ON WEIGHTS OF l-ADIC REPRESENTATION

S.G. Tankeev

Abstract. Let J be an abelian variety over a number field k, [k : ℚ]< ∞. Assume that Cent(End(J ⊗ \overline{k})) = ℤ. If the division ℚ-algebra End^{0}(J ⊗ \overline{k}) splits at a prime number l, then the l-adic representation is defined by the minuscule weights (microweights) of simple classical Lie algebras of types A_{m}, B_{m}, C_{m} or D_{m}.

Introduction

0.1. Let J be an abelian variety over a number field k ⊂ ℂ, [k : ℚ] < ∞. For a given prime number l consider the natural l-adic representation

ρ_{l} : Gal(\overline{k}/k) → GL(H_{et}^{1}(J ⊗ \overline{k}, ℚ_{l})).

We recall some conjectures and results concerning ρ_{l}.

0.2. Serre - Mumford - Tate conjecture [5], [9]. There exists a canonical isomorphism of Lie algebras

Lie Im(ρ_{l}) ≃ Lie[MT(J ⊗_{k} ℂ)(ℚ_{l})],

where MT(J ⊗_{k} ℂ) is the Mumford - Tate group associated to the canonical Hodge ℚ-structure on H^{1}(J ⊗_{k} ℂ, ℚ).

0.3. P.Deligne theorem [14, Theorem 0.5.1b]. Each simple factor g of the reductive Lie algebra Lie MT(J ⊗_{k} ℂ) ⊗ ℂ is a classical Lie algebra of type A_{m}, B_{m}, C_{m} or D_{m}, and the highest weight of any irreducible g-submodule E ⊂ H^{1}(J ⊗_{k} ℂ, ℚ) ⊗_{ℚ} ℂ is a minuscule weight (microweight).

We recall the list of minuscule weights: for type A_{m} (m ≥ 1) : ω_{1}, ω_{2}, ..., ω_{m}; for type B_{m} (m ≥ 2) : ω_{m}; for type C_{m} (m ≥ 3) : ω_{1}; for type D_{m} (m ≥ 4) : ω_{1}, ω_{m−1}, ω_{m};

Typeset by AMS-TEX

Supported by the Russian Foundation for Fundamental Research (grant 97-01-00647). Research at MSRI is supported in part by NSF grant DMS-9022140. This paper was written at the period of my stay at the Institut des Hautes Études Scientifiques (Bures-sur-Yvette) and at the Mathematical Sciences Research Institute (Berkeley) in Spring of 1997. I would like to thank the members of both institutions for their warm hospitality and excellent working conditions.

for type E_{6} : ω_{1}, ω_{6};

for type E_{7} : ω_{7}, where ω_{1}, ω_{2}, ..., ω_{m} are fundamental weights [2, Chapter 8, §7, Section 3].

In virtue of the Deligne theorem the following conjecture may be considered as a weak form of the Serre - Mumford - Tate conjecture:

0.4. Conjecture on minuscule weights [14, Conjecture 0.4]. Let J be an abelian variety over a number field k ⊂ ℂ. Then each simple factor g of the reductive Lie algebra Lie Im(ρ_{l}) ⊗ \overline{ℚ}_{l} is a classical Lie algebra of type A_{m}, B_{m}, C_{m} or D_{m}, and the highest weight of any irreducible g-submodule E ⊂ H_{et}^{1}(J ⊗ \overline{k}, ℚ_{l}) ⊗ \overline{ℚ}_{l} is a minuscule weight (microweight).

0.5. If End(J ⊗ \overline{k}) = ℤ and dim_{k} J is an odd integer, then by the well known J.-P.Serre theorem [3, Section 6.1] there exists a canonical isomorphism of the l-adic Lie algebras

Lie Im(ρ_{l}) ≃ Lie [MT(J ⊗_{k} ℂ)(ℚ_{l})].

In particular, the Serre - Mumford - Tate conjecture and the conjecture on minuscule weights hold for J. The analogous results are obtained for some abelian varieties of another types [10], [11], [12].

In this paper we prove the following main theorem.

