Automorphism Groups of Shimura Varieties

In this paper, we determine the scheme automorphism group of the reduction modulop of the integral model of the connected Shimura variety (of prime-to-p level) for reductive groups of type A and C. The result is very close to the characteristic 0 version studied by Shimura, Deligne and Milne-Shih.

They are equivalent by duality, and the first is generalized by Langlands in non-abelian setting.Geometric reciprocity in non-abelian setting would be via Tannakian duality; so, it involves Shimura varieties.
Iwasawa theory is built upon the geometric reciprocity law.The cyclotomic field Q(µ p ∞ ) is the maximal p-ramified extension of Q fixed by + removing the p-inertia toric factor Z × p .We then try to study arithmetically constructed modules X out of Q(µ p ∞ ) ⊂ Q ab .The main idea is to regard X as a module over over the Iwasawa algebra (which is a completed Hecke algebra relative to GL1(A (∞) ) ), and ring theoretic techniques are used to determine X.
If one wants to get something similar in a non-abelian situation, we really need a scheme whose automorphism group has an identification with G(A (∞) )/Z(Q) for a reductive algebraic group G.If G = GL(2) /Q , the tower V /Q ab of modular curves has Aut(V /Q ) identified with GL 2 (A (∞) )/Z(Q) as Shimura proved.The decomposition group of (p) is given by B(Q p ) × SL 2 (A (p∞) )/{±1} for a Borel subgroup B, and I have been studying various arithmetically constructed modules over the Hecke algebra of , relative to the unipotent subgroup U (Z p ) ⊂ B(Z p ) (removing the toric factor from the decomposition group).Such study has yielded a p-adic deformation theory of automorphic forms (see [PAF] Chapter 1 and 8), and it would be therefore important to study the decomposition group at p of a given Shimura variety, which is basically the automorphism group of the mod p Shimura variety.
Iwasawa theoretic applications (if any) are the author's motivation for the investigation done in this paper.However the study of the automorphism group of a given Shimura variety has its own intrinsic importance.As is clear from the construction of Shimura varieties done by Shimura ([Sh]) and Deligne ([D1] 2.4-7), their description of the automorphism group (of Shimura varieties of characteristic 0) is deeply related to the geometric reciprocity laws generalizing classical ones coming from class field theory and is almost equivalent to the existence of the canonical models defined over a canonical algebraic number field.Except for the modulo p modular curves and Shimura curves studied by Y. Ihara, the author is not aware of a single determination of the automorphism group of the integral model of a Shimura variety and of its reduction modulo p, although Shimura indicated and emphasized at the end of his introduction of the part I of [Sh] a good possibility of having a canonical system of automorphic varieties over finite fields described by the adelic groups such as the ones studied in this paper.
We shall determine the automorphism group of mod p Shimura varieties of PEL type coming from symplectic and unitary groups.

Statement of the theorem
Let B be a central simple algebra over a field M with a positive involution ρ (thus Tr B/Q (xx ρ ) > 0 for all 0 = x ∈ B).Let F be the subfield of M fixed by ρ.Thus F is a totally real field, and either M = F or M is a CM quadratic extension of F .We write O (resp.R) for the integer ring of F (resp.M ).We fix an algebraic closure F of the prime field F p of characteristic p > 0. Fix a proper subset Σ of rational places including ∞ and p.Let F × + be the subset of totally positive elements in F , and O (Σ) denotes the localization of O at Σ (disregarding the infinite place in Σ) and O Σ is the completion of O at Σ (again disregarding the infinite place).We write O × Let O B be a maximal order of B. Let L be a projective O B -module with a non-degenerate F -linear alternating form , : such that bx, y = x, b ρ y for all b ∈ B. Identifying L Q with a product of copies of the column vector space D r on which M r (D) acts via matrix multiplication, we can let σ ∈ Aut alg (D) act component-wise on L Q so that σ(bv) = σ(b)σ(v) for all σ ∈ Aut alg (D).
Let C be the opposite algebra of Then C is a central simple algebra and is isomorphic to M s (D), and hence Out(C) ∼ = Out(D) = Out(B).We write The algebra C has involution * given by cx, y = x, c * y for c ∈ C, and this involution " * " of C extends to an involution again denoted by " * " of End Q (L Q ) given by Tr F/Q ( gx, y ) = Tr F/Q ( x, g * y ) for g ∈ End Q (L Q ).The involution * (resp.ρ) induces the involution * ⊗ 1 (resp.ρ ⊗ 1) on C A (resp. on B A ) which we write as * (resp.ρ) simply.Define an algebraic group G /Q by and an extension G of G by the following subgroup of the opposite group Aut • A (L A ) of the A-linear automorphism group Aut A (L A ): and from this we find that gBg −1 = B ⇔ gCg −1 = C for g ∈ Aut Q (L Q ), and if this holds for g, then gg * ∈ F × ⇔ gx ρ g −1 = (gxg −1 ) ρ for all x ∈ B and gy * g −1 = (gyg −1 ) * for all y ∈ C. Then G is a normal subgroup of G of finite index, and All the four groups are equal if G /Q is quasi-split but are not equal in general.We put P G = G/Z and P G = G/Z for the center Z of G.
We write G 1 for the derived group of G; thus, equal to the kernel of the similitude map g → ν(g); so, in this case, we ignore the right factor Res M/Q G m and regard ) combined with an observation in [Sh] II (4.2.1), the automorphism group Aut A-alg (C A , * ) of the algebra C A preserving the involution * is given by P G(A).In other words, we have an exact sequence of Q-algebraic groups We write for the projection.
The automorphism group of the Shimura variety of level away from Σ is a quotient of the following locally compact subgroup of G(A (Σ) ): where we embed Out We suppose to have an R-algebra homomorphism h : C → C R such that h(z) = h(z) * and (h1) (x, y) = x, h(i)y induces a positive definite hermitian form on L R .
We define X to be the conjugacy classes of h under G(R).Then X is a finite disjoint union of copies of the hermitian symmetric domain isomorphic to G(R) + /C h , where C h is the stabilizer of h and the superscript "+" indicates the identity connected component of the Lie group G(R).Then the pair (G, X) satisfies the three axioms (see [D1] 2.1.1.1-3)specifying the data for defining the Shimura variety Sh (and its field of definition, the reflex field E; see [Ko] Lemma 4.1).In [D1], two more axioms are stated to simplify the situation: (2.1.1.4-5).These two extra axioms may not hold generally for our choice of (G, X) (see [M] Remark 2.2).
