Construction of elements in the Lie algebra

Let p be a prime number, F a finite extension of Qp, and GF the Galois group of Q_p over F. We suppose that we are given a continuous p-adic representation (12) ρ: GF→GL(V),

where V is a Q_p-vector space of finite dimension d. As before, we write GV=ρ(GF). If we are to successfully apply the Lie algebra criteria of §2 to study the GV-cohomology of V, we must be able to construct elements in the Lie algebra

L(GV)_Q

p=L(GV)⊗

Q_p

Q_p.

The aim of this section is to take an important first step in this direction when V is a semistable Galois representation in the sense of Fontaine [14].

Let F0 denote the maximal unramified extension ofQ_p contained in F. We recall that V is said to be semistable if

D(V)⁼(Bst⊗

Q_pV)^GF

has dimension d over F0, where d⁼dim_Q

p(V); here Bst is Fontaine’s ring for semistable representations (see [14]). Assume from now on that V is semistable. Then D(V) is a filtered (ϕ, N)-module in the sense of [13], [14]. We recall briefly the main properties of D(V). Firstly, D(V) is endowed with an F0-linear endomorphism N, called the monodromy operator. Recall that the representation is said to be crystalline if N⁼0. Secondly, D(V) is endowed with an isomorphism of additive groups ϕ : D(V) → D(V), which is σ-linear in the sense that ϕ(av)⁼σ(a)ϕ(v) for a in F0 and v in D(V); here σ denotes the arithmetic Frobenius in the Galois group of F0

over Qp (i.e. σ operates on the residue field of F0 by raising to the p-th power). Let kF denote the residue field of F, and suppose that kF has cardinality q⁼p^f. Hence

(13) Φ⁼ϕ^f

will be an F0-linear automorphism of D(V), and we shall refer to it as the Frobenius endomorphism associated to the filtered module. Recall that we have

NΦ⁼qΦN.

By definition, Φ is an automorphism of a vector space on which GF acts trivially. It is therefore somewhat surprising that we shall show, using the Tannakian formalism of [9] and an elementary fact about Fontaine’s theory for unramified representations, that Φ gives rise to interesting elements in the Lie algebra L(GV)_Q

p of GV=ρ(GF) of our original Galois representation.

We recall the extension of the p-adic logarithm to the multiplicative group of Q_p. Let O denote the ring of integers of Q_p, O^× the group of units of O, and m the maximal ideal of O. The usual series for logz converges on 1 +m. Let µ denote the group of all roots of unity of Q_p of order prime to p. Then O ^×=µ×(1 +m). We extend logz to O^× by specifying that log(µ)⁼0. Now fix any non-zero element π of Q_p whose absolute value is less than one. If x is any element of Q_p^×, we can write x⁼π^ay, where a ∈ Q and y ∈ ^O×. We then define log_π(x)⁼ log( y). Note that this is well-defined as the ratio of any two such y must be a root of unity.

Theorem 3.1. — Assume that V is a semistable Galois representation, and let D(V) be the associated filtered (ϕ,N)-module. Let λ1, ...,λd denote the roots in Qp of the characteristic polynomial of the automorphism Φ⁼ϕ^f of D(V). Then there exists X in the Lie algebra L(GV)_Q

p

ofGV=ρ(GF) such that the characteristic polynomial of X on V_Q

p has roots log_π(λ1), ..., log_π(λd).

We will also need an analogue of this result for the Lie algebra L(HV)_Q

p, where, as before, HV=GV∩SL(V). We recall from [13], [14] that the F-vector space D(V)F=F⊗

F₀D(V) is endowed with a canonical decreasing filtration FilⁱD(V)F (i ∈ Z) of F-subspaces such that FilⁱD(V)F=D(V)F for i sufficiently small and FilⁱD(V)F=0 for i sufficiently large. This filtration enables us to define the so called Hodge-Tate weights of D(V). These are a family of integers

(14) i16i26...6id,

which are defined as follows. Any integer h occurs in (14) if Fil^hD(V)F=| Fil^h+1D(V)F, and when h does occur its multiplicity in (14) is the F-dimension of the quotient Fil^hD(V)F/Fil^h+1D(V)F. We define

(15) t⁼

Xd

h= 1

ih.

