A refinement of the Hodge stratification for connected reductive groups
aa r X i v : . [ m a t h . N T ] J u l A refinement of the Hodge stratificationfor connected reductive groups
Stephan NeupertOctober 8, 2018
Abstract
For connected reductive groups G over a finite extension F of Q p and L the maximal unram-ified extension of F we study the sets H µ,N ( G ) of elements b ∈ G ( L ) with given Hodge pointsof ( bσ ) , ( bσ ) , . . . , ( bσ ) N . We explain the relationship to stratifications of some moduli scheme ofabelian varieties defined by Goren and Oort respectively Andreatta and Goren. We show that forsufficiently large N the Newton point is constant on the sets H µ,N ( G ) and compute such N forcertain classes of groups. Let M be a moduli scheme over F p of abelian varieties of PEL-type. To get a better understanding ofthe geometry of such moduli schemes one aims for a refinement of the stratification of M according tothe p -rank of the abelian variety’s p -torsion.One way to do this is the stratification via Newton points. This invariant of any abelian varietyover an algebraically closed field k of positive characteristic is constant on isogeny classes and can beconstructed in the following way: To an abelian variety A of dimension g associate its p -divisible group A [ p ∞ ] = lim −→ A [ p n ] where A [ p n ] denotes the group of p n -torsion points in A . It is a limit of finite locallyfree group schemes over k . The category of such group schemes is equivalent to the opposite categoryof Dieudonn´e modules. These are free modules D over the Witt-ring W ( k ) over k , endowed with aFrobenius-linear morphism F and a Frobenius-antilinear morphism V satisfying F V = V F = p (where p denotes the multiplication with p ∈ W ( k )). Note that the PEL-data on the abelian variety givecorresponding structures on the Dieudonn´e module. Denoting the Frobenius on k by σ , the morphism F can be identified with a σ -conjugacy class in the reductive subgroup of GL g of morphisms respectingthese additional structures. Kottwitz gave a purely group-theoretic description of the σ -conjugacyclasses extending the notion of classical Newton points for GL g (which classify F after forgetting aboutthe PEL-structure) to all connected reductive groups (cf. [Kot85] and [Kot97]).Another refinement of the p -rank stratification was introduced by Goren and Oort in [GO00] (for theunramified case) respectively by Andreatta and Goren in [AG03] (for the totally ramified case). In thisintroduction let us consider only the second situation: Let M ( F p , µ N ) be the fine moduli scheme over F p of polarized abelian varieties with real multiplication by the ring of integers O ˜ F of a totally real extension˜ F / Q of degree g and totally ramified at p and a µ N -level structure (with N ≥
4) satisfying the Deligne-Pappas condition. This forces those abelian varieties to have dimension exactly g . Andreatta andGoren notice that, contrary to the unramified case, the Ekedahl-Oort-stratification (using some discreteinvariant of the p -torsion of abelian varieties, cf. [Oor01]) does not even define a stratification in thissituation. Instead they introduce two invariants: To get a definition allowing explicit computations,associate to the abelian variety A over k its display ( P, Q, V − , F ). Then one can define a normal formfor such displays in which one can find both invariants j and n as exponents (cf. [AG03, § § j and n comes from the following description, which is pretty similarto the way the invariants in the unramified setting are defined in [GO00]: The first deRham-cohomology H ( A ) is a free k [ T ] / ( T g )-module of rank 2. Then we can find generators α and β of H ( A ) suchthat: H ( A, O A ) = ( T i α ) ⊕ ( T j β ) , i ≥ j, i + j = g H ( A, O A ) as a quotient of H ( A )). This gives the first invariant j . For the second let the a -number a ( A ) be the nullity of the Hasse-Witt matrix of A . Then put n = a ( A ) − j . Now decompose M ( F p , µ N ) into the loci M j,n (with 0 ≤ j ≤ n ≤ g ) of abelian varieties with these invariants (cf. [AG03, § M j,n indeed define a stratification with many niceproperties, e.g. that j and n determine the Newton point corresponding to the abelian varieties in M j,n and that the action of the Hecke correspondence on the strata can be described explicitly.The aim of this work is to give a purely group-theoretic description of both the stratification defined byGoren and Oort and the one of Andreatta and Goren and to extend it to all connected reductive groups G in such a way that it has similar properties with respect to Newton points.To define this partition for a reductive group G over O F (the ring of integers of a finite field extension F of Q p ) with maximal torus T and Weyl group Ω let L be the maximal unramified extension of F with ring of integers O L and define µ : G ( O L ) \ G ( L ) /G ( O L ) → X ∗ ( T ) / Ω to be the map associating toeach element its Hodge point (cf. § N and a N -tuple µ = ( µ , µ , . . . , µ N )of Hodge points. Then let H µ,N ( G ) ⊂ G ( L ) be the set of elements b ∈ G ( L ) with µ (( bσ ) i ) = µ i for i = 1 , . . . , N (again denoting the Frobenius by σ ).The sections 3 and 4 discuss the connection between the partition H µ,k ( G ) and Newton points first forthe case G = GL n and then for arbitrary connected reductive groups. The main result here states thatthere is a constant C µ,G depending on a Hodge point µ (and of course on G ) such that the Newton pointis constant on each H µ,C µ,G ( G ) where µ is a C µ,G -tuple with first entry µ . To show this we analyze theconvergence of the sequence of Hodge points ( i µ (( bσ ) i )) i to the Newton point ν ( b ).The partition of G ( L ) in the H µ,k ( G ) is most useful in the case of small constants C µ,G , in particularwhen one can take C µ,G = 2. In section 5 we give some explicit constants C µ,G for special classes ofgroups. In particular we obtain a result which turns out to be a slight generalization of Andreatta andGoren’s theorem that the Newton point is constant on the strata M j,n .Finally we study the relationship with the stratifications mentioned above: We see in section 6.1 thatthe strata defined by Goren and Oort in [GO00] can be described as the loci where the Frobenius ofthe Dieudonn´e module associated to the abelian variety lies in H µ, (Res F/ Q p ( GL )) for suitable µ . Insection 6.2 the same is done in the totally ramified case. Acknowledgements
I thank Eva Viehmann for her suggestion to study this stratification and many helpful discussions. I alsothank the referee for all the comments, especially for mentioning the possible relationship to [GO00].The author was partially supported by ERC starting grant 277889 ’Moduli spaces of local G -shtukas’. Throughout the whole paper we will fix the following: • k an algebraically closed field of characteristic p > • K = Frac( W ( k )) the fraction field of the ring of Witt vectors over k • F a finite extension of Q p inside a fixed algebraic closure K of K • L = K · F the composite of K and F inside K • σ the Frobenius automorphism of L/F induced by the Frobenius on their residue fields • Γ =
Gal ( F , F ) • G a connected reductive linear algebraic group over F • T ⊂ G a maximal torus containing a maximal L -split torus • Ω the Weyl group associated to T B ( G ) the set of σ -conjugacy classes of G ( L ) • Λ a special vertex in the Bruhat-Tits building of G over L Remark 2.1. i) All valuations of fields are assumed to have value group Z .ii) A similar theory with analogous proofs should work in the function field case, i.e. K = k (( t )).iii) In section 6 F will also denote the Frobenius morphism of a Dieudonn´e module. We hope that itwill always be clear what is meant from the context. We will recall the definition of the Hodge and Newton points for arbitrary connected reductive groups G over F .For the Newton point we state the explicit description given in Kottwitz [Kot85, § D be the pro-algebraic torus over Q p with character group Q . For b ∈ G ( L ) let ν ∈ Hom L ( D , G ) bethe unique element for which there is an integer n >
0, a uniformizer π ∈ L and an element c ∈ G ( L )such that the following three conditions hold:i) nν ∈ Hom L ( G m , G )ii) Int( c ) ◦ ( nν ) is defined over the fixed field of σ n on L iii) c ( bσ ) n c − = c · ( nν )( π ) · c − · σ n (where Int( c ) denotes the conjugation by c ). This defines a map ν : G ( L ) → Hom L ( D , G ). Two σ -conjugate elements in G ( L ) are mapped to elements which are conjugate under G ( L ). Furthermorethe elements in the image of the map ν are invariant under the action of the Frobenius σ . Hence thisinduces a map defining the Newton point ν : B ( G ) → (Int G ( L ) \ Hom L ( D , G )) σ . As all maximal tori of G are conjugate, fixing one of them, namely T , gives an isomorphism(Int G ( L ) \ Hom L ( D , G )) σ ∼ = ( X ∗ ( T ) Q / Ω) Γ . This allows us to view ν as a map ν : B ( G ) → ( X ∗ ( T ) Q / Ω) Γ . Remark 2.2. If G = GL n then the conditions i), ii) and iii) give the classical Dieudonn´e-Manin classification ofisocrystals after choosing some basis and representing the isocrystal by a matrix: For m ∈ N , h ∈ Z with gcd ( m, h ) = 1, let B m,h ∈ GL m with B m,h ( e i ) = e i +1 for i = 1 , . . . , m − B m,h ( e m ) = π h e (where { e i } i =1 ,...,m are basis vectors). Then any b ∈ GL n ( L ) can be σ -conjugated to a block-matrix b with blocks of the form B m,h . In the following b is called the normal form of b ∈ GL n for the Newtonpoint. This block-matrix b is unique up to permutation of the blocks and each block B m,h gives m slopes hm .To compare this to the general Newton point assume that T is the diagonal torus in GL n . Then ν ( b ) isthe (unique up to Ω = S n -action) rational cocharacter such that ( n ! ν ( b ))( π ) = b n !0 .For the Hodge point we follow [Tit77, § G ( O L ) ⊂ G ( L ) be the stabilizer of the fixed special vertex Λ in the Bruhat-Tits building (inanalogy to the classical case G = GL n ). Let S be a maximal L -split torus of G contained in T (thisexists by our choice of T ) and denote the centralizer of S by Z . Denoting the valuation on L by val L define v : Z ( L ) → X ∗ ( S ) Q ⊆ X ∗ ( T ) Q via the equality h χ, v ( z ) i = − val L ( χ ( z )) for all z ∈ Z ( L ) , χ ∈ X ∗ ( Z )(as in the definition of an apartment in the Bruhat-Tits building, cf. [Tit77, § § µ := µ Λ : G ( O L ) \ G ( L ) /G ( O L ) −→ X ∗ ( T ) Q / Ω G ( O L ) zG ( O L ) v ( z )3here z ∈ Z ( L ).For an alternative way to describe µ for unramified reductive groups, cf. [RR96, §
4] and [Kot84].
Remark 2.3. i) The Hodge point depends on the choice of the special vertex. If Λ ′ = g Λ with g ∈ G ( L ) is anothersuch vertex, then the Hodge points compare via µ Λ ′ ( b ) = µ Λ ( g − bσ ( g )) . ii) For G = GL n identify G = GL ( V ) for some F -vector space V . Then the G ( L )-orbit V of Λ in theBruhat-Tits building consists of the O L -lattices in V . Thus the stabilizer of Λ ∈ V can be identifiedwith the set of matrices GL n ( O L ) invertible over O L (after fixing a basis of the lattice Λ ). Furthermore T = S = Z . Then one computes:( − v )(diag( t , . . . , t n ))( λ ) = diag( λ val L ( t ) , . . . , λ val L ( t n ) ) for all λ ∈ L × Thus the Hodge slopes of an element b ∈ G ( O L ) · diag( t , . . . , t n ) · G ( O L ) are exactly the valuations of t , . . . , t n . But these are by definition the elementary divisors of b . Hence for G = GL n the constructionabove recovers the classical definition of Hodge points.Furthermore we define the following partial ordering ≺ on X ∗ ( T ) R / Ω (cf. e.g. [RR96, lemma 2.1]):For a fixed basis of the set of roots in X ∗ ( T ) let C ⊂ X ∗ ( T ) R be the Weyl chamber and C ∨ ⊂ X ∗ ( T ) R be the obtuse Weyl chamber. Then say x ≺ x ′ for x, x ′ ∈ X ∗ ( T ) R / Ω if for the representatives ˜ x resp.˜ x ′ of x resp. x ′ in C ⊂ X ∗ ( T ) one of the following equivalent conditions is satisfied:i) x lies in the convex hull of { ωx ′ ; ω ∈ Ω } .ii) ˜ x ′ − ˜ x ∈ C ∨ .iii) ˜ x ′ − ω ˜ x ∈ C ∨ for all ω ∈ Ω.iv) For any representation ρ : G → GL ( V ) and maximal torus T ′ ∈ GL ( V ) containing ρ ( T ) we have ρ ( x ) ≺ ρ ( x ′ ) (using one of the other three conditions or the reformulation in the following remark todefine ≺ for GL n ). Remark 2.4.
