Automorphism Groups of Shimura Varieties

Haruzo Hida

Received: April 14, 2003 Revised: February 5, 2006

Communicated by Don Blasius

Abstract. In this paper, we determine the scheme automorphism group of the reduction modulo p 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.

2000 Mathematics Subject Classification: 11G15, 11G18, 11G25 Keywords and Phrases: Shimura variety, reciprocity law

There are two aspects of the Artin reciprocity law. One is representation theoretic, for example,

Hom_{cont}(Gal(ℚ^{ab}/ℚ), ℂ^{×}) ≅ Hom_{cont}((𝔸^{(∞)})^{×}/ℚ_{+}^{×}, ℂ^{×})

via the identity of L–functions. Another geometric one is:

Gal(ℚ^{ab}/ℚ) ≅ GL_{1}(𝔸^{(∞)})/ℚ_{+}^{×}.

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 ℤ̂^{(p)} ⊂ 𝔸^{×}/ℚ^{×}ℝ_{+}^{×} 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 \frac{GL_{1}(𝔸^{(∞)})}{GL_{1}(ℤ̂^{(p)})ℚ_{+}^{×}}), 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^{(∞)})/\overline{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 \frac{GL_{2}(𝔸^{(∞)})}{GL_{2}(ℤ̂^{(p)})U(ℤ_{p})ℚ_{+}^{×}}, 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.

1 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_{(Σ)+}^{×} = F_{+}^{×} ∩ O_{(Σ)}. We have an 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.

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 ⟨ , ⟩ : L_{ℚ} × L_{ℚ} → F for L_{A} = L ⊗_{ℤ} A 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 C^{∘} = End_{B}(L_{ℚ}). Then C is a central simple algebra and is isomorphic to M_{s}(D), and hence Out(C) ≅ Out(D) = Out(B). We write C_{A} = C ⊗_{ℚ} A, B_{A} = B ⊗_{ℚ} A and F_{A} = F ⊗_{ℚ} A. 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

G(A) = {g ∈ C_{A}|ν(g) := gg^{∗} ∈ (F_{A})^{×}} for ℚ-algebras A  (1.1)

and an extension G̃ of G by the following subgroup of the opposite group Aut_{A}^{∘}(L_{A}) of the A-linear automorphism group Aut_{A}(L_{A}):

G̃(A) = {g ∈ Aut_{A}^{∘}(L_{A})|gC_{A}g^{−1} = C_{A} and ν(g) := gg^{∗} ∈ (F_{A})^{×}} . (1.2)

Since C^{∘} = End_{B}(L_{ℚ}), we have B^{∘} = End_{C}(L_{ℚ}), 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 G̃ of finite index, and G̃(ℚ)/G(ℚ) = Out_{ℚ-alg}(C, ∗). 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_{ℚ-alg}(C, ∗) ⊂ H^{0}(⟨∗⟩, Out_{ℚ-alg}(C)) ⊂ Out_{ℚ-alg}(C) ⊂ Aut(M/ℚ). 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 G̃ = G̃/Z for the center Z of G.

We write G_{1} for the derived group of G; thus, G_{1} = {g ∈ G|N_{C}(g) = ν(g) = 1} for the reduced norm N_{C} of C over M. We write Z^{G} = G/G_{1} for the cocenter of G. Then g ↦ (ν(g), N_{C}(g)) identifies Z^{G} with a sub-torus of Res_{F/ℚ}𝔾_{m} × Res_{M/ℚ}𝔾_{m}. If M = F, G_{1} is 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 Z^{G} ⊂ Res_{F/ℚ}𝔾_{m}. By a result of Weil ([W]) 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 G̃(A). In other words, we have an exact sequence of ℚ-algebraic groups

1 → P G(A) → Aut_{A-alg}(C_{A}, ∗)(= P G̃(A)) → Out_{A-alg}(B_{A}, ρ) → 1. (1.3)

We write

π : G̃(A) → G̃(A)/G(A) = Out_{A-alg}(B_{A}, ρ) ⊂ Out_{A-alg}(B_{A})

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 G̃(𝔸^{(Σ)}):

G^{(Σ)} = {x ∈ G̃(𝔸^{(Σ)})|π(x) ∈ Out_{ℚ-alg}(B, ρ)} ,  (1.4)

where we embed Out_{ℚ-alg}(B, ρ) into ∏_{ℓ∉Σ} Out_{ℚ_{ℓ}-alg}(B_{ℓ}) = Out_{𝔸^{(Σ)}-alg}(B ⊗_{ℚ} 𝔸^{(Σ)}) by the diagonal map (B_{ℓ} = B ⊗_{ℚ} ℚ_{ℓ}).

We suppose to have an ℝ–algebra homomorphism h : ℂ → C_{ℝ} such that h(\overline{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

Sh(ℂ) = G(ℚ)\ (G(𝔸^{(∞)}) × X) /\overline{Z(ℚ)}.

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_{x} : ℂ → C_{ℝ} given by z ↦ g · h(z)g^{−1} for g ∈ G(ℝ)/C_{h} sending h to x. Then v ↦ h_{x}(z)v for z ∈ ℂ 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 ∈ O_{B} is given by (v mod L) ↦ (b · v mod L).

We suppose

(h2) all rational primes in Σ are unramified in M/ℚ, and Σ contains ∞ and p;

(h3) For every prime ℓ ∈ Σ, O_{B,ℓ} = O_{B} ⊗_{ℤ} ℤ_{ℓ} ≅ M_{n}(R_{ℓ}) for R_{ℓ} = R ⊗_{ℤ} ℤ_{ℓ};

(h4) For every prime ℓ ∈ Σ, ⟨ , ⟩ induces L_{ℓ} = L ⊗_{ℤ} ℤ_{ℓ} ≅ Hom(L_{ℓ}, O_{ℓ});

(h5) The derived subgroup G_{1} is simply connected; so, G_{1}(ℝ) is of type A (unitary groups) or of type C (symplectic groups).

Let G(ℤ_{Σ}) = {g ∈ G(ℚ_{Σ})|g · L_{Σ} = L_{Σ}} for ℚ_{Σ} = ℚ ⊗_{ℤ} ℤ_{Σ} and L_{Σ} = L ⊗_{ℤ} ℤ_{Σ}. We define Sh^{(Σ)} = Sh/G(ℤ_{Σ}). 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,

Sh^{(Σ)}(ℂ) = G(ℚ)\ (G(𝔸^{(∞)}) × X) /\overline{Z(ℚ)}G(ℤ_{Σ})

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 ℚ_{Σ}^{(p)} = ℚ_{Σ}/ℚ_{p} and ℤ_{Σ}^{(p)} = ℤ_{Σ}/ℤ_{p}.

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 ⊂ \overline{ℚ} of ℤ_{(p)}. Thus W is an unramified valuation ring with residue field 𝔽 = \overline{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_{S}^{(p)} = 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.

Since p is unramified in M/ℚ, O_{E,(p)} is contained in W. We fix a geometrically connected component V_{/\overline{ℚ}} of Sh^{(Σ)} ×_{E} \overline{ℚ} and write V_{/W} for the schematic

closure of V_{/\overline{ℚ}} in Sh_{/W}^{(Σ)} := Sh_{/O_{E,(p)}}^{(Σ)} ⊗_{O_{E,(p)}} W. By Zariski’s connectedness theorem combined with the existence of a normal projective compactification (either minimal or smooth) of Sh_{/W}^{(Σ)}, the reduction V_{/𝔽} = V ×_{W} 𝔽 is a geo-

metrically irreducible component of Sh_{/W}^{(Σ)} ⊗_{W} 𝔽. The scheme Sh_{/S}^{(Σ)} classifies, for any S–scheme T, quadruples (A, \overline{λ}, i, ϕ^{(Σ)})_{/T} defined as follows: A is an abelian scheme of dimension \frac{1}{2} rank_{ℤ} L for which we define the Tate module T (A) = \underleftarrow{lim}_{N} A[N], T^{(Σ)}(A) = T (A) ⊗_{ℤ} ℤ̂^{(Σ)}, T_{Σ}(A) = T (A) ⊗_{ℤ} ℤ_{Σ} and V^{(Σ)}(A) = T (A) ⊗_{ℤ} 𝔸^{(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 ϕ^{(p)} : L ⊗_{ℚ} 𝔸^{(p)} ≅ V^{(p)}(A) modulo G(ℤ_{Σ}), where we require that ϕ_{Σ}^{(p)} : L ⊗_{ℤ} ℚ_{Σ}^{(p)} ≅ T_{Σ}(A) ⊗_{ℤ} ℚ_{Σ}^{(p)} send L ⊗_{ℤ} ℤ_{Σ}^{(p)} isomorphically onto T_{Σ}(A) ⊗_{ℤ} ℤ_{Σ}^{(p)}; \overline{λ} 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).