Let det : GF → Z^×_p denote the character of GF obtained by composing the representation ρ with the determinant map. We write ξ: GF→Z^×_p for the cyclotomic character of GF, i.e. the character giving the action of GF on Tp(µ). As V is a semistable GF-module, it is known (see [14, Prop. 5.4.1]) that the restriction of det to the inertial subgroup IF is equal to the restriction of ξ^−t to IF, where t is given by (15). Further,

thep-adic valuation of the determinant of Φ is equal to that of q^t. If we set detΦ⁼q^tu, with u ∈ F0 a unit, then det : GF → Z^×_p is ηξ^−t with η an unramified character and, moreover, η is of finite order if and only if u is a root of unity.

Theorem3.2. — Assume that Vis a semistable Galois representation, and that its determinant character det: GF → Z^×_p coincides on an open subgroup of GF with ξ^−t where ξ is the cyclotomic character, and t is given by (15). Let λ1, ...,λd denote the roots in Q_p of the characteristic polynomial of the automorphism Φ⁼ϕ^f of D(V). Then there exists X in the Lie algebra L(HV)_Q

p of HV=ρ(GF)∩SL(V) such that, for a suitable ordering of λ1, ...,λd, the roots of the characteristic polynomial of X on V_Q

p are given by log_π(λ1q⁻ⁱ¹), ..., log_π(λdq^−id), where q⁼p^f, and i1, ...,id

denote the Hodge-Tate weights (14).

Before embarking on the proof of these two theorems, we recall some basic definitions about algebraic groups, and the p-adic logarithms of their points. Let W be any finite dimensional vector space over some finite extension K of Q_p. We write GLW for the general linear group of W, considered as an algebraic group over K.

Thus, for each finite extension M of K contained in Q_p, the group GLW(M) of M-points of W is the group of M-automorphisms of W⊗

KM. If J denotes any algebraic subgroup of GLW, we write L( J) for its Lie algebra, which coincides with the Lie algebra L( J(K) ) of the p-adic Lie subgroup J(K). Now take θ to be any element of GLW(K). We can write θ⁼su, where s is semisimple, u is unipotent, and s and u commute. As u is unipotent, some power of u−1 is zero, and so we can define log(u) by the usual series log(u)^=P∞_{n= 1}(−1)^{n−1 (u−}_n¹⁾ⁿ. We define log_π(s) as follows. We can write Q_p⊗KW⁼ ⊕Wi, where Wi is some subspace on which the semisimple element s acts via some eigenvalue αi. We then define log_π(s) to be the endomorphism of Q_p⊗KW, which operates on Wi by log_π(αi). In fact, it is clear that if π belongs to K, log_π(s) belongs to the endomorphism ring of W over the original base field K. We finally define

log_π(θ)⁼log_π(s) + log(u).

The automorphism θ topologically generates a compact subgroup of GLW(K) if and only if its eigenvalues αi are units. If this is the case, then log_π(αi), and therefore log_π(θ), do not depend of the choice of the αi. In fact, we can define log(θ) in a more natural manner. Let r denote the cardinality of GLm(k), where m is the dimension of W over K, and k is the residue field of K, and put β⁼θ^r. Our hypothesis on θ shows that θ must stabilize a lattice in W, and so it is clear that the matrix A of β relative to a K-basis of W coming from this lattice must satisfy A ≡1 modπK, where πK is any

local parameter of K. We can then define log(θ)⁼1

r

X∞ n= 1

(−1)ⁿ⁻¹(A−1)ⁿ

n .

The following lemma is well-known (cf. [3, Chap. III, §7.6, Propositions 10 and 13]).

Lemma 3.3. — Let W be a finite dimensional vector space over K, where K is a finite extension of Qp, and let θ be any element of GLW(K). If θ topologically generates a compact subgroup of GLW(K), then log_π(θ)⁼ log(θ). If J is any algebraic subgroup of GLW, and θ belongs to J(K), then log_π(θ) belongs to the Lie algebra L( J) of J.