Consider the special case G = GL n , B ⊂ G the Borel subgroup of upper triangular matrices, T ⊂ B the diagonal torus and the usual identification X ∗ ( T ) R ∼ = R n via the coefficients on the diagonal. Forthe usual choice of simple roots, a representative ˜ x ∈ X ∗ ( T ) R lies in the dominant chamber if and onlyif its corresponding element ( λ , . . . , λ n ) ∈ R n fulfills λ ≥ λ ≥ . . . ≥ λ n Then two elements x, x ′ ∈ X ∗ ( T ) R / Ω satisfy x ≺ x ′ if and only if the corresponding dominant represen-tatives ( λ , . . . , λ n ) resp. ( λ ′ , . . . , λ ′ n ) fulfill: h X i =1 λ ≤ h X i =1 λ ′ i for h = 1 , . . . , n with equality for h = n . Definition 2.5.
Fix a basis { α i } i =1 ,...,n ⊂ X ∗ ( T ) Q . Let x, x ′ ∈ X ∗ ( T ) Q / Ω with representatives ˜ x resp. ˜ x ′ in C . Thendefine: | x, x ′ | = max I ⊂{ ,...,n } ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* ˜ x − ˜ x ′ , X α i ∈ I α i +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) Remark 2.6. i) The definition of | x, x ′ | depends on the α i . But as all such norms are equivalent it will not matterwhich basis we actually fix. 4i) If G = GL n , T ⊂ G the diagonal torus and α i the projection to the i th coordinate of T . Then h x, α i i is the i -th slope of x . Assume now that the sum of all slopes for x and x ′ coincide (this will be the onlycase of interest for us) and let ( λ , . . . , λ n ) resp. ( λ ′ , . . . , λ ′ n ) be representatives of x resp. x ′ in C , i.e.the λ i resp. λ ′ i are monotonically decreasing. Then | x, x ′ | = max I ⊂{ ,...,n } ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ I ( λ i − λ ′ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) = X λ i >λ ′ i ( λ i − λ ′ i )In particular | x, x ′ | is at least the maximal vertical distance between the corresponding concave polygonsfor x resp. x ′ . µ, k ( G ) Before defining the stratification H µ,k ( G ), recall the classical Hodge and Newton stratifications: Definition 2.7. i) The Hodge stratum for µ ∈ X ∗ ( T ) Q / Ω is: H µ ( G ) = { b ∈ G ( L ) | µ ( b ) = µ } ii) The Newton stratum for ν ∈ X ∗ ( T ) Q / Ω is: M ν ( G ) = { b ∈ G ( L ) | ν ( b ) = ν } Definition 2.8.
Let k ≥ and µ = ( µ , µ , . . . , µ k ) ∈ ( X ∗ ( T ) Q / Ω) k . Then let H µ,k ( G ) = { b ∈ G ( L ) | µ (( bσ ) i ) = µ i for i = 1 , , . . . , k } where ( bσ ) i = b · σ ( b ) · σ ( b ) · . . . · σ i − ( b ) . We call H µ,k ( G ) a refined Hodge stratum in G ( L ) . Remark 2.9.
Although we name these sets ’strata’, they only form a partition of G ( L ) but miss the topologicalproperties of actual stratifications. Lemma 2.10.
For any two Hodge points µ, µ ′ ∈ X ∗ ( T ) Q / Ω there is a σ -invariant bounded open subgroup K µ,µ ′ ⊂ G ( L ) contained in G ( O L ) and depending only on µ and µ ′ such that for all b ∈ H µ ( G ) , b ′ ∈ H µ ′ ( G ) and g ∈ K µ,µ ′ : µ ( b · g · b ′ ) = µ ( b · b ′ ) Proof.
Fix for now any representation ρ : G ( L ) −→ GL n ( L ) mapping G ( O L ) into GL n ( O L ) (the O L -matriceswhich are invertible over O L ). Then ρ ( H µ ′ ( G )) ⊂ H ρ ( µ ′ ) ( GL n ) for some Hodge point ρ ( µ ′ ). Choosea positive integer N with N > h ρ ( µ ′ ) , α i for any root α of GL n and consider K ρ ( µ ) ,ρ ( µ ′ ) = { ˜ g ∈ GL n ( O L ) | ˜ g ≡ π N } . Then K ρ ( µ ) ,ρ ( µ ′ ) is indeed a σ -invariant open bounded normal subgroupcontained in GL n ( O L ). Furthermore by choice of N for any ˜ b ′ ∈ H ρ ( µ ′ ) ( GL n ) and ˜ g ∈ K ρ ( µ ) ,ρ ( µ ′ ) wehave ˜ b ′− · ˜ g · ˜ b ′ ∈ G ( O L ). This implies for any b, b ′ and g as in the statement: b · g · b ′ = b · b ′ · ( b ′− gb ′ ) ∈ b · b ′ · G ( O L )and in particular the equality of Hodge points.Now choose a finite set of representations { ρ i } i such that any two of the finitely many Hodge pointsappearing as µ ( bgb ′ ) for b ∈ H µ ( G ), b ′ ∈ H µ ′ ( G ) and g ∈ G ( O L ) can be distinguished by their imageunder at least one ρ i . The existence of such a set follows from part iv) of the definition of the partialordering ≺ (cf. § K µ,µ ′ = G ( O L ) ∩ \ i ρ − i ( K ρ i ( µ ) ,ρ i ( µ ′ ) ) ⊂ G ( O L )5his is an open bounded subgroup contained in G ( O L ) such that for any b ∈ H µ ( G ), b ′ ∈ H µ ′ ( G ) and g ∈ K µ,µ ′ the element bgb ′ and bb ′ have the same Hodge points after applying any of the ρ i . Hence theyhave the same Hodge points in G . Note that σ ( K µ,µ ′ ) = K µ,µ ′ as it fixes all of the K ρ i ( µ ) ,ρ i ( µ ′ ) . (cid:3) Proposition 2.11.
Let G be an arbitrary connected reductive group.i) G ( L ) = S µ H µ,k ( G ) defines a partition of G ( L ) for fixed k .ii) For each k ∈ N and µ there is a bounded open subgroup K µ ⊂ G ( L ) with b · K µ ⊂ H µ,k ( G ) for every b ∈ H µ,k ( G ) .Proof. i) trivial.ii) For k = 1 one may take K µ = G ( O L ). For k > µ = ( µ , . . . .µ k ) define K µ = \ i,j> i + j ≤ k K µ i ,σ i ( µ j ) ⊂ G ( O L )with the subgroups K µ i ,σ i ( µ j ) as in the Lemma 2.10. As a finite intersection of bounded open subgroups, K µ has these properties, too. For any b ∈ H µ,k ( G ) and g ∈ K µ we have as g ∈ G ( O L ) µ ( bgσ ) = µ ( bσ ) = µ . Thus assume by induction µ (( bgσ ) j ) = µ j for any j < i . Then µ (( bgσ ) i ) = µ ( b · g · σ (( bgσ ) i − )) with g ∈ K µ ,σ ( µ i − ) = µ ( b · σ (( bgσ ) i − ))= µ (( bσ ( b )) · σ ( g ) · σ (( bgσ ) i − ) with σ ( g ) ∈ K µ ,σ ( µ i − ) = µ (( bσ ) · ( bgσ ) i − ))= . . . = µ (( bσ ) i ) = µ i and indeed b · g ∈ H µ,k ( G ). (cid:3) Remark 2.12.
In general there is no constant N (not even depending on the first Hodge point) such that µ (( bσ ) N +1 )is determined by µ (( bσ ) i ) for i = 1 , . . . , N . In particular the partition into the sets H µ,k ( G ) will not getstationary for growing k . This phenomena already appears for G = GL , although the computations insection 5.1 show that (for k ≥
2) this can happen only in the case where H µ,k ( GL ) is contained in aNewton stratum with basic Newton point. At several points especially in section 5 we will need to pass to field extensions of L . We will nowexplain in which cases we may do so. Although no additional conditions are necessary for G = GL n ,we have to impose further restriction concerning Hodge points in the general setting as special pointsdo not behave very well under field extensions.In this section the index at the map giving Hodge or Newton points indicates the field we use to definethe points in the reductive group. Proposition 2.13.
Let G = GL n and T ⊂ GL n a maximal split torus. Let L ⊂ L ′ be a finite field extension with ramification ndex e ∈ N . Finally let b ∈ GL n ( L ) some matrix. Then b defines canonically an element b ′ ∈ GL n ( L ′ ) ,i.e. b ′ is the matrix with the same entries as b , which are now viewed as elements in L ′ . Then ν L ( b ) = 1 e ν L ′ ( b ′ ) µ L ( b ) = 1 e µ L ′ ( b ′ ) where each value in the n -tuple of ν L ′ ( b ′ ) resp. µ L ′ ( b ′ ) is divided by e .Proof. Wlog. let T be the diagonal torus. For the Newton point choose c ∈ GL n ( L ) such that cbσ ( c ) − is thenormal form of b (cf. Remark 2.2) over L . Identifying both matrices b and c with elements in GL n ( L ′ )the same equation computes the Newton slope of b ′ ∈ GL n ( L ′ ). The only difference is that the matrixcoefficients π h in the normal form are no longer the h -th power but the eh -th power of a uniformizerof the ground field. Hence the Newton slopes of b ′ ∈ GL n ( L ′ ) are the e th multiples of the slopes of b ∈ GL n ( L ).For Hodge points choose c , c ∈ GL n ( O L ) such that c bc = diag( π µ ( b ) , . . . , π µ ( b ) n ). Again the samecomputation gives the Hodge point of b ′ ∈ GL n ( L ′ ) but with the difference as above that π ∈ L is nolonger a uniformizer of L ′ . (cid:3) Proposition 2.14.
Consider again the general setting as defined in § L ⊂ L ′ be a finite field extension withramification index e ∈ N and fix an element b ∈ G ( L ) . This b defines canonically an element b ′ ∈ G ( L ′ ) .Then ν L ( b ) = 1 e ν L ′ ( b ′ ) ∈ X ∗ ( T ) Q / Ω . If there is a maximal L -split torus S L ⊂ T L and a maximal L ′ -split torus S L ′ ⊂ T L ′ already defined over L and containing S L , then identify the apartment A ( G L , S L , L ) with the invariants of the apartment A ( G L ′ , S L ′ , L ′ ) under the Galois group (cf. [Tit77, § Λ over L stays special when considered in A ( G L ′ , S L ′ , L ′ ) . Then µ L ( b ) = 1 e µ L ′ ( b ′ ) ∈ X ∗ ( T ) Q / Ω . Proof.