0.6. Theorem . Let J be a simple abelian variety over a number field k, [k : ℚ] < ∞. Assume that Cent(End(J ⊗ \overline{k})) = ℤ. If the division ℚ-algebra End^{0}(J ⊗ \overline{k}) splits at a prime number l, then each simple factor g of the reductive Lie algebra Lie Im(ρ_{l}) ⊗ \overline{ℚ}_{l} is a simple Lie algebra of type A_{m}, B_{m}, C_{m} or D_{m}, and the highest weight of any irreducible g-submodule E ⊂ H_{et}^{1}(J ⊗ \overline{k}, ℚ_{l}) ⊗ \overline{ℚ}_{l} is a minuscule weight (microweight).

I am grateful to J.-P.Serre, D.Bertrand and Y.André for the stimulating discussions on this topic.

§1. Some remarks on l-adic representation

1.1. In what lollows V_{l} = H_{et}^{1}(J ⊗ \overline{k}, ℚ_{l}). Let G_{V_{l}} denote the algebraic envelope of the compact l-adic Lie group Im(ρ_{l}). It is well known that G_{V_{l}} coincides with the closure of Im(ρ_{l}) in the Zariski topology of the group GL(V_{l}).

Fix a non-Archimedean place v of the field k. Let k(v) is its residue field of characteristic p_{v}, k_{v} is the completion of the field k relative to the topology induced by the place v, \overline{v} is any extension of v to the field \overline{k}, D(\overline{v}) is the decomposition group (an element σ ∈ Gal(\overline{k}/k) belongs to D(\overline{v}) ⇔ σ(\overline{v}) = \overline{v} ⇔ ∀α ∈ \overline{k} \overline{v}(σα) = \overline{v}(α)), I(\overline{v}) ⊂ Gal(\overline{k}/k) is the inertia subgroup (an element σ ∈ Gal(\overline{k}/k) belongs to I(\overline{v}) ⇔ ∀α ∈ \overline{k} the relation \overline{v}(α) ≥ 0 implies \overline{v}(σα − α) > 0)).

It is well known that I(\overline{v}) is a normal subgroup in D(\overline{v}), and D(\overline{v})/I(\overline{v}) ≃ Gal(\overline{k(v)}/k(v)). We may identify Gal(\overline{k_{v}}/k_{v}) with the decomposition group D(\overline{v}).

1.2. Assume that v is a non-Archimedean place of k at which J has a good reduction. Let F\overline{v} ∈ Gal(\overline{k}/k) be the Frobenius element.

It is well known that the characteristic polynomial of ρ_{l}(F −1 ; \overline{v}) coincides with the characteristic polynomial of the Frobenius endomorphism π_{v} of the reduction J_{v} of J at v. We denote by Δ the set of all eigenvalues of ρ_{l}(F −1 ; \overline{v}) (counting multiplicities). Let Γ_{v} be a multiplicative subgroup of \overline{ℚ}^{×} generated by Δ.

By the Honda - Tate theory [13] ℚ[π_{v}] = ∏ K_{i}, K_{i} are number fields. The multiplicative group ℚ[π_{v}]^{×} determines a ℚ-torus T_{π_{v}} = ∏ R_{K_{i}/ℚ}(G_{mK_{i}}), where R_{K_{i}/ℚ} are the Weil restrictions of scalar functors. Let H_{v} be the smallest algebraic subgroup of T_{π_{v}} defined over ℚ, such that π_{v} ∈ H_{v}(ℚ). As is well known, H_{v} is a group of multiplicative type. The connected component of the identity in H_{v} is called the Frobenius torus T_{v}. Moreover, T_{v} is a ℚ-torus in G_{V_{l}} [3, Section 3b].

1.3. We denote by ℂ_{p_{v}} = (\overline{ℚ}_{p_{v}})^{∧} the completion of the algebraic closure of the field ℚ_{p_{v}}. According to Tate and Raynaud there exists the canonical Hodge - Tate decomposition

V_{ℂ_{pv}} = V_{p_{v}} ⊗_{ℚ_{pv}} ℂ_{p_{v}} = V_{ℂ_{pv}} (0) ⊕ V_{ℂ_{pv}} (−1),

and dim V_{ℂ_{pv}} (0)=dim V_{ℂ_{pv}} (−1) ([3, Section 4.3]).

