assumption that f : X → Y is surjective. Sof(X)∩ S is nonemtpy. Takex∈ X such thatf(x)∈ S. Then by [3, 9.1.3], we have

dimX ≥d(H(x)/k)≥s(H(x)/k)≥s(H(f(x))/k) = dimY.

Proposition 0.8. Let ϕ_t : R(ρ_O) → O be an O-algebra homomorphism, let p_t = kerϕ_t, and let (ρt,(Pt,s)_s∈S)be the tuple obtained by pushing forward the universal tuple(ρuniv,(Puniv,s)_s∈S)through the homomorphismϕ_t.

(i) The completion R(ρO)^∧_pt of the local ring R(ρO)p_t is canonically isomorphic to the universal deformation ringR(ρt⊗_OE)for the functor F :CE→(Sets) defined by

F(A) ={(ρ,(Ps)s∈S) | ρ:π1(X−S,η)¯ →GL(A^r)is a representation, Ps∈GL(A^r), ρ modmA=ρt, Ps modmA=Pt,s,

P_s⁻¹ρ|Gal( ¯η_s/η_s)Ps=ρt|Gal( ¯η_s/η_s) for alls∈S}/∼, for any local ArtinianE algebraA∈obC_E with residue field E.

(ii) Lett be the point inF^rig corresponding toϕt. We haveOb_Frig,t∼=R(ρO)^∧_pt.

We will prove this proposition in§4. We are now ready to prove the main result of this paper:

Theorem 0.9. Suppose that X = P¹, that FE is physically rigid, and that one of the following condition holds:

(i)End(F0)consists of scalar multiplications.

(ii) End(FE)consists of scalar multiplications and FE has finite monodromy.

Thenχ(X, j_∗End(FE)) = 2.

Proof. To prove this theorem, we may assumeS is nonempty. Indeed, replacingS byS∪ {x}for any closed pointxinX−Shas no effect on the theorem. Letϕ0:R(ρO)→ Obe the local homomorphism ofO-algebras corresponding to the pointt0in F^rig, and letp0= kerϕ0. By Proposition 0.8, we have

R(ρ_O)^∧_p

0

∼=Ob_Frig,t₀.

The condition that End(F0) consists of scalar multiplications implies that End(FE) consist of scalar multiplications. For a proof of this fact, see Lemma 3.1 (iii). By Proposition 0.1 applied to the case Λ =E and Proposition 0.8,R(ρ_O)^∧_p

0

∼=R(ρ_E) is a formally smoothE-algebra, and we have dimR(ρO)^∧_p0 =−χ(X, j∗End⁽⁰⁾(FE)) +X

s∈S

dimH⁰(Gal(¯ηs/ηs),Ad(ρE))−1.

So we have

dimOb_Frig,t0 =−χ(X, j∗End⁽⁰⁾(FE)) +X

s∈S

dimH⁰(Gal(¯ηs/ηs),Ad(ρE))−1.

On the other hand, by Proposition 0.5, Conjecture 0.3 holds under the assumption (i) or (ii). So by Lemmas 0.6 and 0.7, we have

dimOb_Frig,t0≤dimV ≤dimf⁻¹(V)≤dimG^rig=X

s∈S

dimH⁰(Gal(¯ηs/ηs),Ad(ρE))−1.

Comparing the above expressions for dimOb_Frig,t0, we get χ(X, j_∗End⁽⁰⁾(FE))≥0.

AsEnd(F_E)∼=End⁽⁰⁾(F_E)⊕E,we have

χ(X, j∗End(FE)) =χ(X, j∗End⁽⁰⁾(FE)) +χ(X, E)≥2,

Here we use the fact thatχ(X, E) = 2 sinceX =P¹. By our assumption, the spaceH⁰(X, j_∗End(FE))∼= End(F_E) consists of scalar multiplications and hence has dimension 1. The pairing

End(FE)× End(FE)→E, (φ, ψ)7→Tr(ψ◦φ)

defines a self-duality onEnd(FE). By the Poincar´e duality ([8, 1.3 and 2.2]), we have a perfect pairing H²(X, j_∗End(FE))×H⁰(X, j_∗End(FE)(1))→E.

It follows thatH²(X, j_∗End(FE)) also has dimension 1. So χ(X, j_∗End(FE)) =

2

X

i=0

(−1)ⁱdimHⁱ(X, j_∗End(FE))

= 2−dimH¹(X, j_∗End(FE))

≤ 2.

Compared with the previous opposite inequality, we getχ(X, j_∗End(FE)) = 2. This proves that FE

is cohomologically rigid.

In Lemma 3.1 (iii), we prove that the condition that End(F₀) consists of scalar multiplications implies that End(FE) consist of scalar multiplications. If Conjecture 0.3 is true, then Theorem 0.9 holds under the weaker condition that End(FE) consists of scalar multiplications.

The paper is organized as follows. In §1, we introduced a family of functors of framed deforma- tions of representations ofπ1(X −S,η). These functors are pro-representable and we calculate the¯ dimensions of their Zariski tangent spaces. In§2, we study the obstruction to lifting a deformation.