The complex points of Sh are given by Documenta Mathematica 11 (2006)   This variety can be characterized as a moduli variety of abelian varieties up to isogeny with multiplication by B. For each x ∈ X, we have h gives rise to a complex vector space structure on L R , and X x (C) = L R /L is an abelian variety, because by (h1), , induces a Riemann form on L. The multiplication by b We suppose (h2) all rational primes in Σ are unramified in M/Q, and Σ contains ∞ and p; ).This moduli interpretation (combined with (h1-4)) allows us to have a well defined p-integral model of level away from Σ (see below for a brief description of the moduli problem, and a more complete description can be found in [PAF] 7.1.3).In other words, has a well defined smooth model over O E,(p) := O E ⊗ Z Z (p) which is again a moduli scheme of abelian varieties up to prime-to-Σ isogenies.We write Sh (p)  for Sh (Σ) when Σ = {p, ∞}.We also write We have taken full polarization classes under scalar multiplication by O × (Σ)+ in our moduli problem (while Kottwitz's choice in [Ko] is a partial class of multiplication by Z × (Σ)+ ).By our choice, the group G is the full similitude group, while Kottwitz choice is a partial rational similitude group.Our choice is convenient for our purpose because G has cohomologically trivial center, and the special fiber at p of the characteristic 0 Shimura variety Sh/G(Z Σ ) gives rise to the mod p moduli of abelian varieties of the specific type we study (as shown in [PAF] Theorem 7.5), while Kottwitz's mod p moduli is a disjoint union of the reduction modulo p of finitely many characteristic 0 Shimura varieties associated to finitely many different pairs (G i , X i ) with G i locally isomorphic each other at every place ( [Ko] Section 8).
We fix a strict henselization W ⊂ Q of Z (p) .Thus W is an unramified valuation ring with residue field F = F p .Under these five conditions (h1-5), combining (and generalizing) the method of Chai-Faltings [DAV] for Siegel modular varieties and that of [Ra] for Hilbert modular varieties, Fujiwara ([F] Theorem in §0.4) proved the existence of a smooth toroidal compactification of Sh (p) S = Sh (p) /S for sufficiently small principal congruence subgroups S with respect to L in G(A (p) ) as an algebraic space over a suitable open subscheme of Spec(O E ) containing Spec(W ∩ E).Although our moduli problem is slightly different from the one Kottwitz considered in [Ko], as was done in [PAF] 7.1.3,following [Ko] closely, the p-integral moduli Sh (p) over O E,(p) is proven to be a quasi-projective scheme; so, Fujiwara's algebraic space is a projective scheme (if we choose the toroidal compactification data well).If the reader is not familiar with Fujiwara's work, the reader can take the existence of the smooth toroidal compactification (which is generally believed to be true) as an assumption of our main result.
/O E,(p) ⊗ O E,(p) W. By Zariski's connectedness theorem combined with the existence of a normal projective compactification (either minimal or smooth) of Sh /S classifies, for any S-scheme T , quadruples (A, λ, i, φ (Σ) ) /T defined as follows: A is an abelian scheme of dimension 1 2 rank Z L for which we define the Tate module Σ) ; The symbol i stands for an algebra embedding i : O B → End(A) taking the identity to the identity map on A; φ (Σ) is a level structure away from Σ, that is, an O B -linear , where we require that φ The group G(A (Σ) ) acts on Sh (Σ) by φ (Σ) → φ (Σ) • g.We can extend the action of G(A (Σ) ) to the extension G (Σ) .Each element g ∈ G (Σ)  with projection π(g) = σ g in Out Q-alg (B, ρ) acts also on G(A) for a suitable finite extension field A of M , and the norm map N C factoring through the right factor G(A) descends to N C : G(Q) → M × (which determines We now state the main result: Theorem 1.1.Suppose (h1-5).
Then the field automorphism group Aut(F(V )/F) of the function field F(V ) over F is given by the stabilizer the connected component V (in π 0 (Sh ).The stabilizer is given by In particular, this implies that the scheme automorphism group Aut(V /F ) coincides with the field automorphism group Aut(F(V )/F) and is given as above.
This type of theorems in characteristic 0 situation has been proven mainly by Shimura, Deligne and Milne-Shih (see [Sh] II, [D1] 2.4-7 and [MS] 4.13), whose proof uses the topological fundamental group of V and the existence of the analytic universal covering space.Our proof uses the algebraic fundamental group of V and the solution of the Tate conjecture on endomorphisms of abelian varieties over function fields of characteristic p due to Zarhin (see [Z], [DAV] Theorem V.4.7 and [RPT]).The characteristic 0 version of the finiteness theorem due to Faltings (see [RPT]) yields a proof in characteristic 0, arguing slightly more, but we have assumed for simplicity that the characteristic of the base field is positive (see [PAF] for the argument in characteristic 0).We shall give the proof in the following section and prove some group theoretic facts necessary in the proof in the section following the proof.Our original proof was longer and was based on a density result of Chai (which has been proven under some restrictive conditions on G), and Ching-Li Chai suggested us a shorter proof via the results of Zarhin and Faltings (which also eliminated the extra assumptions we imposed).The author is grateful for his comments.

Proof of the theorem
We start with Proposition 2.1.Suppose (h1-5).Let σ ∈ Aut(F(V )/F).Let U ⊂ V be a connected open dense subscheme on which σ ∈ Aut(F(V )/F) induces an isomorphism U ∼ = σ(U ).For x ∈ (U ∩ σ(U ))(F), the two abelian varieties X x and X σ(x) are isogenous over F, where X x is the abelian variety sitting over x.
Proof.We recall the subgroup in the theorem: .
By characteristic 0 theory in [D] Theorem 2.4 or [MS] p.929 (or [PAF] 7.2.3), the action of G(A (Σ) )/Z(Z (Σ) ) on π 0 (Sh Since each geometrically connected component of Sh (Σ) is defined over the field K of fractions of W, by the existence of a normal projective compactification (either smooth toroidal or minimal) over W (and Zariski's connectedness theorem), we have a bijection between geometrically connected components over K and over the residue field F induced by reduction modulo p. Then the stabilizer in The scheme theoretic automorphism group Aut(V /F) is a subgroup of the field automorphism group Aut(F(V )/F).By a generalization due to N. Jacobson of the Galois theory (see [IAT] 6.3) to field automorphism groups, the Krull topology of Aut(F(V )/F) is defined by a system of open neighborhoods of the identity, which is made up of the stabilizers of subfields of F(V ) finitely generated over F. For an open compact subgroups ) give a fundamental system of open neighborhoods of the identity of Aut(F(V )/F) under the Krull topology.In other words, the scheme theoretic automorphism group Aut By the above description of the stabilizer of V , the image of G 1 (A (Σ) ) in the scheme automorphism group Aut(Sh ) by the center of G 1 (Z (Σ) )).We take a sufficiently small open compact subgroup S of G(A (Σ) ).We write , we hereafter identify the two groups.
Let m be the maximal ideal of W and we write κ defined over the finite field κ, the Galois group Gal(F/κ) acts on the underlying topological space of Sh (Σ) /S.Since π 0 ((Sh (Σ) /S) /Q ) is finite, π 0 ((Sh (Σ) /S) /F ) is finite, and we have therefore a finite extension F q of the prime field F p such that Gal(F/F q ) gives the stabilizer of V S in π 0 ((Sh (Σ) /S) /F ).We may assume that (as varieties) U S and Uσ S are defined over F q and that U S × Fq F and Uσ S × Fq F are irreducible.Since σ ∈ Hom(U S , Uσ S ), the Galois group Gal(F/F q ) acts on σ by conjugation.By further extending F q if necessary, we may assume that σ is fixed by Gal(F/F q ), x ∈ U S (F q ) and σ(x) ∈ Uσ S (F q ).Thus σ descends to an isomorphism σ S : U S ∼ = Uσ S defined over F q .
