Asymptotic semi-stability of nilpotent orbits in p-adic Hodge theory
aa r X i v : . [ m a t h . N T ] D ec NILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALECOHOMOLOGY MOHAMMAD REZA RAHMATI
Abstract.
We formulate the analogue of the nilpotent orbits and nilpotent orbittheorem for variation of p -adic ´etale cohomology or crystalline cohomology withrespect to the slope filtration. Specifically we show that any such orbit convergesto semistable filtration. In continuation we discuss a generalization of SL -orbittheorem. Introduction
In complex Hodge theory a variation of Hodge structure VHS, V over the complexmanifold S locally defines a period map(1) Φ : (∆ ∗ ) n → D/ Γwhere D is the flag variety of the Hodge filtrations parametrized by the points in S and Γ is the monodromy group. ∆ ∗ is the puncture disc in dimension one. D receives a complex structure by an embedding D ֒ → ˇ D . By factoring with Γ onemeans the period map is a Γ-equivariant map into D . Then a nilpotent orbit in ˇ D can be written as(2) ψ ( z , ..., z n ) = exp( z N + ... + z n N n ) exp( n ( z , ...z n )) F on U n where U is the upper half plane, and n takes values in the nilpotent cone of g = End ( V ). F is the Hodge filtration on V and is a point in ˇ D . The study of theasymptotics of this map is an important subject in Hodge theory, which basicallyleads to the nilpotent and SL -orbit theorems. Key words and phrases.
Variation of Hodge structure, ´etale cohomology, Crystalline cohomology,Hodge-Tate decomposition, Slope filtration, Semistability, Nilpotent Orbits, p-adic Nilpotent orbit,Quasi-unipotent F-Isocrystals.
In the p -adic case one has the Hodge-Tate decompositions for the ´etale and crys-talline cohomologies when the coefficient system is extended to C p = c Q p . TheHodge-Tate (HT)-decomposition defines the slope filtration for these vector spacesover C p and one can consider the period space for these filtrations with same theFrobenius-Hodge numbers, similar to the complex case. In this case a variation ofHT-structure defines a G = Gal ( Q p / Q p )-equivariant map(3) Θ : S −→ F ( ν ) rig,ss called also the period map. The flag variety F ( ν ) rig,ss is assumed to be embeddedby the Berkovich functor as a rigid analytic space over C p . ν = ( ν , ..., ν n ) ∈ Q n arethe set of slopes. It is the flag variety of semi-stable HT-filtrations embedded as anopen subset in the whole flag manifold F ( ν ). The group G = Gal ( Q p / Q p ) plays therole of the monodromy group in this case. A well known theorem by S. Sen saysthat, for L a p -adic field, a finite dimensional C p -representation ρ of G L = Gal ( ¯
L/L )onto V can locally be written as(4) ρ ( σ ) = exp[ θ log χ ( σ )]where χ is a cyclotomic character. The operator θ ∈ End ( V C p ) is called the Sen op-erator associated to the representation ρ . We will use this terminology followed by aclassical Fourier analysis on the p -adic disc due to Amice-Schneider [ST]. Y. Amiceintroduces a p -adic Fourier correspondence which allows to identify the Q p -adic discwith its character variety; i.e a space of cyclotomic-type characters. This correspon-dence later was generalized by P. Schneider and Teitelbaum. We consider the disc asthe space of characters via the p -adic Fourier theory of Amice and Schneider. Theidentification will help us to shorten the proof of orbit theorems and also use theestimates in [ST] for the convergence in p -adic case. It follows that we can write a p -adic nilpotent orbit as(5) exp[ N log( κ z )]Θ( κ z ) = exp[ N log( κ z )] exp[ n ( χ z ) log χ z ] F where κ z and χ z are cyclotomic type characters and F is a reference point. Thedifference between the orbit in (4) and that in (2) is that the latter is parametrizedby characters. In other words we reparametrize the period map of p -adic isocrystalsin terms of cyclotomic (Lubin-Tate) characters labeled by the points of the p -adicdisc. Then what we need is that the pointwise parametrization is analytic. Thisis the point we use the theory in [ST], [BSX]. The application concerns the studyof semistability in the limit of a nilpotent orbit of p -adic Hodge-Tate structures.The theorem states that the limit filtration associated to a nilpotent element in the ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 3 nilpotent cone of the representation V is semistable. We will call it the limit slopefiltration. It will play the role of the limit Hodge filtration in complex Hodge theory.This concept will open the way of doing standard Hodge geometry in the p -adicHodge theory.The motivation after nilpotent orbit theorem is the SL -orbit theorem. In complexHodge theory it is stated as follows. Given any nilpotent orbit exp( zN ) F , it ispossible to choose a homomorphism of complex Lie groups ψ : SL (2 , C ) → G C , anda holomorphic, horizontal, equivariant embedding ˜ ψ : P → ˇ D , which is related to ψ by(6) ˜ ψ ( h . i ) = ψ ( h ) ◦ F, Moreover, one can also find a holomorphic mapping z g ( z ) of a neighborhood of ∞ ∈ P into the complex Lie group G C such that(7) exp( zN ) ◦ F = g ( − iz ) ˜ ψ ( z )Starting from a p -adic nilpotent orbit exp( N log κ z ) one can similarly embed N into an sl -triple in g = End ( V ) where V is an isocrystal. There is 1-1 correspon-dence between the Nilpotent orbits in g and equivalence class of ǫ -Hermitian Youngtableaux, ǫ = ±
1. By a Gram-Schmid type argument one can choose an orthog-onal frame of the p -adic local system in a way that it remains orthogonal along anilpotent orbit. This will lead us in the first step to attach an SL -orbit to a givennilpotent orbit, such that they are asymptotic to each other. We screen a sketch ofthe proof in [S] and [P] and a general p -adic Lie group argument to explain a partialgeneralization of SL -orbit theorem. Organization of the text:
Section (2) involves the explanation of Hodge-Tatestructure on the ´etale and crystalline cohomologies. In section (3) we generalize thislanguage to p -adic isocrystals and define semistability. Section (4) involves a basicdiscussion on the period domains of isocrystals, in order to introduce the notationsand give some sense on the nature of these spaces. We mainly express the strongstratification property of these spaces following the texts [R], [R1], [R2]. In section(5) we review the Amice-Schneider-Teitelbaum p -adic Fourier theory. Our use of thistheory is formal and it only makes the proof of the main result in the next sectionshorter, or more solid. It is possible to skip this language, in the proof of limittheorem however the proof would need adhoc technical facts to be proved. Thusthe use of the theory in [ST] will provide a solid concentration in the proof of thenilpotent orbit theorem in p -adic case. Section (6) contains the definition of the p -adic nilpotent orbits and the analogous theorem for the Schmid nilpotent orbittheorem in the p -adic case. The Hilbert-Mumford criterion for semistability has MOHAMMAD REZA RAHMATI been used together with several ideas in the Schmid proof in [S] together with someapplications of the Schneider-Teitelbaum Fourier theory in the proof. In section (7)we sketch the ideas of SL -orbit theorem, and provide a partial generalization ofthe work of W.Schmid to the nilpotent orbits of p -adic isocrystals. Section 8 is anexpository description of the monodromy transformation in p -adic Hodge theory. InSection 9 we have discussed the mixed case.2. Hodge decomposition and classical nilpotent orbit theorem
The cohomology group H m ( X, Q ) of a compact Ka¨hler manifold X has a canonicaldecomposition(8) H k ( X, C ) = M p + q = k H p,q (= M p + q = k H q ( X, Ω pX ) ) , H q,p = H p,q for all 0 ≤ k ≤ n = dim( X )) where Ω qX is the sheaf of q -differentials on X ,known as Hodge decomposition. This decomposition is equivalent to the existenceof the Hodge filtration defined by F p H k C := L r ≥ p H r,s ( X ). The combination of thisfiltration with the symmetry H p,q = H q,p is called a Hodge structure. ComplexHodge theory associates to a smooth projective morphism f : X → S of complexquasi-projective varieties, the variation of Hodge structure V s := H k ( X s = f − ( s ))over S , for each 0 ≤ k ≤ n and defines a map φ k : S → D . The set D is theclassifying space of polarized Hodge structure, which inherits a complex structure byembedding D ֒ → ˇ D into its compact dual. If S ֒ → S is a partial compactification,then V degenerates along the boundary S \ S . The data of such a variation of Hodgestructure can also be encoded in a semisimple representation(9) ρ : π ( S, s ) → Aut ( V = V s )It is convenient to take S = ∆ ∗ n and consider unipotent monodromies T j =exp( − N j ). Then the period map is(10) φ : (∆ ∗ ) r → D/ Γ , s F • s where Γ = Im ( ρ ). The period space D/ Γ should be understood as a moduli of Hodgeflags moduli the equivalences coming from the automorphisms of the ambient (thefibers) variety. By the Monodromy Theorem, [S], the eigenvalues of the generatorsof the monodromy group Γ are roots of unity. Therefore possibly after passing to afinite cover of the punctured disc, we have the lift of the period map
ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 5 (11) F : U r → D where U is the upper half plane, and the covering map U r → (∆ × ) r is defined by(12) s j := exp(2 πiz j )The map(13) Ψ( z , ..., z r ) = exp( X z j N j ) F ( z , ..., z r ) , Ψ : U r → ˇ D is holomorphic. A classical question in Hodge theory is to study the asymptotic ofthe Hodge decomposition along a degenerating limit. This idea can be described bythe notion of nilpotent orbit. Definition 2.1.