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 Sh^{(Σ)} by (A, \overline{λ}, i, ϕ^{(Σ)})_{/T} ↦ (A, \overline{λ}, i ∘ σ_{g}, ϕ^{(Σ)} ∘ g)_{/T}.

The reduced norm map N_{C} : C^{×} → M^{×} extends uniquely to a homomorphism N_{C} : G̃ → Res_{M/ℚ}𝔾_{m} of algebraic groups. Indeed, we have an isomorphism G̃(A) ≅ Out_{A-alg}(C_{A}, ∗) ⋉ 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} : G̃(ℚ) → M^{×} (which determines N_{C} : G̃ → Res_{M/ℚ}𝔾_{m} independently of the choice of A). The diagonal map μ : G̃ ∋ g ↦ (ν(g), N_{C}(g)) ∈ (Res_{F/ℚ}𝔾_{m} × Res_{M/ℚ}𝔾_{m}) factors through the cocenter Z^{G} of G; so, we have a homomorphism μ : G̃ → Z^{G} of algebraic ℚ-groups. We write Z^{G}(ℝ)^{+} for the identity connected component of Z^{G}(ℝ) and put Z^{G}(ℤ_{(Σ)})^{+} = Z^{G}(ℝ)^{+} ∩ (O_{(Σ)}^{×} × R_{(Σ)}^{×}); so, Z^{G}(ℤ_{(Σ)})^{+} = O_{(Σ)+}^{×} if

M = F. Similarly, we identify Z with Res_{R/ℤ}𝔾_{m} so that Z(ℤ_{(Σ)}) = R_{(Σ)}^{×}.

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_{/𝔽}^{(Σ)})) inside G^{(Σ)}/\overline{Z(ℤ_{(Σ)})}. The stabilizer is given by

G_{V} = \frac{\{g ∈ G^{(Σ)}|μ(g) ∈ \overline{Z^{G}(ℤ_{(Σ)})^{+}}\}}{\overline{Z(ℤ_{(Σ)})}},

where \overline{Z^{G}(ℤ_{(Σ)})^{+}} (resp. \overline{Z(ℤ_{(Σ)})}) is the topological closure of Z^{G}(ℤ_{(Σ)})^{+} (resp. Z(ℤ_{(Σ)}))in Z^{G}(𝔸^{(Σ)}) (resp. in G(𝔸^{(Σ)})). 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.

2 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:

G_{V} = \frac{\{g ∈ G^{(Σ)}|μ(g) ∈ \overline{Z^{G}(ℤ_{(Σ)})^{+}}\}}{\overline{Z(ℤ_{(Σ)})}}.

By characteristic 0 theory in [D] Theorem 2.4 or [MS] p.929 (or [PAF] 7.2.3), the action of G̃(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})} on π_{0}(Sh_{/\overline{ℚ}}^{(Σ)}) factors through the homomor-

phism G̃(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})} → Z^{G}(𝔸^{(Σ)})/\overline{Z^{G}(ℤ_{(Σ)})^{+}} induced by μ. The idele class group of cocenter Z^{G}(𝔸^{(Σ)})/\overline{Z^{G}(ℤ_{(Σ)})^{+}} acts on π_{0}(Sh_{/\overline{ℚ}}^{(Σ)}) faithfully.

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 G̃(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})} of V in π_{0}(Sh_{/𝔽}^{(Σ)}) is given by G_{V}.

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 K in G(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})}, we consider the image V_{K} of V in Sh^{(Σ)}/K. Then we have V_{K} = V/\overline{K}_{V} for \overline{K}_{V} = K ∩ G_{V}, and \overline{K}_{V} is isomorphic to the scheme theoretic Galois group Gal(V/V_{K/𝔽}), which is in turn isomorphic to Gal(𝔽(V)/𝔽(V_{K})). Since all sufficiently small open compact subgroups of G_{V} are of the form \overline{K}_{V} for open compact subgroups K of G(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})}, the \overline{K}_{V} ’s for open compact subgroups K of G(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})} 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(V_{/𝔽}) is an open subgroup of Aut(𝔽(V)/𝔽). If we choose K sufficiently small depending on σ ∈ Aut(𝔽(V)/𝔽), we have ^{σ}\overline{K}_{V} = σ\overline{K}_{V} σ^{−1} still inside G_{V} in G(𝔸^{(Σ)})/\overline{Z(ℤ_{(Σ)})}. We write U_{K} (resp. U_{^{σ}K}) for the image of U (resp. of σ(U)) in V_{K} (resp. in V_{^{σ}K}).

By the above description of the stabilizer of V, the image of G_{1}(𝔸^{(Σ)}) in the scheme automorphism group Aut(Sh_{/𝔽}^{(Σ)}) is contained in the stabilizer Aut(V_{/𝔽}) and hence in the field automorphism group Aut(𝔽(V)/𝔽). Let \overline{G}_{1}(𝔸^{(Σ)}) be the image of G_{1}(𝔸^{(Σ)}) in Aut(𝔽(V)/𝔽) (so \overline{G}_{1}(𝔸^{(Σ)}) is isomorphic to the quotient of G_{1}(𝔸^{(Σ)}) by the center of G_{1}(ℤ_{(Σ)})). We take a sufficiently small open compact subgroup S of G(𝔸^{(Σ)}). We write S_{1} = G_{1}(𝔸^{(Σ)})∩S and \overline{S}_{1} for the image of S_{1} in \overline{G}_{1}(𝔸^{(Σ)}) with \overline{G}_{1}(𝔸^{(Σ)}). Shrinking S if necessary, we may assume that S_{1} ≅ \overline{S}_{1}, that V/V_{S} is étale, that Gal(V/V_{S/𝔽}) = \overline{S}_{1}, that S = ∏_{ℓ} S_{ℓ} with S_{ℓ} = S ∩ G(ℚ_{ℓ}) for primes ℓ ∉ Σ and that ^{σ}\overline{S}_{1} := σ\overline{S}_{1}σ^{−1} ⊂ \overline{G}_{1}(𝔸^{(Σ)}). We identify Sh^{(p)}/S̃ = Sh^{(Σ)}/S for S̃ = S ×G(ℤ_{Σ}^{(p)}), where ℤ_{Σ}^{(p)} = ∏_{ℓ∈Σ−{p,∞}} ℤ_{ℓ}. Since S_{1} ≅ \overline{S}_{1}, we hereafter identify the two groups.

Let 𝔪 be the maximal ideal of W and we write κ = O_{E}/(O_{E} ∩ 𝔪) for the reflex field E. Since Sh_{/𝔽}^{(Σ)} is a scalar extension relative to 𝔽/κ of the model Sh_{/κ}^{(Σ)} 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)_{/\overline{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 U_{^{σ}S} are defined over 𝔽_{q} and that U_{S} ×_{𝔽_{q}} 𝔽 and U_{^{σ}S} ×_{𝔽_{q}} 𝔽 are irreducible. Since σ ∈ Hom(U_{S}, U_{^{σ}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) ∈ U_{^{σ}S}(𝔽_{q}). Thus σ descends to an isomorphism σ_{S} : U_{S} ≅ U_{^{σ}S} defined over 𝔽_{q}.

Let 𝕏_{S/𝔽_{q}} → U_{S/𝔽_{q}} 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 \overline{x} ∈ V (𝔽) above x. The prime-to-p part π_{1}^{(p)}(𝕏_{x}, 0\overline{x}) of π_{1}(𝕏_{x}, 0\overline{x}) is canonically isomorphic to the prime to-p part T^{(p)}(𝕏_{\overline{x}/𝔽}) of the Tate module T (𝕏_{\overline{x}/𝔽}), and the p-part of π_{1}(𝕏_{x}, 0\overline{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_{1}^{\{p\}}(𝕏_{S/𝔽}, 0\overline{x}) by the image of the p-part of π_{1}(𝕏_{x}, 0\overline{x}). Then we have the following exact sequence ([SGA] 1.X.1.4):

T^{(p)}(𝕏\overline{x}) ⟶^{i} π_{1}^{\{p\}}(𝕏_{S/𝔽_{q}}, 0\overline{x}) → π_{1}(U_{S/𝔽_{q}}, \overline{x}) → 1.