Consider first the case of Newton points. Let D be the pro-algebraic torus over Q p with character group Q . Then we may consider ν L ( b ) ∈ Int G ( L ) \ Hom L ( D , G ) and ν L ′ ( b ′ ) ∈ Int G ( L ′ ) \ Hom L ′ ( D , G ) as theidentification of Int G ( L ) \ Hom L ( D , G ) with X ∗ ( T ) Q / Ω is functorial.Now choose ν ∈ Hom L ( D , G ), n > c ∈ G ( L ) s.th. the following three conditions hold for someuniformizer π ∈ L : i ) nν ∈ Hom L ( G m , G ) ii ) Int( c ) ◦ ( nν ) is defined over the fixed field of σ n on Liii ) c ( bσ ) n c − = c · ( nν )( π ) · c − · σ n Note that ν and ν L ( b ) coincide by definition in the quotient Int G ( L ) \ Hom L ( D , G ).We will now check that ν ′ = e · ν (where we view ν via base-change as an element in Hom L ′ ( D , G ))has the same properties with respect to L ′ : For this let n ′ = n and c ′ the L ′ -valued point defined by c .Then i ) As nν lies in the integral cocharacter group of G , its base-change to L ′ lies in the integral cocharactergroup of G ′ . But n ′ ν ′ = n · ( eν ) = e · ( nν ). Hence n ′ ν ′ ∈ Hom L ′ ( G m , G ). ii ) Int( c ′ ) ◦ ( n ′ ν ′ ) is the e -th power of the base-change of Int( c ) ◦ ( nν ) to L ′ . Thus it is defined over thefixed field of σ n on L ′ . iii ). First note that for some uniformizer π ′ ∈ L ′ we have:( n ′ ν ′ )( π ′ ) = ( nν )( π ′ e ) = ( nν )( π )(the rightmost element considered via G ( L ) ⊂ G ( L ′ )). Thus we have c ′ ( b ′ σ ) n ′ c ′− = c ′ · ( n ′ ν ′ )( π ′ ) · c ′− · σ n ′
7s both sides lie in G ( L ) and are equal there.This shows that ν ′ is exactly ν L ′ ( b ′ ) as desired.The case of Hodge points is easier: As val L ′ | L = e · val L for the valuations, we get by definition v L = 1 e v L ′ | Z L ( L ′ ) : Z L ( L ′ ) → X ∗ ( T ) Q (denoting the centralizer of S L by Z L ). As the Hodge points are essentially the map v L respectively v L ′ one has: µ L ( b ) = 1 e µ L ′ ( b ′ ) (cid:3) Remark 2.15. i) Note that we change only the field L and do not pass to some base-changed group.ii) The additional assumptions for Hodge points are satisfied for unramified groups G with hyperspe-cial vertices: As G is quasi-split, the assumption on the existence of S L ′ is automatic. Furthermorehyperspecial vertices are by definition stable under finite field extensions. µ, N ( G ) and Newton points for G = GL n The main result of this section, Theorem 3.17, states that for sufficiently large N (depending on GL n and the first Hodge point in µ ) the stratum H µ,N ( GL n ) lies inside one Newton stratum, i.e. the fixedHodge points determine the Newton point.To show this we will find an element b min,ν ∈ GL n ( L ) for each b ∈ H µ,N ( GL n ) such that one can boundthe two differencesi) | ν (( b min ,ν σ ) k ) , µ (( b min ,ν σ ) k ) | ii) | µ (( b min ,ν σ ) k ) , µ (( bσ ) k ) | independently of k .Throughout this paragraph fix |· , ·| as in Remark 2.6, i.e. for elements with the same endpoint of theirpolygons: | ( λ , . . . , λ n ) , ( λ ′ , . . . , λ ′ n ) | = X λ i >λ ′ i ( λ i − λ ′ i )where P ni =1 λ i = P ni =1 λ ′ i and λ ≥ . . . ≥ λ n , λ ′ ≥ . . . ≥ λ ′ n are in dominant order. In this section we will bound | ν (( b min ,ν σ ) k ) , µ (( b min ,ν σ ) k ) | . The b min ,ν will be the minimal elements ina Newton stratum corresponding to the minimal p -divisible groups as introduced in [Oor05]. Definition 3.1. i) A Newton point ν = ( hm , . . . , hm ) ∈ Q m is called superbasic if gcd ( h, m ) = 1 .ii) Let ν be superbasic with slope hm . Let e , . . . , e m − be the standard basis of L m and define inductively e k + m = πe k (for k ∈ Z ). Then define b min,ν ∈ GL m ( L ) by b min,ν ( e k ) = e k + h for k = 0 , . . . , m − .iii) Let ν = ( ν , ..., ν n ) be an arbitrary Newton point with dominantly ordered slopes. Let ν =( ν , . . . , ν i ) , ν = ( ν i +1 , . . . , ν i ) , . . . , ν j = ( ν i j − +1 , . . . , ν n ) be its superbasic parts. Then define b min,ν to be the block matrix b min,ν = b min,ν . . . b min,ν . . . ... ... . . . ... . . . b min,ν j The element b min,ν is called a minimal element in the Newton stratum corresponding to ν . emark 3.2. In general a σ -conjugacy class [ b ] ∈ B ( G ) is called superbasic if no representative of [ b ] lies in a properLevi subgroup defined over F (cf. [GHKR06, § G = GL n this happens if the Newtonpoint of [ b ] is superbasic according to the definition given above. Lemma 3.3. i) b min,ν has indeed Newton point ν .ii) ( b min,ν σ ) k = b min,kν if gcd ( k, m ) = 1 . Otherwise one can conjugate ( b min,ν σ ) k by a σ -invariantpermutation matrix into b min,kν .Proof. Obviously it suffices to check both statements for superbasic Newton points ν .i) As gcd ( h, m ) = 1 consider the basis e ′ i = e hi for i = 0 , . . . , m −
1. Then b min,ν ( e ′ i ) = e ′ i +1 for i = 0 , . . . , m − b min,ν ( e ′ m − ) = e hm = π h e ′ . The base-change matrix from ( e i ) to ( e ′ i ) is σ -invariant as it has entries 0 or π k ( k ∈ Z ). Thus one can σ -conjugate b min,ν into the standard form forthe basic Newton-point ν .ii) b min,ν is σ -invariant, thus ( b min,ν σ ) k ( e i ) = b kmin,ν ( e i ) = e i + kh . If gcd ( k, m ) = 1, this is the definition of b min,kν .If gcd ( k, m ) = s > t = ms , then let ν ′ = ( khm , . . . , khm ) ∈ Q t be superbasic with slope khm . Denote by { e ′ j } j ∈ Z the basis vectors used in the definition of b min,ν ′ . For fixed i = 0 , . . . s − ϕ i : L t → L m of vector spaces with ϕ i ( e ′ j ) = e i + sj . Then b kmin,ν ( ϕ i ( e ′ j )) = b kmin,ν ( e i + sj ) = e i + sj + kh = ϕ i ( e ′ j + khs ) = ϕ i ( b min,ν ′ ( e ′ j )) . Thus the restriction of b min,ν to each subspace h e i + sj i j =0 ,...,t − is via ϕ i equal to b min,ν ′ . Hence aftera permutation of the basis vectors b kmin,ν and b min,kν are equal. (cid:3) Proposition 3.4.
For each b min,ν ∈ GL n ( L ) and k ∈ N we have: | ν (( b min,ν σ ) k ) , µ (( b min,ν σ ) k ) | ≤ n Proof.
By part ii) of the previous lemma, it suffices to show this for k = 1. Note for this that a permutationof the basis vectors does not change |· , ·| .Let us first consider the superbasic case. Then the Hodge slopes are exactly (cid:4) i + hn (cid:5) for i = n − , n − , . . . ,
0. It follows that if hn = a + bn with a ∈ Z , 0 ≤ b < n , b min,ν has Hodge slopes µ i ( b min,ν ) = (cid:26) a + 1 if i ≤ ba if i > b Thus to compute | ν ( b min,ν ) , µ ( b min,ν ) | one has to sum up the differences of the first b slopes: | ν ( b min,ν ) , µ ( b min,ν ) | = (cid:12)(cid:12)(cid:12)(cid:12) b ( a + 1) − b hn (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) b − b n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n − n ( n − b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n ν = ( ν , . . . , ν n ) be the dominant representative of an arbitrary Newton point. Then the slopesof µ ( b min,ν ) = ( µ , . . . , µ n ) can be permuted by some ω ∈ Ω = S n such that for each superbasic part( ν i , ν i +1 , . . . , ν j ) the tuple ( µ ωi , . . . , µ ωj ) is the Hodge point of the corresponding superbasic minimalelement (again dominantly ordered). Let now I be the index set where µ i ≥ ν i and J be the indexset where µ ωi ≥ ν i . First note that ω preserves the sets { i ∈ { , , . . . , n }| k ≤ ν i < k + 1 } . Thus µ ω − j − ν ω − j ≥ µ ω − j − ν j ≥
0. As the converse is obvious we have I = ωJ and see using ν ( b min,ν ) ≺ µ ( b min,ν ): | ν ( b min,ν ) , µ ( b min,ν ) | = X i ∈ I ( µ i − ν i ) ≤ X i ∈ I ( µ i − ν ω − i ) = X j ∈ J ( µ ωj − ν j ) ≤ n ν . (cid:3) .2 Bounding the σ -conjugating elements Now we will show that any two elements in GL n with the same Newton and Hodge point are σ -conjugatesby some element with bounded Hodge point. To do so we use the main result of [RZ99] in the case of G = GL n . In this case it can be seen as a direct consequence of the Rapoport-Zink lemma (cf. [RZ96,2.17-2.19] or [RZ99, prop. 1.6]). The general theorem will be stated in § Definition 3.5.
Let G be any reductive group defined over F , b ∈ G ( L ) . Then let J b be the connected linear algebraicgroup over F with J b ( R ) = { g ∈ G ( R ⊗ F L ) | σ ( g ) = b − gb } for any F -algebra R . Remark 3.6.
That J b is indeed representable as a connected linear algebraic group is shown in [RZ96, prop. 1.12]. Definition 3.7. (cf. [RZ96, def. 1.8] or loc. cit. def. 4.4)An element b ∈ GL n ( L ) is said to fulfill a decency equation if there is an s ∈ N and a uniformizer π ∈ L with ( bσ ) s = diag( π sν , π sν , . . . , π sν n ) with Newton slopes ν , . . . , ν n of b (not necessarily in dominant order). Remark 3.8.
Minimal elements always fulfill decency equations.
Proposition 3.9. (cf. [RZ99, thm. 1.4] for G = GL n )Let b ∈ GL n ( L ) fulfilling a decency equation relative to s ∈ N and fix µ ∈ X ∗ ( T ) Q / Ω . Then there is afinite set S ⊂ X ∗ ( T ) Q / Ω with the following property: For any g ∈ GL n ( L ) with µ ( g − b σ ( g )) = µ thereis a j ∈ J b ( F s ) with µ ( jg ) ∈ S .Here F s denotes the unramified extension of degree s of F . Proposition 3.10.
Fix ν ∈ X ∗ ( T ) Q / Ω , µ ∈ X ∗ ( T ) Q / Ω . Then there is a finite set S ⊂ X ∗ ( T ) Q / Ω such that for any b ∈ GL n ( L ) with µ ( b ) = µ and ν ( b ) = ν there is a g ∈ GL n ( L ) and a minimal element b min,ν with µ ( g ) ∈ S and b = g − · b min,ν · σ ( g ) .Proof. Choose a finite set
S ⊂ X ∗ ( T ) Q / Ω for b = b min,ν as in the previous proposition. Let now b ∈ GL n ( L )with µ ( b ) = µ and ν ( b ) = ν . As all elements are conjugate to some minimal element there is some g ′ ∈ GL n ( L ) and a minimal element b min,ν with b = g ′− · b min,ν · σ ( g ′ ). Then g ′ fulfills all conditionsof the previous proposition and there is a j ∈ J b min,ν ( F s ) ⊂ GL n ( L ) with g := jg ′ ∈ S . Finally notethat by definition of J b min,ν g − · b min,ν · σ ( g ) = g ′− · j − · b min,ν · σ ( j ) · σ ( g ′ ) = g ′− · b min,ν · σ ( g ′ ) = b. (cid:3) The main remaining part is to show that for b, g ∈ GL n ( L ) one can give bounds on the Hodge points of g − bσ ( g ) in terms of the Hodge point of b and g . Definition 3.11.