A nilpotent orbit is a holomorphic horizontal map θ : C r → ˇ D givenby (14) θ ( z ) = exp( X z i N i ) F where N j ∈ g R := End ( V ) R are nilpotent and F ∈ ˇ D such that there is a constant α such that θ ( z ) ∈ D for Im ( z ) > α . We have the following well known fact.
Theorem 2.2. (Nilpotent Orbit Theorem - W. Schmid, [S] ) Let N , ..., N r be mon-odromy logarithms and ψ : (∆ ∗ ) r → ˇ D be the untwisted period map. Then • The limit F ∞ := lim s → ψ ( s ) exists and ψ ( s ) extends holomorphically to ∆ r . • The map θ ( z ) = exp( P z j N j ) F ∞ is a nilpotent orbit. • For any G -invariant distance on D , there exists positive constants β, K suchthat for Im ( z j ) > α , such that d ( θ ( z ) , F ( z )) ≤ K P j ( Im ( z j )) β e − πIm ( z j ) . The theorem says; there exists an associated nilpotent orbit(15) θ ( z ) = exp( X z j N j ) F ∞ MOHAMMAD REZA RAHMATI which is asymptotic to F ( z ) with respect to a suitable metric on D . The map θ in (15) is horizontal in the sense by Griffiths. The nilpotent orbit theorem forperiod mapping is closely related to regularity of the Gauss-Manin connection. For aproof using regularity see [GS]. We propose to generalize this theorem on the p -adicdomains. The correspondence between the local system of Hodge structures and flatconnections on S satisfying Griffiths transversality conditions is explained by the socalled Riemann-Hilbert correspondence, [KP], [P], [S].When X is defined over an l or p -adic field L , ( p is the characteristic of theground residue field and l = p a prime), the groups H kdR ( X/L ) of algebraic de Rhamcohomology is defined similar to the complex case. If X is proper smooth then theHodge to de Rham spectral sequence(16) H q ( X, Ω pX/L ) ⇒ H p + qdR ( X )degenerates at E page and there exists a filtration F • on each H kdR ( X/L ) namelyHodge filtration such that(17) gr iF H kdR ( X/L ) = H k − i ( X, Ω iX/L )where Ω iX/L is the sheaf of i -th algebraic differential forms on X .The p -adic ´etale cohomologies of X are defined by(18) H ket ( X, Q p ) = Q p ⊗ Z p lim ← H k ( X, Z /p n ) , ≤ k ≤ n The ´etale cohomology with C p coefficient also satisfy a decomposition property . Theorem 2.3. (J. Tate) [BX]
Let X be a projective, non-singular variety over L .The p -adic ´etale cohomologies H ket ( X Q p , Q p ) satisfy the Hodge-Tate decomposition (19) H ket ( X Q p , Q p ) ⊗ C p = k M i =0 H k − i ( X, Ω iX/L ) ⊗ Q p C p ( − i ) for ≤ k ≤ n . The theorem states that the ´etale cohomology of smooth projective varieties de-composes over C p . The filtration(20) F p H ket ( X Q p , C p ) := M r ≥ p H k − r ( X, Ω rX/L ) ⊗ C p ( − i ) ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 7 corresponds to (8) and is called the Hodge-Tate filtration. The integers {− i } appearing as the twists in the decomposition are called the Hodge-Tate weights.The groups H ket ( X Q p , Q p ) are Q p -vector spaces endowed with continuous action(21) ̺ : Gal ( Q p /L ) → Gl ( H ket ( X Q p , Q p ))In general any representation of the G L satisfying a decomposition like (19) iscalled Hodge-Tate representation with HT-weights {− i } appearing in the sum. Thetheorem says that after tensoring with C p the Galois action is diagonalizable as L i C p ( − i ) ⊕ h i,m − i with eigenvalues given by the twists of cyclotomic character. If X is proper smooth over L , there are functorial comparison morphisms(22) γ k : H kdR ( X ) ⊗ K B dR ∼ = H ket ( X Q p , Q p ) ⊗ Q p B dR which respects the Hodge filtrations and the Galois actions. We have(23) F i ( B dR ⊗ H kdR ( X )) = X a + b = i F a B dR ⊗ F b H kdR ( X )The field B dr is the quotient of the Fontaine ring(24) B + dR = W ( quot ( lim x → x p O K /p ) )It is equipped with an action of a Frobenius. B + dR is a complete non-discrete valuedring with perfect residue field. The filtration F i is defined as(25) F i B dR = B + dR .ξ i , ξ a uniformizer , i ∈ Z It is known that any non-constant polynomial in Q p [ t ] has a root in B dR , i.e Q p ⊂ B dR , [BX], [FO].Assume L = quot ( W = W ( k )) for a perfect field k of char ( k ) = p and we havea proper smooth variety X /W whose generic fiber is X . P. Berthelot defines thecrystalline functor(26) H kcrys : X H kcrys ( X/W ) = lim ← H k ( X/W n ) , W n := W/p n W The crystalline cohomology groups H kcris ( X/W ) are finite dimensional W -vectorspaces with an action of a σ -linear map Φ called Frobenius ( σ is the Frobenius of L ). They have the expected dimension dim Q l H ket ( X k , Q l ) , l = p . The W -structure MOHAMMAD REZA RAHMATI of H kcrys ( X/W ) inside H kdR ( X ) endowed with Frobenius action does not depend onchoice of proper smooth model X of X . The crystalline cohomology is endowed witha canonical cup product H i × H j → H i + j and also a trace map tr : H n ( X/W ) → W .It is related to the de Rham cohomology of X/k by the long exact universal coefficientsequence(27) ... → H icrys ( X/W ) p → H icrys ( X/W ) → H idR ( X/k ) → H i +1 crys ( X/W ) → ... If X has a smooth proper lifting X then one has H kdR ( X /W ) ∼ = → H kcrys ( X/W ).In general this isomorphism will hold true after extension of the scalars to a finiteextension of W , [O2]. We have similar decomposition and functorial comparisonisomorphisms(28) H kcrys ( X/W ) ⊗ C p = M r + s = k H r,scrys ( X )( − r ) γ crysk : H kcrys ( X ) ⊗ L B cris ∼ = H ket ( X Q p , Q p ) ⊗ Q p B cris The twist means twisting with the cyclotomic character χ : Gal( Q p / Q p ) → Z × p tothe power − r , (See [FO] or [BX] for the definition of B cris ). The comparison mapsrespect the Hodge filtration and Galois action. It follows that the p -adic ´etale andcrystalline cohomology are essentially the same and determine each other. Definethe descending filtration(29) F i := { x ∈ H | Φ( x ) = p i .x } namely Frobenius Hodge filtration on the crystalline cohomology, [BX], [O2], [H],[Y]. Theorem 2.4. [O2]
Assume the Hodge spectral sequence of
X/L degenerates at E and the H ∗ crys ( X ) are torsion free. Then the image of the natural map (30) F i H kcrys ( X/W ) → H kdR ( X/L ) is the Hodge filtration. In this case the Frobenius viewed as a linear map Φ : σ ∗ H kcrys ( X ) → H kcrys ( X ) isdiagonalizable, where σ is the Frobenius of L . If H kcrys is torsion free there is a basisof this vector space with respect to which the matrix of Φ is diagonalizable. Then ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 9 the Frobenius Hodge numbers h i,j (Φ) are defined as the number of diagonal termsin the matrix whose p -adic ordinal is i . This number can also be defined as(31) h i,j (Φ) = gr iF gr jF H kcrys ( X/W )where F is the opposite filtration defined by F i := p i F n − i , [O1]. Theorem 2.5. (B. Mazure) [KL]
Assume
X/W is smooth, then the Frobenius Hodgenumbers are equal to geometric Hodge numbers. (32) h i,j (Φ) = h i,j ( X/k ) = dim H j ( X, Ω iX/k )The similarity of the behavior of the Frobenius Hodge numbers motivates the factthat analogous period domains may be defined for ´etale or crystalline cohomolo-gies with C p -coefficients. The strategy we follow concerns a representation theoryuniformization of the p -adic disc to explain the variations of ´etale cohomology orcrystalline cohomology. As a final remark lets add that the decompositions (19) and(28) can also been stated for open non-smooth separated variety over L with somelittle modifications, (see [Y] for details).3. Isocrystals
The ´etale and crystalline cohomologies are examples of more general objects namely(overconvergent) Φ-isocrystals.