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^{(p)}(𝕏_{x}) injects into π_{1}^{\{p\}}(𝕏_{S/𝔽}, 0\overline{x}). We make the quotient π_{1}^{Σ}(𝕏_{S/𝔽_{q}}, 0\overline{x}) = π_{1}^{\{p\}}(𝕏_{S/𝔽_{q}}, 0\overline{x})/i(T_{Σ}^{(p)}(𝕏\overline{x})), and we get a split short exact sequence:

0 → T^{(Σ)}(𝕏\overline{x}) → π_{1}^{Σ}(𝕏_{S/𝔽_{q}}, 0\overline{x}) → π_{1}(U_{S/𝔽_{q}}, \overline{x}) → 1.  (2.1)

By this exact sequence, π_{1}(U_{S/𝔽_{q}}, \overline{x}) acts by conjugation on T^{(Σ)}(𝕏\overline{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/𝔽_{q}}, \overline{x}) ↠ Gal(U/U_{S}). We write S_{V} = Gal(U/U_{S}), which is an extension of \overline{S}_{V} by Gal(𝔽/𝔽_{q}) generated by the Frobenius automorphism over 𝔽_{q}. Since 𝕏\overline{x}[N] for all integers N outside Σ gets trivialized over U, the action of π_{1}(U_{S/𝔽_{q}}, \overline{x}) on T^{(Σ)}(𝕏\overline{x}) factors through π_{1}(U_{S/𝔽_{q}}, \overline{x}) ↠ S_{V}.

We now have another split exact sequence:

0 → T^{(Σ)}(𝕏_{σ(\overline{x})}) → π_{1}^{Σ}(𝕏_{^{σ}S/𝔽_{q}}, 0_{σ(\overline{x})}) → π_{1}(U_{^{σ}S/𝔽_{q}}, σ(\overline{x})) → 1.  (2.2)

Again the action of π_{1}(U_{^{σ}S/𝔽_{q}}, σ(\overline{x})) on T^{(Σ)}(𝕏_{σ(\overline{x})}) factors through Gal(U/U_{^{σ}S}) = ^{σ}S_{V}. We fix a path in U_{^{σ}S} from σ(\overline{x}) to \overline{x} and lift it to a path from σ(0\overline{x}) to 0\overline{x} in 𝕏_{^{σ}S}, which induces canonical isomorphisms ([SGA] V.7):

ι^{σ} : π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ≅ π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0\overline{x}) and ι_{σ} : π_{1}(U_{^{σ}S}, 0_{σ(\overline{x})}) ≅ π_{1}(U_{^{σ}S}, \overline{x}).

The isomorphism ι^{σ} in turn induces an isomorphism ι : T^{(Σ)}(𝕏_{σ(\overline{x})}) → T^{(Σ)}(𝕏\overline{x}) of ^{σ}S_{1}–modules.

We want to have the following commutative diagram

T^{(Σ)}(𝕏\overline{x}) ⟶^{↪} π_{1}^{Σ}(𝕏_{S/𝔽}, 0\overline{x}) ⟶^{↠} π_{1}(U_{S/𝔽}, \overline{x})
?↓  ↓^{?}  ^{σ_{∗}}↓
T^{(Σ)}(𝕏_{σ(\overline{x})}) ⟶^{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶^{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x}))
ι↓  _{ι^{σ}} ↓  ↓^{ι_{σ}}
T^{(Σ)}(𝕏\overline{x}) ⟶_{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0\overline{x}) ⟶_{↠} π_{1}(U_{^{σ}S/𝔽}, \overline{x}),

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 L ∈ End_{𝔸^{(Σ)}}(T^{(Σ)}(𝕏\overline{x}) ⊗ ℚ) such that L(s · v) = ^{σ}s · L(v) for all s ∈ S_{1} and v ∈ T^{(Σ)}(𝕏\overline{x}), where ^{σ}s = σsσ^{−1} is the image of ι_{σ}(σ_{∗}(s)) in ^{σ}S_{1} for any lift s ∈ π_{1}(V_{S}, \overline{x}) inducing s ∈ S_{1}. Since Hom(G_{1}(ℤ_{ℓ}), G_{1}(ℤ_{ℓ′})) is a singleton made of the zero-map (taking the entire G_{1}(ℤ_{ℓ}) to the identity of G_{1}(ℤ_{ℓ′})) if two primes ℓ and ℓ′ are large and distinct (see Section 3 (S3) for a proof of this fact), s ↦ ^{σ}s sends S_{1,ℓ} into ^{σ}S_{1,ℓ} for almost all primes ℓ, where S_{1,ℓ} = G_{1}(ℚ_{ℓ}) ∩ S_{ℓ}. If we shrink S further if necessary for exceptional finitely many primes, we achieve that S_{ℓ} is ℓ–profinite for exceptional ℓ and the logarithm log_{ℓ} : S_{1,ℓ} → Lie(S_{1,ℓ}) given by log_{ℓ}(s) = ∑_{n=1}^{∞}(−1)^{n+1} \frac{(s−1)^{n}}{n} is an ℓ-adically continuous isomorphism. Then by a result of Lazard [GAN] IV.3.2.6 (see Section 3 (S1)), σ induces by log_{ℓ} ∘σ = [σ]_{ℓ} ∘ log_{ℓ} an automorphism [σ]_{ℓ} of the Lie algebra Lie(S_{ℓ}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} over ℚ_{ℓ}. Note that

Lie(S_{1,ℓ}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} = {x ∈ C_{ℓ}|ρ(x) = −x and Tr(x) = 0} ,

where Tr : C_{ℓ} → M_{ℓ} is the reduced trace map. Extending scalar to a finite Galois extension of K/ℚ_{ℓ}, Lie(S_{ℓ}) ⊗_{ℤ_{ℓ}} K becomes split semi-simple over K, and therefore [σ]_{ℓ} is induced by an element of P G̃(K) (the Lie algebra version of (S2) in the following section), which implies by Galois descent that [σ]_{ℓ} is induced by an element of P G̃(ℚ_{ℓ}). Thus for all ℓ ∉ Σ, s ↦ ^{σ}s sends S_{1,ℓ} into σS_{1,ℓ} and that the isomorphism: s ↦ ^{σ}s is induced by an element \overline{L} of the group fitting into the middle term of the exact sequence (1.3):

1 → P G(𝔸^{(Σ)}) → P G̃(𝔸^{(Σ)}) → Aut(M_{𝔸}^{(Σ)}/𝔸^{(Σ)}).

The element \overline{L} in Aut(G_{1}(𝔸^{(Σ)})) is in turn induced by an endomorphism L ∈ End_{𝔸^{(Σ)}}(T^{(Σ)}(𝕏\overline{x}) ⊗ ℚ). Define g(σ) = ι^{−1} ∘ L. Then g(σ) is an element of Hom_{\overline{S}_{1}}(T^{(Σ)}𝕏_{x}, T^{(Σ)}𝕏_{σ(x)}) invertible in Hom_{\overline{S}_{1}}(T^{(Σ)}𝕏_{x}, T^{(Σ)}𝕏_{σ(x)})⊗_{ℤ} ℚ, and g(σ) is S_{1}–linear in the sense that g(σ)(sx) = ^{σ}s·g(σ)(x) for all s ∈ \overline{S}_{1}. 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

\begin{matrix}T^{(p)}(𝕏_{x}) ⊗_{ℤ} 𝔸^{(p)} & ⟶^{g(σ)} & T^{(p)}(𝕏_{σ(x)}) ⊗_{ℤ} 𝔸^{(p)} \\ ^{≀}↑^{ϕ_{x}^{(p)}} & & ^{≀}↑^{ϕ_{σ(x)}^{(p)}} \\ L ⊗_{ℤ} 𝔸^{(p)} & ⟶_{g_{σ}} & L ⊗_{ℤ} 𝔸^{(p)} \end{matrix}  (2.3)

for g_{σ} ∈ G̃(𝔸^{(p)}). Thus g_{σ}^{(Σ)} has the projection π(g_{σ}^{(Σ)}) ∈ Out_{𝔸^{(Σ)}-alg}(B_{𝔸}^{(Σ)}, ρ).