Let X S/Fq → U S/Fq be the universal abelian scheme with the origin 0. We write (X x , 0 x ) for the fiber of (X S , 0) over x and fix a geometric point x ∈ V (F) above x.The prime-to-p part π (p) 1 (X x , 0 x ) of π 1 (X x , 0 x ) is canonically isomorphic to the prime to-p part T (p) (X x/F ) of the Tate module T (X x/F ), and the p-part of π 1 (X x , 0 x ) is the discrete p-adic Tate module of X x/F which is the inverse limit of the reduced part of X x [p n ](F) (e.g.[ABV] page 171).We can make the quotient π {p} 1 (X S/F , 0 x ) by the image of the p-part of π 1 (X x , 0 x ).Then we have the following exact sequence ([SGA] 1.X.1.4): This sequence is split exact, because of the zero section 0 : U S → X S .The multiplication by N : X → X (for N prime to p) is an irreducible étale covering, and we conclude that T , and we get a split short exact sequence: By this exact sequence, π 1 (U S/Fq , x) acts by conjugation on T (Σ) (X x ).Recall that we have chosen S sufficiently small so that V V S is étale.We have a canonical surjection π 1 (U K/Fq , x) Gal(U/U S ).We write S V = Gal(U/U S ), which is an extension of S V by Gal(F/F q ) generated by the Frobenius automorphism over F q .Since X x [N ] for all integers N outside Σ gets trivialized over U , the action of π 1 (U S/Fq , x) on T (Σ) (X x ) factors through π 1 (U S/Fq , x) S V .
We now have another split exact sequence: Again the action of We fix a path in Uσ S from σ(x) to x and lift it to a path from σ(0 x ) to 0 x in Xσ S , which induces canonical isomorphisms ([SGA] V.7): The isomorphism ι σ in turn induces an isomorphism ι : We want to have the following commutative diagram and we will find homomorphisms of topological groups fitting into the spot indicated by "?".In other words, we ask if we can find a linear endomorphism ) is a singleton made of the zero-map (taking the entire G 1 (Z ) to the identity of G 1 (Z )) if two primes and are large and distinct (see Section 3 (S3) for a proof of this fact), s → σ s sends S 1, into σ S 1, for almost all primes , where If we shrink S further if necessary for exceptional finitely many primes, we achieve that S is -profinite for exceptional and the logarithm log : is an -adically continuous isomorphism.Then by a result of Lazard [GAN] where Tr : C → M is the reduced trace map.Extending scalar to a finite Galois extension of K/Q , Lie(S ) ⊗ Z K becomes split semi-simple over K, and therefore [σ] is induced by an element of P G(K) (the Lie algebra version of (S2) in the following section), which implies by Galois descent that [σ] is induced by an element of P G(Q ).Thus for all ∈ Σ, s → σ s sends S 1, into σ S 1, and that the isomorphism: s → σ s is induced by an element L of the group fitting into the middle term of the exact sequence (1.3): Though L may depends on the choice of the path from x to σ(x), the isomorphism g(σ) (modulo the centralizer of S 1 ) is independent of the choice of the path; so, we will forget about the path hereafter.Applying this argument to Σ = {p, ∞}, we have the following commutative diagram has the projection π(g Consider the relative Frobenius map π S : U S → U S over F q .Since σ : U S ∼ = Uσ S is defined over F q by our choice, σ satisfies σ S • π S = πσ S • σ S .If X → U S is an étale irreducible covering, X × US ,πS U S → U S is étale irreducible, and π S : U S → U S induces an endomorphism π S, * : π 1 (U S , x) → π 1 (U S , x).We have a diagram: where π x is the relative Frobenius endomorphism of X x over F q .The middle horizontal three squares of the above diagram are commutative, because (π x π S, * ) is induced by the relative Frobenius endomorphism of X S/Fq .The top and the bottom three squares are commutative by construction; so, the entire diagram is commutative.In short, we have the following commutative diagram: Since π σ(x) also gives a similar commutative diagram: we find out that g(σ)π x g(σ) −1 π −1 σ(x) commutes with the action of σ S 1 , and hence it is in the center of Aut R (T (Σ) (X σ(x) )).In other words, g(σ)π x g(σ) −1 = zπ σ(x) for z ∈ ( R (Σ) ) × .Taking the determinant with respect to g T (X σ(x) ) for the rank g = rank R T (X σ(x) ) with a prime ∈ Σ, we find that det(π x ) = z g det(π σ(x) ).Since det(π x ) = N (π x ) r with a positive integer r for the reduced norm map N : B → M , we find that det(π x ) = det(π σ(x) ), and hence z is a g-th root of unity in ( R (Σ) ) × (purity of the Weil number π x ).Then g(σ) ∈ Hom(T (p) X x , T (p) X σ(x) ) satisfies g(σ) • π g x = π g σ(x) • g(σ), and hence g(σ) is an isogeny of Gal(F/F q g )-modules.Then by a result of Tate ([T]), Hom Gal(F/F q g ) (T (Σ) X x , T (Σ) X σ(x) ) = Hom(X x/F q g , X σ(x)/F q g ) ⊗ Z A (Σ) , we find that X x and X σ(x) are isogenous over F q g .We have a canonical projection Aut top group (G (M ) which will be written as σ B = π(g 3); so, its ) is given by σ B .By the proof of Proposition 2.1, g(σ) is induced by ξ ∈ Hom OB (X x , X σ(x) ) modulo Z(Z (Σ) )S.Choose a rational prime q outside Σ.We have and that the conjugation by ξ sends B q ∩ (End(X x ) ⊗ Z Q) = B into itself.Since the image of the conjugation by g(σ) in Out Qq-alg (B q ) and the image of the conjugation by Corollary 2.3.For the generic point η of V S , X η and X σ(η) are isogenous.In particular, if σ B is the identity in Out Q-alg (B, ρ), we find a S ∈ G(A (Σ) ) inducing σ on F(V S ) for all sufficiently small open compact subgroups S of G(A (Σ) ).
Proof.We choose S sufficiently small as in the proof of Proposition 2.1.We replace q in the proof of Proposition 2.1 by q g at the end of the proof in order to simplify the symbols.Suppose that σ S induces U S ∼ = Uσ S for an open dense subscheme U S ⊂ V S .Again we use the exact sequence:0 → T (Σ) X η → π Σ 1 (X /Fq , 0 η ) → π 1 (U S/Fq , η) → 1.By the same argument as above, we find g η (σ) ∈ Hom π1(U S/F ,η) (T (Σ) X η , T (Σ) X σ(η) ).Since X η [ ∞ ] gets trivialized over U for a prime ∈ Σ, fixing a path from η to x for a closed point x ∈ U S (F q ) and taking its image from σ(η) to σ(x), we may identify π(U S/Fq , x) (resp.T (Σ) X x and T (Σ) X σ(x) ) with the Galois group Gal(F( U )/F q (U S )) for the universal covering U (resp. with the generic Tate modules T (Σ) X η and T (Σ) X σ(η) ).By the universality, σ : U ∼ = σ U extends to σ : U ∼ = σ U .Writing D x for the decomposition group of the closed point x ∈ U S (F q ), the points x : Spec(F q ) → U S and σ(x) : Spec(F q ) → Uσ S induce isomorphisms D x ∼ = Gal(F/F q ) ∼ = D σ(x) = σD x σ −1 (choosing the extension σ suitably) and splittings: Then by a result of Zarhin (see [RPT] Chapter VI, [Z] and also [ARG] Chapter II), X η/Fq(VS ) and X σ(η)/Fq(Vσ S ) are isogenous.Here we note that the field F q (V S ) = F q (U S ) is finitely generated over F p (which has to be the case in order to apply Zarhin's result).Thus we can find an isogeny α η : X η → X σ(η) , which extends to an isogeny X S → σ * Xσ S = Xσ S × Uσ S ,σ U S over U S .We write α : X S → Xσ S for the composite of the above isogeny with the projection Xσ S × Uσ S ,σ U S → Xσ S .