Definition 3.1. [DOP]
Assume k is a field of char ( k ) = p > , W ( k ) the associatedring of Witt vectors and L its field of quotients. Let σ ∈ Aut ( K/ Q p ) be the Frobenius. • A Φ -crystal M over k is a free W ( k ) -module of finite rank with a σ -linearendomorphism Φ : M → M such that M/ Φ M has finite length. • An Φ -isocrystal E over k is a finite dimensional L -vector space with a σ -linearbijective endomorphism of Φ : E → E . A morphism f : E → E ′ of Φ-isocrystals is called an isogeny if there exists g : E ′ → E such that f g = gf = p n for some n . The category of Φ-isocrystals is obtained fromthe category of Φ-crystals by formally inverting isogenies. If the ground field k isalgebraically closed we have the following structural fact. Theorem 3.2. (J. Dieudonne) [R] , [DOP] Let k be algebraically closed. Then thecategory of φ -isocrystals over k is semisimple and the simple objects are parametrisedby the rational numbers. The simple object E λ corresponding to λ ∈ Q is (33) E λ = ( K s , Φ λ = p r . . .
00 0 1 0 .σ ) , λ = r/s, s > , ( r, s ) = 1 Furthermore,
End ( E λ ) = D λ where D λ is the division algebra with center Q p andinvariant λ ∈ Q / Z . The rational number λ is called the slope of E λ . It follows that there exists aunique decomposition(34) E = M λ ∈ Q E λ where E λ has slope λ = r/s . This means; there exists a W ( k )-lattice M ⊂ V α suchthat Φ s M = p r M . The filtration F β = ⊕ λ ≤− β E λ is called the slope or Newtonfiltration.By ordering the slopes λ ≥ λ ≥ ... ≥ λ n one can define the Newton polygon of E as the graph of the function(35) i λ + λ + ... + λ i , k is the algebraic closure of F q , q = p h the Newton polygon is the Newtonpolygon of the polynomial det(1 − Φ h .t ). The Newton polygon describes the com-binatorics of the slopes of an isocrystal in a simple way. It is possible to classifyΦ-isocrystals in low dimensions with the aid of their Newton polygons.Another fact we are going to deal with is an analogue of monodromy operator for p -adic isocrystals (see also Section 8). It will play an important role to us to statethe result of this paper. Theorem 3.3. (S. Sen) [S] , [FO] Suppose L is a p -adic field. If ρ : G L → Aut C p ( V ) is a C p -representation of G L on V , then there exists a unique operator Θ ∈ End ( V ) such that for every ω ∈ G L , there exists an open subset U ω ∈ G L . (36) ρ ( σ ) = exp[Θ log χ ( σ )] , σ ∈ U ω ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 11 where χ is a cyclotomic character. The operator Θ in this theorem is called Sen operator. We have stated this theoremin its simplest form that is over C p and is sufficient for our purpose. For more detailssee the reference. One can prove that two C p representations are isomorphic if andonly if their corresponding Sen operators defined by Theorem 3.3 are similar. It is afamous result by Sen that the Lie algebra g = Lie ( ρ ( G L )) is the smallest Q p -space S ⊂ End Q p V such that Θ ∈ S ⊗ Q p C p . The Sen theorem philosophically explainsour method to state nilpotent orbits in the p -adic Hodge theory.The Category of isocrystals is equipped with the data of a nilpotent transformation N : E → E which satisfies(37) N ◦ φ = p r .φ ◦ N, some r ∈ Z > This transformation may be derived from the Theorems 3.3, but it can be indepen-dent. In this case a Φ-isocrystal may also be called (Φ , N )-modules.
Example 3.4.
Take L = Q p ( p √ , and V = L ∼ = Le ⊕ Le . Consider the followingdata φ ( e ) = e , φ ( e ) = p k − e and F k − = L ( e + π i e ) for i ∈ { , ..., p − } and set N = 0 , and the Galois action g.e = e , g.e = χ L ( g ) − i .e for g ∈ Gal ( L/ Q ) . Where χ L is Teichmuller lift of the cyclotomic char-acter. Then V is a crystalline representation of G L , i.e defines Φ -isocrystals withHT-weights { , k − } , [B] . A basic notion related to isocrystals is that of semi-stability.
Definition 3.5. [DOP] if ( V, F ) is a filtered L -vector space, then define the degreeand the slope of the filtration F (38) deg( V, F ) = X i i. dim gr iF ( V ) , µ ( V, F ) = deg(
V, F ) /rank ( V, F ) respectively. A filtration F • of an isocrystal M is called semistable if µ ( N, F • ) ≤ µ ( M, F • ) for any subisocrystal N of M . The degree is an additive function on the category of filtered vector spaces andalso one has deg( V ) = deg( V max V ). The slope satisfies µ ( V ⊗ W ) = µ ( V ) + µ ( W ) , µ ( V ∗ ) = − µ ( V ). We purpose to study semistability in a variation of Φ-isocrystals. Particularly we are interested to check out this property in the limits ofspecial one parameter family of p -adic isocrystals. The concept of (semi)-stability can also be defined for vector bundles which playsa central role in the theory of moduli spaces. The next theorem called the Hilbert-Mumford inequality for a semistable vector bundles in GIT explains a criteria forthis. Assume that the filtration F of V is split semisimple, that is V = ⊕ i V i with V i ofpure type. This decomposition correspond to a 1-parameter subgroup of G = Gl ( V )namely λ : G m → Sl ( V ). By considering a suitable basis { s , ..., s n } of V we canwrite λ diagonally as(39) λ ( t ) = diag [ t ρ , ..., t ρ n ] t − ρ , ρ ≥ ... ≥ ρ n where ρ = P ρ i / ( n + 1). Mumford introduced a subsheaf L ( ν ) = ( t ρ s , ..., t ρ n s n ) ⊂O X × A (1) generated by the sections in the parenthesis. The semistable subset F ( ν ) ss of F ( ν ) can also be described in terms of geometric invariant theory. Theorem 3.6. [DOP] , [R2] A point F ∈ F ( ν ) is semistable if and only if (40) µ L ( ν ) ( F, λ ) ≤ L ( ν ) can also be defined as having the fiber N ri =1 det ( F i /F i − ) ⊗ ν ( i ) .We will use this criterion in order to check out the semistability in the limit of anilpotent orbit, in Section (6).4. Filtration by slopes and Period domains over p -adic fields In this section we briefly review the structure of period domains over p -adic fieldsfrom the references [R], [R1] and [DOP]. A short discussion in terms of the languageof p -divisible groups and their deformation spaces has been added at the end. Webegin with A direct consequence of the Theorem 2.2 that is if k is algebraically closed,there is an injection(41) { Isomorphism classes of Φ − isocrystals of rank n } ֒ → ( Q n ) + E ν ( E )called Newton map where ( Q n ) + := { ( ν , ..., ν n ) ∈ Q n | ν ≥ ... ≥ ν n } where λ occursin ν ( E ) with multiplicity the dimension of the isocrystal (isotypic or isoclinic alsoused) component of type λ . The image of the Newton map can be described asfollows. Write ν ∈ ( Q n ) + as(42) ν = ( ν n , ..., ν n r r ) , ν > ν > ... > ν r ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 13 Then(43) ν ∈ Image ⇔ ν i .n i ∈ Z , ∀ i Assume for example we have a family of abelian varieties
A/S over a base scheme ofcharacteristic p >
0. When l = p the family of Tate modules T l ( A s ), for s ∈ S definesa local system of Z l -modules on S . If l = p , the Tate modules T l ( A s ) are replacedby Dieudonne modules M ( A s ), which are Φ-crystals. In this case the Dieudonnemodules are not constant as s varies. There exists a stratification of S with locallyclosed strata’s where the isomorphism class of the Dieudonne module M ( A s ) ⊗ Z p Q p are constant, [R], [DOP]. Theorem 4.1. [R] , [DOP] Let E be Φ -isocrystals over a scheme S of characteristic p . Then the Newton vector of E s goes down under specialization, that is when E isof constant rank the function s
7→ k ν ( E s ) k is locally constant on S and for any ν ,the set { s ∈ S | ν ( E s ) ≤ ν } is locally closed in S . A non-trivial consequence of this theorem is the following stratification propertyof period domains of Φ-isocrystals.