Consider the relative Frobenius map π_{S} : U_{S} → U_{S} over 𝔽_{q}. Since σ : U_{S} ≅ U_{^{σ}S} is defined over 𝔽_{q} by our choice, σ satisfies σ_{S} ∘ π_{S} = π_{^{σ}S} ∘ σ_{S}. If X → U_{S} is an étale irreducible covering, X ×_{U_{S},π_{S}} U_{S} → U_{S} is étale irreducible, and π_{S} : U_{S} → U_{S} induces an endomorphism π_{S,∗} : π_{1}(U_{S}, \overline{x}) → π_{1}(U_{S}, \overline{x}). We have a diagram:

T^{(Σ)}(𝕏_{σ(\overline{x})}) ⟶^{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶^{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x}))
g(σ)^{−1}↓  ↓^{g(σ)^{−1}⋉σ_{S,∗}^{−1}}  ↓^{σ_{S,∗}^{−1}}
T^{(Σ)}(𝕏\overline{x}) ⟶^{↪} π_{1}^{Σ}(𝕏_{S/𝔽}, 0\overline{x}) ⟶^{↠} π_{1}(U_{S/𝔽}, \overline{x})
π_{x}↓  ↓^{π_{x}⋉π_{S,∗}}  ↓^{π_{S,∗}}
T^{(Σ)}(𝕏\overline{x}) ⟶^{↪} π_{1}^{Σ}(𝕏_{S/𝔽}, 0\overline{x}) ⟶^{↠} π_{1}(U_{S/𝔽}, \overline{x})
g(σ)↓  ↓^{g(σ)⋉σ_{S,∗}}  ↓^{σ_{S,∗}}
T^{(Σ)}(𝕏_{σ(\overline{x}})) ⟶_{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶_{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x})),

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/𝔽_{q}}. 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:

T^{(Σ)}(𝕏_{σ(\overline{x})}) ⟶^{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶^{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x}))
g(σ)π_{x}g(σ)^{−1}↓  ↓  ^{π_{^{σ} S,∗}}↓
T^{(Σ)}(𝕏_{σ(\overline{x}})) ⟶_{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶_{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x})),

because σ_{S}π_{S}σ_{S}^{−1} = π_{^{σ}S}. Since π_{σ(x)} also gives a similar commutative diagram:

T^{(Σ)}(𝕏_{σ(\overline{x})}) ⟶^{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶^{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x}))
π_{σ(x)}↓  ↓  ↓^{π_{^{σ} S,∗}}
T^{(Σ)}(𝕏_{σ(\overline{x}})) ⟶_{↪} π_{1}^{Σ}(𝕏_{^{σ}S/𝔽}, 0_{σ(\overline{x})}) ⟶_{↠} π_{1}(U_{^{σ}S/𝔽}, σ(\overline{x})),

we find out that g(σ)π_{x}g(σ)^{−1}π_{σ(x)}^{−1} commutes with the action of^{σ}\overline{S}_{1}, and hence it is in the center of Aut_{R}(T^{(Σ)}(𝕏_{σ(\overline{x})})). In other words, g(σ)π_{x}g(σ)^{−1} = zπ_{σ(x)} for z ∈ (R̂^{(Σ)})^{×}. Taking the determinant with respect to ⋀^{g} T_{ℓ}(𝕏_{σ(\overline{x})}) for the rank g = rank_{R_{ℓ}} T_{ℓ}(𝕏_{σ(\overline{x})}) with a prime ℓ ∉ 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 (R̂^{(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_{x}^{g} = π_{σ(x)}^{g} ∘ 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}(\overline{G}_{1}(𝔸^{(Σ)})) → Out_{𝔸^{(Σ)}-alg}(B_{𝔸}^{(Σ)}, ρ) (induced by π) whose kernel is given by P G(𝔸^{(Σ)}). Thus σ ∈ Aut(𝔽(V)_{/𝔽}) has projection π(g_{σ}^{(Σ)}) (for g_{σ} in (2.3)) in

Out_{𝔸^{(Σ)}-alg}(B_{𝔸}^{(Σ)}, ρ) ⊂ Aut_{𝔸^{(Σ)}-alg}(M_{𝔸}^{(Σ)}) = ∏_{ℓ∉Σ} Aut_{ℚ_{ℓ}-alg}(M_{ℓ})

which will be written as σ_{B} = π(g_{σ}^{(Σ)}).

Corollary 2.2. If σ ∈ Aut(𝔽(V)/𝔽), we have σ_{B} ∈ Out_{ℚ-alg}(B, ρ), where the group Out_{ℚ-alg}(B, ρ) is diagonally embedded into ∏_{ℓ∉Σ} Aut_{ℚ_{ℓ}-alg}(M_{ℓ}).

Proof. The element g(σ) = ι^{−1} ∘ L ∈ Hom_{S}(T^{(Σ)}𝕏_{x}, T^{(Σ)}𝕏_{σ(x)}) in the proof of Proposition 2.1 acts on G_{1}(𝔸^{(Σ)}) by conjugation of g_{σ} ∈ G̃(𝔸^{(Σ)}) in (2.3); so, its projection π(g_{σ}) in Out_{𝔸^{(Σ)}-alg}(B_{𝔸}^{(Σ)}, ρ) inside Aut(M_{𝔸}^{(Σ)}/𝔸^{(Σ)}) is given by σ_{B}. By the proof of Proposition 2.1, g(σ) is induced by ξ ∈ Hom_{O_{B}}(𝕏_{x}, 𝕏_{σ(x)}) modulo Z(ℤ_{(Σ)})S. Choose a rational prime q outside Σ. We have B_{q} ∩ End(𝕏_{x}) = O_{B}. Note that b ↦ g(σ)^{−1} ∘ b ∘ g(σ) sends B_{q} into itself 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_{ℚ_{q}-alg}(B_{q}) and the image of the conjugation by ξ in Out(End(𝕏_{x}) ⊗_{ℤ} ℚ) coincide, we conclude σ_{B} ∈ Out_{ℚ-alg}(B). Since σ_{B} ∈ Out_{𝔸^{(Σ)}-alg}(B_{𝔸}^{(Σ)}, ρ) ∩ Out_{ℚ-alg}(B), we get σ_{B} ∈ Out_{ℚ-alg}(B, ρ).

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} ≅ U_{^{σ}S} for an open dense subscheme U_{S} ⊂ V_{S}. Again we use the exact sequence:0 → T^{(Σ)}𝕏\overline{η} → π_{1}^{Σ}(𝕏_{/𝔽_{q}}, 0\overline{η}) → π_{1}(U_{S/𝔽_{q}}, \overline{η}) → 1. By the same argument as above, we find g_{η}(σ) ∈ Hom_{π_{1}(U_{S/𝔽},\overline{η})}(T^{(Σ)}𝕏_{η}, T^{(Σ)}𝕏_{σ(η)}). Since 𝕏_{η}[ℓ^{∞}] gets trivialized over U for a prime ℓ ∉ 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/𝔽_{q}}, \overline{x}) (resp. T^{(Σ)}𝕏_{x} and T^{(Σ)}𝕏_{σ(x)}) with the Galois group Gal(𝔽(Ũ)/𝔽_{q}(U_{S})) for the universal covering Ũ (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 σ̃ : Ũ ≅ ^{σ}Ũ. 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}) ↪ U_{^{σ}S} induce isomorphisms D_{x} ≅ Gal(𝔽/𝔽_{q}) ≅ D_{σ(x)} = σ̃D_{x}σ^{−1} (choosing the extension σ̃ suitably) and splittings: Gal(𝔽(Ũ)/𝔽_{q}(U_{S})) = D_{x} ⋉ Gal(𝔽(Ũ)/𝔽(U_{S})) and Gal(𝔽(^{σ}Ũ)/𝔽_{q}(U_{^{σ}S})) = D_{σ(x)} ⋉ Gal(𝔽(^{σ}Ũ)/𝔽(U_{^{σ}S})). The morphism g(σ) : T^{(Σ)}𝕏_{x} → T^{(Σ)}𝕏_{σ(x)} induces a morphism g_{η}(σ) : T^{(Σ)}𝕏_{η} → T^{(Σ)}𝕏_{σ(η)} satisfying g_{η}(σ)(sx) = ^{σ}s · g_{η}(σ)(x) for all s ∈ S_{V}. Thus g_{η}(σ) is a morphism of π_{1}(U_{S/𝔽_{q}}, \overline{η})–modules (not just that of π_{1}^{Σ}(U_{S/𝔽}, \overline{η})–modules). Then by a result of Zarhin (see [RPT] Chapter VI, [Z] and also [ARG] Chapter II), 𝕏_{η/𝔽_{q}(V_{S})} and 𝕏_{σ(η)/𝔽_{q}(V_{^{σ} 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^{∗}𝕏_{^{σ}S} = 𝕏_{^{σ}S} ×_{U_{^{σ} S},σ} U_{S} over U_{S}. We write α : 𝕏_{S} → 𝕏_{^{σ}S} for the composite of the above isogeny with the projection 𝕏_{^{σ}S} ×_{U_{^{σ} S},σ} U_{S} → 𝕏_{^{σ}S}.