Let G be a reductive group. For x, x ′ ∈ X ∗ ( T ) R / Ω let ˜ x resp. ˜ x ′ be the dominant representatives of x resp. x ′ in C ⊂ X ∗ ( T ) R . Then leti) x ⊕ x ′ be the image of ˜ x + ˜ x ′ ∈ X ∗ ( T ) R in the quotient X ∗ ( T ) R / Ω .ii) x ⊕ ω x ′ be the image of ˜ x + ω ˜ x ′ ∈ X ∗ ( T ) R in the quotient X ∗ ( T ) R / Ω , where ω ∈ Ω is the longestelement. emark 3.12. In the case of G = GL n , ˜ x = ( λ , . . . , λ n ), ˜ x ′ = ( λ ′ , . . . , λ ′ n ) (both dominant) x ⊕ x ′ = ( λ + λ ′ , λ + λ ′ , . . . , λ n + λ ′ n ) x ⊕ ω x ′ = ( λ + λ ′ n , λ + λ ′ n − , . . . , λ n + λ ′ )Note that the last element need not lie in the dominant chamber. Lemma 3.13.
Let b, b ′ ∈ GL n ( L ) . Then µ ( b ) ⊕ ω µ ( b ′ ) ≺ µ ( bb ′ ) ≺ µ ( b ) ⊕ µ ( b ′ ) . Proof.
We will first consider only the smallest Hodge slope: To simplify the situation use the Cartan decom-position to write b ′ = c b ′ c with b ′ = diag( π µ ′ , . . . , π µ ′ n ) ∈ T ( L ), µ ′ ≥ . . . ≥ µ ′ n , c , c ∈ GL n ( O L ).Now decompose bc = c b c with c ∈ GL n ( O L ), b = diag( π µ τ (1) , . . . , π µ τ ( n ) ) ∈ T ( L ), τ ∈ S n apermutation, µ ≥ . . . ≥ µ n , c ∈ U ( O L ) (where U ⊂ B denotes the unipotent radical). Then replacing b by b c and b ′ by b ′ does not change any of the considered Hodge points. The algorithm to computeelementary divisors implies, that the smallest Hodge slope equals the smallest valuation of a matrixcoefficient of b c b ′ . Hence (denoting the coefficients of c by ( c ) i,j ) µ ( bb ′ ) n = µ ( b c b ′ ) n = min i,j ∈{ ,...,n } ( µ τ ( i ) + µ ′ j + v (( c ) i,j )) . In particular one has µ ( bb ′ ) n ≥ µ n + µ ′ n . For the upper bound note that for each j ∈ { , . . . , n } there is a i ∈ { , . . . , n } with j ≤ i and n − j + 1 ≤ τ ( i ) (otherwise the n − j + 1 elements j, . . . , n would be mapped under τ to the n − j elements 1 , . . . , n − j ). Hence µ n − j +1 + µ ′ j ≥ µ τ ( i ) + µ ′ i ≥ µ ( bb ′ ) n and µ ( bb ′ ) n ≤ min i ( µ ( b ) ⊕ ω µ ( b ′ )) i .For further Hodge slopes let GL n ∼ = GL ( V ) for some vector space V and consider b ∧ k ∈ GL ( V k V ). If theslopes of b are µ ≥ . . . ≥ µ n (as above) then the slopes of b ∧ k are { P i ∈ S µ i | S ⊆ { , , . . . , n } , | S | = k } .Similarly for b ′ and bb ′ . Thus we get for all k = 1 , . . . , n n X i = n − k +1 µ ( bb ′ ) i = µ (( bb ′ ) ∧ k )( nk ) = µ ( b ∧ k b ′∧ k )( nk ) ≥ µ ( b ∧ k ) n + µ ( b ′∧ k ) n = n X i = n − k +1 µ ( b ) i + n X i = n − k +1 µ ( b ′ ) i = n X i = n − k +1 ( µ ( b ) ⊕ µ ( b ′ )) i thus the right hand side of the inequality.Similarly fix any S ⊂ { , . . . n } , | S | = k . Then by an analogue argument as for the smallest slope, thereis a j ∈ { , . . . , (cid:0) nk (cid:1) } such that X i ∈ S µ ( b ) i ≥ µ ( b ∧ k ) j and X i ∈ S µ ( b ′ ) n − i +1 ≥ µ ( b ′∧ k )( nk ) − j +1 . Hence X i ∈ S µ ( b ) i + X i ∈ S µ ( b ′ ) n − i +1 ≥ µ ( b ∧ k ) j + µ ( b ′∧ k )( nk ) − j +1 ≥ min i ∈ { ,..., ( nk ) } ( µ ( b ∧ k ) i + µ ( b ′∧ k )( nk ) − i +1 ) ≥ µ ( b ∧ k b ′∧ k )( nk ) = µ (( bb ′ ) ∧ k )( nk ) = n X i = n − k +1 µ ( bb ′ ) i . P ni = n − k +1 µ ( bb ′ ) i is smaller that the sum of the k least elements in µ ( b ) ⊕ ω µ ( b ′ ). Thisgives the left hand side of the inequality. (cid:3) Remark 3.14.
When only considering G = GL n the right hand side of the inequality would suffice for our purposes.Only in the next chapter we will really need both sides to derive similar statements for arbitraryconnected reductive groups. Proposition 3.15.
Let ν ∈ X ∗ ( T ) Q / Ω be any Newton point and µ ∈ X ∗ ( T ) Q / Ω any Hodge point. Then there is a constant C such that for any b ∈ GL n ( L ) with ν ( b ) = ν and µ ( b ) = µ and any k ∈ N | µ (( bσ ) k ) , µ (( b min,ν σ ) k ) | < C. Proof. b min,ν satisfies a decency equation and we may apply Proposition 3.10. Thus we can write b = g − · b min,ν · σ ( g ) for some g ∈ GL n ( L ) with Hodge point in some finite set S . Applying Proposition 3.13gives together with µ ( g ) = µ ( σ ( g )) µ ( g − ) ⊕ ω µ (( b min,ν σ ) k ) ⊕ ω µ ( g )= µ ( g − ) ⊕ ω µ (( b min,ν σ ) k ) ⊕ ω µ ( σ k ( g )) ≺ µ ( g − · ( b min,ν σ ) k · g ) = µ (( g − b min,ν σ ( g ) σ ) k ) = µ (( bσ ) k ) ≺ µ ( g − ) ⊕ µ (( b min,ν σ ) k ) ⊕ µ ( σ k ( g ))= µ ( g − ) ⊕ µ (( b min,ν σ ) k ) ⊕ µ ( g ) .µ ( g ) and µ ( g − ) are bounded because S is finite. Hence there is some constant C > | µ (( bσ ) k ) , µ (( b min,ν σ ) k ) | < C. (cid:3) Proposition 3.16.
For any two b, b ′ ∈ GL n ( L ) with ν ( b ) = ν ( b ′ ) , there is some value at which the (concave) Newtonpolygons of ν ( b ) and ν ( b ′ ) differ by at least n . In particular | ν ( b ) , ν ( b ′ ) | ≥ n . Proof. As ν ( b ) = ν ( b ′ ) one can find at least one vertex of the Newton polygon of either ν ( b ) or ν ( b ′ ) which doesnot lie on the Newton polygon of the other. At this point, the difference of both polygons is at least n .But | ν ( b ) , ν ( b ′ ) | is at least as big as any vertical difference between those polygons. (cid:3) Theorem 3.17.
Let µ be a Hodge point. Then there is a constant C ′ only depending on µ , such that each stratum H µ,N with N > C ′ and µ = ( µ , µ , . . . , µ N ) lies inside a Newton stratum.Proof. Let µ be a Hodge point. Fix a Newton point ν for now. Let b ∈ GL n ( L ) with Newton point ν andHodge point µ . Then with the constant C of Proposition 3.15 | kν ( b ) , µ (( bσ ) k ) | = | ν (( bσ ) k ) , µ (( bσ ) k ) |≤ | ν (( bσ ) k ) , µ (( b min,ν σ ) k ) | + | µ (( b min,ν σ ) k ) , µ (( bσ ) k ) |≤ n C and for N > C ′ ν := n ( n + C ): (cid:12)(cid:12)(cid:12)(cid:12) ν ( b ) , N µ ( bσ ) N (cid:12)(cid:12)(cid:12)(cid:12) ≤ N ( n C ) < n . ν ( b ) lies below the polygon of N µ ( bσ ) N by Mazur’s inequalityand differs from it by less than n at every point. Hence by Proposition 3.16 µ ( bσ ) C ′ ν determines ν ( b ).Now let C ′ = max ν ≺ µ ( C ′ ν ) and fix some µ = ( µ , . . . , µ N ) for N > C ′ . Let b, b ′ ∈ H µ,N . Then ν ( b ) , ν ( b ′ ) ≺ µ by Mazur’s inequality. By definition of C ′ , the Hodge point µ N determines both ν ( b )and ν ( b ′ ). Thus ν ( b ) = ν ( b ′ ), i.e. H µ,N lies inside a Newton stratum. (cid:3) µ, n and Newton points for general G We use the same strategy as for GL n in the general context. Nevertheless additional arguments areneeded at several points. There is no well-established notion of ’minimal elements’ for arbitrary connected reductive groups (al-though there is some generalization under additional assumptions). Hence we use some generalizationof the normal form of elements in GL n instead. Lemma 4.1.
Let b, b ′ ∈ G ( L ) . Then µ ( b ) ⊕ ω µ ( b ′ ) ≺ µ ( bb ′ ) ≺ µ ( b ) ⊕ µ ( b ′ ) . Proof.
It suffices to check this for the images under all representations G → GL ( V ) (cf. [RR96, lemma 2.2]).But for GL ( V ) this was shown in Lemma 3.13. (cid:3) Proposition 4.2.
For each b ∈ G ( L ) there is a b ∈ G ( L ) in the σ -conjugacy class of b and a constant C > such thatfor all k ∈ N we have | ν ( b σ ) k , µ ( b σ ) k | ≤ C . Proof.
To any b ∈ G ( L ) there is a representative ν ∈ Hom L ( D , G ) of the Newton point ν ( b ) (viewed as anelement in Int G ( L ) \ Hom L ( D , G )), n > π ∈ L a uniformizer and an element c ∈ G ( L ) such that c ( bσ ) n c − = c · ( nν )( π ) · c − · σ n . Let b = cbσ ( c ) − and ν = cνc − . Then( b σ ) n = ( cbσ ( c ) − σ ) n = c ( bσ ) n c − = c · ( nν )( π ) · c − · σ n = ( nν )( π ) · σ n . Hence ( b σ ) n lies in some torus and trivially µ (( b σ ) n ) = ν (( b σ ) n ) = ν = ν ∈ Int G ( L ) \ Hom L ( D , G ) . For general k = xn + y , x ∈ N , 0 ≤ y < n we have µ (( b σ ) xn ) ⊕ ω µ (( b σ ) y ) = µ (( b σ ) xn ) ⊕ ω µ ( σ xn ( b σ ) y ) ≺ µ (( b σ ) xn + y ) ≺ µ (( b σ ) xn ) ⊕ µ ( σ xn ( b σ ) y )= µ (( b σ ) xn ) ⊕ µ (( b σ ) y ) . Thus there is a constant C > k = xn + y | µ (( b σ ) xn + y ) , µ (( b σ ) xn | < C . Furthermore as ν (( bσ ) k ) = kν ( bσ ) one has a constant C with | ν (( b σ ) xn + y ) , ν (( b σ ) xn | = | ν (( b σ ) y ) , | < C | ν (( b σ ) xn + y ) , µ (( b σ ) xn + y ) | < C + C + | ν (( b σ ) xn ) , µ (( b σ ) xn ) | = C + C =: C . (cid:3) Remark 4.3.