We define a morphism of algebraic ℂ_{p_{v}}-groups

h_{V_{pv}} : G_{mℂ_{pv}} → GL(V_{ℂ_{pv}})

by the formula

h_{V_{pv}} (c)(x) = \{ \begin{matrix}x if x ∈ V_{ℂ_{pv}} (0), \\ c^{−1}x if x ∈ V_{ℂ_{pv}} (−1). \end{matrix}

The algebraic envelope of ρ_{p_{v}}(I(\overline{v})) is an algebraic subgroup of G_{V_{pv}}. By the theorem of S.Sen [7], the 1-parameter subgroup h_{V_{pv}} is contained in the algebraic envelope of ρ_{p_{v}}(I(\overline{v})). In particular, one has Im(h_{V_{pv}}) ⊂ G_{V_{pv}} (ℂ_{p_{v}}).

1.4. Lemma . Assume that Cent End^{0}(J ⊗ \overline{k}) is a totally real field, and J(k) contains all points of order p_{v}^{2} in J(\overline{k}). Let M ⊂ G_{V_{pv}} be the smallest normal algebraic subgroup for which Im(h_{V_{pv}}) ⊂ M(ℂ_{p_{v}}). Then the canonical representation of the semisimple Lie algebra [Lie (M)]^{ss} ⊗ \overline{ℚ}_{p_{v}} in V_{p_{v}} ⊗ \overline{ℚ}_{p_{v}} is defined by the minuscule weights of the classical Lie algebras (of types A_{m}, B_{m}, C_{m}, D_{m}).

Proof . By F.A.Bogomolov theorem [1] Lie Im(ρ_{p_{v}})=Lie(G_{V_{pv}}) and G_{V_{pv}} containes the group G_{m} of homotheties. By G.Faltings theorems [4] G_{V_{pv}} is reductive. In the case under consideration G_{V_{pv}} is connected [3, Proposition 3.6], and

G_{V_{pv}} = G_{m} · S_{V_{pv}} ,

where S_{V_{pv}} =[G_{V_{pv}} , G_{V_{pv}}] is the commutator subgroup of G_{V_{pv}} [3, Section 1.2.2b]. It is clear that M is a connected reductive algebraic group over ℚ_{p_{v}} with a nontrivial semisimple part M^{ss} = [M, M] (because Im (h_{V_{pv}}) is not contained in G_{m}(ℂ_{p_{v}})); the canonical representation V_{p_{v}} of M is faithful; if N is a normal connected algebraic subgroup of M, defined over ℚ_{p_{v}}, such that Im (h_{V_{pv}}) ⊂ N(ℂ_{p_{v}}), then N is a normal algebraic subgroup of G_{V_{pv}} ( indeed, let f : S_{V_{pv}}^{sc} → S_{V_{pv}} be the universal covering morphism; since M^{ss} is a normal algebraic subgroup of S_{V_{pv}}, the group (f^{−1}(M^{ss}))^{0} is a normal algebraic subgroup of S_{V_{pv}}^{sc}, and so we get the decomposition

S_{V_{pv}}^{sc} = (f^{−1}(M^{ss}))^{0} × H′′,

where (f^{−1}(M^{ss}))^{0} and H′′ are simply connected semisimple ℚ_{p_{v}}-groups; on the other hand, (f^{−1}(N^{ss}))^{0} is a normal connected subgroup of (f^{−1}(M^{ss}))^{0}, so we get the decompositions

(f^{−1}(M^{ss}))^{0} = (f^{−1}(N^{ss}))^{0} × H′

and

S_{V_{pv}}^{sc} = (f^{−1}(N^{ss}))^{0} × H′ × H′′;

consequently N^{ss} is a normal algebraic subgroup of S_{V_{pv}}, and N is a normal algebraic subgroup of G_{V p_{v}}), so N = M; the action of G_{mℂ_{pv}} on V_{ℂ_{pv}} = V_{p_{v}} ⊗_{ℚ_{pv}} ℂ_{p_{v}} defined by h_{V_{pv}} is of weights 0 and −1. In virtue of the well known results of J.-P.Serre [8, §3] the representation V_{p_{v}} of M is defined by minuscule weights of the classical Lie algebras. Lemma 1.4 is proved.

1.5. Corollary . Assume that Cent End^{0}(J ⊗\overline{k}) is a totally real field. Then for each prime number l the reductive Lie algebra Lie Im(ρ_{l}) ⊗ \overline{ℚ}_{l} has a simple factor g such that the canonical representation of g in V_{l} ⊗ \overline{ℚ}_{l} is defined by the minuscule weights of the classical Lie algebras.

Indeed, we can choose a non-Archimedean place v such that p_{v} = l. Replacing k by some finite algebraic extension, we may assume that J(k) containes all points of order p_{v}^{2} in J(\overline{k}).