Corollary 4.2. [R] , [DOP] When E be Φ -isocrystals over a noetherian scheme S ofcharacteristic p , then the set of points of S where the Newton vector is constant islocally closed in S and defines a finite decomposition of S . Suppose V is an L -vector space of dimension n , and let ( ν (1) > ... > ν ( r )) ∈ Q r be arbitrary. Let(44) ν = ( ν (1) ( n ) , ..., ν ( r ) ( n r ) )Define(45) F ( ν ) := { ⊂ V ⊂ ... ⊂ V r = V | rank( gr iF ( V )) = n i } Berkovich defines a natural analytification functor F ( ν )
7→ F ( ν ) an into a smoothcompact K -manifold, which satisfies all the expected compatibility properties op.cit, [DOP]. The group Gl ( V ) acts transitively on F ( ν ). Let F ( ν ) ss be the locusof semistable filtrations. F ( ν ) ss is the period domain associated to ( V, ν ), and isZariski open in F ( ν ) for basic reasons. The structure theory of flag variety F ( ν ) asa symmetric space can be studied via the root system of the Lie group G = Gl ( V ) andits Lie algebra. The basics of this approach proceeds almost the same as the complexcase. In this case F ( ν ) has the structure of Schubert variety, that its stratification is through the Schubert cells corresponding to closure of torus orbits, via the momentmap, see [DOP] for details. Theorem 4.3. [DOP]
Let E be an isocrystal, and ν ∈ ( Q n ) + . • The subset { x ∈ F ( ν ) an | F x ∈ F ( ν )( k ( x )) ss } , is open in F ( ν ) an , and henceis the underlying space of an analytic space. (The field k ( x ) is the residuefield of the local ring at the point x and bar means completion). • The map (46) F ( ν ) → R , x µ F x ( E ⊗ k ( x )) is upper semi-continuous and locally constant. • There exists a rigid analytic space F ( ν ) ss,rig and a fully faithful functor F ( ν ) ss,an → F ( ν ) ss,rig . By rigid space we mean it is locally isomorphic to an affinoid space on which theGalois group G acts. An affinoid space is a space isomorphic to a subset of p -adicdisc defined by the zero set of some ideal of convergent power series. A rigid analyticspace is the analogue of the complex analytic space over a non-archimedean field.We will always assume that F ( ν ) is embedded into its rigid analytification via theBerkovich functor and Theorem (4.3), without mentioning it, see [DOP], [SW] fordetails.By Theorem 4.1, it follows that(47) F ν ⊂ [ ν ′ ≤ ν F ν ′ The property (47) distinguishes the p -adic period domains from the complex ones.This inclusion is refereed to a strong stratification property (also called Newtonstratification) of the period domains over (finite and) p -adic fields. It raises manystructural open questions about these spaces. For instance a major open questionconcerns if the closure of each strata F ( ν ′ ) meets F ( ν ) for ν ′ ≤ ν . Example 4.4. [R]
Let g be a positive integer, and m ≥ prime to p an auxiliaryinteger. The Siegel moduli space M g,m of genus g over F p classifies the isomorphismclasses of triples ( A, λ, η ) where A is an abelian scheme of relative dimension g over S , and λ : A → ˇ A is a polarization, and η is a level m structure. The existence ofpolarization implies that the Newton vector of the fibers lie in (48) ( Q g ) = { ( ν , ..., ν g ) ∈ ( Q g ) + | ν i + ν g − i +1 = 1 , ≤ ν i ≤ } ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 15 Let B g be the subset satisfying the integrality condition, partially ordered. The vector ̺ = (1 g , g ) is the maximal element called ordinary Newton vector. In contrary σ = ((1 / g ) is the minimal one called supersingular Newton vector. One can showthat (49) S ν = [ ν ′ ≤ ν S ν ′ Let ∆( ν ) = { ( i, j ) ∈ Z | ≤ i ≤ g, P il =1 ν g − l +1 ≤ j < i } and d ( ν ) = ♯ ∆( ν ) . Thenit is known that S ν is equidimensional of dimension d ( ν ) . One should notice that theinclusion in (47) is in general strict, rather (49). There exists analogous (dual) machinery of isocrystals in terms of the language of p -divisible groups. A p -divisible group may be defined as an inverse system of finitegroup schemes { G p hn } n over a base formal scheme S . In this case h is called theheight and p is called the characteristic of the group. • There is an anti-equivalence between the category of p -divisible groups over spec ( k ) and the subcategory of Φ-crystals over spec ( k ) consisting ( E, Φ) suchthat p.E ⊂ F.M . • There exists an anti-equivalence between the category of p -divisible groupsover spec ( k ) up to isogeny and the full subcategory of Φ-isocrystals over spec ( k ) with slope between 0 and 1.In this way the classifying space of the Φ-isocrystals is interpreted as deformationspace of the p -divisible group. In the following we briefly introduce this concept.Assume O L is a complete discrete valuation ring with uniformizer π , we denote by N ilp ( O L ) the category of locally noetherian schemes S over Spec ( O L ) such that theideal sheaf π. O L is locally nilpotent. Let S be the closed subscheme defined by π O L .The following theorem describes a typical deformation space for p -divisible groups. Theorem 4.5. [R1]
Let X be a p -divisible group over spec ( k ) . Consider the functor M : N ilp ( W ( k )) → p -DIV /S which associates to a S ∈ N ilp ( W ( k )) the pair ( X, ̺ ) consisting of a p -divisible group over Spec ( k ) and a quasi-isogeny ̺ : X × k S → X × S S . Then M is representable by a formal scheme locally formally of finite typeover the formal scheme Spf ( W ( k )) . The structure sheaf of a formal scheme Spf( A ) where A is complete with ideal ofdefinition m is lim ← A/ m n . This representability implies that certain functors on the category of p -divisible groups with endomorphisms or polarizations are also repre-sentable. We have the following analogue of the Theorems 4.1 and 4.2. Theorem 4.6. [R]
Assume S is a regular scheme of char = p > , and X is a p -divisible group over S with constant Newton vector ν . Then X is isogenous to a p -divisible group Y which admits a filtration by closed embeddings Y ⊂ Y ⊂ ... ⊂ Y r = Y satisfying integrality conditions (43). There also exists natural numbers r i ≥ , s > such that ν ( i ) = r i /s i and (50) p − r i F r s i : Y i → Y ( σ si ) i and (51) p − r i F r s i : Y i /Y i − → ( Y i /Y i − ) ( σ si ) are isomorphisms. The superfix in Theorem means the fixed element under the action. If X is a p -divisible group of height h over S of characteristic p s.t (43) holds, we can associatea lisse p -adic sheaf of W ( F p )-modules V X = lim ← V X,n for the ´etale topology on S by(52) V X,n = { x ∈ M/p n M | p − r F s ( x ) = x } where M is the Dieudonne crystal of X . The fibers of V X are free W ( F p )-modulesof rank h . The corresponding W ( F p ) ⊗ Z p Q p -adic sheaf depends on the isogeny classof X and corresponds to a representation of the fundamental group(53) ̺ X : π ( S ) → Gl h ( W ( F p ) ⊗ Z p Q p )The stratification in Theorem 4.6 decomposes the representation ̺ = ⊕ i ̺ i by theblocks of size h i = height( Y i /Y i − ), cf. [R].5. Representation interpretation of the p -adic disc( p -adic Fourier theory) In the following we give a brief description on representation theory interpretationof the disc in p -adic theory, using a method due to Amice in 60s, [A], [ST], [BSX].Let B be the unit disc over Q p . Let O ( B ) (resp. O b ( B )) be ring of holomorphic(resp. bounded holomorphic) functions on B . Denote by(54) R ( B ) = functions holomorphic on the anulus r < | z | < r ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 17 Let E † ( B ) be the ring of bounded elements in R ( B ). It is equipped with the norm k P a n z n k := sup n | a n | and we call E ( B ) its norm completion. All of these rings carrya monoid action of Z p r a ∗ ( z ) = [ a ]( z ) := (1 + z ) a − , Γ)-module with Γ = Z × p , φ = p ∗ . A (Φ , Γ)-modulewith a semilinear action of Z p r
0. Let(56) χ : G Q p → Z × p , H Q p := ker( χ )be the cyclotomic character. Its action is the same as (55). Theorem 5.1. (Fontaine) [FO] , [ST] There is a one to one correspondence (57) { Z p representations of G Q p } ∼ = −→ { ´etale ( φ, Γ) − modules over E ( B ) } T ( E ( B ) ⊗ Z p T ) H Q p where the superscript means fixed elements under action of H Q p . Here Γ =