Proof of Theorem 3.1. — If H is any subgroup of GLV(Qp), we define H^alg to be the Zariski closure of H in GLV i.e. the intersection of all algebraic subgroups J of GLV such that J is defined over Qp and J(Qp) contains H. As earlier, let IF denote the inertial subgroup of the Galois group GF. We put

GV=ρ(GF), IV=ρ(IF).

We then have the algebraic groups G^alg_V and I^alg_V in GLV, and we can consider the four Lie algebras L(GV), L(IV), L(G^alg_V ), L(I^alg_V ). By a basic result of Serre and Sen, [23], we have

(16) L(IV)⁼L(I^alg_V).

Hence we have the inclusions

(17) L(I^alg_V )⊂ L(GV)⊂L(G^alg_V ).

By definition, the image of the Galois representation ρ of GF is contained in G^alg_V (Qp).

Thus, for each representation α of the algebraic group G^alg_V in a finite dimensional Qp-vector space Vα, we obtain a new Galois representation:

(18) ρα: GF →G^alg_V (Qp)→GLV_α(Qp)⁼GL(Vα).

The main idea of the proof of Theorem 3.1 is to work with this new Galois representation ρα for a suitable choice of α. We first note that, for every such α, the Galois representation ρα is also semistable. Indeed, it is known [9, Proposition 2.20] that the representation α of G^alg_V is a subquotient of a finite direct sum of copies of tensor products of the tautological representation of G^alg_V and its dual. But it is also known that the category of semistable representations of GF is stable under the Tannakian operations in the category of finite dimensional p-adic representations of GF and so ρα must be semistable, because ρ is semistable. For simplicity, we write Dα

for the filtered (ϕ, N)-module associated with ρα. In general, we write a subscript α on each object associated with the original representation ρ to denote the corresponding object attached to ρα.

We put GV_α=ρα(GF). The algebraic groups I^alg_V , G^alg_V clearly act on V_α, and we denote the images of these groups in GLV_α by I^alg_Vα, G^alg_Vα, respectively.

For the rest of the proof, we fix the following representation of G^alg_V . As I^alg_V is clearly normal in G^alg_V , there exists a representation α0 of G^alg_V in GLV_α0 whose kernel is precisely I^alg_V . For simplicity, we write V0 instead of Vα₀, ρ0 instead of ρα₀, and so on for all objects attached to the representation ρα₀. We write Fr for the arithmetic Frobenius in GF/IF. Thus ρ0(Fr) topologically generates GV₀=ρ0(GF). Hence the Lie algebra L(GV₀) of GV₀ is the one dimensional Q_p-subspace of End (V0) generated by log(ρ0(Fr) ); note that log(ρ0(Fr) ) is defined as explained earlier because ρ0(Fr) generates a compact group. Clearly G^alg_V0 is the smallest algebraic subgroup of GLV₀ containing ρ0(Fr), and is abelian. Let π : L(G^alg_V ) → L(G^alg_V0) denote the natural surjection. Since L(I^alg_V )⁼L(IV), we evidently have

(19) L(GV)⁼π⁻¹(L(GV₀) ).

There are two basic steps in the proof of Theorem 3.1. The first uses all the representations α of G^alg_V to construct, via the Tannakian formalism, an element in the Lie algebra L(G^alg_V )_Q

p having the same eigenvalues as the endomorphism log_π(Φ) of D(V). The second step exploits the unramified Galois representation ρ0 arising from α0 to show, using Fontaine’s theory in the modest case of unramified Galois representations, together with the key fact (19), the existence of the desired element X in L(GV)_Q

p.