Contrary to the case G = GL n this proof yields a constant depending on the σ -conjugacy class and notonly on G itself. σ -conjugating elements for general G and final estimates Recall the definition of J b , F s as given in section 3.2 and generalize Definition 3.7 as follows: Definition 4.4. ([RZ96, def. 1.8])An element b ∈ G ( L ) is said to fulfill a decency equation if there is a s ∈ N and a uniformizer π ∈ L with ( bσ ) s = sν b ( π ) · σ s ∈ G ( L ) ⋊ h σ i . The main tool to bound the difference between Hodge points of σ -conjugated elements is again themain theorem of [RZ99], now in full generality: Theorem 4.5. ([RZ99, thm. 1.4])Fix b ∈ G ( L ) and s > such that a decency equation holds for b relative to s . For c > there existsa bound C > with the following property: If x ∈ B ( G, L ) is an element in the extended Bruhat-Titsbuilding over L such that d ( x, bσ ( x )) < c , then there exists a x ∈ B ( J, F s ) with d ( x, x ) < C . Proposition 4.6.
Let ν ∈ X ∗ ( T ) Q / Ω be any Newton point and µ ∈ X ∗ ( T ) Q / Ω any Hodge point. Then there is a constant C such that for any b ∈ G ( L ) with ν ( b ) = ν and µ ( b ) = µ and any k ∈ N | µ ( bσ ) k , µ ( b σ ) k | < C. Proof.
Copy the proof of Proposition 3.10 and 3.15, replace GL n by G and note that by definition b satisfiesa decency equation. (cid:3) Theorem 4.7.
Let µ be a Hodge point of G . Then there is a constant C ′ only depending on µ , such that each stratum H µ,N with N > C ′ and µ = ( µ , µ , . . . , µ N ) lies inside a Newton stratum.Proof. Fix a Hodge point µ . For now fix a Newton point ν , too. Let b ∈ G ( L ) with Newton point ν and Hodgepoint µ . Then for b ∈ G ( L ) and C as in Proposition 4.2 and C as in Proposition 4.6 | kν ( b ) , µ (( bσ ) k ) | = | ν (( bσ ) k ) , µ (( bσ ) k ) |≤ | ν (( bσ ) k ) , µ (( b σ ) k ) | + | µ (( b σ ) k ) , µ (( bσ ) k ) | ≤ C + C. As there are only finitely many Newton points below µ (cf. [RR96, prop. 2.4 iii)]) one can findsome constant ε > ν ′ , ν ′′ ≺ µ one has | ν ′ , ν ′′ | > ε . Then for N > C ′ ν = 2 ε − ( C + C ) (cid:12)(cid:12)(cid:12)(cid:12) ν ( b ) , N µ ( bσ ) N (cid:12)(cid:12)(cid:12)(cid:12) ≤ N ( C + C ) < ε . Thus ν ( b ) is uniquely determined by any µ ( bσ ) N .Now let C ′ = max ν ≺ µ ( C ′ ν ) (again we take the maximum over a finite set) and fix some µ = ( µ , . . . , µ N )for N > C ′ . Then for any two elements in H µ,N , the Hodge point µ N determines the Newton point forboth of them. Hence H µ,N lies inside a Newton stratum. (cid:3) onjecture 4.8. Let G be a reductive group. Then there is a constant C ′ depending only on G , such that each stratum H µ,N with N > C ′ and any N -tuple of Hodge points µ lies inside a Newton stratum. Remark 4.9.
The proof of this conjecture needs a different approach. Even for GL and any k ∈ N there are matrices b ∈ GL ( L ) where µ (( bσ ) k ) differs greatly from kν ( b ).However we prove the conjecture for G = GL in Proposition 5.1 and even for any scalar restriction of GL in Proposition 5.13. In the last paragraph it was shown that H µ,N lies in a Newton stratum for sufficiently large N . But inmost cases the constants derived in the general proof are far from being optimal. This problem will betreated for some small groups to establish some small bounds, preferably 2. Proposition 5.1.
For any pair of Hodge points µ = ( µ , µ ) of GL ( L ) the stratum H µ, ( GL ) lies in some Newtonstratum.Proof. Let b ∈ GL ( L ). If b has non-integral Newton slopes, let L ′ be a totally ramified extension of L ofdegree 2 and view b as an element b ′ ∈ GL ( L ′ ). The Newton point of b ′ over L ′ lies now in the integralcocharacter group and we can recover the Newton point ν L ( b ) out of the Newton point ν L ′ ( b ′ ) (cf.Proposition 2.13). Hence we may replace L by L ′ and b by b ′ and can assume wlog. that ν ( b ) lies inthe (integral) cocharacter group.Then there are elements c ∈ GL ( L ), ν , ν ∈ Z with ν ≤ ν and some uniformizer π ∈ L such that b = c · (cid:18) π ν π ν (cid:19) · σ ( c ) − . Applying the algorithm to compute elementary divisors to c we can write c = (cid:18) λ ′ π d ′ (cid:19) · (cid:18) (cid:19) e ′ · (cid:18) a a (cid:19) · (cid:18) π x π y (cid:19) · (cid:18) (cid:19) e · (cid:18) λπ d (cid:19) for some x, y ∈ Z , y ≤ x , d, d ′ ≥ e, e ′ ∈ { , } , λ, λ ′ ∈ O × L ∪ { } and a , a ∈ O × L . As neither Newtonnor Hodge points change when σ -conjugating with some element of GL ( O L ), we may assume λ ′ = 0, e ′ = 0 and a = a = 1. Then b = (cid:18) π x π y (cid:19) · (cid:18) (cid:19) e · (cid:18) λπ d (cid:19) · (cid:18) π ν π ν (cid:19) ·· (cid:18) − σ ( λ ) π d (cid:19) · (cid:18) (cid:19) e · (cid:18) π − x π − y (cid:19) . Case 1: e = 0Computing b we get with δ = d − x + y : b = (cid:18) π ν λπ ν + δ − σ ( λ ) π ν + δ π ν (cid:19) ( bσ ) = (cid:18) π ν λπ ν + δ − σ ( λ ) π ν + δ π ν (cid:19) If δ ≥ λ = 0 then µ ( b ) = ( ν , ν ), µ (( bσ ) ) = (2 ν , ν ).If δ < λ = 0 we have to distinguish two further cases:15 ase 1.1: δ < , λ = 0 and ν = ν Then µ ( b ) = ( ν + δ, ν − δ ), µ (( bσ ) ) = (2 ν + δ, ν − δ ). Case 1.2: δ < , λ = 0 and ν = ν Now let δ ′ , δ ′ ≥ λ − σ ( λ ) ∈ ( π δ ′ ) \ ( π δ ′ +1 ) and λ − σ ( λ ) ∈ ( π δ ′ ) \ ( π δ ′ +1 )Obviously δ ′ ≤ δ ′ .If δ + δ ′ ≥
0, then µ ( b ) = ( ν , ν ), µ (( bσ ) ) = (2 ν , ν ).If δ + δ ′ < δ + δ ′ ≥
0, then µ ( b ) = ( ν + ( δ + δ ′ ) , ν − ( δ + δ ′ )), µ (( bσ ) ) = (2 ν , ν ).If δ + δ ′ <
0, then µ ( b ) = ( ν + ( δ + δ ′ ) , ν − ( δ + δ ′ )), µ (( bσ ) ) = (2 ν + ( δ + δ ′ ) , ν − ( δ + δ ′ )). Case 2: e = 1Computing b again with δ = d + x − y ≥ b = (cid:18) π ν λπ ν + δ − σ ( λ ) π ν + δ π ν (cid:19) ( bσ ) = (cid:18) π ν λπ ν + δ − σ ( λ ) π ν + δ π ν (cid:19) Thus µ ( b ) = ( ν , ν ), µ (( bσ ) ) = (2 ν , ν ).Hence in all cases we can recover the Newton point out of µ ( b ) and µ (( bσ ) ) via the following procedure:If µ ( b ) = ( µ , , µ , ) and µ (( bσ ) ) = ( µ , , µ , ) are dominant representatives of the Hodge points, thencompute 2 µ , − µ , . This value is negative if and only if we are in case 1 . δ + δ ′ <
0. But then theNewton point is basic and can be computed as ν ( b ) = ( µ , + µ , , µ , + µ , ). But if 2 µ , − µ , ≥ ν ( b ) = ( µ , − µ , , µ , − µ , ) holds. (cid:3) Remark 5.2.
The same case-by-case analysis but for ( bσ ) n and ( bσ ) n (with n > n >
0) instead of b and ( bσ ) shows that the Newton point can be recovered from µ (( bσ ) n ) and µ (( bσ ) n ) whenever n | n . The lastdivisibility condition is necessary to ensure a similar inequality between the δ ′ i as above. In this section we compare scalar restrictions for totally ramified extensions of some connected reductivegroup G to the group itself.For this comparison fix a tower of fields Q p ⊂ F ′ ⊂ F ⊂ Q p with F ′ and F finite over Q p and F/F ′ totally ramified. Let as in § L = F · K and L ′ = F ′ · K . Then L/L ′ is again a totally ramifiedextension of the same degree as F/F ′ .Let G be a connected reductive group over F . Then by definition of scalar restrictions there is anisomorphism of abstract groups φ : (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) ∼ = G ( F ⊗ F ′ L ′ ) = G ( L ) . Fix as usual a maximal torus T ⊂ G . Then T ′ = Res F/F ′ ( T ) ⊂ Res
F/F ′ ( G ) is again a maximal torus.We will denote their base-change to L resp. L ′ by T L resp. T ′ L ′ . The universal property of scalarrestrictions yields a canonical isomorphism Hom F ′ ( G m , T ′ ) ∼ = Hom F ( G m , T ). Tensoring with K over Q p gives an isomorphism α : Hom L ′ ( G m , T ′ L ′ ) ∼ = Hom L ( G m , T L ) . Note that a priori this is an isomorphism between morphism sets of schemes, but it restricts to a bijectionbetween the sets of group scheme morphisms α : X ∗ ( T ′ L ′ ) ∼ = X ∗ ( T L ) . Then α can be extended to a morphism between the rational cocharacter groups. Lemma 5.3.
Let ψ ∈ X ∗ ( T ′ L ′ ) and π ∈ L ′ be any uniformizer. Then α ( ψ )( π ) = φ ( ψ ( π )) . roof. Although this lemma should not come as a surprise, we have to consider the maps on the level ofmorphism between the underlying algebras, if we want to be precise at this point. Fist of all fix anisomorphism O T ( T ) ∼ = F [ { x i } i ] / ( { f j } j ) and a basis { ε k } k of F over F ′ . Set x i = P k y i,k ε k (with furthervariables y i,k ) and let f j,k be polynomials in the variables y i,k such that P k f j,k ( { y i,k } i,k ) ε k = f j ( { x i } i ).Then the explicit description of scalar restrictions gives O T ′ L ′ ( T ′ L ′ ) = F ′ [ { y i,k } i,k ] / ( { f j,k } j,k ) ⊗ F ′ L ′ = L ′ [ { y i,k } i,k ] / ( { f j,k } j,k ) . Then consider ψ : G m,L ′ → T ′ L ′ and its image of π : ψ : L ′ [ { y i,k } i,k ] / ( { f j,k } j,k ) = F ′ [ { y i,k } i,k ] / ( { f j,k } j,k ) ⊗ F ′ L ′ −→ F ′ [ t ± ] ⊗ F ′ L ′ = L ′ [ t ± ]( { y i,k − δ i,k } i,k ) = ( { y i,k ⊗ − ⊗ δ i,k } i,k )= ( ψ ) − ( t ⊗ − ⊗ π )= ( ψ ) − ( t − π )(for certain elements δ i,k ∈ L ′ ). Under the given bijection we get for α ( ψ ) : G m,L → T L : α ( ψ ) : F [ { x i } i ] / ( { f j } j ) ⊗ F L −→ F [ t ± ] ⊗ F L = L [ t ± ]( { x i ⊗ − X k ε k ⊗ δ i,k } i ) =( { X k y i,k ε k ⊗ − ε k ⊗ δ i,k } i ) = ( α ( ψ ) ) − ( t ⊗ − ⊗ π )= ( α ( ψ ) ) − ( t − π )But the point ( { y i,k − δ i,k } i,k ) ∈ T ′ ( L ′ ) ⊂ (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) is exactly mapped to ( { x i ⊗ − P k ε k ⊗ δ i,k } i ) ∈ T ( L ) ⊂ G ( L ) under the map φ : (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) → G ( L ) (as both points have the samecoordinates but the first one has them as a linear combination of the ε k ). Thus α ( ψ )( π ) = φ ( ψ ( π )) , where we identify as usual the element π ∈ L ′ ⊂ L with the ideal ( t − π ) in Spec L ′ [ t ± ] respectively inSpec L [ t ± ]. (cid:3) Proposition 5.4.