Gal ( Q p ( µ p ∞ ) / Q p ). In p -adic Hodge theory we are involved with se-ries of natural 1-1 correspondences between (Φ , Γ)-modules and different Z p -linearrepresentations satisfying de Rham, crystalline or semistability conditions. Thesecorrespondences become compatible with Theorem 5.1 according to [BSX]. Theo-rem 5.1 can also be stated with the rings R ( B ) or E † ( B ) as well. Under all theseisomorphisms the ´etale modules correspond to each other. When Q p is replaced bya finite extension L , there is a major difficulty appearing. The character in (56) isreplaced by its analogue(58) χ L : G L → O × L , H L = ker( χ L )called Lubin-Tate character. Taking a uniformizer π ∈ O L the inverse system ofresidual quotients of O L by powers of π defines a formal group law or a p-divisiblegroup denoted also by T . Then analogue of the Fontaine theorem holds with Γ = O × L and φ = π ∗ . In case we have an isomorphism O L ∼ = Z p × ... × Z p as Q p -Lie groupswhich implies(59) B L ֒ → B Q p × ... × B Q p , O L ( B L ) և O L ( B Q p × ... × B Q p ) as analytic subvarieties. We have O L ( B L ) ֒ → B dR . The geometry of the disc B canbe explained by the analytic characters of O L as the following. Theorem 5.2. [ST]
There is an isomorphism (60) B ( C p ) ∼ = −→ { L-analytic characters O L → C p } =: c O L z κ z ( x ) := (1 + z ) x The space c O L ⊂ D ( O L , C p ) is the subspace of locally analytic characters on O L .It is also equal to the continuous dual to the space of C p -valued analytic functions C an ( O L , C p ). A simple way to stress this isomorphism for L to be Q p is by lookingat its converse given as(61) c Z p ( C p ) → B ( C p ) , χ χ (1)Theorem 5.2 is crucial to us, as it parametrizes the points in the p -adic disc withthe Lubin-Tate characters (cyclotomic in case L = Q p ). This with the formalism ofthe Sen Theorem 3.3 allows us to express the period morphism in terms points inthe character variety c O L . Another observation concerning (60) is the possibility towrite down a power series expansion for κ z . The map κ ( z ) = κ z is given by a formalpower series of the form(62) κ ( z ) = 1 + F t ′ ( z ) , F t ′ ( z ) = Ω( t ′ ) .z + ... ∈ z. O L [[ z ]]depending to a choice of t ′ ∈ T ′ , where T ′ is the (Cartier) dual of T . The elementΩ( t ′ ) ∈ O L is a fixed period depending on t ′ . different choices of t ′ will correspondto multiplication by an element in O L on Ω. Theorem 5.3. [ST] , [BSX] There exists a Fourier transform (63) F : D ( O L , C p ) ∼ = −→ c O L , λ F λ where F λ is defined by (64) F λ : c O L ( C p ) → C p , χ λ ( χ )The monoid action of O L r a ∗ ( κ z ) := κ z ( a. − ) ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 19 The isomorphism (60) at the first glance seems natural that the characters areparametrized by the disc around one, but the power series is of course a power serieswhich only converges on a disc around zero, [ST], [BSX].The isomorphisms (60) and (63) imply a pairing(66) { , } : O ( B/ C p ) × C an ( O L ) → C p The following formulas hold for { , } : • { , f } = f (0). • { F, κ z } = F ( z ). • { F, κ z .f } = { F ( z + ) , f } . • { F, f ( a. ) } = { F ◦ [ a ] , f } . • { F, f ′ } = { Ω log
F, f }• {
F, xf ( x ) } = { Ω ∂F, f }• { F, P m ( Ω) } = (1 /m !) d m f /dz m (0)The space of analytic functions f : O L → C p has the structure of a Hilbert spacewith respect to the above pairing as the following theorem states. Theorem 5.4. [ST]
Any analytic function f : O L → C p has a unique representationin the form (67) f = ∞ X n =0 c n P n ( . Ω) where c n = { f, z n } . Furthermore, such a series is convergent provided that thereexists a real number r such that | c n | r n → as n → ∞ . We will use this theorem in an argument on convergence for nilpotent orbits inthe next section.6.
Nilpotent orbits and the limit slope filtration
In this section we entirely assume that the coefficient system is extended to C p .When we have a variation of p -adic ´etale cohomology (or crystalline cohomology)over the scheme S , the filtrations(68) F is = M r ≥ i H r,set ( X )( C p ) , resp. ( H r,scrys ( C p )) on the ´etale cohomologies H ket ( X Q p , Q p ) ⊗ C p (resp. H kcrys ( X/W ) ⊗ W C p ) explainedin Section 1, namely relations (19), (28) define the period space of Hodge-Tate struc-tures having the same Frobenius-Hodge numbers, namely F ( ν ) (here ν = ( ν i ) where ν i = P r ≥ i h r,s , h r,s := dim H r,set ( X )) and a period map(69) Θ : S → F ( ν ) , s F is where F ( ν ) is assumed to be embedded by the Berkovich functor as a rigid analyticspace, that is locally looks like an affinoid space. We call (69) the period mapassociated to the variation of ´etale or crystalline cohomology. We also replace S bythe disc B later on and assume that its radius is small enough. Theorem 6.1. [DOP] , [HG] , [HA] , [DG] , [SW] The period map
Θ : B L → F ( ν ) isa G L -equivariant rigid analytic map of rigid analytic spaces over L . By this map a Hodge-Tate structure should be understood as an invariant of analgebraic manifold defined over a number field. In order to make it depend on theisomorphism class of the manifold one must identify any two HT-structure that arerelated by a isomorphism of the manifold. We use the analyticity of the period mapto write down a generalized Mahler expansion for Θ as in Theorem 5.4, (67). Wedefine the p -adic nilpotent orbit as(70) η ( κ z ) = exp[ N log( κ z )] F , F ∈ F ( ν )where as before N is a nilpotent transformation on the isocrystal satisfying N ◦ φ = p r .φ ◦ N , and F ∈ F ( ν ). Definition 6.2.
The map η : B ( C p ) → F ( ν ) given by (70) is a nilpotent orbit if • η ( κ z ) ∈ F ( ν ) ss for κ z → . • N ◦ φ = p r .φ ◦ N , for some r . Making the limit makes sense because the ground variety is an analytic spaceand carries a natural metric. The definition can be alternatively be stated with thevariable κ z with its logarithm, when its radius is chosen sufficiently enough. Thenthe parameter should be considered to tend to infinity. Theorem 6.3.
Let
Θ : B → F ( ν ) be the period map of a variation of ´etale coho-mology. Define η ( κ z ) = exp[ N log( κ z )]Θ(log κ z ) . Then the followings are true ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 21 (1) The limit F ∞ := lim κ z → η ( κ z ) exists and is a semistable filtration of V . (2) ξ ( κ z ) = exp[ N log( κ z )] F ∞ is a nilpotent orbit. (3) For each non-archimedean metric d on F ( ν ) an,rig , there exists constants C , k , l and r > such that (71) d ( ξ ( κ z ) , Θ(log κ z )) < C.p − k.l/q ( n ) r n , ∀ n is true, where q ( n ) → ∞ as n → ∞ . By non-archimedean metric we mean a metric that satisfies the following propertyknown as triangle axiom(72) d ( ξ, θ ) ≤ max [ d ( ξ, ζ ) , d ( ζ , θ )]for any 3 points ξ, ζ , θ in the space. Using the triangle axiom we can assume thatthe estimate in item (3) is enough to be considered for each component. The proofis based on several structural facts which we list as lemmas. Lemma 6.4. [ST]
The function (73) H ( x, y ) = exp( y log x ) is a rigid analytic function on B ( r ) for r sufficiently small. In fact the Fourier expansion for H ( x, y ) has a simple form that looks like theTaylor expansion in analysis. Define the norm k . k on the space of power series by(74) k X i c i ( z − a ) i k a,n := max i {| c i π ni |} , Alt. k f k a,n = max z ∈ a + π n O L | f ( z ) | Lemma 6.5. [ST]
In the Mahler expansion of the function exp( . Ω log z ) = P P m ( . Ω) z m for all a ∈ O L and all n (75) k P n ( y. Ω) k ,n ≤ max ≤ i ≤ n k P i ( y ) k ,n Furthermore, there are constants C and k such that (76) k P l ( y. Ω) k ,n < C p − k.l/q ( n ) , n ≥ where q ( n ) → ∞ as n → ∞ . We have the power series expansion(77) exp( y log( z )) = ∞ X n =0 P n ( y ) z n where the polynomials p n ( y ) satisfy the following properties, • p ( y ) = 1 , p ( y ) = y . • p n (0) = 1 , n ≥ • deg( p n ) = n and the leading coefficient of p n is 1 /n !. • p n ( y + y ′ ) = P i + j = n p i ( y ) p j ( y ′ ) • p n ( ∂ ) .f ( x ) | x =0 = (1 /n !)( d n f /dx n ) | x =0 , where f ( x ) ∈ C p [[ x ]]The last property can be obtained by a comparison to the formal Taylor series; let δ = d/dx then the Taylor formula reads as(78) exp( δb ) h ( a ) = X n δ n n ! h ( a ) b n = h ( a + b )Inserting a = log( x ) , b = log( y ) , h = f ◦ exp one gets(79) exp( ∂ log( y )) f ( x ) = f ( x + y )Therefore the orbit limit we are going to compute with is an analogue of formalTaylor series expansion, [ST].If F and F be two filtrations on the same vector space V , the pairing betweenthem is defined by(80) < F , F > := X x,y xy. dim gr xF ( gr yF ( V )))The GIT slope defined in 3.6 is related to this pairing as follows. ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 23 Lemma 6.6. [DOP]