Documenta Mathematica 11 (2006) 25-56
We then have the commutative diagram: Assume that σ B = 1.Then α is B-linear.Suppose we have another B-linear isogeny α : X S → σ * Xσ S inducing g η (σ).Then α −1 α commutes with the action of Gal(F( U )/F(U S )) and hence with the action of S. Thus we find ).This implies B-linear ξ commutes with the action of C, and hence in the center of Σ )S.We consider the commutative diagram similar to (2.3): (2.4) The prime-to-Σ component of g −1 σ eventually gives a S in the corollary.By the above fact, g Note that σ * (Xσ S , λσ S , iσ S , φ σ ) /US is a quadruple classified by Sh /S (proven under (h2-4)), we have a morphism τ : U S → U S ⊂ Sh (Σ) /S with a prime-to-Σ and B-linear isogeny β : σ * Xσ S → τ * X S over U S .Identifying Gal(U/U S ) with a subgroup 2).Thus the effect of τ (and βα η ) on T (Σ) (X η ) commutes with the action of S V , and the action of B-linear βα η on the Tate module T (Σ) (X η ) commutes with the action of S V .Therefore it is in the center Z(Q).Thus the isogeny α between σ * (Xσ S ) and X S can be chosen (after modification by a central element) to be a prime-to-Σ isogeny.This τ could be non-trivial without the three assumptions (h2-4), and if this is the case, the action of τ is induced by an element of G(Q Σ ).Under (h2-4), τ is determined by its effect on T (Σ) (X η ) and is the identity map (see the following two paragraphs), and we may assume that α is a prime-to-Σ isogeny (after modifying by an element of Z(Q)).Thus g We add here a few words on this point related to the universality of Sh (Σ) .Without the assumptions (h2-4), the effect of σ on the restriction of each Documenta Mathematica 11 ( 2006) 25-56 level structure φ (p) to L A (Σ) may not be sufficient to uniquely determine σ.In other words, in the definition of our moduli problem, we indeed have the datum of φ Σ ), which we cannot forget.To clarify this, take a (characteristic 0) geometrically connected component V 0 of Sh /E whose image /E = Sh/G(Z Σ ) giving rise to V /F after extending scalar to W and then taking reduction modulo p.By the description of π 0 (Sh /Q ) at the beginning of the proof of Proposition 2.1, the stabilizer in Inside this group, every element g ∈ G(A (∞) ) inducing an automorphism of V (Σ) 0 has its -component g for a prime ∈ Σ in the normalizer of G(Z ).Under (h2-4), as is well known, the normalizer of [Ko] Lemma 7.2, which is one of the key points of the proof of the universality of Sh (Σ) ).By (h2-4), G(Q ) is quasi-split over Q , and we have the Iwasawa decomposition G(Q ) = P 0 (Q )G(Z ) for ∈ Σ with a minimal parabolic subgroup P 0 of G, from which we can easily prove that the normalizer of G(Z ) is Z(Q )G (Z ).An elementary proof of the Iwasawa decomposition (for a unitary group or a symplectic group acting on M r keeping a skew-hermitian form relative to M /F ) can be found in [EPE] Section 5, particularly pages 36-37.By (h3-4), G(Q ) is isomorphic to a unitary or symplectic group acting on M n = εL Q for an idempotent ε (for example, ε = diag[1, 0, . . ., 0] fixed by ρ) of O B ∼ = M n (R ) with respect to the skew-hermitian form on εL Q induced by •, • ; so, the result in [EPE] Section 5 applies to our case.
Suppose that g ∈ G(A (∞) ) preserves the quotient V (Σ) 0 of V 0 .If Σ is finite, we can therefore choose ξ ∈ Z(Q) so that (ξg) is in G(Z ) for all ∈ Σ, and the action of (ξg σ ) −1 even if Σ is infinite.This fact can be also shown in a group theoretic way as in the case of finite Σ: Modifying g by an element in G(Z Σ ), we may assume that g ∈ Z(Q ) for all ∈ Σ. Taking an increasing sequence of finite sets Σ i so that Σ = i Σ i and choosing ξ i ∈ Z(Q) so that the action of (ξ i g) (Σi) induces the action of g on V is identical to that of g.We write F i for the closed subset of elements in G(Z Σi )G(A (Σi) ) whose action on V (Σ) 0 is identical to that of g.In the locally compact group G(A (∞) ), the filter {F i } i has a nontrivial intersection i F i = ∅.Thus the action of g on V (Σ) 0 is represented by an element in G(A (Σ) ).In other words, an element of G(A (∞) ) in the stabilizer of the connected component V Writing the prime-to-Σ level structures of X η and X σ(η) as φ (Σ) η and φ (Σ) σ(η) , respectively, we now find that α η • φ Since the effect of σ on T (Σ) (X η ) determines σ, we have a −1 S (σ(η)) = η, which implies that a S = σ on the Zariski open dense subset U S of V S , and hence, they are equal on the entire V S .
By the smoothness of Sh (Σ) over W, Zariski's connectedness theorem (combined with the existence of a projective compactification normal over W), we have a bijection π 0 (Sh /Q ) as described at the beginning of the proof of Proposition 2.1.Since our group G has cohomologically trivial center (cf., [MS] 4.12), the stabilizer of has a simple expression given by the subgroup G V in the theorem (see [D1] 2. 1.6,2.1.16,2.6.3 and [MS] Theorem 4.13), and the above corollaries finish the proof of the theorem because σ on V is then induced by a = lim S→1 a S in G(A (Σ) )

Z(Z (Σ) )
. Since /F ), we conclude a ∈ G V .The description of the stabilizer of V in the theorem necessitates the strong approximation theorem (which follows from noncompactness of G 1 (R) combined with simply connectedness of G 1 : [Kn]).

Automorphism groups of quasi-split classical groups
In the above proof of the theorem, we have used the following facts: (S2) For a prime at which G 1/Z is smooth quasi-split (so for a sufficiently large rational prime ), we have ) For sufficiently large distinct rational primes p and , any group homomorphism φ : The assertion (S1) follows directly from a result of Lazard on -adic Lie groups (see [GAN] IV.3.2.6), because the automorphism of the Lie algebra of S (hence of S ) are all inner up to the automorphism of the field (in our case).The assertion (S2) for finite fields is an old theorem of Steinberg (see [St] 3.2), and as remarked in [CST] in the comments (in page 587) on [St], (S2) for general infinite fields follows from a very general result in [BT] 8.14.Since the paper [BT] is a long paper and treats only algebraic groups over an infinite field (not over a valuation ring like Z ), for the reader's convenience, we will give a self-contained proof of (S2) restricting ourselves to unitary groups and symplectic groups.
Since the assertions (S3) concerns only sufficiently large primes, we may always assume (QS) G 1 (Z p ) and G 1 (Z ) are quasi-split.