Using the notation as above, let e be the ramification index of F over F ′ . Then the following diagramscommute: (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) φ / / ν L (cid:15) (cid:15) G ( L ) ν L (cid:15) (cid:15) (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) φ / / µ L (cid:15) (cid:15) G ( L ) µ L (cid:15) (cid:15) X ∗ ( T ′ L ′ ) Q / Ω eα / / X ∗ ( T L ) Q / Ω X ∗ ( T ′ L ′ ) Q / Ω eα / / X ∗ ( T L ) Q / Ω Proof.
Consider first the Newton points:Let b ′ ∈ (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) and choose c ′ ∈ (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ), ν ′ ∈ X ∗ ( T ′ L ′ ) Q , n ′ ∈ N and π ′ ∈ L ′ suchthat:i) n ′ ν ′ ∈ X ∗ ( T ′ L ′ ).ii) Int( c ′ ) ◦ ( n ′ ν ′ ) is defined over the fixed field L ′h σ n ′ i of σ n ′ on L ′ .17ii) c ′ · ( b ′ σ ) n ′ · c ′− = c ′ · ( n ′ ν ′ )( π ′ ) · c ′− · σ n ′ .Applying φ to all elements and denoting b = φ ( b ′ ), c ′′ = φ ( c ′ ), n = n ′ one gets c ′′ · ( bσ ) n · c ′′− = c ′′ · φ (( n ′ ν ′ )( π ′ )) · c ′′− · σ n . Using the lemma for ψ = n ′ ν ′ we have c ′′ · ( bσ ) n · c ′′− = c ′′ · ( α ( n ′ ν ′ ))( π ′ ) · c ′′− · σ n = c ′′ · ( nα ( ν ′ ))( π ′ ) · c ′′− · σ n . Now there is a uniformizer π ∈ L and an element γ ∈ L such that π e = γ · π ′ · σ ( γ ) − . Then conjugating b further with ( α ( n ′ ν ′ ))( γ ), i.e. setting c = c ′′ · ( α ( n ′ ν ′ ))( γ ) − gives c · ( bσ ) n · c − = c · ( α ( n ′ ν ′ ))( π e ) · c − · σ n = c · ( neα ( ν ′ ))( π ) · c − · σ n . This checks that the elements c , ν = e · α ( ν ′ ), n and π fulfill the third condition used to describe theNewton point of b .It remains to show that these elements also satisfy the two remaining conditions:The first condition is obvious as α is a map between the integral cocharacter groups and so is e · α .Hence neα ( ν ′ ) = eα ( n ′ ν ′ ) ∈ X ∗ ( T L ).To show the second condition note that Int(( α ( n ′ ν ′ ))( γ )) ◦ ( neα ( ν ′ )) = neα ( ν ′ ) as we conjugate insidethe torus T L . As rising to the e th power is even defined over F p it remains to show that Int( c ′′ ) ◦ α ( n ′ ν ′ )is defined over the fixed field of σ n on L . But the universal property of scalar restrictions respect beingdefined over the fixed field of σ n on the respective ground field. Hence a cocharacter already definedover L ′h σ n i is mapped via α to a cocharacter defined over L h σ n i . As conjugating with c ′′ only changesthe chosen maximal torus, Int( c ′′ ) ◦ α ( nν ) is defined over L h σ n i if Int( c ′ ) ◦ ( n ′ ν ′ ) is defined over L ′h σ n i .But this was assumed.This shows that ν = eα ( ν ′ ) is indeed the Newton point of b = φ ( b ′ ).The case of Hodge points is easier:Let b ′ ∈ (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ′ ) as before and choose c ′ , c ′ ∈ (cid:0) Res
F/F ′ ( G ) (cid:1) ( O L ′ ), a representative of itsHodge point µ ′ ∈ X ∗ ( T ′ L ′ ) Q and a uniformizer π ′ ∈ L ′ such that b ′ = c ′ · µ ′ ( π ′ ) · c ′ . Applying φ and denoting b = φ ( b ′ ), c = φ ( c ′ ) and c = φ ( c ) gives together with the lemma for ψ = µ ′ : b = c · φ ( µ ′ ( π ′ )) · c = c · α ( µ ′ )( π ′ ) · c . Note that φ maps (cid:0) Res
F/F ′ ( G ) (cid:1) ( O L ′ ) to G ( O L ), hence c and c lie in the correct group.Pick now any uniformizer π ∈ L . Because of π e · π ′− ∈ O L there is an element c ∈ T ( O L ) such that α ( µ ′ )( π ′ ) = α ( µ ′ )( π e ) · c = eα ( µ ′ )( π ) · c . Hence b = c · eα ( µ ′ )( π ) · c c and eα ( µ ′ ) is the Hodge point of b = φ ( b ′ ). (cid:3) Theorem 5.5.
Let F ′ , F , L ′ , L and T ⊂ G be as above and fix a Hodge point µ ′ ∈ X ∗ ( T ′ L ′ ) Q . Let µ = eα ( µ ′ ) ∈ X ∗ ( T L ) Q be the image of µ ′ . Then choose a constant C > such that for any N > C and any µ = ( µ , µ , . . . , µ N ) ∈ ( X ∗ ( T L ) Q ) N (where the first entry is fixed) the set H µ,N ( G ) ⊂ G ( L ) lies insidea Newton stratum.Then for any N > C and any µ ′ = ( µ ′ , µ ′ , . . . , µ ′ N ) ∈ ( X ∗ ( T ′ L ′ ) Q ) N with first entry the fixed Hodgepoint µ ′ the set H µ ′ ,N (Res F/F ′ ( G )) ⊂ (Res F/F ′ ( G ))( L ′ ) lies inside a Newton stratum. roof. Fix
N > C and µ ′ = ( µ ′ , µ ′ , . . . , µ ′ N ) and choose any L ′ -valued point b ′ ∈ H µ ′ ,N (Res F/F ′ ( G )). Thenthe commutativity of the right diagram in the proposition gives for b = φ ( b ′ ) and any k = 1 , . . . , N (using the ramification index e as in the proposition): µ L (( bσ ) k ) = µ L ( φ (( b ′ σ ) k )) = eα ( µ L ′ (( b ′ σ ) k )) = eα ( µ ′ k ) . For µ k = eα ( µ ′ k ) we have thus b ∈ H ( µ ,µ ,...,µ N ) ,N ( G ). By assumption this set lies inside a Newtonstratum. Hence µ ′ determines the Newton point ν L ( b ). But using the commutativity of the left diagram ν L ( b ) = ν L ( φ ( b ′ )) = eα ( ν L ′ ( b ′ )) . As eα is injective, ν L ′ ( b ′ ) depends only on µ ′ but not on the chosen element b ′ . (cid:3) Proposition 5.6.
Let
F/F ′ be as above. Then for any pair of Hodge points µ ′ = ( µ ′ , µ ′ ) of the group Res
F/F ′ ( GL ) thestratum H µ ′ , (Res F/F ′ ( GL )) lies in some Newton stratum.Proof. Use the previous proposition for C = , N = 2, G = GL . Then our assumptions are met by Proposition5.1. (cid:3) Remark 5.7.
A slightly weaker version of this proposition was already shown by Andreatta and Goren [AG03, thm.9.2]. We will explain how their theorem compares to our result in section 6.2.
The goal of this section is similar to the previous one but for scalar restrictions via unramified extensions.Similarly to above fix a tower of fields Q p ⊂ F ′ ⊂ F ⊂ Q p with F ′ and F finite over Q p but now F/F ′ unramified of degree f . Let as in § L = F · K = F ′ · K .Let G be a connected reductive group over F . Then by definition of scalar restrictions there is anisomorphism of abstract groups φ : (cid:0) Res
F/F ′ ( G ) (cid:1) ( L ) ∼ = G ( F ⊗ F ′ L ) = Y τ ∈ Hom F ′ ( F,L ) G ( L ) . Fixing a maximal torus T ⊂ G as in section 2.1 gives a maximal torus T ′ = Res F/F ′ ( T ) ⊂ Res
F/F ′ ( G ).Then φ restricts to φ : T ′ ( L ) → Q τ T ( L ).Note that σ cyclically permutes the τ -factors. Lemma 5.8.
Fix a Hodge point µ ∈ X ∗ ( T ′ ) Q , an integer d > and an element ϑ ∈ Hom F ′ ( F, L ) . Then there is aconstant C ( µ , d ) depending only on µ , d and ϑ with the following property: For any element b ′ ∈ G ( L ) such that there is an element b ∈ Res
F/F ′ ( G )( L ) with Hodge point µ and µ ( b ′ ) = µ ((( bσ ) d ) ϑ ) (denotingthe ϑ -component of ( bσ ) d under φ by (( bσ ) d ) ϑ ), the Hodge points of ( b ′ σ ) i for i = 1 , . . . , C ( µ , d ) determine the Newton point of b ′ .Proof. By Lemma 4.1 there are only finitely many possibilities for µ (( bσ ) d ). As φ maps Res F/F ′ ( G )( O L ) into Q τ G ( O L ), µ (( bσ ) d ) determines µ ( φ (( bσ ) d )) and in particular µ ((( bσ ) d ) ϑ ). Applying Theorem 4.7 tothe Hodge strata defined by each of them and taking the maximum over all appearing constants givesthe desired C ( µ , d ). (cid:3) Theorem 5.9.
Let the C ( µ , d ) be the constants of the previous lemma.i) If G is unramified and has a hyperspecial vertex, then any stratum H µ,C ( µ ,f ) · f (Res F/F ′ ( G )) wherethe first entry of µ equals µ lies in some Newton stratum.ii) For general G choose an integer ξ > such that ξν ( b ) lies in the integral cocharacter group for each lement b ∈ Res
F/F ′ ( G )( L ) . Then any stratum H µ,C ( µ ,ξf ) · ξf (Res F/F ′ ( G )) where the first entry of µ equals µ lies in some Newton stratum.Proof. i) Let ξ be some integer such that the Newton points of all elements in Res F/F ′ ( G ) lie in the ξ X ∗ ( T ′ ).Then by passing to some totally ramified field extension of L of degree ξ , we may wlog. assume thatthe Newton points of all appearing elements lie in X ∗ ( T ′ ) (similarly to the argument in Proposition 5.1but now using Proposition 2.14).Consider now an element b ∈ H µ,C ( µ ,f ) · f (Res F/F ′ ( G )) and let b ′ τ := (( bσ ) f ) τ ∈ G ( L ). As σ f is anendomorphism of Q τ G ( L ) which fixes each component, we have (( bσ ) fi ) τ = ( b ′ τ σ f ) i for each i > τ . In particular the fixed Hodge points of b determine the Hodge points µ ( b ′ τ σ ) , µ (( b ′ τ σ ) ) , . . . , µ (( b ′ τ σ ) C ( µ ,f ) ). By choice of C ( µ , f ) this information suffices to determine the Newton point of b ′ ϑ ∈ G ( L ) for some ϑ .As the Frobenius σ cyclically permutes the τ -factors, we may choose c ∈ Res
F/F ′ ( G )( L ), ˜ b ∈ G ( L ) with˜ b = ν ′ ( π ) for some ν ′ ∈ X ∗ ( T ) and some uniformizer π ∈ L such that φ ( cbσ ( c ) − ) = (˜ b ) τ . Write φ ( b ) = ( b τ ) τ and φ ( c ) = ( c τ ) τ . Then c τ b τ σ ( c − στ ) = ˜ b in G ( L ). Thus as f = [ F : F ′ ] equals thenumber of embeddings τ (recall L is the maximal unramified extension of F ) b ′ τ = (( bσ ) f ) τ = c − τ · (˜ bσ ) f · c τ ∈ G ( L ) . In particular the Newton point of b ′ ϑ determines the Newton point of (˜ bσ ) f , hence the one of ˜ b . But bydefinition of ˜ b , ν (˜ b ) gives directly the Newton point of b itself. But this was our goal.ii) Instead of changing the base field replace b by ( bσ ) ξ . Then we find a ˜ b with the same properties asin ii) and the same arguments show that the Hodge points µ (( bσ ) ξf ) , µ (( bσ ) ξf ) , . . . , µ (( bσ ) C ( µ ,ξf ) · ξf )determine ν ((( bσ ) ξf ) ϑ ) and hence ν ( b ). (cid:3) Corollary 5.10.