We now give the first step. Denote by Rep(G^alg_V ) (respectively, Rep(G^alg_Vα) ) the category of all finite dimensional Qp-representations of the algebraic group G^alg_V (respectively, G^alg_Vα). Since α gives rise to a homomorphism from G^alg_V to G^alg_Vα, we can identify Rep(G^alg_Vα) with a sub-category of Rep(G^alg_V ). We refer to [9] for the following basic facts about Tannakian categories and their associated formalism. We note that Rep(G^alg_V ) is a Tannakian category, and that Rep(G^alg_Vα) is a sub-Tannakian category. Let Vec_F

0 denote the category of all finite dimensional vector spaces over the field F0. We have two fibre functors over F0,

ωG: Rep(G^alg_V )→Vec_F

0, ωD: Rep(G^alg_V )→Vec_F

0

which are defined by ωG(α)⁼Vα⊗

Q_pF0, ωD(α)⁼Dα;

we recall that D_α is the filtered (ϕ, N)-module attached by Fontaine’s theory to the semistable Galois representation ρα. It is proven in [9, Proposition 2.8] that we can identify G^alg_Vα over F0 with the group of ⊗-automorphisms of ωG restricted to the sub-category Rep(G^alg_Vα) of Rep(G^alg_V ). We define G^alg_D to be the algebraic group of

⊗-automorphisms of the fibre functor ωD. Let G^alg_Dα denote the algebraic group of

⊗-automorphisms of ωD restricted to the sub-category Rep(G^alg_Vα). Both G^alg_D and G^alg_Dα are defined over F0, and G^alg_Dα is the image of G^alg_D in GLD_α.

Write =s for the affine algebraic variety of ⊗-isomorphisms from ωD to ωG, and

=sα for the variety of ⊗-isomorphisms from ωD and ωG restricted to Rep(G^alg_Vα). The variety =sα is a right torsor under G^alg_Dα, and a left torsor under G^alg_Vα. Now the choice of a point i in =s(Q_p) gives, for each α, a point i_α in =s_α(Q_p). This point i_α gives rise to an isomorphism

(20) ηi_α : Dα⊗

F₀Q_p'→(Vα)_Q

p=Vα⊗

Q_pQ_p which induces an isomorphism of algebraic groups

G^alg_Dα×

F₀Q_p' G^alg_Vα×

Q_pQ_p and an isomorphism of Lie algebras

L(G^alg_Dα)⊗

F₀Q_p'L(G^alg_Vα)_Q

p.

Moreover, a different choice of the point i has the effect of changing these two isomorphisms by an inner automorphism of G^alg_Dα(Q_p) or an inner automorphism of G^alg_Vα(Q_p).

For each α in Rep(G^alg_V ), we recall that the associated Galois representation ρα

is semistable, and that the filtered (ϕ, N)-module D_α attached to ρα has the F0 -auto-morphism Φα=ϕ_α^f. Thus the family of all Φα define an automorphism of the functor ωD, and so we obtain a canonical element of G^alg_D(F0), which we again denote by Φ. Recalling the definition of log_π(Φ) given earlier, it follows that it belongs to the Lie algebra L(G^alg_D) of the algebraic group G^alg_D. Now fix any point i in =s(Q_p), and let ηi

denote the isomorphism (20) for the tautological representation of G^alg_V in GLV. We define the endomorphism Xi of V_Q

p by

(21) Xi=ηi◦log_π(Φ)◦η⁻_i ¹.

As log_π(Φ) belongs to L(G^alg_D), it follows that Xi belongs to L(G^alg_D)_Q

p. Moreover, if λ1, ...,λd denote the eigenvalues of Φ with multiplicities, it is clear that the eigenvalues

of Xi with multiplicities are log_π(λ1), ..., log_π(λd). This construction of Xi completes the first basic step in the proof of Theorem 3.1.

To finish the proof of Theorem 3.1, we must show that Xi belongs to the Lie subalgebra L(GV)_Q

p of L(G^alg_V )_Q

p. Here we make use of the representation α0 of G^alg_V . Let L(α0) be the map from L(G^alg_V )_Q

p to L(GLV₀)_Q

p induced by α0. In view of (19), it suffices to show that

(22) L(α0)(Xi)∈L(GV₀)_Q

p.