We now prove the assertion (S3).Let φ : G 1 (Z p ) → G 1 (Z ) be a homomorphism.Since G 1 (Z p ) is quasi split, G 1 (Z p ) is generated by unipotent elements (see Proposition 3.1), and its unipotent radical U is generated by an additive subgroup U α corresponding to a simple root α.
If G 1 = SL(n) /Z , for example, we may assume that U α is made of diagonal matrices with u ∈ Z p for an index j (with 1 ≤ j ≤ n), where 1 j is the j × j identity matrix.
In general, U and U α are p-profinite.We consider the normalizer N (U α ) and the centralizer where The quotient N/Z in this case is isomorphic to the subgroup of permutation matrices preserving Z.
)) grows at least on the order of p as p grows.In the above example of for large enough p is generated by U α for all simple roots α, φ has to be trivial for p large enough.
Since we assume to have the strong approximation theorem, we need to assume that G 1 is simply connected; so, we may restrict ourselves to symplectic and unitary groups (those groups of types A and C).We shall give a detailed exposition of how to prove (S2) for general linear groups and split symplectic groups and give a sketch for quasi split unitary groups.
Write χ : G → Z G = G/G 1 for the projection map for the cocenter Z G .In the following subsection, we assume that the base field K is either a number field or a nonarchimedean field of characteristic 0 (often a p-adic field).When K is nonarchimedean, we suppose that the classical group G is defined over Z p , and if K is a number field, G is defined over Q.We write O for the maximal compact ring of K if K is nonarchimedean (so, O is the p-adic integer ring if K is p-adic).We equip the natural locally compact topology (resp.the discrete topology) on G(A) for A = K or O if K is a local field (resp.a number field).Then we define, for A = K and O, Aut χ (G(A)) by the group of continuous automorphisms of the group G(A) which preserve χ up to automorphisms induced on Z G by the field automorphisms of K.
For a subgroup H with G 1 (A) ⊂ H ⊂ G(A) and a section s of χ : H → Z G (A), we write Aut s (H) for the group of continuous automorphisms of H preserving s up to field automorphisms and inner automorphisms (in [PAF] 4.4.3, the symbol Aut det (GL 2 (A)) means the group Aut s here for a section s of det : Documenta Mathematica 11 (2006) 25-56

General linear groups
Let L j ⊂ P n−1 be the hyperplane of the projective space P n−1 (K) defined by the vanishing of the j-th homogeneous coordinate x j .We start with the following well known fact: Proposition 3.1.Let P be the maximal parabolic subgroup of GL(n) fixing the infinity hyperplane L n of P n−1 .For an infinite field K, SL n (K) is generated by conjugates of U P (K), where U P is the unipotent radical of P .
Proof.Let H be a subgroup generated by all conjugates of U P .Thus H is a normal subgroup of SL n (K).Since SL n (K) is almost simple, we find that For an open compact subgroup S of SL n (K) for a p-adic local field K, the unipotent radical U of a Borel subgroup B 1 of SL n (K) and S generate SL n (K).Similarly S and a Borel subgroup B of GL n (K) generate GL n (K).
Proof.We may assume that U is upper triangular.Thus U ⊃ U P for the maximal parabolic subgroup P in Proposition 3.1.We consider the subgroup H generated by S and U P .The group U P acts transitively on the affine space A n−1 (K) = P n−1 (K) − L n .For any g ∈ S − B 1 for the upper triangular Borel subgroup B 1 , gU g −1 acts transitively on P n−1 (K) − g(L n ).Note that g∈S g(L n ) is empty, because intersection of n transversal hyperplanes is empty.Thus we find that H acts transitively on P n−1 (K).Since P n−1 (K) is in bijection with the set of all unipotent subgroups conjugate to U P in SL n (K), H contains all conjugates of U P in SL n (K); so, H = SL n (K) by Proposition 3.1.From this, generation of GL n (K) by B and S is clear because In this case of GL n , we have χ = det and Z G = G m .Thus, for A = K or O, Aut det (GL n (A)) is the automorphism group of the group GL n (A) preserving the determinant map up to field automorphisms of K, that is, σ ∈ Aut det (GL n (A)) satisfies det(σ(g)) = τ (det(g)) for a field automorphism τ ∈ Aut(K).More generally, for a subgroup H ⊂ GL n (A) containing SL n (A), we define Aut det (H) for the automorphism group of H preserving det : H → A × up to field automorphisms of K. Fixing a section s : for some g ∈ GL n (A) and τ ∈ Aut(K).Similarly, we define Aut s (H) for a section s of the determinant map det : H → A × .We write Z(SL n (A)) for the center of SL n (A), which is the finite group µ n (A) of n-th roots of unity.
We now prove (S2) for SL n (A): Proposition 3.3.If A = K, assume that K is either a local field of characteristic 0 or a number field.If A = O, assume that K is a local field of characteristic 0. Then we have 1.The continuous automorphism groups Aut(P GL n (A)), Aut(P SL n (A)) and Aut(SL n (A)) are all canonically isomorphic to where where τ g indicates the projection of g to Aut(K).
3. We have a canonical split exact sequence In other words, for σ ∈ Aut det (GL n (A)), there exists g ∈ If K is a local field, we put the natural locally compact topology on the group, and if K is a global field, we put the discrete topology on the group.We shall give a computational proof for GL n , because it describes well the mechanism of how an automorphism is determined entry by entry (of the matrices involved).
Proof.We first deal with the case where A is the field K.We first study P GL n (K).We have an exact sequence: where i(x)(g) = xgx −1 .We write B (resp.U ) for the upper triangular Borel subgroup (resp.the upper triangular unipotent subgroup) of GL n (K).Their image in P GL n (K) will be denoted by B and U .
Let A be a subgroup of GL n (K) isomorphic to the additive group K; so, we have an isomorphism a : K ∼ = A. Consider the image a(1) of 1 ∈ K in A. Replacing K by a finite extension containing an eigenvalue α of a(1), let V α ⊂ K n be the eigenspace of a(1) with eigenvalue α.Then a( 1 m ) acts on V α and a( 1m ) m = a(1) = α ∈ End(V α ).Thus we have an algebra homomorphism: Thus if K is a non-archimedean local field or a number field, we find that α has to be 1.Thus A is made up of commuting unipotent elements; so, by conjugation, we can embed A into U .