Let
F/F ′ be as in the proposition. Then any stratum H µ, f (Res F/F ′ ( GL )) lies in some Newton stratum.Proof. Apply part i) of the previous proposition for G = GL and note that due to Proposition 5.1 we maychoose C ( µ , f ) = 2. (cid:3) Remark 5.11. i) Note that the Newton points are not contained in X ∗ ( T ′ ), but in f X ∗ ( T ′ ).ii) Contrary to the totally ramified situation the Hodge points of b and ( bσ ) do not suffice to determinethe Newton point. To see this consider Res F/ Q p ( GL ) with F/ Q p unramified of degree 3 and µ =((1 , , (1 , , (1 , µ = ((1 , , (2 , , (2 , b = (cid:18)(cid:18) π
00 1 (cid:19) , (cid:18) π (cid:19) , (cid:18) π (cid:19)(cid:19) ∈ Y τ ∈ Hom Q p ( F,K ) GL ( K ) ∼ = Res F/ Q p ( GL )( K )has Newton point ν ( b ) = (( , ) , ( , ) , ( , )). But the results of section 6.1 imply that the ’generic’element in H ( µ ,µ ) , (Res F/ Q p ( GL )) has Newton point (( , ) , ( , ) , ( , )). We will connect the previous two sections 5.2 and 5.3 to get corresponding statements for arbitraryscalar restrictions.
Theorem 5.12.
Let Q p ⊂ F ′ ⊂ F ⊂ Q p be finite field extensions inside Q p . Let G be a connected reductive group over F with maximal torus T and fix a Hodge point µ ′ ∈ X ∗ ( T ′ ) Q where T ′ = Res F/F ′ ( T ) ⊂ Res
F/F ′ ( G ) .Then one can give an explicit bound C depending only on the constants for G itself and on invariants ofthe extension F/F ′ , such that for every tuple µ ′ = ( µ ′ , . . . , µ ′ N ) with N > C and first entry the chosenHodge point µ ′ the stratum H µ ′ ,N (Res F/F ′ ( G )) lies in some Newton stratum. roof. Let F ′′ be the maximal unramified extension of F ′ inside F and f = [ F ′′ : F ]. Denote L ′ = F ′ · K = F ′′ · K and L = F · K . Then Res F/F ′ ( G ) = Res F ′′ /F ′ (Res F/F ′′ ( G ))and we have the following isomorphisms of groups: φ : (Res F/F ′ ( G ))( L ′ ) = (Res F ′′ /F ′ (Res F/F ′′ ( G )))( L ′ ) φ ′′ −→ Y τ ∈ Hom F ′ ( F ′′ ,L ′ ) (Res F/F ′′ ( G ))( L ′ ) φ ′ −→ Y τ ∈ Hom F ′ ( F ′′ ,L ′ ) G ( L )where φ ′′ resp. φ ′ are the isomorphisms of section 5.3 resp. section 5.2. Now fix the Hodge point µ ′ ∈ X ∗ ( T ′ ) Q , an element ϑ ∈ Hom F ′ ( F ′′ , L ′ ) and ξ > F/F ′ ( G )are contained in ξ X ∗ ( T ′ ). As in Lemma 5.8 we may find a constant C ( µ ′ , ξf ) satisfying the very sameproperties as stated there, but now for the morphism φ considered here. Then applying Theorem 5.5we see that this C ( µ ′ , ξf ) also satisfies the property of Lemma 5.8 for the group Res F/F ′′ ( G ), i.e. whenconsidering only the morphism φ ′′ . But this means that all assumptions of Theorem 5.9 are met for theunramified extension F/F ′′ , and we may take C = C ( µ ′ , ξf ) · ξf .Note that if G is unramified and contains a hyperspecial vertex the same argument works with ξ = 1. (cid:3) Proposition 5.13.
Let Q p ⊂ F ′ ⊂ F ⊂ Q p be finite field extensions inside Q p and let f be the degree of the maximalunramified extension of F ′ inside F (as in the proof of the previous theorem). Then for any tuple ofHodge points µ = ( µ , µ , . . . , µ f ) of Res
F/F ′ ( GL ) the stratum H µ, f (Res F/F ′ ( GL )) lies in someNewton stratum.Proof. Use the previous theorem in the special case G = GL . Then we may choose C = 2 f by Proposition5.1. (cid:3) n for n ≥ Proposition 5.14.
For each n ≥ and arbitrary L , there are strata H µ,n − ⊂ SL n ( L ) which are not contained in anyNewton stratum.Proof. Consider for a chosen uniformizer π the matrices: b = . . . − n − π n − π − . . . π − . . . . . . π − . . . π − b = . . . − n π n − π − . . . π − . . . . . . π − . . . π − T ⊂ SL n be the diagonal torus. We will give the Hodge and Newton points via giving the image of π on some representative in X ∗ ( T ) Q (for 1 ≤ i ≤ n − µ (( b σ ) i )( π ) = diag ( π n − i , π n − i , . . . , π n − i | {z } i , π − i , . . . , π − i | {z } n − i ) = µ (( b σ ) i )( π ) µ (( b σ ) n )( π ) = (1 , , . . . , = µ (( b σ ) n )( π ) = ( π n , , , . . . , , π − n ) ν ( b )( π ) = (1 , , . . . , = ν ( b )( π ) = ( π n − , π n − , . . . , π n − , π − )Note that the last two inequality signs hold even after taking the quotient by Ω = S n . (cid:3) Remark 5.15. i) The same statement holds for G = GL n or G = P GL n using the same matrices.ii) These examples restrict the cases where one can hope that the first two Hodge points already definethe Newton point: No connected reductive group containing a subgroup SL or P GL has this property. Fix a totally real extension ˜ F over Q such that the corresponding extension F over Q p is unramifiedof degree g . We consider quadruples ( A, λ, ι, α ) consisting of an abelian variety A over k (which weassume to be algebraically closed), a principal polarization λ , ι : O ˜ F → End ( A ) fixed by the Rosatiinvolution associated to λ and a full symplectic level- n -structure α . The moduli space of such tuples isrepresentable by a regular irreducible variety M n over k .To such quadruples Goren and Oort associate in [GO00] a discrete invariant τ ( A ), the type of A andstudy the corresponding stratification W τ on M n . We will explain here a group theoretic descriptionof the type and give a conceptual rather than computational proof of a weaker version of [GO00, thm.5.4.11] about the generic Newton point on the stratum defined by a fixed type τ .Throughout this section κ ( · ) always denotes the residue field of a local field and we fix isomorphisms Z /g Z ∼ = Hom F q ( κ ( F ) , k ) ∼ = Hom Z p ( O F , O K ) ∼ = Hom Q p ( F, K )(where K = Frac( W ( k )) as in section 2.1) such that the homomorphism associated to i + 1 is obtainedby composing the homomorphism associated to i with σ . Definition 6.1. ([GO00, def. 2.1.1])Let ( A, λ, ι, α ) ∈ M n ( k ) . Then κ ( F ) acts via ι on H (Ω A ) and hence on the k -vector space ker ( V : H (Ω A ) → H (Ω A )) where V denotes the Verschiebung map. This kernel decomposes as a direct sumof subspaces on which the κ ( F ) -action is given via a character in Hom F q ( κ ( F ) , k ) = Z /g Z . Then the type τ ( A ) of ( A, λ, ι, α ) is the subset of Z /g Z consisting of all characters for which the correspondingsubspace is non-trivial. The Frobenius morphism F on the rational Dieudonn´e-module corresponding to ( A, λ, ι, α ) is givenby an element (or rather a Res F/ Q p ( GL )( O K )- σ -conjugacy class) b ∈ Res F/ Q p ( GL )( K ). Using as insection 5.3 Res F/ Q p ( GL )( K ) ∼ = Y τ ∈ Hom Q p ( F,K ) GL ( K ) = Y i ∈ Z /g Z GL ( K )we will view b as a tuple ( b i ) i on the right-hand side.Wlog. let T ⊂ GL be the maximal torus of diagonal elements. Then we use again T ′ = Res F/ Q p ( T ) as22 maximal torus of Res F/ Q p ( GL ). But rather than working with Hodge or Newton points in X ∗ ( T ′ ) Q ,we use their image under the identification X ∗ ( T ′ ) Q = Hom K ( G m,K , T ′ × Spec Q p Spec K ) Q = Hom K ( G m,K , Y i ∈ Z /g Z ( T × Spec F Spec K )) Q = Y i ∈ Z /g Z X ∗ ( T ) Q Then an easy computation gives for b ∈ Res F/ Q p ( GL )( K ) with components ( b i ) i ∈ Q i ∈ Z /g Z GL ( K )the equality µ ( b ) i = µ ( b i ) under the isomorphism between cocharacter groups above. For Newton pointsthe situation is slightly different as σ permutes all GL -factors. This implies that the image of ν ( b ) ineach X ∗ ( T ) Q -factor is the same. Proposition 6.2.
Let ( A, λ, ι, α ) ∈ M n ( k ) and let b ∈ Res F/ Q p ( GL )( K ) represent the action of the Frobenius on theDieudonn´e module associated to A . Then i ∈ τ ( A ) if and only if µ (( bσ ) ) i = (1 , .Proof. Let (
D, F, V ) be the Dieudonn´e module of A . Then there is a canonical isomorphism H (Ω A ) ∼ = V D/pD as k = κ ( K )-module. Furthermore D decomposes as a direct sum of O K -modules D i on which the O F -action induced by ι is given by the element i ∈ Z /g Z = Hom Z p ( O F , O K ) (cf. [GO00, § D i is free of rank 2. As the Verschiebung map V is σ − -linear its restriction V | D i : D i → D factors via V i : D i → D i − . Similarly the restriction F | D i of the Frobenius factors via F i : D i → D i +1 . Dividingout p we get 2-dimensional k -vector spaces D i /pD i and morphisms V i : D i +1 /pD i +1 → D i /pD i , F i : D i /pD i → D i +1 /pD i +1 . By [GO00, lemma 2.3.1] each of the V i and F i has cokernel isomorphic to O K /pO K = k . In particular each V i has a 1-dimensional kernel and a 1-dimensional image.Now we can reformulate the condition for i ∈ τ ( A ): By definition this happens if and only if V i − : V i ( D i +1 /pD i +1 ) = V i D i +1 /pD i → V i − ( D i /pD i ) = V i − D i /pD i − ֒ → D i − /pD i − has non-trivial kernel. As each V i has 1-dimensional kernel, this happens exactly if V i − ◦ V i : D i +1 /pD i +1 → D i − /pD i − is the zero morphism. Using that the V i are injective with the statedcokernel, we may rewrite this as V i − ◦ V i = pσ − : D i +1 → D i − . With V i ◦ F i = p : D i → D i for all i this is equivalent to F i ◦ F i − = ( p − σ ) ◦ V i − ◦ V i ◦ F i ◦ F i − = pσ : D i − → D i +1 But this condition can be reformulated in group theoretic terms: Fixing a suitable O K -basis of D we canview the Frobenius as F = bσ for some element b ∈ Res F/ Q p ( GL )( K ). Then the group decompositionRes F/ Q p ( GL )( K ) = Q i ∈ Z /g Z GL ( K ) corresponds exactly to the decomposition with respect to theaction on the eigenspaces D i , i.e. we have F i = b i σ : D i → D i +1 . Thus F i ◦ F i − = pσ if and only if( b i σ ) · ( b i − σ ) ∈ GL ( O K ) · pE · GL ( O K )(denoting the unit matrix by E ) or in other words µ ( b i · σ ( b i − )) = (1 , b i · σ ( b i − ) is nothingelse than the i -th component of the element ( bσ ) ∈ Res F/ Q p ( GL )( K ). (cid:3) Corollary 6.3.