Let i0 denote the point in =sα₀(Q_p) arising from i. We have (23) L(α0)(Xi)⁼ηi₀ ◦log_π(Φ0)◦η⁻i₀¹,

whereΦ0=ϕ₀^f comes from the filtered (ϕ, N)-module D0. Since the representation ρ0 of GF is unramified, it is well-known that Φ0 fixes a lattice (see [13]). Hence Φ0 generates a compact subgroup of GLD₀(F0). Thus log_π(Φ0)⁼ log(Φ0) is independent of the choice of π. Moreover, GV₀ is topologically generated byρ0(Fr), and so G^alg_V0 is abelian, whence G^alg_D0×

F₀Q_p is also abelian. In view of the remark made earlier, we see that the right hand side of (23) does not depend on the choice of i in =s(Q_p) nor on the choice of π. In fact, we see that the right hand side of (23) is the same for any choice of i0

in =sα₀(M), where M is any extension field of F0. We now make a suitable choice of M and i0, which will enable us to compute explicitly the right hand side of (23).

We will use the following lemma about Fontaine’s theory for arbitrary unramified Galois representations ψ : GF → GL(W), where W is a Q_p-vector space of finite dimension. Let K denote the maximal unramified extension of Q_p, and write K for^b the completion of K. In the following, GF acts on K^b ⊗

Q_pW by τ(a⊗w)⁼τ(a)⊗τ(w) for a in K and^b w in W.

Lemma 3.4. — Assume that the Galois representation ψ is unramified. Consider the F0 -subspace of K^b ⊗

Q_pW given by U⁼(K^b ⊗

Q_pW)^GF. Then W is crystalline in the sense of Fontaine [13], [14] and the associated filtered (ϕ,N)-module is D(W)⁼U. Moreover, the F0-automorphism ΦW=ϕW^f of D(W) is given by

(24) β◦ΦW◦β⁻¹⁼ψ(Fr⁻¹),

where Fr is the arithmetic Frobenius in GF/IF, and β : K^b ⊗

F₀U'→ Kb ⊗

Q_pW is the isomorphism obtained by extending scalars on the F0-subspace U of W.

Proof. — The fact that β is an isomorphism is, of course, not obvious, and we refer to the Appendix of Chapter 3 of [25] for a proof. Also, K is naturally included^b in Fontaine’s ring Bcris, and so U is an F0-subspace of D(W) (see [13], [14]). But the fact that β is an isomorphism shows that the F0-dimension of U must be equal to the Q_p-dimension of W, and hence D(W)⁼U. Let ϕB denote the action of Frobenius on the Fontaine ring Bcris (see [13], [14]). We write σ for the arithmetic Frobenius in the Galois group of K over Qp. The restriction of ϕB to K is the arithmetic Frobenius^b σ. Let 1W denote the identity map of W. By definition, ϕW is the restriction to U of the map σ⊗

Q_p1W. Hence the σ-linear extension σ⊗

Q_pϕW of ϕW to K^b ⊗

F₀U satisfies β◦(σ⊗

F₀ϕW)◦β⁻¹⁼σ⊗

Q_p1W.

Raising both sides to the power f and extending ΦW linearly to the whole of K^b⊗

F₀U, we conclude that

(25) β◦(σ^f⊗

F₀1U)◦ΦW◦β⁻¹⁼σ^f⊗

Q_p1W.

As the restriction to K of the action of G^b F on Bcris is the usual action, U is the F0-subspace of K^b ⊗

Q_pW fixed by the σ^f-linear extension of ψ(Fr) to K^b ⊗

Q_pW. Extending ψ(Fr) linearly to K^b ⊗

Q_pW, it follows that (26) β◦(σ^f⊗

F₀1U)◦β⁻¹⁼(σ^f⊗

Q_p1W)◦ψ(Fr).

The equation (24) follows on comparing (25) and (26). This completes the proof of Lemma 3.4.

We can at last complete the proof of Theorem 3.1. We apply Lemma 3.4 to the unramified Galois representation ρ0 of GF in V0. We have the isomorphism

(27) β0:K^b ⊗

F₀D0 'K^b ⊗

Q_pV0,

and the analogous isomorphisms for all the unramified Galois representations in the Tannakian category generated by ρ0. Thus these isomorphisms define a point i0 in

=s0(K). On applying log to both sides of (24), we deduce from (23) that^b (28) L(α0)(Xi)⁼logρ0(Fr⁻¹).

As the right hand side of (28) is clearly an element of L(GV₀), the proof of Theorem 3.1 now follows from (19).