We then have the commutative diagram:

𝕏_{S} ⟶^{α} 𝕏_{^{σ}S}
↓  ↓
U_{S} ⟶_{σ} U_{^{σ}S}.

Assume that σ_{B} = 1. Then α is B-linear. Suppose we have another B-linear isogeny α′ : 𝕏_{S} → σ^{∗}𝕏_{^{σ}S} inducing g_{η}(σ). Then α^{−1}α′ commutes with the action of Gal(𝔽(Ũ)/𝔽(U_{S})) and hence with the action of S. Thus we find ξ = ϕ^{−1}α^{−1}α′ϕ ∈ End_{S}(L_{𝔸^{(Σ)}}) for the level structure ϕ = ϕ^{(Σ)} : L_{𝔸^{(Σ)}} → V^{(Σ)}(𝕏_{η}). This implies B-linear ξ commutes with the action of C, and hence in the center of B ⊂ End_{C}(L_{ℚ}). We thus find ξ ∈ Z(ℚ)G(ℤ_{Σ}^{(p)})S. We consider the commutative diagram similar to (2.3):

\begin{matrix}T^{(p)}(𝕏_{η}) ⊗_{ℤ} 𝔸^{(p)} & ⟶^{α_{η}} & T^{(p)}(𝕏_{σ(η)}) ⊗_{ℤ} 𝔸^{(p)} \\ ^{≀}↑^{ϕ_{η}^{(p)}} & & ^{≀}↑^{ϕ_{σ(η)}^{(p)}} \\ L ⊗_{ℤ} 𝔸^{(p)} & ⟶_{g_{σ}} & L ⊗_{ℤ} 𝔸^{(p)}. \end{matrix}  (2.4)

The prime-to-Σ component of g_{σ}^{−1} eventually gives a_{S} in the corollary. By the above fact, g_{σ}^{(Σ)} is uniquely determined in G̃(𝔸^{(Σ)})/Z(ℚ)S.

Note that σ^{∗}(𝕏_{^{σ}S}, \overline{λ}_{^{σ}S}, i_{^{σ}S}, ϕ_{σ(η)}^{(p)} ∘ g_{σ}^{(Σ)})_{/U_{S}} is a quadruple classified by

Sh_{/S}^{(Σ)} = Sh^{(p)}/G(ℤ_{Σ}^{(p)})S. By the universality of Sh_{/S}^{(Σ)} (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^{∗}𝕏_{^{σ}S} → τ^{∗}𝕏_{S} over U_{S}. Identifying Gal(U/U_{S}) with a subgroup S_{V} of S ⊂ G(𝔸^{(Σ)}), the actions of s ∈ S_{V} on ϕ_{1} = ϕ_{σ(η)}^{(Σ)} ∘ g_{σ} and on ϕ_{2} = ϕ_{η}^{(Σ)} have identical effect: s ∘ ϕ_{j} = ϕ_{j} ∘ s (j = 1, 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^{∗}(𝕏_{^{σ}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}^{(p)}) normalizing G(ℤ_{Σ}^{(p)}). 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_{σ}^{(Σ)} is uniquely determined in G_{V} modulo S.

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 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}^{(p)} modulo G(ℤ_{Σ}^{(p)}), which we cannot forget. To clarify this, take a (characteristic 0) geometrically connected component V_{0} of Sh_{/E} whose image V_{0}^{(Σ)} in Sh_{/E}^{(Σ)} = 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_{/\overline{ℚ}}) at the beginning of the proof of Proposition 2.1, the stabilizer in G(𝔸^{(∞)})/\overline{Z(ℚ)} of V_{0/\overline{ℚ}} ∈ π_{0}(Sh_{/\overline{ℚ}}) is given by

G_{0} = \frac{\{g ∈ G(𝔸^{(∞)})|μ(g) ∈ \overline{Z^{G}(ℚ)^{+}}\}}{\overline{Z(ℚ)}}.

Inside this group, every element g ∈ G(𝔸^{(∞)}) inducing an automorphism of V_{0}^{(Σ)} has its ℓ-component g_{ℓ} for a prime ℓ ∈ Σ in the normalizer of G(ℤ_{ℓ}). Under (h2-4), as is well known, the normalizer of G(ℤ_{ℓ}) in G(ℚ_{ℓ}) is Z(ℚ_{ℓ})G(ℤ_{ℓ}) (see [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_{ℓ}) is quasi-split over ℚ_{ℓ}, and we have the Iwasawa decomposition G(ℚ_{ℓ}) = P_{0}(ℚ_{ℓ})G(ℤ_{ℓ}) for ℓ ∈ Σ with a minimal parabolic subgroup P_{0} of G, from which we can easily prove that the normalizer of G(ℤ_{ℓ}) is Z(ℚ_{ℓ})G(ℤ_{ℓ}). An elementary proof of the Iwasawa decomposition (for a unitary group or a symplectic group acting on M_{ℓ}^{r} keeping a skew-hermitian form relative to M_{ℓ}/F_{ℓ}) can be found in [EPE] Section 5, particularly pages 36-37. By (h3-4), G(ℚ_{ℓ}) is isomorphic to a unitary or symplectic group acting on M_{ℓ}^{n} = εL_{ℚ_{ℓ}} for an idempotent ε (for example, ε = diag[1, 0, …, 0] fixed by ρ) of O_{B_{ℓ}} ≅ M_{n}(R_{ℓ}) with respect to the skew-hermitian form on εL_{ℚ_{ℓ}} induced by ⟨·, ·⟩; so, the result in [EPE] Section 5 applies to our case.

Suppose that g ∈ G(𝔸^{(∞)}) preserves the quotient V_{0}^{(Σ)} of V_{0}. If Σ is finite, we can therefore choose ξ ∈ Z(ℚ) so that (ξg)_{ℓ} is in G(ℤ_{ℓ}) for all ℓ ∈ Σ, and the action of (ξg)^{(Σ)} ∈ G_{V} on V_{0}^{(Σ)} induces the action of g. Suppose that Σ is infinite. Since α_{η} is a prime-to-Σ isogeny, (g_{σ})_{Σ} is contained in G(ℤ_{Σ}^{(p)}). Thus σ is induced by (g_{σ}^{(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_{ℓ} ∈ Z(ℚ_{ℓ}) for all ℓ ∈ 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)^{(Σ_{i})} induces the action of g on V_{0}^{(Σ_{i})}, we find ξ_{i}g ∈ G(ℤ_{Σ_{i}})G(𝔸^{(Σ_{i})}) whose action on V_{0}^{(Σ)} is identical to that of g. We write F_{i} for the closed subset of elements in G(ℤ_{Σ_{i}})G(𝔸^{(Σ_{i})}) whose action on V_{0}^{(Σ)} 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_{0}^{(Σ)} 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_{0}^{(Σ)} ∈ π_{0}(Sh_{/\overline{ℚ}}^{(Σ)}) is represented by an element in the group G(𝔸^{(Σ)}) without Σ-component. Since π_{0}(Sh_{/\overline{ℚ}}^{(Σ)})

is in bijection with π_{0}(Sh_{/𝔽}^{(Σ)}), any element in the stabilizer in G̃(𝔸^{(p)}) of

V ∈ π_{0}(Sh_{/𝔽}^{(Σ)}) is represented by an element in G(𝔸^{(Σ)}). By this fact, under

(h2-4), the effect of σ on ϕ_{Σ}^{(p)} is determined by g_{σ}^{(Σ)} outside Σ. Thus we can really forget about the Σ-component.

Writing the prime-to-Σ level structures of 𝕏_{η} and 𝕏_{σ(η)} as ϕ_{η}^{(Σ)} and ϕ_{σ(η)}^{(Σ)},

respectively, we now find that α_{η} ∘ϕ_{η}^{(Σ)} = ϕ_{σ(η)}^{(Σ)} ∘a_{S}^{−1} for a_{S}^{−1} = g_{σ}^{(Σ)} ∈ G̃(𝔸^{(Σ)}). Since the effect of σ on T^{(Σ)}(𝕏_{η}) determines σ, we have a_{S}^{−1}(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_{/𝔽}^{(Σ)}) ≅ π_{0}(Sh_{/\overline{ℚ}}^{(Σ)}) 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 V_{0}^{(Σ)} ∈ π_{0}(Sh_{/\overline{ℚ}}^{(Σ)}) in \frac{G(𝔸^{(Σ)})}{\overline{Z(ℤ_{(Σ)})}} 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 \frac{G(𝔸^{(Σ)})}{\overline{Z(ℤ_{(Σ)})}}. Since

a fixes V ∈ π_{0}(Sh_{/𝔽}^{(Σ)}), 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]).