Using results in§1 and§2, we prove Proposition 0.1 at the end of§2. We prove Proposition 0.5 in§3, and Lemma 0.8 in§4.

1 Calculation of dimensions of tangent spaces

In this section, we suppose thatS is nonempty and that we are given a representationρΛ :π1(X− S,η)¯ → GL(Λ^r) and a frame P0,s ∈ GL(κ^r) for each s ∈ S. Let λ : π1(X −S,η)¯ → Λ^∗ be the determinant ofρ_Λ and let ρ₀ : π₁(X−S,η)¯ →GL(κ^r) be ρ_Λ modm_Λ. For anyA∈obCΛ, denote the composite

π₁(X−S,η)¯ →^λ Λ^∗→A^∗ also byλ. Define

D_S(A) = {(ρ,(P_s)_s∈S)|ρ:π₁(X−S,η)¯ →GL(A^r) is a representation, P_s∈GL(A^r), ρ modmA=ρ0, Ps modmA=Ps,0}/∼,

D_S^,λ(A) = {(ρ,(Ps)_s∈S)∈D_S(A)|det(ρ) =λ},

where two tuples (ρ⁽ⁱ⁾,(Ps⁽ⁱ⁾)s∈S) (i= 1,2) are equivalent if there exists P∈GL(A^r) such that (ρ⁽¹⁾,(P_s⁽¹⁾)_s∈S) = (P⁻¹ρ⁽²⁾P,(P⁻¹P_s⁽²⁾)_s∈S).

For anys∈S, define

D_s(A) = {ρ|ρ: Gal(¯ηs/ηs)→GL(A^r) is a representation, ρ modmA=ρ0|_{Gal( ¯ηs/ηs)}}, D_s^,λ(A) = {ρ∈D_s(A)|det(ρ) =λ|Gal( ¯η_s/η_s)}.

For any morphismA⁰→AinC_Λ, defineD(A⁰)→D(A) to be the map induced by GL(A^0r)→GL(A^r) for D = D_S, D_S^,λ, D_s, D_s^,λ. Then D_S, D_S^,λ, D_s, D_s^,λ are functors from the category CΛ to the category of sets. We have canonical morphisms of functors

D_S →Y

s∈S

D_s, D_S^,λ→Y

s∈S

D_s^,λ

given by

(ρ,(Ps)s∈S)→(P_s⁻¹ρ|Gal( ¯η_s/η_s)Ps)s∈S. Finally, letF (resp. F^λ) be the subfunctor of D_S (resp. D_S^,λ) defined by

F(A) = {(ρ,(Ps)s∈S)∈D_S(A)|P_s⁻¹ρ|Gal( ¯η_s/η_s)Ps=ρΛ|Gal( ¯η_s/η_s)}, (resp. F^λ(A) = {(ρ,(P_s)_s∈S)∈D_S^,λ(A)|P_s⁻¹ρ|Gal( ¯η_s/η_s)P_s=ρ_Λ|Gal( ¯η_s/η_s)}).

Note thatF (resp. F^λ) can be thought as the fiber of the morphism of functors D_S →Y

s∈S

D_s (resp. D_S^,λ→Y

s∈S

D_s^,λ)

over (ρΛ|_{Gal( ¯ηs/ηs)})_s∈S. For any morphism A⁰ → A and A⁰⁰ → A in CΛ, one can verify that the canonical map

D(A⁰×AA⁰⁰)→D(A⁰)×_D(A)D(A⁰⁰)

is bijective ifA⁰⁰ →Ais surjective for each of the functors D =D_S, D_S^,λ, D_s, D_s^,λ, F, F^λ. Propo- sition 1.1 below shows thatD(κ[]) is finite dimensional. So the functorsD_S, D_S^,λ, D_s, D_s^,λ, F, F^λ are pro-representable by Schlessinger’s criteria [17, Theorem 2.11].

Proposition 1.1. Letκ[] be the ring of dual numbers. We have

dimD_S(κ[]) = −χ(π1(X−S,η),¯ Ad(ρ0)) +|S|r²

= −χ(X−S,End(F0)) +|S|r²,

dimD_S^,λ(κ[]) = −χ(π1(X−S,η),¯ Ad⁽⁰⁾(ρ₀)) +|S|r²−1

= −χ(X−S,End⁽⁰⁾(F0)) +|S|r²−1, dimD_s(κ[]) = −χ(Gal(¯η_s/η_s),Ad(ρ₀)) +r², dimD_s^,λ(κ[]) = −χ(Gal(¯η_s/η_s),Ad⁽⁰⁾(ρ₀)) +r²−1,

where Ad(ρ₀) is the κ-vector space of r×r matrices with entries in κ on which π₁(X −S,η)¯ and Gal(¯ηs/ηs) act by the composition ofρ0 with the adjoint representation of GL(κ^r), Ad⁽⁰⁾(ρ0) is the subspace ofAd(ρ₀)consisting of matrices of trace 0, F₀ is the lisse κ-sheaf on X −S corresponding to the representationρ0:π1(X−S,η)¯ →GL(κ^r), andEnd⁽⁰⁾(F0)is the subsheaf of End(F0)formed by sections of trace0.