We shall need the following lemma for the proof of Theorem 3.2. Recall that ξ : GF →Z^×_p is the cyclotomic character.

Lemma 3.5. — Assume that V is a semistable representation, and let det : GF → Z^×_p be its determinant character. Then the following statements are equivalent:

(i) The map det coincides on an open subgroup of GF with ξ^−t, where t is given by (15).

(ii) If δ⁼δ(Φ) denotes the determinant of the endomorphism Φ⁼ϕ^f of D(V), then (29) log_π(δ)⁼log_π(q^t), where q⁼p^f.

In particular, when t=|0, these equivalent assertions imply that the image of det is infinite.

Proof. — Recall that d is the Q_p-dimension of V, and put W⁼(Λ^dV)(t). As the restriction of the determinant character of GF to IF is equal to ξ^−t, we see that W is an unramified representation of GF of dimension 1 over Q_p. Let ψ : GF → Q^×_p denote the character giving the action of GF on W. We can compute ΦW in terms of δ as follows. We have that D(W)⁼Z[−t], where Z⁼Λ^dD(V), and Z[−t] means that the automorphism ΦZ of Z is replaced by the automorphism q^−tΦZ. Hence ΦW is multiplication by δ.q^−t, and we recall that δ.q^−t is a p-adic unit. Now assertion (i) is equivalent to saying that GF/IF acts on W via a finite quotient. Hence the arithmetic Frobenius Fr in GF/IF must act on W via a root of unity, and therefore we have ψ(Fr)ⁿ⁼1 for some integer n. Applying Lemma 3.4 to W, we conclude that Φⁿ_W⁼1.

Hence (i) is equivalent to δ.q^−t=ζ,

for some n-th root of unity ζ. The lemma is now obvious as log_π(ζ)⁼0.

Proof of Theorem 3.2. — Assume now that the hypotheses of Theorem 3.2 hold for V. Thenλ1, ...,λd=δ, where δis the determinant ofΦ. By Theorem 3.1, there exists an element X in L(GV)_Q

p, whose characteristic polynomial has roots log_π(λ1), ..., log_π(λd).

Recall that HV=GV ∩SL(V), so that L(HV)_Q

p consists of the elements of L(GV)_Q

p of trace zero. But, by Lemma 3.5,

(30) tr(X)⁼ _Σ^d

i= 1

log_π(λi)⁼ log_π(q^t).

Hence, ast⁼ _Σ^d

i= 1

ij, it suffices to find another element Y in L(GV)_Q

p, whose characteristic polynomial has roots

(31) log_π(λ1q⁻ⁱ¹), ..., log_π(λdq^−id),

for a suitable ordering of λ1, ...,λd. Since V is semistable, it is of Hodge-Tate type (see [28,§2]), and we now exploit this fact. Let C denote the completion ofQ_p. We consider VC=C⊗Q_pV, endowed with the semilinear action of GF given byτ(c⊗v)⁼τ(c)⊗τ(v) for all τ in GF, c in C, and v in V. For each m in Z, we write VC{m} for the F-subspace of VC consisting of all v such that τ(v)⁼ξ^m(τ)v for τ in GF. Put VC(m)⁼C⊗FVC{m}, again endowed with the semilinear action of GF. To say that V is of Hodge-Tate type means that we have a direct sum decomposition

(32) VC= ⊕

m_∈JVC(m),

where J is some finite set of integers. It is well-known [13], [14, Theorem 3.8] that the integers in J are the negatives of the distinct integers occuring in the sequence of Hodge-Tate weights (14), and that the C-dimension of VC(m) is equal to the F-dimension of Fil^−mD(V)F/Fil^−m+1D(V)F. Moreover, the direct sum decomposition (32) allows us to define a homomorphism

(33) µ : G_m →I^alg_V ×

Q_pC

of algebraic groups over C, where, for c ∈C^×, µ(c) is the automorphism of VC given by the formula

µ(c)(x)⁼c^mx for all x in VC(m) (m∈J).

As is explained in [28, §1.5], the image of µ is contained in I^alg_V ×Q_pC.