3 Automorphism groups of quasi-split classical groups

In the above proof of the theorem, we have used the following facts:

(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) ∈ G̃(𝔸^{(Σ)}) as in (1.2).

We may modify σ by g ∈ G̃(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_{ℓ} and S′_{ℓ} of G_{1}(ℚ_{ℓ}) (for every prime ℓ), an isomorphism σ_{ℓ} : S_{ℓ} ≅ S′_{ℓ} is induced by conjugation s ↦ g_{ℓ}(σ)sg_{ℓ}(σ)^{−1} for g_{ℓ}(σ) ∈ G(ℚ_{ℓ}) after replacing S_{ℓ} by an open subgroup of S_{ℓ} and replacing S′_{ℓ} by the image of the new S_{ℓ}; (S2) For a prime ℓ at which G_{1/ℤ_{ℓ}} is smooth quasi-split (so for a sufficiently

large rational prime ℓ), we have

Aut(G_{1}(ℤ_{ℓ})) = Aut(M_{ℓ}/ℚ_{ℓ}) ⋉ P G(ℤ_{ℓ})

and

Aut(G_{1}(ℚ_{ℓ})) = Aut(M_{ℓ}/ℚ_{ℓ}) ⋉ P G(ℚ_{ℓ}),

(S3) For sufficiently large distinct rational primes p and ℓ, any group homomorphism ϕ : G_{1}(ℤ_{p}) → G_{1}(ℤ_{ℓ})) is trivial (that is, Ker(ϕ) = G_{1}(ℤ_{p})).

The assertion (S1) follows directly from a result of Lazard on ℓ–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_{ℓ}), 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_{ℓ}) are quasi-split.

We now prove the assertion (S3). Let ϕ : G_{1}(ℤ_{p}) → G_{1}(ℤ_{ℓ}) 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

diag[1_{j}, ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) , 1_{n−j−2}] := ( 0 ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) 0_{0 0 1_{n−j−2}}^{1_{j} 0 0} )

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 Z(U_{α}) of U_{α} in G_{1}. Then by conjugation, N(U_{α})/Z(U_{α}) acts on U_{α}. Since ϕ is a group homomorphism, the quotient N(U_{α})/Z(U_{α}) keeps acting on the image ϕ(U_{α}) in G_{1}(ℤ_{ℓ}) through conjugation by elements in ϕ(N(U_{α})). If p ≠ ℓ, every element of ϕ(U_{α}) is semi-simple (because unipotent radical of G_{1}(ℤ_{ℓ}) is ℓ-profinite). Thus the centralizer (resp. the normalizer) of ϕ(U_{ℓ}) is given by Z(ℤ_{ℓ}) (resp. N(ℤ_{ℓ})) for a reductive subgroup Z (resp. N) of G_{1}. Then N(ℤ_{ℓ})/Z(ℤ_{ℓ}) is a finite subgroup of the Weyl group W_{1} of G_{1/ℂ} which is independent of ℓ.

For example, if G_{1} = SL(n)_{/ℤ}, and if ϕ(U_{α}) is made of diagonal matrices diag[ζ_{1}1_{m_{1}}, ζ_{2}1_{m_{2}}, …, ζ_{r}1_{m_{r}}] for generically distinct ζ_{j}, Z is given by the subgroup

SL(n) ∩ (GL(m_{1}) × GL(m_{2}) × ⋯ × GL(m_{r})),

where GL(m_{1}) × GL(m_{2}) × ⋯ × GL(m_{r}) is embedded in GL(n) diagonally. The quotient N/Z in this case is isomorphic to the subgroup of permutation matrices preserving Z.

If ϕ(U_{α}) is nontrivial, the image of N(U_{α})/Z(U_{α}) in Aut(ϕ(U_{α})) grows at least on the order of p as p grows. In the above example of G_{1} = SL(n)_{/ℤ}, if ϕ(U_{α}) ≅ ℤ/p^{m}ℤ for m > 0, all elements in Aut(ϕ(U_{α})) ≅ (ℤ/p^{m}ℤ)^{×} come from N(U_{α})/Z(U_{α}) ≅ ℤ_{p}^{×}. This is impossible if p ≫ |W_{1}|. Thus ϕ(U_{α}) = 1. Since G_{1}(ℤ_{p}) 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. Thus

Aut_{χ}(G(A)) = {σ ∈ Aut(G(A))|χ(σ(g)) = τ(χ(g)) for ∃ τ ∈ Aut(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 : GL(2) → 𝔾_{m}).

3.1 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 SL_{n}(K) = H.

Proposition 3.2. 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}, gUg^{−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 GL_{n}(K) = B · SL_{n}(K).

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 : A^{×} → GL_{n}(A) of the determinant map, that is, det(s(x)) = x, we recall Aut_{s}(GL_{n}(A)) = {σ ∈ Aut(GL_{n}(A))|σ(s(x)) = g · s(τ(x))g^{−1}} 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

\begin{cases}(Aut(K) × ⟨J⟩) ⋉ P GL_{n}(A) & if n ≥ 3, \\ Aut(K) ⋉ P GL_{n}(A) & if n = 2, \end{cases}

where J(x) = w_{0}^{t}x^{−1}w_{0}^{−1} for w_{0} = (δ_{i,n+1−j}) ∈ GL(n) and Aut(K) is the continuous field automorphism group of K.

2. If H ⊃ SL_{n}(A) is a subgroup of GL_{n}(A), we have

Aut_{s}(H) = {g ∈ Aut(P GL_{n}(A))|τ_{g}(det(H)) = det(H)},

where τ_{g} indicates the projection of g to Aut(K).

3. We have a canonical split exact sequence

1 → Hom(A^{×}, Z(SL_{n}(A))) → Aut_{det}(GL_{n}(A)) → Aut(SL_{n}(A)) → 1.

In other words, for σ ∈ Aut_{det}(GL_{n}(A)), there exists g ∈ GL_{n}(A) and τ ∈ Aut(K) with σ(x) = ζ(det(x))gτ(x)g^{−1} for ζ ∈ Hom(A^{×}, Z(SL_{n}(A))).

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:

1 → P GL_{n}(K) ⟶^{i} Aut(P GL_{n}(K)) → Out(P GL_{n}(K)) → 1,

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 \overline{B} and \overline{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(\frac{1}{m}) acts on V_{α} and a(\frac{1}{m})^{m} = a(1) = α ∈ End(V_{α}). Thus we have an algebra homomorphism: K[x]/(x^{m} − α) → End_{K}(V_{α}) for all 0 < m ∈ ℤ. If K is a p–adic local field, ⋂_{m}(K^{×})^{m!} = {1}. By using this, we find ⋂_{m}(K^{×})^{m} = {1} for a number field K. 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 \overline{U} ≅ U is generated by unipotent subgroups isomorphic to K, by the above argument, σ(\overline{U}) for σ ∈ Aut(P GL_{n}(K)) is again a unipotent subgroup of P GL_{n}(K). Since \overline{B} is the normalizer of \overline{U}, again σ(\overline{B}) is the normalizer of σ(\overline{U}); so, σ(\overline{B}) is a Borel subgroup. We find g ∈ GL_{n}(K) such that σ(\overline{B}) = g\overline{B}g^{−1}. Thus we may assume that σ fixes \overline{B}. Applying the same argument to \overline{U}, we may assume that σ fixes \overline{U}. Since we have a unique filtration:

\overline{U} = U_{1} ⊃ U_{2} ⊃ U_{3} ⊃ ⋯ ⊃ U_{n−1} ⊃ \{1\} = U_{n}

with [U_{j}, U_{j}] = U_{j+1} and U_{j}/U_{j+1} ≅ K^{n−j}, σ preserves this filtration. We fix an isomorphism a_{j} : K^{n−j} ↪ U_{j} given by

a_{j}(α_{1}, …, α_{n−j}) = 1 + α_{1}E_{1,j+1} + α_{2}E_{2,j+2} + ⋯ + α_{n−j}E_{n−j,n},

where E_{i,j} is the matrix having non-zero entry 1 only at the (i, j)-spot. Then a_{j} induces K^{n−j} ≅ U_{j}/U_{j+1}. Since σ([u, u′]) = [σ(u), σ(u′)] for u, u′ ∈ U_{j} and [σ(u), σ(u′)] mod U_{j+2} is uniquely determined by the cosets uU_{j+1} and u′U_{j+1}, σ : \overline{U} ≅ \overline{U} is uniquely determined by σ_{1} : U_{1}/U_{2} ≅ U_{1}/U_{2} induced by σ.