The loci in M n with constant type τ are exactly the loci where µ ( b ) and µ (( bσ ) ) are constant, i.e. thevariety W τ is the preimage of H µ, (Res F/ Q p ( GL )) for a suitable µ under the map M n → Res F/ Q p ( GL )( K ) / (change of basis) associating to each quadruple the element defining the action of the Frobenius. roof. Let (
A, λ, ι, α ) ∈ M n ( k ) and choose a representative b ∈ Res F/ Q p ( GL ) of the Frobenius morphism onits Dieudonn´e module. By [GO00, lemma 2.3.1] µ ( b ) := ((1 , , . . . , (1 , M n ,hence does not give any information at all. But as µ (( bσ ) ) i is either (1 ,
1) or (2 ,
0) for each i , theprevious proposition states that the type encodes precisely the same information as µ (( bσ ) ). (cid:3) To treat the generic Newton point of W τ = S τ ′ ⊂ τ W τ ′ ⊂ M n (c.f. [GO00, def. 2.3.5]), we recall itsdescription found by Goren and Oort: Definition 6.4.
A subset τ ′ ⊆ Z /g Z is called spaced if it contains no two consecutive elements. For any τ ⊂ Z /g Z set λ ( τ ) = (cid:26) if g odd and τ = Z /g Z g max {| τ ′ | | τ ′ ⊆ τ spaced } else β τ = ((1 − λ ( τ ) , λ ( τ )) , . . . , (1 − λ ( τ ) , λ ( τ ))) ∈ X ∗ ( T ′ ) Q = Y i ∈ Z /g Z X ∗ ( T ) Q Remark 6.5.
Note that the given definitions of λ ( τ ) and β τ differ slightly from the definitions 5 . . . . Proposition 6.6.
Let τ ⊂ Z /g Z be a type. Then the Newton point ν of every geometric point in W τ satisfies ν ≺ β τ .Proof. Note first that for τ ′ ⊆ τ we have β τ ≺ β τ ′ . As W τ = S τ ′ ⊆ τ W τ ′ we are reduced to consider only pointswith type τ .Thus fix any point in W τ and let b ∈ Res F/ Q p ( GL ) represent the action of the Frobenius morphism onthe Dieudonn´e module. If g is odd and τ = Z /g Z then we have µ (( bσ ) ) = ((1 , , . . . , (1 , ν ( b ) = 12 ν (( bσ ) ) ≺ µ (( bσ ) ) = β τ . Consider now the remaining cases and recall µ ( b ) i = (1 ,
0) and µ (( bσ ) ) i = (cid:26) (1 ,
1) if i ∈ τ (2 ,
0) if i / ∈ τ Fix a spaced subset τ ′ ⊆ τ of maximal cardinality. Replacing b by σ k ( b ) for suitable k (which does notchange the Newton point), we may assume that 0 ∈ τ ′ if τ ′ is non-empty. Now consider the partition P τ ′ of Z /g Z defined byi) { i } ∈ P τ ′ if and only if i / ∈ τ ′ and i + 1 / ∈ τ ′ .ii) { i − , i } ∈ P τ ′ if and only if i ∈ τ ′ .With Lemma 3.13 we get µ (( bσ ) g ) = µ Y I ∈P τ ′ Y i ∈ I σ g − i ( b i ) ! ≺ M I ∈P τ ′ µ Y i ∈ I σ g − i ( b i )) ! (for a suitable enumeration of the elements in P τ ′ in the expression in the middle). By our choice ofthe partition we have µ Y i ∈ I σ g − i ( b i ) ! = (cid:26) (1 ,
0) if | I | = 1(1 ,
1) if | I | = 2and obtain µ (( bσ ) g ) ≺ (1 , ⊕ g − | τ ′ | ⊕ (1 , ⊕| τ ′ | = ( g − | τ ′ | , | τ ′ | ) . i ∈ Z /g Z and any n > µ (( bσ ) ng ) i = µ (( bσ ) i · (( bσ ) g ) n − · ( bσ ) g − i ) i ≺ M j = i,i − ,..., µ ( b ) j ⊕ µ (( bσ ) g ) ⊕ n − ⊕ M j = g,g − ,...,i +1 µ ( b ) j = (1 , ⊕ g ⊕ ( g − | τ ′ | , | τ ′ | ) ⊕ n − = ( ng − ( n − | τ ′ | , ( n − | τ ′ | ) . Hence we have again by Mazur’s inequality ν ( b ) ≺ lim n →∞ gn µ (( bσ ) gn ) = 1 g (( g − | τ ′ | , | τ ′ | ) , . . . , ( g − | τ ′ | , | τ ′ | )) = β τ . (cid:3) Remark 6.7.
This proposition is a weaker form of [GO00, thm. 5.4.11]. There an explicit calculation using the matrix A τ = (cid:18)(cid:18) a i − π (cid:19)(cid:19) i with a i = (cid:26) i ∈ τ generic and non-zero if i / ∈ τ shows that on each irreducible component of W τ the Newton point β τ indeed occurs and is the genericone. Nevertheless we included the proof above as it gives a more conceptual explanation why one shouldexpect this behavior: The fixed Hodge points µ ( b ) and µ (( bσ ) ) give by using Lemma 4.1 upper boundson n µ (( bσ ) n ) (for n > Fix a totally real extension ˜ F over Q such that the corresponding extension F over Q p is totallyramified of degree g . We consider quadruples ( A, λ, ι, ε ) consisting of an abelian variety A over a field ofcharacteristic p , a polarization λ , ι : O ˜ F → End ( A ) and a level- N -structure ε (with N ≥
4) satisfyingthe Deligne-Pappas condition. For a precise treatment see [AG03, § M ( F p , µ N ) over F p .We recall the definition of the two invariants j and n defined in [AG03] using the display associatedthe tuple ( A, λ, ι, ε ) defined over an algebraically closed field k . For this set as usual K = Frac( W ( k ))and L = F · K . Then L/K is a totally ramified extension of degree g and its ring of integers can beidentified with O L = O ˜ F ⊗ Q W ( k ) ∼ = W ( k )[ T ] / ( h ( T )) for some Eisenstein polynomial h ( T ) of degree g (cf. [AG03, § Definition 6.8. (cf. [AG03, def. 4.1])Let R be some F p -algebra and W ( R ) the Witt vectors over R with the Frobenius morphism σ . An O ˜ F -display (over R ) is a quadruple ( P, Q, V − , F ) with:i) P a projective O ˜ F ⊗ W ( R ) -module of rank .ii) Q ⊆ P a finitely generated O ˜ F ⊗ W ( R ) -submodule.iii) F : P → P linear with respect to O ˜ F and σ -linear with respect to W ( R ) .iv) V − : Q → P linear with respect to O ˜ F and σ -linear with respect to W ( R ) .satisfying several conditions as stated in [Zin02, def. 1]. Proposition 6.9. ([AG03, prop. 4.10])Let ( P, Q, V − , F ) be an O ˜ F -display over k . Let P = P ⊗ W ( k ) k and Q the image of Q under theprojection P → P . Let F : P → P be the reduction of F : P → P . Then there are α, β ∈ P anduniquely determined i, j, m ∈ Z such thati) P = O L α ⊕ O L β .ii) The Hodge filtration Q = ker ( F ) ⊂ P is given by Q = ( T i O L /p ) α ⊕ ( T j O L /p ) β ⊂ P = ( O L /p ) α ⊕ ( O L /p ) β ith i + j = g , ≤ j ≤ i ≤ g .iii) There exists a c ∈ O × L such that F ( α ) = T m α + T j β , F ( β ) = c · T i α with m ≥ j . Definition 6.10.
For an O L -display ( P, Q, V − , F ) define the following two integers:i) j as in the previous proposition.ii) n = min { m, i } , with m and i as in the previous proposition.iii) A display ( P, Q, V − , F ) is of type ( j, n ) if these integers are the constants defined in i) and ii).Furthermore let λ ( n ) = min { ng , } . Note that the display associated to a tuple (
A, λ, ι, ε ) ∈ M ( F p , µ N )( k ) is exactly an O ˜ F -display over k . Thus it makes sense to define M j,n ⊂ M ( F p , µ N ) as the locus where the display associated to theabelian variety has type ( j, n ). Theorem 6.11. ([AG03, thm. 9.2])For any geometric point ( A, λ, ι, ε ) ∈ M j,n ( k ) the slopes of the Newton polygon of A are λ ( n ) and − λ ( n ) . We will give an alternative proof of this theorem and explain how the type ( j, n ) is related tocertain Hodge points. To do so we will denote the elements of Res F/ Q p ( GL )( K ) as matrices via theisomorphism of groups Res F/ Q p ( GL )( K ) ∼ = GL ( L ). Proposition 6.12.
Let ( P, Q, V − , F ) be an O ˜ F -display and view F ∈ Res F/ Q p ( GL )( K ) = GL ( L ) by choice of some basis.Theni) j is the smaller Hodge slope of F (as an element in GL ( L ) ).ii) n + j is the smaller Hodge slope of ( F σ ) (as an element in GL ( L ) ).In particular M j,n is exactly the locus where the Frobenius of the display associated to the abelian varietylies in H ( g ( g − j,j ) , g (2 g − n − j,n + j )) , (Res F/ Q p ( GL )) .Proof. By Proposition 6.9 we may assume that the display is given in its normal form. In particular one has: F = (cid:18) T m cT i T j (cid:19) i) Then: (cid:18) c − − c − T m − j (cid:19) · F = (cid:18) T j T i (cid:19) and the Hodge slopes of F are j and i .ii) One computes ( F σ ) = (cid:18) cT i + j + T m σ ( c ) T i + m T j + m σ ( c ) T i + j (cid:19) Case 1: m ≥ i , i.e. n = i Then the following equation yields the Hodge slopes: (cid:18) − T m − j (cid:19) · ( F σ ) · (cid:18) − σ ( c ) − T m − i (cid:19) = (cid:18) cT j + i σ ( c ) T j + i (cid:19) Thus both Hodge slopes of (
F σ ) are i + j = n + j . Case 2: m < i
Then the following equation yields the Hodge slopes: (cid:18) − cT i − m − T m − j (cid:19) · ( F σ ) · (cid:18) − σ ( c ) T i − m (cid:19) = (cid:18) T j + m − cσ ( c ) T i + j − m (cid:19) F σ ) is m + j = n + j .The last assertion follows directly from this description of j and n . That the slopes get divided by g isdue to the fact that we now consider F ∈ Res F/ Q p ( GL )( K ) and use Proposition 5.4 to compare theHodge points. (cid:3) Proof. (of theorem 6.11)By Proposition 5.13 all elements in H ( g ( g − j,j ) , g (2 g − n − j,n + j )) , (Res F/ Q p ( GL )) have the same Newtonpoint and hence the same holds for M j,n . To actually compute it use the procedure explained at theend of the proof of Proposition 5.1. (cid:3) References [AG03] F. Andreatta and E. Z. Goren. Geometry of Hilbert modular varieties over totally ramifiedprimes.
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