As in the proof of Theorem 3.1, let =s denote the affine algebraic variety of

⊗-isomorphisms from the fibre functor ωD to the fibre functor ωG. Again, we fix a point i in =s(Q_p), and we write ηi for the isomorphism (20), when α is taken to be the tautological representation of G^alg_V in GLV. Put

(34) Ω⁼ηi◦Φ◦η⁻_i ¹,

where Φ is given by (13). Thus Ω belongs to G^alg_V (Q_p). Now it is well-known [2, Theorem 4.4] that there then exist commuting elements s and u in G^alg_V (Q_p) such that s is semisimple, u is unipotent, and

(35) Ω⁼su⁼us.

Let Θ be the smallest algebraic subgroup (over Q_p) which contains s. As s is semisimple, Θ is a multiplicative group (see [10, Chap. IV, §3]). Let Θ⁰ be the connected component of Θ so that Θ⁼Θ⁰×P, where P is a finite group. Suppose that F⁰ is a finite extension of F such that the residue field extension has degree a multiple of #P. Passing to the extension F⁰ and working with the Φ and s associated to F⁰, we may clearly assume that the multiplicative group Θ is a torus.

Let T denote a maximal torus in G^alg_V ×Q_pQ_p containing the torus Θ. Returning to our homomorphism µ coming from p-adic Hodge theory, we choose a maximal torus in G^alg_V ×Q_p C containing the image of µ. But all maximal tori in G^alg_V ×Q_p C are conjugate [B, Prop. 11.3], and so there exists g in G^alg_V (C) such that µ⁰⁼gµg⁻¹ has image in T. Moreover, the induced map µ⁰ : G_m → T is necessarily defined over Q_p (see [2, Proposition 8.11]). Hence µ⁰(q) belongs to T(Q_p).

We recall that we are seeking to construct an element in the Lie algebra L(GV)_Q

p, whose characteristic polynomial has the roots (31). To complete the proof, we use once more the representation α0 of G^alg_V in GLV₀, whose kernel is precisely I^alg_V. The torus T acts on G^alg_V ×Q_p Q_p and its normal subgroup I^alg_V ×Q_pQ_p by inner automorphisms, and so also on the quotient G^alg_V0×Q_pQ_p. As G^alg_V0×Q_pQ_p is abelian, T acts trivially on it by inner automorphisms. Given an algebraic group M over Qp, let M_Q

p=M×Q_pQ_p denote its extension to Q_p. For an algebraic group H defined over Q_p we write Hu for its unipotent radical [2, §11.21] and L(H)u for the corresponding Lie algebra. Now we have the exact sequence of Lie algebras over Q_p

0→L( (I^alg_V )_Q

p)u→L( (G^alg_V )_Q

p)u→L( (G^alg_V0)_Q

p)u→0

[2, Cor. 14.11]. The action of T on the algebraic groups induces the adjoint action of T on the Lie algebras. We denote with the superscript T the maximal Lie subalgebras on which T acts trivially. Then, since representations of a torus are semisimple, we have the exact sequence

(36) 0→L( (I^alg_V )_Q

p)^T_u →L( (G^alg_V )_Q

p)^T_u →L( (G^alg_V0)_Q

p)u→0.

Here we have used the fact that T acts trivially on the Lie algebra L( (G^alg_V0)_Q

p)u, as GV₀ is abelian. Let u0 be the image of u in (G^alg_V0)_Q

p and let n⁰ be an element in L( (G^alg_V )_Q

p)^T_u which is a lift of log(u0). Consider the element u⁰⁼exp(n⁰); clearly u⁰ commutes with elements of T. In particular, we have now arranged that the elements µ⁰(q), s and u⁰ commute. It is plain that the roots of the characteristic polynomial of

Y⁼ log_π(s.µ⁰(q).u⁰)

are as in (31). As X and Y have the same image in L(G^alg_V0)_Q

p, by Theorem 3.1 and

p, by Theorem 3.1 and

Dans le document On the Euler-Poincaré characteristics of finite dimensional <span class="mathjax-formula">$p$</span>-adic Galois representations (Page 11-22)