Since U ∼ = U is generated by unipotent subgroups isomorphic to K, by the above argument, σ(U ) for σ ∈ Aut(P GL n (K)) is again a unipotent subgroup of P GL n (K).Since B is the normalizer of U , again σ(B) is the normalizer of σ(U ); so, σ(B) is a Borel subgroup.We find g ∈ GL n (K) such that σ(B) = gBg −1 .Thus we may assume that σ fixes B. Applying the same argument to U , we may assume that σ fixes U .Since we have a unique filtration: , σ preserves this filtration.We fix an isomorphism a j : K n−j → U j given by where E i,j is the matrix having non-zero entry 1 only at the (i, j)-spot.Then Each subquotient U j /U j+1 is a K-vector space and is a direct sum of onedimensional eigenspaces under the conjugate action of T := B/U .Define an isomorphism t : (K × ) n /K × ∼ = T by t(α 1 , . . ., α n ) = diag[α 1 , . . ., α n ], and we write α j (t) = α j if t = diag[α 1 , . . ., α n ].Then U ij ⊂ U (j > i) generated by u ij = 1 + E i,j is the eigen-subgroup (isomorphic to one dimensional vector space over K) on which t ∈ T acts via the multiplication by χ ij (t) = α i α −1 j (t).The automorphism σ also induces an automorphism σ of T = B/U .Thus σ permutes the eigen-subgroups Then σ induces a k-linear automorphism on U j /U j+1 for all j.We first assume Further by conjugating σ by an element in U , we may assume that σ(a 1 (1)) = a 1 (1).Thus σ(a 1 (r • 1)) = a 1 (r • 1) for all r ∈ Q.By taking commutators of a 1 (r • 1), we have a nontrivial element in U j /U j+1 fixed by σ for all j.In particular, σ fixes U n−1 ∼ = k and hence fixes the character n , we conclude that σ either interchanges the two eigenspaces or fixes each.If σ interchange the two, replacing σ by σ • J for the automorphism J = J n of GL n (K) given by J σ has to fix all eigen-subgroups U ij of U .Since tu ij t −1 = χ ij (t)u ij and σ commutes with the multiplication by χ ij (t) ∈ k on U ij , we find by the . Thus σ acts trivially on T .Since a 1 (1) has non-trivial projection to all T -eigenspaces in U 1 /U 2 , we conclude that σ(u i,i+1 ) = u i,i+1 .Then U is fixed by σ again by the commutator relation [u ij , u jk ] = u ik if i < j < k.Thus, modifying σ further by an inner automorphism and J, we may assume that σ fixes B element-by-element.Now we assume that K k.Modifying σ as above by composing an inner automorphism and the action of J if necessary, we assume that σ preserves the eigen subgroups U ij for all i < j.We are going to show that This field automorphism σ = σ ij does not depend on (i, j) by the commutator relation [u ij , u jk ] = u ik for all i < j < k.Thus modifying σ further by an element of Aut(K), we may assume that σ fixes B.
We are going to prove that σ inducing the identity map on B is the identity on the entire group.For the moment, we suppose that K is p-adic.Then by [GAN] IV.3.2.6, for a sufficiently small open compact subgroup S ⊂ P GL n (K), σ : S ∼ = σ(S) induces an automorphism Φ σ of the Lie algebra G Qp of P GL n (K) over Q p .Since dim Qp G Qp = dim Qp G K for the Lie algebra G K of P GL n (K) over K, we find that Aut K (G K ) ⊂ Aut Qp G Qp has the same dimension over Q p as a Lie group over Q p (cf. [BLI] VIII.5.5).Thus Φ σ ∈ G K is induced by g ∈ GL n (K) through the adjoint action (cf.[BLI] VIII.13).Since σ fixes B, we find that g commutes with B and, hence, g is in the center.Therefore, shrinking S further if necessary, we conclude σ = 1 on S and on B. Since B and S generate P GL n (K) (see Proposition 3.2), we find σ is the identity map over entire P GL n (K).This shows that, under the condition that n ≥ 3, if K is a local p-adic field.If n = 2, we need to remove the factor J from the above formula.If K = R or C, the above fact is well known (see [BLI] III.10.2).
Suppose now that K is a number field.Write O for the integer ring of K. Take a prime p such that O p ∼ = Z p .Since σ fixes B, for the diagonal torus T , σ fixes its normalizer N (T ).Since N (T ) = W T , we find that σ(w) = tw for an element t ∈ T .Since P GL n (K) = w∈W BwB, we find that σ is continuous with respect to the p-adic topology.Thus σ induces Aut(P GL n (K p )) fixing B, and we find that σ = 1, which shows again Out(P GL n (K)) ∼ = (Aut(K) × J ) Documenta Mathematica 11 (2006) 25-56 and Aut(P GL n (K)) = (Aut(K) × J ) P GL n (K) for a number field K.
We can apply the same argument to Aut(P SL n (K)) and Aut(SL n (K)).Modifying σ by inner automorphisms, J and an element in Aut(K), we may assume that σ leaves B 1 fixed.Then by the same argument as above, we conclude σ = 1 and hence we find that, if n > 2, If n = 2, again we need to remove the factor J from the above formulas.
Now we look at Aut s (H) for a section s of det : H → K × .Since σ ∈ Aut det (H) preserves the section s up to field and inner automorphisms, modifying σ by such an automorphism, we may assume that σ fixes Im(s).Then σ is determined by its restriction to SL n (K) ⊂ H and, hence, comes from and element in Aut(K) P GL 2 (K) preserving H.
To see the last assertion (3) for A = K, we consider the restriction map Since Aut(SL n (K)) acts naturally on GL n (K) by the result already proven, the homomorphism Res is surjective.Take σ ∈ Ker(Res), and fix a section s : For any g ∈ GL n (K), we can write uniquely g = s(det(g))u with u ∈ SL n (K).For a homomorphism ζ : gives an endomorphism of GL n (K).It is an automorphism because σ induces the identity on SL n (K) and K × = det(GL n (K)).Thus we get the desired exact sequence.
Let M = K ⊕K be a semi-simple algebra with involution c(x, y) = (y, x).Then we can realize SL n (K) as a special unitary group with respect to the hermitian form (u, v) = Tr( t u c w 0 v) (u, v ∈ M n ): Indeed SL n (K) ∼ = G 1 (K) by x → (x, J(x)).Then we have Aut(M ) ∼ = Aut(K) × J , and the results in Propositions 3.3 for A = O and K can be restated as Aut(G 1 (A)) = Aut(M ) P G(A) for the unitary group G with respect to (•, •).We used in the proof of the theorem this version of the result in this section when K is a completion of the totally real field F at a prime l splitting in the CM field M ; so, M l = K l ⊕ K l and c is induced by complex conjugation c of M .

Symplectic groups
We start with a general fact valid for quasi-split almost simple connected groups G 1 not necessarily a symplectic group.
Proposition 3.4.Let K be a p-adic local field.Let S be an open subgroup of G 1 (K) of a classical almost simple connected group G 1 quasi-split over K. Let P 0 be a minimal parabolic subgroup of G 1 defined over K with unipotent radical U .Then S and U (K) generate G 1 (K).
The proof is similar to that of Proposition 3.2.Here is a sketch.Taking the universal covering of G 1 , we may assume that G 1 is simply connected and is given by a Chevalley group G 1 inside GL(n) for an appropriate n defined over O. Thus G 1 (K) is almost simple.We may assume that P 0 = B ∩ G 1 for the upper-triangular Borel subgroup B of GL(n).Thus G 1 acts on the projective space P n−1 through the embedding G 1 ⊂ GL(n).Take the stabilizer P ⊂ G 1 of the infinity hyperplane L n of P n−1 .Then P is a maximal parabolic subgroup of G 1 containing P 0 .Since G 1 (K) is almost simple, G 1 (K) is generated by conjugates of the unipotent radical U P (K) of P .The flag variety P = G 1 /P is an irreducible closed subscheme of P n−1 , and U P (K) acts transitively on P − L n .Since P is covered by finitely many affine open subschemes of the form P−g(L n ) (on which gU P g −1 acts transitively), the subgroup H generated by S and U P (K) acts transitively on P and hence contains all conjugates of U P (K).This shows that G 1 (K) is generated by S and U P (K).