Proof. Let’s calculate the dimensions of D_S^,λ(κ[]). Fix s0 ∈ S. Any element in D_S^,λ(κ[]) is equivalent to an element (ρ,(P_s)_s∈S) with the property P_s0 =P_0,s0. Two elements (ρ⁽ⁱ⁾,(Ps⁽ⁱ⁾)_s∈S) (i= 1,2) inD_S^,λ(F[]) with the propertyPs⁽ⁱ⁾0 =P0,s₀ are equivalent if any only if (ρ⁽¹⁾,(Ps⁽¹⁾)_s∈S) = (ρ⁽²⁾,(Ps⁽²⁾)_s∈S). Let (ρ,(P_s)_s∈S) be an element inD_S^,λ(F[]) withP_s0 =P_0,s0. We can write

ρ(g) =ρ₀(g) +M(g)ρ₀(g), P_s=P_0,s+Q_sP_0,s

for some (r×r)-matricesM(g) andQs with entries inκ. Thatρis a homomorphism is equivalent to saying the map

π₁(X−S,η)¯ →End⁽⁰⁾(κ^r), g7→M(g)

is a 1-cocycle for Ad(ρ₀). That det(ρ) =λis equivalent to saying Tr(M(g)) = 0 for allg∈π₁(X−S,η).¯ We haveQs₀ = 0, and there is no restriction forQsifs∈S− {s0}. So

dimD_S^,λ(F[]) = dimZ¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + (|S| −1)r²,

whereZ¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) is the group of 1-cocycles. LetB¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) be the group of 1-coboundaries. Its elements are of the form

π₁(X−S,η)¯ →End⁽⁰⁾(κ^r), g7→ρ₀(g)Aρ₀(g)⁻¹−A

for some (r×r)-matrixA of trace 0 with entries inκ. A 1-coboundary of the above form is 0 if any only ifρ0(g)Aρ0(g)⁻¹−A= 0 for allg∈π1(X−S,η), that is,¯

A∈End⁽⁰⁾_π

1(X−S,¯η)(κ^r)∼=H⁰(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)).

So we have exact sequences

0→H⁰(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀))→End⁽⁰⁾(κ^r)→B¹(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀))→0,

0→B¹(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀))→Z¹(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀))→H¹(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀))→0.

It follows that dimD_S^,λ(κ[])

= dimZ¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + (|S| −1)r²

= dimH¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + dimB¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + (|S| −1)r²

= dimH¹(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + dim End⁽⁰⁾(κ^r)−dimH⁰(π1(X−S,η),¯ Ad⁽⁰⁾(ρ0)) + (|S| −1)r²

= −χ(π1(X−S,η),¯ Ad⁽⁰⁾(ρ₀)) +|S|r²−1,

where for the last equality, we use the fact thatH²(π1(X−S, η),Ad⁽⁰⁾(ρ0)) = 0 ([11, Lemma 2.1]).

On the other hand, we have H²(X −S,End⁽⁰⁾(F0)) = 0 since X −S is an affine curve. We have H¹(π₁(X−S, η),Ad⁽⁰⁾(ρ₀))∼=H¹(X−S,End⁽⁰⁾(F0)) by [11, Lemma 1.6]. It follows from the definition thatH⁰(π1(X−S, η),Ad⁽⁰⁾(ρ0))∼=H⁰(X−S,End⁽⁰⁾(F0)). So we have

χ(π₁(X−S,η),¯ Ad⁽⁰⁾(ρ₀)) =χ(X−S,End⁽⁰⁾(F₀)).

We leave it to the read to calculate dimD_S(κ[]), dimD_s(κ[]), dimD_s^,λ(κ[]).

2 Obstruction theory

LetA⁰→Abe an epimorphism in the categoryCΛ such that its kernela has the propertymA⁰a= 0.

We can regardaas a vector space overκ∼= A⁰/mA⁰. Letρ:π1(X−S,η)¯ →GL(A^r) be a representation such thatρ modmA =ρ₀. Fix a set theoretic continuous lifting γ :π₁(X −S,η)¯ →GL(A^0r) of ρ.

Consider the map

c:π₁(X−S,η)¯ ×π₁(X−S,η)¯ → End(κ^r)⊗_κa∼= Ad(ρ₀)⊗_κa, c(g1, g2) = γ(g1g2)γ(g2)⁻¹γ(g1)⁻¹−I.

One can show that c is a 2-cocycle. By [11, Lemma 2.1], c must be a 2-coboundary. Choose a continuous map

δ:π1(X−S,η)¯ →Ad(ρ0)⊗κa

such thatc=d(δγ⁻¹). Then ρ⁰ =γ+δ :π₁(X−S,η)¯ →GL(A^0r) is a representation liftingρ. We conclude thatρcan always be lifted to a representationρ⁰:π1(X−S,η)¯ →GL(A^0r). Similarly, for any s∈S, one can prove that any representation Gal(¯ηs/ηs)→GL(A^r) can be lifted to a representation Gal(¯η_s/η_s)→GL(A^0r). This proves the functorD_s is smooth.