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 \overline{T} := \overline{B}/\overline{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} ⊂ \overline{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 ∈ \overline{T} acts via the multiplication by χ_{ij}(t) = α_{i}α_{j}^{−1}(t). The automorphism σ also induces an automorphism \overline{σ} of T = \overline{B}/\overline{U}. Thus σ permutes the eigen-subgroups U_{ij} of \overline{U}.

Let k = ℚ if K is a number field and k = ℚ_{p} if K is a p-adic field. Then σ induces a k-linear automorphism on U_{j}/U_{j+1} for all j. We first assume K = k. Write σ(a_{1}(1, …, 1)) = a_{1}(α_{1}, …, α_{n−1}) mod U_{2}. Solve a_{j}a_{j+1}^{−1} = α_{j} for j = 1, …, n − 1. Then changing σ by x ↦ tσ(x)t^{−1} for t = diag[a_{1}, a_{2}, …, a_{n}], we may assume that σ_{1}(a_{1}(1)) ≡ a_{1}(1) mod U_{2} for 1 = (1, 1, …, 1) ∈ K^{n−1}. Further by conjugating σ by an element in \overline{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 χ_{1n}. If n ≥ 3, looking at U_{n−2}/U_{n−1} = U_{1,n−1} ⊕ U_{2,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(x) = w_{0}^{t}x^{−1}w_{0}^{−1}, we may assume σ fix each \overline{T} -eigenspace of U_{n−2}/U_{n−1}. By the commutator relation [u_{ij}, u_{jk}] = u_{ik} if i < j < k, we conclude that σ has to fix all eigen-subgroups U_{ij} of \overline{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 K-linearity of σ that χ_{ij}(\overline{σ}(t))σ(u_{ij}) = σ(χ_{ij}(t)u_{ij}) = χ_{ij}(t)σ(u_{ij}). Thus σ acts trivially on \overline{T}. Since a_{1}(1) has non-trivial projection to all \overline{T} -eigenspaces in U_{1}/U_{2}, we conclude that σ(u_{i,i+1}) = u_{i,i+1}. Then \overline{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 χ_{ij} ∘ \overline{σ} = σ̃ ∘ χ_{ij} (for all i < j) for a continuous field automorphism σ̃ of K. Since σ induces a k-linear automorphism of U_{ij} ≅ K and χ_{ij}(t) ∈ Im(χ_{ij}) = K^{×} acts on U_{ij} through the multiplication by χ_{ij}(t) ∈ K^{×}, we find an automorphism σ̃_{ij} ∈ Aut(K) of the field K such that χ_{ij} ∘ \overline{σ} = σ̃_{ij} ∘ χ_{ij}. This field automorphism σ̃ = σ̃_{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_{ℚ_{p}} of P GL_{n}(K) over ℚ_{p}. Since dim_{ℚ_{p}} 𝔊_{ℚ_{p}} = dim_{ℚ_{p}} 𝔊_{K} for the Lie algebra 𝔊_{K} of P GL_{n}(K) over K, we find that Aut_{K}(𝔊_{K}) ⊂ Aut_{ℚ_{p}} 𝔊_{ℚ_{p}} 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,

Out(P GL_{n}(K)) ≅ Aut(K) × ⟨J⟩ and Aut(P GL_{n}(K)) = (Aut(K) × ⟨J⟩) ⋉ P GL_{n}(K)

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⟩) 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 \overline{B}_{1} fixed. Then by the same argument as above, we conclude σ = 1 and hence we find that, if n > 2,

Aut(SL_{n}(K)) = Aut(P SL_{n}(K)) = (Aut(K) × ⟨J⟩) ⋉ P GL_{n}(K).

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

Res : Aut_{det}(GL_{n}(K)) → Aut(SL_{n}(K)).

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 : K^{×} → GL_{n}(K) 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)).

For any g ∈ GL_{n}(K), we can write uniquely g = s(det(g))u with u ∈ SL_{n}(K). For a homomorphism ζ : K^{×} → Z(SL_{n}(K)),

σ(g) = σ(s(det(g))u) = ζ(det(g))s(det(g))u = ζ(det(g))g

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.

We 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, …, b_{j}^{j}, 0, …, 0) ∈ O^{n−1} with b_{j} ∈ O, we have t(α)a_{1}(b_{j})t(α)^{−1} = a_{1}(α_{j}b_{j}α_{j+1}^{−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. Then applying σ to the above formula, we see that \overline{σ} preserves the coordinates α_{j}, and \overline{σ}(t(α))σ(a_{1}(b_{j}))\overline{σ}(t(α)^{−1}) = σ(a_{1}(α_{j}b_{j}α_{j+1}^{−1})) for α = (α_{j}) ∈ (O^{×})^{n}.

If σ(a_{1}(e_{j})) ∈ 𝔪 for e_{j} = (0, …, 1^{j}, 0, …, 0) ∈ O^{n−1}, then

\overline{σ}(t(α))σ(a_{1}(e_{j}))\overline{σ}(t(α))^{−1} = σ(a_{1}(αe_{j}α^{−1}))

has entry in 𝔪 at (j, j + 1). However σ(U/U_{2}) = U/U_{2} ≅ O^{n−1} by a_{1}(b) ↔ b, we find O ⊂ 𝔪, a contradiction. Thus we have σ(a_{1}(1)) = a_{1}(α) for an element in α ∈ (O^{×})^{n−1}. Then t(α) ∈ GL_{n}(O),and modifying σ by the conjugation of t(α), we may assume that σ(a_{1}(1)) = a_{1}(1). Then proceeding in exactly the same way in the case of the field, we find that

Aut(P GL_{n}(O)) = \begin{cases}(Aut(O) × ⟨J⟩) ⋉ P GL_{n}(O) & if n ≥ 3, \\ Aut(O) ⋉ P GL_{2}(O) & if n = 2. \end{cases}

From this, again we obtain the desired result for all other automorphism groups listed in the proposition.

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}):

G_{1}(K) = {α ∈ SL_{n}(M)|(αu, αv) = (u, v)} .

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.

3.2 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 I_{n} = (δ_{n+1−i,j}) ∈ M_{n}(ℚ) and J_{n} = ( \begin{matrix}0 & −I_{g} \\ I_{g} & 0 \end{matrix} ) for n = 2g is an anti-diagonal alternating matrix. In this subsection, we deal with the split symplectic group defined over ℚ given by

G(A) = GSp_{2g}(A) = {α ∈ GL_{n}(A)|αJ_{2g}^{t}α = ν(α)J_{2g} for ν(α) ∈ A^{×}} ,

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

1. Aut(Sp_{2g}(K)) = Aut(P GSp_{2n}(K)) = Aut(K) ⋉ P GSp_{2g}(K), where we define P GSp_{2g}(K) = GSp_{2n}(K)/Z(K).

2. For a section s of ν : H → K^{×} for a closed subgroup H with Sp_{2n}(K) ⊂ H ⊂ GSp_{2n}(K), Aut_{s}(H)is given by

{(τ, g) ∈ Aut(K) ⋉ P GSp_{2g}(K)|τ(ν(H)) = ν(H)}.

3. We have a canonical split exact sequence

1 → Hom(K^{×}, Z(Sp_{2n}(K))) → Aut_{ν}(GSp_{2n}(K)) → Aut(Sp_{2n}(K)) → 1,

where Z(Sp_{2n}(K)) is the center {±1} of Sp_{2n}(K).

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).

Let k = ℚ if K is global and k = ℚ_{p} if K is a p–adic field. Since σ(U) 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 \overline{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∘\overline{σ}}.

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] 3.4.2 or [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(\overline{σ}(t)); so, \overline{σ} is also the identity map.