Let I n be the antidiagonal for n = 2g is an anti-diagonal alternating matrix.In this subsection, we deal with the split symplectic group defined over Q given by and G 1 = Sp 2g = Ker(ν).We write Z for the center of GSp 2g .We write B for the upper triangular Borel subgroup of GSp 2g .We write U for the unipotent radical of B. For the diagonal torus T , we have B = T U , and B is the normalizer of U (K) in GSp 2g (K) for a field extension K of Q.We take a standard parabolic subgroup P ⊃ B of GSp 2g (K) with unipotent radical U P contained in U .
In this symplectic case, χ : G → Z G is the similitude map ν : GSp 2n → G m ; so, we have Aut ν (GSp 2n (A)) and Aut s (H) for a subgroup H with Sp 2n (A) ⊂ H ⊂ GSp 2n (A) and a section s of ν : H → K × .Here A = K or O.
Proposition 3.5.Let K be a local or global field of characteristic 0. Then we have We describe here a shorter argument proving the assertion (1) for GSp 2g (than the computational one for GL(n)) using the theory of root systems (although this is just an interpretation of the computational argument in terms of a slightly more sophisticated language).The assertions ( 2) and (3) follow from the assertion (1) by the same argument as in the case of GL n (K).
Proof.By [BLI] III.10.2, we may assume that K is either p-adic local or a number field.Write simply B = B(K), U = U (K) and T = T (K).Let σ ∈ Aut(G(K)).In the same manner as in the case of GL(n), we verify that σ sends unipotent elements to unipotent elements.Write N = log(U ) which is a maximal nilpotent subalgebra of the Lie algebra G of Sp 2g (K).Suppose K = k.Then σ is K-linear; in particular, σ induces a permutation of roots which has to give rise to a K-linear automorphism of the Lie algebra G.
Modifying σ by the action of Weyl group (conjugation by a permutation matrix), we find that the permutation has to be trivial or an outer automorphism of the Dynkin diagram of Sp 2g (e.g.[Tt] 3.4.2or [BLI] VIII.13).Since the Dynkin diagram of Sp 2g does not have any non-trivial automorphism, we find that the permutation is the identity map.Since on N α , T acts by a character α : T → K × , we find that α(t) = α(σ(t)); so, σ is also the identity map.
We now assume that K = k.For the set of simple roots ∆ of T with respect to N, α∈∆ N α → N induces an isomorphism α∈∆ N α ∼ = N/[N, N].In other words, {N α |α ∈ ∆} generates N over K.The K-vector space structure of N induces an embedding i ), the subalgebra A 1 generated by i 1 (K) and the action of T is a maximal commutative k-subalgebra.The subtorus T 0 given by the connected component of Thus modifying by τ , we may assume that σ : N → N is K-linear.Then σ induces a permutation of roots which has to give rise to an automorphism of the Lie algebra of Sp 2g .Then by the same argument as in the case where K = k, we conclude that σ induces identity map on B/U and U .Taking T to be diagonal, we may assume that σ(T ) = u σ T u −1 σ for u σ ∈ U .Thus by modifying σ by the inner automorphism of u σ , we may assume that σ is the identity on B.
Suppose that K is p-adic, then σ sends an open compact subgroup S to σ(S), which induces an endomorphism of the Lie algebra of Sp 2g (O) for the p-adic integer ring O and induces the identity map on the Lie algebra of B; in particular, σ is a O-linear map on the Lie algebra.Since an automorphism of the Lie algebra is inner induced by conjugation by an element g ∈ Sp 2g (K), we have gbg −1 = b for b ∈ B. Since the centralizer of B is the center of G, we find that σ is the identity on S. Since S and U generate G 1 = Sp 2g , we find that σ is the identity over G.This proves the desired result for G 1 and p-adic fields K.
We can proceed in exactly the same way as in the case of GL(n) when K is a number field and conclude the result.
We then get the following integral analogue in a manner similar to Proposition 3.  where c is the generator of Gal(M/K), ν : G → K × is the similitude map, I n = w 0 if n is odd and I 2m = 1m 0 0 −1m w 0 if n = 2m is even.We may assume that n ≥ 3 because in the case of n = 2, we have P G ∼ = P GL 2 (so, the desired result in this case has been proven already in 3.1).
We write G 1 for the derived group of G. Thus Let us fill in the proof of the assertion (1) with some more details, assuming first for simplicity that n = 3.In this case, by computation, we have U (K) = u(x, y) = By the above argument in the general case, we may assume that σ(B) = B for σ ∈ Aut µ (G(K)).Then we have σ(u(1, y)) = u(a, y ) for a ∈ M × and t(a, 1) −1 σ(u(1, y))t(a, 1) = u(1, y ) for some y , y ∈ M .Thus modifying σ by an inner automorphism of an element in T (K) and identifying N/[N, N] with M by u(x, * ) mod [N, N] → x, we find that σ induces an automorphism of the field M = N/[N, N] and the same automorphism on T (K) = B(K)/U (K) coordinate-wise.Thus again modifying σ by an element in Aut(K) and by an inner automorphism of an element of U (K), we may assume that σ induces the identity map on B. We then conclude that σ is the identity map on G(K) by the same argument as in the case of GL(n) and GSp(2n).
Next, we suppose that n = 4. Again by computation, we have U (K) = u(w, y, x, z) = (Σ)+ = F × + ∩ O (Σ) .We have an Documenta Mathematica 11 (2006) 25-56 exact sequence 1 → B × /M × → Aut alg (B) → Out(B) → 1, and by a theorem of Skolem-Noether, Out(B) ⊂ Aut(M ).Here b ∈ B × acts on B by x → bxb −1 .Since B is central simple, any simple B-module N is isomorphic each other.Take one such simple B-module.Then End B (N ) is a division algebra D • .Taking a base of N over D • and identifying N ∼ = (D • ) r , we have B = End D • (N ) ∼ = M r (D) for the opposite algebra D of D • .Letting Aut alg (D) act on b ∈ M r (D) entry-by-entry, we have Aut alg (D) ⊂ Aut alg (B), and Out(D) = Out(B) under this isomorphism.
is a class of polarizations λ up to scalar multiplication by i(O × (Σ)+ ) which induces the Riemann form •, • on L up to scalar multiplication by O × (Σ)+ .There is one more condition (cf.[Ko] Section 5 or [PAF] 7.1.1(det)) specifying the module structure of Ω A/T over O B ⊗ Z O T (which we do not recall).