Let λ be a one parameter subgroup of Gl ( V ) , that is λ : G m → Gl ( V ) is a group homomorphism. Assume that F is a fixed point of λ . Then (81) µ L ( ν ) ( F, λ ) = − < F, F λ > The filtration F λ is called the filtration corresponding to λ . It corresponds toan eigenspace decomposition V = ⊕ x ∈ Z V x associated to the co-character λ . Then F xλ := ⊕ y ≥ x V x . The projection π x : F xλ → gr xF λ V has a unique G m -equivariant sectionnamely i x : gr xF λ ∼ = → V x . The G m -family t → t − x λ ( t ) of endomorphisms of F xλ has alimit at 0 given by i x ◦ π x . The property in (81) is one of the structural propertiesof the 1-parameter subgroups in Gl ( V ) and Mumford invariant for semi-stability. Lemma 6.7. ( [DOP] page 39) Assume F is a Q -filtration on V . Let λ be a 1-parameter subgroup of GL ( V ) . Let F ∞ = lim t → λ ( t ) .F . Then (82) < F ∞ , F λ > = < F , F λ > The lemma asserts that the function < ., F λ > is constant along the orbit λ corresponded to F λ . This property is general for 1-parameter orbits on flag domains,see [VRS], and remark 6.8 below.For the following assume r the radius of the disc B ( r ) is small enough such thatlog : B ( r ) → B ( r ) is an isomorphism and we have(83) B ( r ) ∼ = −−−→ B ( r ) z → κ z y y z Ω .z b O L ( r Ω) −−−−−−−→ κ z → log κ z (1) B ( r | Ω | )The log is the logarithm of the associated formal group law. To explain this diagramfirst note that the map z κ z maps B ( r ) → c O L ( r | Ω | ) and is a rigid analyticisomorphism. Thus the diagram says under the isomorphism B ( C p ) ∼ = → b O L ( C p ) thelogarithm functions are compatible, see [ST] and [BSX] for details.Before starting the proof of the Theorem 6.3 lets note a difference of the parametersappearing in the functions η and Θ. The variable of the first function is supposedto belong to the p -adic disc B ( r ) via Theorem 5.2. The second is assumed to takea variable on the cover of the disc by the O L -logarithm. It is a similar situation inTheorem 2.2. We have also assumed the radius of the disc namely r is small enough. Proof. (proof of Theorem 6.3) According to Theorem (6.1) and Lemma (6.4) themap η is a rigid analytic map of the variable κ z . We use the Sen theorem to writeΘ(log κ z ) = exp[ n (log χ z ) log χ z ] F for a fixed point F ∈ F ( ν ). Then the function n (log χ z ) is locally rigid analytic on the variable by (76). The orbit function(84) η ( κ z ) = exp[ N log( κ z )]Θ( κ z ) = exp[ N log( κ z )] exp[ n ( χ z ) log χ z ] F can be written as the product of two series of the form P m ˜ P m ( . Ω) z m with co-efficients to be polynomial functions in matrices. The polynomials appearing ascoefficient of z m in both of these series satisfy the estimates mentioned in Lamma6.5 when the matrix in variable may be regarded as a single variable. Thereforethe criteria in Theorem 5.4 will hold for their convergence. Thus if we restrict theparameter in a sufficiently small disc B ( r ), (84) will converge. We denote the limitby F ∞ (an analogue of Hartogs theorem in complex analysis implies that η extendsanalytically to 0, but we do not need this fact here). The linear group G = Gl ( V )acts transitively on F ( ν ). Therefore F ( ν ) = G/P where P is a parabolic subgroupof G consisting of the elements preserving a fixed flag F . For κ z ∈ B ( r )( C p ) chooseelements g, g Θ ∈ G lifting ξ ( κ z ) , Θ(log κ z ) respectively. Then if k κ z k ≤ r for suitable r both of the functions g, g Θ are confined with a compact subset of G . Therefore if d is a non-archimedean metric on the flag variety, then for a suitable constant C (85) d ( ξ ( κ z ) , Θ(log κ z )) ≤ C.max ( k exp[ N log( κ z )] k , k exp[ N log( κ z )] exp[ n (log χ z ) log χ z ] k )Now the estimates for Fourier coefficients in the Lemma 6.5 will hold for each factorin the right hand side, which implies the inequality in item (3). Let g ∈ Gl ( V ) andSet(86) ℏ := lim κ z → exp[ N log κ z ] .g. exp[ − N log κ z ]This limit exists. If g is chosen such that g.F = F ∞ we have ℏ F = lim κ z → exp[ N log κ z ] .g. exp[ − N log κ z ] .F = lim κ z → exp[ N log κ z ] .gF = lim κ z → ξ ( κ z )Let λ (log κ z ) = exp[ N log κ z ] (in the situation of Lemmas 6.6, 6.7 and Theorem 3.6),which defines F λ . Then the estimates in item (3) says thatlim κ z → λ (log κ z ) F ∞ = F ∞ ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 25 i.e. F ∞ is the fixed point of this orbit. Now assume g ∈ Gl ( V ) of (79) be the specificsuch that F ∞ = g.F λ . Using Lemma 6.7 we have(87) < F ∞ , F λ > = lim κ z → < F ∞ , g.F ∞ > = < ℏ F ∞ , ℏ − g ℏ F ∞ > = < F ∞ , F ∞ > ≥ F ∞ is semistable. (cid:3) The filtration F ∞ may be called the limit slope filtration in analogy to the complexcase of limit Hodge filtration. We briefly remind this as follows. First fix s ∈ S ofthe ground scheme and consider the isomorphisms u s : V s → V s of isocrystals by theparallel transport. We equip V s with the semistable filtration F ∞ and define a newfiltration on each V s by transferring F ∞ via the isomorphism u s . This constructionequips the local system with a canonical filtration that fits in natural exact sequencesin cohomology theories involved.From the proof it can be understood that a similar limit(88) η ( κ z , ..., κ z r ) := exp[ N log( κ z ) + ... + N r log( κ z r )]Θ( κ z , ..., κ z r )also exists and satisfies similar estimates. The different limits corresponding to thechoice of elements in the nilpotent cone define boundary points in the period space F ( ν ) ss,rig . Remark 6.8.
One can show that the function ψ ( t ) = < F t , F > is a Morse functionand is convex along the flows of the 1-parameter family F t , if may be considered over R . Then the R -orbit θ ( t ) = exp( it.N ) F satisfies (89) θ ′ ( t ) = ∇ ψ ( t ) Thus λ ( t ) defines the gradient flow line of ψ ( t ) . This implies that the limit (90) lim t →±∞ ( θ ( t ) = exp( it.N ) F ) always exists and should correspond to some critical points of the function ψ . In [VRS] the properties of the orbit function θ ( t ) has been studied via a moment map µ : F → g , where F is the flag variety and g is the Lie algebra of the Lie group G acting on F . Then to a point F ∈ F one can associate the invariant (91) w µ ( F, u ) := lim t →∞ < µ (exp( itu ) F ) , u >, u ∈ g for a symplectic Riemanninan metric < ., . > on F , called Mumford numerical in-variant of F . It is a Theorem by Mumford that (92) w µ ( gF, gug − ) = w µ ( F, u ) , g ∈ G This invariant plays a crucial role in geometric invariant theory on the semi-stability, [VRS] . Defining the limit slope filtration in a variation of ´etale or crystalline cohomologywith C p -coefficients is a major step in doing geometric Hodge theory. For instancebeginning from a proper smooth map f : X → S , one can associate the p -adic(Hodge-Tate) local system(93) V = R k f ∗ Q p ⊗ C p = [ s ( H ket,s = H ket ( X s , Q p ) ⊗ C p )We equip this local system with the limit slope filtration corresponding to somechoice of nilpotent elements in the Lie algebra of Gl ( V ), as explained above. It is acanonical semistable filtration by Theorem 6.3. Using this one can define standardHodge theory objects similar to the complex case. For instance one may define thenormal functions as(94) ν : S → J et = [ s ( J et,s = Ext HT ( Z p (0) , H ket,s )where the extension is taken in the category of Hodge-Tate isocrystals. The localsystem V , defines a Gauss-Manin connection(95) ∇ : V ⊗ O ( B )( C p ) → V ⊗ Ω B/ C p which satisfies the Griffiths transversality with respect to the limit slope filtration.Normally the Hodge-Tate structure degenerates along singular locus, then under-standing the limit behavior of them is an important question in Hodge theory. Oneway to conduct with this question is by the Deligne nearby cycle functor. Considera diagram of schemes(96) X η j −−−→ X i ←−−− X s y f y y Spec ( L ) = η −−−→ S = Spec ( O L ) ←−−− s = spec ( k ) ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 27 where k is the residue field. The Deligne nearby functor is defined by R k ψ f = i ∗ R k j ∗ .Applying this functor to the constant sheaf C p on X η defines a sheaf concentratedon s . Therefore this sheaf may be regarded as an extenstion of V in (93) over thespecial fiber. There are canonical ways to equip this extension with a new filtrationbuild up from the limit slope filtration. This fact is a major step in the theory offiltered D -modules, op. cit. Remark 6.9.