We now assume that K ≠ k. For the set of simple roots Δ of T with respect to 𝔑, ⨁_{α∈Δ} 𝔑_{α} ↪ 𝔑 induces an isomorphism ⨁_{α∈Δ} 𝔑_{α} ≅ 𝔑/[𝔑, 𝔑]. In other words, {𝔑_{α}|α ∈ Δ} generates 𝔑 over K. The K–vector space structure of 𝔑 induces an embedding i_{1} : K ↪ End_{k}(𝔑) of k–algebras. Since σ induces σ_{𝔑} ∈ End_{k}(𝔑), we have another embedding i_{2} = σ_{𝔑}^{−1}i_{1}σ_{𝔑} of K into End_{k}(𝔑). In End_{k}(𝔑/[𝔑, 𝔑]), 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

{t ∈ T |α(t) = β(t) for all α, β ∈ Δ}

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 \overline{σ}, we have \overline{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 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 = i_{2}^{−1} ∘ i_{1} ∈ Aut(K). Hence, σ is τ–linear for τ ∈ Aut(K) (that is, σ(ξv) = ξ^{τ} σ(v) for ξ ∈ K). 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_{σ}^{−1} 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 3.3:

Proposition 3.6. If K is a finite extension of ℚ_{p} for p > 2 with integer ring O, we have

Aut(Sp_{n}(O)) = Aut(P GSp_{n}(O)) = Aut(K) ⋉ P GL_{n}(O),

and a canonical split exact sequence

1 → Hom(O^{×}, Z(Sp_{2n}(O))) → Aut_{ν}(GSp_{n}(O)) → Aut(Sp_{2n}(O)) → 1,

where Z(Sp_{2n}(O)) = {±1} is the center of Sp_{2n}(O).

3.3 Quasi-split unitary groups

Let M/K be a p–adic quadratic extension with p–adic integer rings R/O, and consider the quasi split unitary group

G(K) = {α ∈ GL_{n}(M)|αI_{n}^{t}α^{c} = ν(α)I_{n}} and G(O) = GL_{n}(R) ∩ G(K),

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} = ( \begin{matrix}1_{m} & 0 \\ 0 & −1_{m} \end{matrix} ) 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

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/ℚ_{p}}𝔾_{m}×Res_{K/ℚ_{p}}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 closure M^{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 have

1. Aut(G_{1}(A)) = Aut(P G(A)) = Aut(M) ⋉ P G(A),

2. We have a canonical split exact sequence:

1 → Hom(Z^{G}(A), Z(G_{1}(A))) → Aut_{μ}(G(A)) → Aut(G_{1}(A)) → 1,

where Z(G_{1}(A)) is the center of G_{1}(A) and is isomorphic to μ_{n}(A).

Proof. We start with a brief sketch of the argument. A standard Borel subgroup of G (i.e., a standard minimal parabolic subgroup) is given by the subgroup B made up of all upper triangular matrices. We consider the subgroup U ⊂ B made up of upper unipotent matrices. If σ ∈ Aut_{μ}(G(K)), σ(U) is again generated by unipotent elements. Thus by [B] 6.5, σ(U) is a conjugate of U in G(K). Then by the same argument in the case of GL(n), modifying σ by an element in Aut(M) ⋉ G(K), we may assume that σ fixes B. Again by the same argument as in the case of GL(n), we conclude that σ = 1. Thus Aut(P G(K)) and Aut(G_{1}(K)) are given by Aut(M) ⋉ P G(K), and further assuming that p is odd and unramified in M/ℚ_{p}, Aut(P G(O)) and Aut(G_{1}(O)) are given by Aut(M) ⋉ P G(O). If σ ∈ Aut_{μ}(G(A)), then we can write σ(g) = h_{σ}τ(g)h_{σ}^{−1} 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_{σ}^{−1} 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).

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) = ( \begin{matrix}1 & x & y \\ 0 & 1 & −x^{c} \\ 0 & 0 & 1 \end{matrix} ) ∈ 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). 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) = ( \begin{matrix}1 & w & x & z \\ 0 & 1 & y & x^{c}−yw^{c} \\ 0 & 0 & 1 & −w^{c} \\ 0 & 0 & 0 & 1 \end{matrix} ) ∈ GL_{4}(M)| \begin{matrix}y=y^{c} and \\ z^{c}−z=xw^{c}−wx^{c} \end{matrix} \} .

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 that σ preserves the root space decomposition M ⊕ K of 𝔑/[𝔑, 𝔑]. We write σ_{M} (resp. σ_{K}) for the ℚ_{p}-linear map induced by σ on M (resp. K). Thus modifying σ by conjugation of t ∈ T (K), we may assume that σ(1, 1) = (1, 1) for (1, 1) ∈ M ⊕ K. Writing the character of T giving the action of T on M (resp. K) by χ_{M} (resp. χ_{K}), we have χ_{M}(t(a, b, ν)) = ab^{−1} and χ_{K}(t(a, b, ν)) = bb^{c}ν. Then on M (resp. on K), the action of T gives rise to the multiplication by elements of M^{×} = Im(χ_{M}) (resp. in K^{×} = Im(χ_{K})), which is preserved by σ_{M} (resp. by σ_{K}); that is, we have σ_{M}(χ_{M}(t)w) = χ_{M}(σ(t))σ_{M}(w) and σ_{K}(χ_{K}(t)y) = χ_{K}(σ(t))σ_{K}(y). Since σ fixes (1, 1), we find that σ_{M} ∈ Aut(M) and σ_{K} ∈ Aut(K) satisfying χ_{M} ∘ σ = σ_{M} ∘ χ_{M} and χ_{K} ∘ σ = σ_{K} ∘ χ_{K}. By the commutator relation [u(w, y, ∗, ∗), u(w′, y′, ∗, ∗)] = u(0, 0, wy′ − w′y, ∗), we find that σ_{M}|_{K} = σ_{K}. Then modifying σ by the element σ_{M} ∈ Aut(M), we find that σ fixes \overline{T} = B/U, and again modifying σ by the conjugation by an element in U(K), we bring σ to preserve T ⊂ B; so, σ induces the identity map on B. Out of this, we conclude that σ is the identity map on G(K) by the same argument as in the case of GL(n) and GSp(2n), because σ coincides with an inner automorphism on an open neighborhood of the identity in G(K) (by the argument at the level of the Lie algebra).

In the general case of n > 4, if n = 2m + 1 is odd, we may identify 𝔑/[𝔑, 𝔑] = M^{m} as M-vector space by (u_{i,j}) ∈ 𝔑 ↦ (u_{1,2}, …, u_{m,m+1}). On the j-th factor, t = diag[a_{1}, …, a_{n}] ∈ T (K) acts through the multiplication by a_{j}a_{j+1}^{−1} ∈ M^{×}. If n = 2m is even, 𝔑/[𝔑, 𝔑] ≅ M^{m−1} ⊕ K by sending upper unipotent matrices (u_{i,j}) ∈ U(K) to (u_{1,2}, …, u_{m,m+1}). On the first j-th factors with j < m, t = diag[νa_{1}, …, νa_{m}, a_{m}^{−c}, …, a_{1}^{−c}] ∈ T (K) (a_{j} ∈ M^{×} and ν ∈ K^{×} acts through the multiplication by a_{j}a_{j+1}^{−1} ∈ M^{×} and on the m-th factor, t acts through the multiplication by νa_{m}a_{m}^{c} ∈ K^{×}). More generally, writing 𝔑_{j} = [𝔑_{j−1}, 𝔑_{j−1}] starting with 𝔑_{1} = [𝔑, 𝔑], 𝔑_{j}/𝔑_{j+1} is a T (K)-module under the conjugation action. We go in the same way as in the case of GL(n): modifying σ by an inner automorphism of an element of T, we may assume that σ fixes 1 = (1, …, 1) ∈ M^{m} if n = 2m + 1 and that if n = 2m, σ fixes 1 = (1, …, 1, 1) ∈ M^{m−1} ⊕ K. Once σ is normalized in this way, we see that σ on B/U and σ on 𝔑/[𝔑, 𝔑] are the action of an element of Aut(M) coordinate-wise. The action of Aut(M) is faithful if n ≥ 3 because we have a factor M in 𝔑/[𝔑, 𝔑]. We modify σ therefore by an element of Aut(M); then, σ is T (K)-linear on 𝔑/[𝔑, 𝔑]. Once this is established, we verify, using commutator relations, that σ commutes with the conjugation action of T on 𝔑_{j}/𝔑_{j+1} for all j. Then modifying again by conjugation of an element of U(K), we conclude that σ is the identity on B, and the rest is the same as the proof in the case of GL(n) and GSp(2n).

If p is odd and unramified in M/ℚ_{p}, the nilpotent Lie algebra 𝔑_{/O} is the direct sum of its root spaces as T (O)-modules. Then the above argument done over the field K can be checked word-by-word over O, and we get the same assertion for A = O.

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Haruzo Hida

Department of Mathematics UCLA Los Angeles, CA 90095-1555 hida@math.ucla.edu