(S) For an open compact subgroups S, S ⊂ G 1 (A (Σ) ), if σ : S ∼ = S is an isomorphism of groups, replacing S by an open subgroup and replacing S accordingly by the image of σ, σ is induced by the conjugation by an element g(σ) ∈ G(A (Σ) ) as in (1.2).We may modify σ by g ∈ G(A (Σ) ) so that σ B = 1.Then this assertion (S) follows from the following three assertions for σ with σ B = 1: (S1) For open subgroups S and S of G 1 (Q ) (for every prime ), an isomorphism σ : S ∼ = S is induced by conjugation s → g (σ)sg (σ) −1 for g (σ) ∈ G(Q ) after replacing S by an open subgroup of S and replacing S by the image of the new S ; Documenta Mathematica 11 (2006) 25-56 of the determinant map.Then for x ∈ SL n (K), we have s(a)xs(a) −1 = σ(s(a)xs(a) −1 ) = σ(s(a))xσ(s(a)) −1 , because Res(σ) is the identity map.Thus σ(s(a))s(a) −1 commutes with SL n (K).Taking the determinant of σ(s(a))s(a) −1 , we find that σ(s(a))s(a) −1 ∈ Z(SL n (K)) and a → ζ(a) = σ(s(a))s(a) −1 is a homomorphism of the group K × into Z(SL 2 (K)).
now assume A = O.Since the argument is the same as in the case of the field K, we only indicates some essential points.Let U (O) = U ∩SL n (O) for the subgroup U of upper unipotent matrices.Since P n−1 (O) = P n−1 (K), all Borel subgroups of SL n (O) are conjugate each other.Since B 1 (O) = SL n (O)∩B is a semi-direct product of T 1 (O) and U (O), all unipotent subgroups are conjugate each other.By the same argument in the case of the field, we may assume that σ ∈ Aut(P GL n (O)) leaves U (O) stable.Writing b j for (0, . . ., j b j , 0, . . ., 0) ∈ Documenta Mathematica 11 (2006) 25-56O n−1 with b j ∈ O, we have t(α)a 1 (b j )t(α) −1 = a 1 (α j b j α −1 j+1 ) for a 1 : O n−1 ∼ = U (O)/[U (O), U(O)] and t : (O × ) n ∼ = T (O) as in the proof of Proposition 3.3.
is generated by unipotent matrices, we have log(σ(U )) (which we write σ(N)) is a nilpotent subalgebra of G, and dim k σ(N) = dim k (N).Thus σ(N) is a maximal nilpotent subalgebra of G; so, it is a conjugate of N by a ∈ Sp 2g (K).This implies σ(N) = aNa −1 .Conjugating back by a, we may assume that σ(N) = N.Then σ(U ) = U and hence σ(B) = B because B is the normalizer of U .Thus σ induces an automorphism σ of B/U ∼ = T .We have weight spaces N α and N = α N α .From this, we conclude that σ permutes N α : σ(N α ) = N α•σ .
{t ∈ T |α(t) = β(t) for all α, β ∈ ∆} is dimension 1 and acts on N/[N, N] by scalar multiplication.This property characterizes T 0 .Since the fact that T 0 acts by scalar multiplication on N/[N, N] does not change after applying σ, we have σ(T 0 ) = T 0 .The image of i j (K) in End k (N/[N, N]) is generated over k by the action of T 0 ; so, they coincide.Since σ ∈ Aut k (N) is an automorphism of the Lie algebra, the Documenta Mathematica 11 (2006) 25-56 action of σ on N/[N, N] determines the action of σ on N. In particular, we conclude i 1 (K) = i 2 (K), and we can think of τ

G 1 (
K) = {g ∈ G(K)| det(g) = ν(g) = 1}.Define the cocenter Z G = G/G 1 and write µ : G → Z G for the projection.We may identify µ with det ×ν and Z G with its image in Res M/Qp G m ×Res K/Qp G m .We consider Aut µ (G(A)) for A = O and K made up of group automorphisms σ of G(A) satisfying µ • σ = µ.We suppose that the nontrivial automorphism c of M over K is induced by an order 2 automorphism of the Galois closureM gal of M/Q p in the center of Gal(M gal /Q p ). Write G A for the Lie algebra of G 1 (A) for A = K, O, M and R. Since G 1 (M ) ∼ = SL n (M ), by Proposition 3.3, the Lie algebra automorphism group Aut(G M ) is isomorphic to (Aut(M ) × J ) P G(M ).Since G K ⊗ K M = G M , any automorphism of G K extends to an automorphism of G M ; so, Aut K (G K ) ⊂ Aut(G M ),and by this inclusion sends σ ∈ Aut(M ) ⊂ Aut(G K ) to an element (σ, 1) ∈ (Aut(M ) × J ).By this fact, at the level of the Lie algebra, all automorphisms of G A for A = O and K are inner up to automorphism of M , and we have Aut(G A ) = Aut(A) P G(A).We now study the automorphism group of the p-adic Lie group G(K) and G(O).Proposition 3.7.Let A = O or K for a p-adic field K.We assume that p > 2 and K/Q p is unramified if A = O.Then we havethen we can write σ(g) = h σ τ (g)h −1 σ with a unique h σ ∈ P G(A) and τ ∈ Aut(M ) for all x ∈ G 1 (A).Since Aut(P G(A)) = Aut(M ) P G(A), we find σ(g) = ζ(g)h σ τ (g)h −1 σ with ζ(g) ∈ Z(G(A)) for all g ∈ G(A).Applying µ and noting that µ(σ(g)) = τ (µ(g)), we find τ (µ(g)) = µ(ζ(g))τ (µ(g)); so,ζ(g) ∈ Z(G 1 (A)).Since σ is an automorphism, ζ : G(A) → Z(G 1 (A)) is a homomorphism.By our assumption on p, G 1 (A) is the derived group of the topological group G(A), and hence, ζ factors through Z G (A) = G(A)/G 1 (A), since Z(G 1 (A)) is abelian.This shows that the assertion (2) follows from the assertion(1).

∈
GL 3 (M ) xx c + (y + y c ) = 0 .The diagonal torus T (K) ⊂ G is made of t(a, b) = diag[a, b, a −c bb c ] for a ∈ M × and b ∈ M × .Thus writing N = log(U (K)), we have N/[N, N] ∼ = M by u(x, y) → x and N/[N, N] is a one-dimensional vector space over M (so, it is two-dimensional over the field of definition K) on which t(a, b) acts through the multiplication by ab c : t(a, b)u(x, y)t(a, b) −1 = u(ab c x, (ab −1 )(ab −1 ) c y).

∈
GL 4 (M ) y=y c andz c −z=xw c −wx c .The diagonal torus T (K) ⊂ G is made of t(a, b, ν) = diag[aν, bν, b −c , a −c ] for a, b ∈ M × and ν ∈ K × .We have t(a, b, ν)u(w, y, x, z)t(a, b, ν) −1 = u(ab −1 w, bb c νy, ab c νx, aa c νz).Thus writing N = log(U (K)), we have N/[N, N] ∼ = M ⊕ K by u(w, y, x, z) → (w, y) and N/[N, N] is a three-dimensional vector space over K on which t(a, b, ν) acts through (w, y) → (ab −1 w, bb c νy).By the above argument at the level of the Lie algebra, we may assume that σ(B) = B for σ ∈ Aut µ (G(K)) and Here Out Q-alg (C, * ) is the outer automorphism group of C commuting with * ; in other words, it is the quotient of the group of automorphisms of C commuting with * by the group of inner automorphisms commuting with * .Thus we have Out Thus the centralizer (resp.thenormalizer) of φ(U ) is given by Z(Z ) (resp.N (Z )) for a reductive subgroupZ (resp.N ) of G 1 .Then N (Z )/Z(Z ) is a finite subgroup of the Weyl group W 1 of G 1/C which is independent of .For example, if G 1 = SL(n) /Z , and if φ(U α ) is made of diagonal matrices diag[ζ 1 1 m1 , ζ 2 1 m2, . . ., ζ r 1 mr ] for generically distinct ζ j , Z is given by the subgroup