Let L be a p-adic field. The Harder-Narasimhan (HN)-filtration ofthe isocrystal E is the unique filtration such that gr α ( E ) is semistable of slope α whenever nonzero. Note that E is semistable iff the HN-filtration has only one jump.There is an obvious functor (97) gr • : F ilIsoc ( L ) → Grad ( F ilIsoc ( L )) , ( E, F ) M α ( gr Fα E )[ − α ] Then F is the HN-filtration iff gr F • E is semistable with slope . Considering this cri-terion one can state the nilpotent orbit theorem in terms HN-filtrations. The functor gr • is an additive tensor functor which commutes with symmetric and exterior prod-uct and with duality. The Harder-Narasimhan polygon is also defined similar to theNewton polygon in (35), and analogous of Theorems 4.1 and 4.2 will hold true inthis case. SL -orbit Theorem for p -adic Hodge structures In the complex one variable case the SL -orbit theorem roughly states that, anynilpotent orbit is asymptotic to an equivariant embedded copy of the upper halfplane. More specifically, to any nilpotent orbit e zN .F of a polarized pure Hodgestructure, there can associate an SL -orbit e zN . ˆ F and a real analytic function g :(0 , ∞ ) → G R such that(98) e iyN .F = g ( y ) e iyN . ˆ F The G R -valued functions g ( y ) and g − ( y ) have convergent series expansions about ∞ of the form (1 + P ∞ k =1 A k y − k ) with A k ∈ ker(ad N ) k +1 . The precise statementgoes as follows. Theorem 7.1. [S]
It is possible to choose • a homomorphism of complex Lie groups ψ : SL (2 , C ) → G C , • a holomorphic, horizontal, equivariant embedding ˜ ψ : P → ˇ D , which isrelated to ψ by (99) ˜ ψ ( g ◦ i ) = ψ ( g ) ◦ F • a holomorphic mapping z g ( z ) of a neighborhood W of ∞ ∈ P into thecomplex Lie group G C such that (a) exp( zN ) ◦ a = g ( − iz ) ˜ ψ ( z ) , z ∈ W − {∞} ; (b) ψ ( SL (2 , R ) ⊂ G R , and ˜ ψ ( U ) ⊂ D ; (c) ψ ∗ : sl ( C ) → g = Lie ( G ) is of type (0 , for Hodge maps, that is (100) ψ ∗ ( X + ) ∈ g − , , ψ ∗ ( Z ) ∈ g , , ψ ∗ ( X − ) ∈ g , − where X + , Z, X − are (101) Z = (cid:18) − ii (cid:19) , X + = (cid:18) − i i (cid:19) , X − = (cid:18) i − i (cid:19) the generators generators of sl ; – g ( y ) ∈ G R for iy ∈ W ∩ i R ; – Ad g ( ∞ ) − ( N ) is the image under ψ ∗ of (cid:18) (cid:19) – for iy ∈ W ∩ i R , y > , let h ( y ) be defined by (102) h ( y ) = g ( y ) exp( −
12 log yψ ∗ ( Y )) then (103) h ( y ) − ddy h ( y ) ∈ ( g , − ⊕ g − , ) ∩ g – the linear transformation ψ ∗ ( Y ) ∈ Hom ( H C , C ) acts semisimply withintegral eigenvalues; let (104) g ( z ) = g ( ∞ )(1 + g z − + ... , g ( z ) − = g ( ∞ ) − (1 + f z − + ... be the power series expansion of g ( z ) and g ( z ) − around ∞ . Then themaps f n and g n map the l -eigenspace of ψ ∗ ( Y ) into linear span of thethe eigenspaces corresponding eignvalues equal or less l + n − . ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 29 – if the base point F may be chosen suitable then g ( ∞ ) = 1 , and N is theimage of (cid:18) (cid:19) ; and ( Ad N ) n +1 g n = 0 and ( ad N ) n +1 f n = 0 . We sketch some details from [P] and [S]. The subgroup G R acts transitively on theclassifying space D and the G R -valued function h ( y ) satisfies a differential equationof the form(105) h − dhdy = − LAd ( h − ( y )) N relative to a suitable L ∈ h = Lie ( G R ). After some change of variables (105) iswritten in the Lax form(106) − ddy X + ( y ) = [ Z ( y ) , X + ( y )] , ddy X − ( y ) = [ Z ( y ) , X − ( y )] ddy Z ( y ) = [ X + ( y ) , X − ( y )]A solution for this system is given as X − ( y ) = Φ( y ) x − , Z ( y ) = Φ( y ) z , X + ( y ) =Φ( y ) x + where(107) Φ : ( a, ∞ ) → Hom ( sl ( C ) , g C ) , Φ( y ) = X n Φ n y − − n/ and(108) x − = 12 (cid:18) − i − i − (cid:19) , z = (cid:18) − ii (cid:19) , x + = 12 (cid:18) ii − (cid:19) Now lets check out that how much can be stated from the above in the p -adiccase. The existence of the map ψ in Theorem 7.1 is based on a general theoremof Jacobson-Morosov; that is: given any nilpotent element x ∈ g of a semisimpleLie algebra g (over any field of char = 0) can be embedded in a three-dimensionalsubalgebra.Lets start with a nilpotent orbit exp( N log κ z ) ◦ F , where F ∈ F ( ν ) (may be notsemistable). As we explained before, we have a homomorphism of lie groups(109) ψ p : SL (2 , C p ) → G C p which determines an analytic equivariant embedding ˜ ψ : P → F ( ν ) with ˜ ψ ( g ◦ o ) = ψ ( g ) ◦ F , where o is any fixed point of P ( C p ). Then (110) ψ p ∗ : sl (2 , C p ) → g C p , ψ p ∗ ( sl (2 , C p )) ⊂ b ⊕ g − where g j := { x ∈ g | ad ( Z ) x = j.x } . Then define the function g by the following(111) exp( N log κ z ) F = g (log κ z ) ˜ ψ (log κ z )This is well defined. We can state the following analogue of Theorem 7.1. Theorem 7.2.
It is possible to choose • a homomorphism of complex Lie groups ψ p : SL (2 , C p ) → G C p , • an analytic, horizontal, equivariant embedding ˜ ψ p : P ( C p ) → F ( ν ) , which isrelated to ψ p by (112) ˜ ψ p ( g ◦ i ) = ψ p ( g ) ◦ F • ψ p, ∗ : sl ( C p ) → g = Lie ( G ) satisfies (113) ψ p, ∗ ( X + ) ∈ g − , ψ p, ∗ ( Z ) ∈ g , ψ p, ∗ ( X − = N ) ∈ g where X + , Z, X − = N are sl -triples. • an analytic mapping κ z g (log κ z ) of a neighborhood W of into the p -adic Lie group G C p such that (a) exp( N log κ z ) ◦ F = g ( κ z ) ˜ ψ p ( κ z ) ; (b) Ad g (0) − ( N ) is the image under ψ p, ∗ of (cid:18) (cid:19) (c) Let h ( κ z ) be defined by (114) h ( κ z ) = g ( κ z ) exp( −
12 log κ z ψ ∗ ( Z )) then h ( κ z ) − ddy h ( κ z ) ∈ g . The G -valued functions g ( κ z ) and g − ( κ z ) haveconvergent series expansions about of the form (1+ P ∞ k =1 A k κ − k/ z ) with A k ∈ ker( ad N ) k +1 .Proof. (sketchy adoption of [S]) The existence of the of the map ψ p , ψ p, ∗ and ˜ ψ p is based on general sl -theory as explained in the introduction and above. We alsodefined g by (111). Property can always be fixed by conjugating the representation ψ p by an element in G . We proceed to prove the properties of the functions h and g . The point exp( N log κ z ) ◦ F defines the Hodge Tate decomposition ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 31 (115) H ket ( C p ) = k M i =0 H i,k − i ( − i )( y )of the ´etale cohomology with C p -coefficients. we have H i,k − i ( − i )( κ z ) = h ( κ z ) H i,k − i ( − i )).As we mentioned in the introduction the nilpotent orbit exp( N log κ z ) ◦ F providesan ǫ -Hermitian class ( H ket ( C p ) , B ) for the representation H ket ( C p ). We will choosea basis ( w , ..., w r ) such that they successively generate the Frobenius-Hodge piecesin the slope filtration. By the same method as in [S], page 298, It is possible tomodify the given basis such that ( w ( κ z ) , ..., w r ( κ z )) represent a basis with similarproperty and remain orthogonal with respect to the Hodge filtration. Then w j ( κ z )are rational functions of κ z . The function h ( κ z ) is a lifting of the 1-parameter familyexp( N log κ z ). Its logarithmic derivative A = − h − h ′ takes values in g C p . Setting(116) A ( κ z ) = − h − h ′ , F ( κ z ) = Ad h ( κ z ) − N, E ( κ z ) = − θF ( κ z )where θ is a Cartan involution of g . One is needed to solve the similar Lax systemin analytic coordinates(117)2 E ′ ( κ z ) = − [ A ( κ z ) , E ( κ ( z )] , F ′ ( κ z ) = [ A ( κ z ) , F ( κ z )] , A ′ ( κ z ) = − [ E ( κ z ) , F ( κ z )]In order to prove the last assertion, according to the Iwasawa decomposition G = U AK the G = GL ( H ket )-valued function h ( κ z ) can be written as(118) h ( κ z ) = u ( κ z ) a ( κ z ) k ( κ z )In suitable orthonormal basis u becomes lower triangular with diagonal to be only 1,and a a diagonal matrice, however these entries depend rationally to the parametervariable. Then a similar argument as [S] page 299 proves that the entries of u and a have the series expansions(119) u ij ( κ z ) = a m (log( κ z ) − m + ..., a ij ( κ z ) = b n (log( κ z ) ( − n +1) / + ... being convergent near 0. The situation for the function k ( κ z ) is separated. Howevera repeated argument similar that for a and u shows that k can be extended analyt-ically to 0, see [S] pages 300-305 for detatils. Using (114) proves the result for thepower series expansion of g . Property (b) can always be fixed by conjugating therepresentation ψ p by an element in G . We are done. (cid:3) Monodromy in p -adic Hodge theory Suppose V l is an l -adic representation of the absolute Galois group G K over p -adicfield K namely(120) ̺ : G K → V l We have the short exact sequence(121) 1 → I → G K → Gal (¯ k/k ) → k is the residue field and I is the inertia group. As always our example is the´etale cohomology groups H n ( X K , Q l ). If l = p the semistability of V l means thatthe inertia subgroup I ( p ) at the prime p of K acts unipotently on V l . Moreover thisaction is given by the exponential of a morphism(122) N : V l (1) → V l of l -adic representations of G K . It follows that(123) ̺ ( g ) = exp( N.t l ( g )) , g ∈ I ( p )This is always the case for the l -adic ´etale cohomologies. The endomorphism N satisfies(124) N F = q.F N where q is the order of the residue field of K . The endomorphism N defines the localmonodromy filtration: That is the unique increasing filtration ...M i ⊂ M i +1 ⊂ ... such that • N.M i V (1) ⊂ M i − V • gr Mk V ( k ) ∼ = −→ gr M − k V By the work of Deligne the eigenvalues α of the Frobenius F are algebraic integers,Moreover there exists integers w ( α ) such that all complex conjugates of α haveabsolute value q w ( α ) / . Let W j be the sum of all the generalized eigenspaces of F with eigenvalue α such that w ( α ) = j . The filtration P • := {⊕ j ≤ i W j } is called theweight filtration. One can show that the trace of the Frobenius F q is independent of l in this case, see [D].When l = p in the log-cristalline setting there is an analogue of this map namely(125) N : V p (1) → V p ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 33 of p -adic representations or the local Galois group G K that is called the p -adic mon-odromy morphism. There is no hope to have such a p -adic monodromy morphismfor every semistable p -adic Galois representation. This analogy is explained by thenotion of Fontaine or (Φ , N )-modules. A (Φ , N )-module is a K -vector space D equipped with • a σ -linear map (called Frobenius) Φ : D → D • a K -endomorphism (called monodromy) N : D → D • N Φ = p Φ N • D has an exhaustive separated decreasing filtration ( F il i D K ) i ∈ Z .Denote by M K (Φ , N ) the category of these modules. Let V be a continuous finitedimensional representation of G K . Define(126) D st : Rep st ( G k ) → M K (Φ , N ) , D st ( V ) := ( B st ⊗ Q p V ) G K where the superfix means the fixed points, and B st is the Fontaine period ring. Theimage of this functor are called admissible filtered (Φ , N )-module. We always havedim Q p V ≥ dim K D st ( V ) and V is called semistable if(127) dim Q p V = dim K D st ( V )If this equality holds for K replaced by a finite extension then it is called potentiallysemistable. We say an object D ∈ M K (Φ , N ) has transverse monodromy if(128) N.F il i D K ⊂ F il i − D K If D is admissible then the monodromy N : D → D induces a K -linear map(129) N st : D (1) → D that depends on the trivialization of the Tate twist. The morphism N st is calledthe monodromy morphism of D . One can show that N st is a morphism of filtered(Φ , N )-modules if and only if D has transverse monodromy. Lets M a,tmK (Φ , N ) bethe category of such modules. It is a tannakian subcategory of M aK (Φ , N ). Thecorresponding representations would be denoted by Rep st,tm ( G K ). The map N st corresponds to(130) N p : V (1) → V via this correspondence.The category V ect Q p ( N ) of Q p -vector spaces V and a nilpotent transformation N : V → V is a tannakian category with fiber functor η : ( V, N ) → V . Theassociated group scheme Aut ⊗ ( η ) is the additive group G a, Q p . This gives a tensorequivalence (131) η : V ect Q p ( N ) ∼ = −→ Rep ( G a, Q p )Using the dicussion of the former paragraph one obtains a functor(132) w : Rep st,tm ( G K ) → V ect Q p ( N ) , V ( V, N p )which depends to a fixed trivialization of the Tate twist. This gives a morphism(133) e N : G a, Q p → Aut ⊗ ( w p )where Aut ⊗ ( w p ) is the tannakian fundamental group of the fiber functor w p = η ◦ w .The map e N is called the exponential of the p -adic monodromy up to the Tate twist.It can be presented as(134) e N : G a, Q p ( Q p [ T ]) → Gl ( V p )( Q p [ T ]) , T X i> N i T i i !where T = id G a , see [Pa].9. Nilpotent orbits in the mixed case
The definitions of period domain and period map can also be similarly stated forvariation of mixed Hodge structure (MHS). However the asymptotic behavior of theperiod maps and the nilpotent orbits are essentially different from the pure case.Also the period map can have essential (non-removable) singularities which is neverthe case in pure HS. We try to discuss this notion for the mixed Hodge-Tate structurewhich we define in the proceeding paragraphs.let (
V, F • , W • , Q ) be a polarized MHS with F • and W • be the Hodge and weight fil-trations respectively. The classifying space M , for this MHS consists of all filtrations F • such that ( F • , W • ) is a MHS on V which is polarized by Q . The isomorphismclass of a variation of polarized MHS V → S is determined by its period map(135) φ : S → M/ Γ , Γ =
Image ( ρ )and its monodromy representation ρ : π ( S, s ) → Gl ( V ) on a fixed reference fiber V = V s .Unlike the pure case, the existence of the limiting mixed Hodge structure is notguaranteed for MHS. This means that a nilpotent orbit may not be in general as-ymptotic to the period map. Thus one requires to make this assumption at thebeginning. More specifically a variation of MHS on the punctured disc ∆ ∗ withunipotent monodromy is admissible if ILPOTENT ORBITS IN VARIATION OF p -ADIC ´ETALE COHOMOLOGY 35 • the limiting MHS F ∞ exists. • the relative weight filtration W rel ( N, W • ), i.e the filtration induced by thenilpotent transformation N on the Gr Wk exists.where N is the logarithm of the monodromy. In this case ( F ∞ , W rel ( N, W • )) is calledthe limiting MHS. One can analogously define admissible nilpotent orbits, [P]. Theorem 9.1. (Nilpotent Orbit Theorem, G. Pearlstein [P] ) If
V → ∆ ∗ is an ad-missible variation of MHS with unipotent monodromy, then, • η ( z ) = e zN .F ∞ is an admissible nilpotent orbit. • There are constants α, β and K such that (136) d M ( F ( z ) , η ( z )) < K Im ( z ) β e − π Im ( z ) and η ( z ) ∈ M whenever Im ( z ) > α . Now lets go back to the p -adic geometry. We make the following definition. Definition 9.2. (Mixed Hodge-Tate (MHT) structure) A Q p -vector space is said tohave a mixed Hodge-Tate structure if it is endowed an increasing filtration P • definedover Q p (indexed over integers called weight filtration) and a decreasing filtration F • defined over C p (indexed over rational numbers called slope filtration) such that thegraded pieces Gr Pj V together with the induced filtration by F • are pure Hodge-Tatestructure in the sense explained in Section 2. If N : V → V is a nilpotent transformation defined over Q p then we call the MHTstructure ( V, F • , P • ) to be N -admissible if • the limiting slope filtration F ∞ exists. • the relative weight filtration P rel ( N, P • ) existsIn this case ( F ∞ , P rel ( N, P • )) is called the limit mixed Hodge-Tate structure. Westate the following analogue for the Theorem 9.1. Theorem 9.3. (Nilpotent orbit theorem for mixed Hodge-Tate structure) Assume
V → B ( r ) ∗ is a variation of mixed Hodge-Tate structure over the punctured p -adicdisc and N : V → V is a nilpotent transformation on V s = V . Then • η ( κ z ) = exp( N log( κ z )) F ∞ is an N -admissible nilpotent orbit. • For each non-archimedean metric d on F an,rig , we have the following distanceestimate (137) d (Θ(log κ z ) , η ( κ z )) < C.p − k.l/q ( n ) r n , ∀ n, ( r << where q ( n ) → ∞ as n → ∞ .Proof. The first assertion is a consequence of admissibility, and the second followsfrom (85) and the uniform bound in (76). (cid:3)
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