§2. On Gal(\overline{ℚ}/ℚ)-invariant subsets of Δ · Δ

2.1. Proposition . Let J be an abelian variety over a number field k. Suppose that Cent End(J ⊗ \overline{k}) = ℤ and the division ℚ-algebra End^{0}(J ⊗ \overline{k}) splits at the prime number l.

After replacing k by some finite extension we may assume that for abelian variety J and for each non-Archimedean place v of k from some set S_{non−Arch}^{generic}(J, k, l) of places of Dirichlet’s density 1 the following conditions hold:

1) for a fixed integer n ≥ 2 such that l^{n} > (2 · dim_{k}J)^{2}, the l^{n}-torsion points of J(\overline{k}) are rational points over k, and k containes all the (l^{n})^{th} roots of unity;

2) p_{v} =char(k(v)) > (2 · dim_{k}J)^{2};

3) Norm_{k/ℚ}(v) = p_{v}; p_{v} is unramified in k;

4) J has a good reduction at v;

5) Γ_{v} is torsion-free, G_{V_{l}} is connected and ρ_{l}(F −1 ; \overline{v}) ∈ T_{v}(ℚ);

6) the Frobenius torus T_{v} is a maximal torus of G_{V_{l}} and

rank(Γ_{v}) = dim(T_{v}) = rank(G_{V_{l}}).

Proof . Cent End(J ⊗ \overline{k}) = ℤ by the condition of Proposition 2.1. By Albert’s classification [6, §21] End(J ⊗\overline{k}) = ℤ or End^{0}(J ⊗\overline{k}) is a quaternion division algebra over ℚ which splits at l. So

End(J ⊗ \overline{k}) ⊗ ℚ_{l} = \{ \begin{matrix}ℚ_{l} if J is of type I by Albert’s classification, \\ M_{2}(ℚ_{l}) if J is of type II or III by Albert’s classification. \end{matrix}

After replacing k by some finite extension we may assume that for abelian variety J and for k the condition (2.1.1) holds. Then the Dirichlet density of the set S_{non−Arch}^{generic}(J, k, l) of all non-Archimedean places of k satisfying the conditions (2.1.2) - (2.1.6) is equal to 1 [3, Lemma (2.1), Proposition (3.6), Corollary (3.8)]. On the other hand,

G_{V_{l}} = G_{m} · S_{V_{l}},

where S_{V_{l}}=[G_{V_{l}}, G_{V_{l}}] is the commutator subgroup of G_{V_{l}} [3, Section 1.2.2b]. Moreover, V_{l} is a direct some of 1 or 2 copies of absolutely irreducible G_{V_{l}}-module U_{l}, and U_{l} ⊗ ℂ_{l} has the Hodge - Tate decomposition with usual properties [3, Section 4.3]. Proposition 2.1 is proved.

From now on we assume that v ∈ S_{non−Arch}^{generic}(J, k, l).

2.2. Let f : S_{V_{l}}^{sc} → S_{V_{l}} be the universal covering morphism. An isogeny f extends to an isogeny f_{G_{m}} : G_{m} × S_{V_{l}}^{sc} → G_{m} · S_{V_{l}} = G_{V_{l}}, which is defined by the formula f_{G_{m}}(a × s) = a · f(s) for a ∈ G_{m}, s ∈ S_{V_{l}}^{sc}.

Consider the exact sequence

1 → Ker(f_{G_{m}}) → G_{m} × S_{V_{l}}^{sc} → G_{m} · S_{V_{l}} = G_{V_{l}} → 1.

It determines the exact sequence of commutative algebraic ℚ_{l}-groups

1 → Ker(f_{G_{m}}) ∩ [f_{G_{m}}^{−1}(T_{v})]^{0} → [f_{G_{m}}^{−1}(T_{v})]^{0}⟶^{φ}T_{v} → 1,  (2.2.1)

where [f_{G_{m}}^{−1}(T_{v})]^{0} is a maximal ℚ_{l}-subtorus of G_{m} × S_{V_{l}}^{sc}, which coincides with the connected component of the unity of f_{G_{m}}^{−1}(T_{v}), φ is an isogeny of ℚ_{l}-tori, Ker(f_{G_{m}}) ∩ [f_{G_{m}}^{−1}(T_{v})]^{0} = Ker(φ) is a finite group, which is isomorphic to some subgroup of Cent(S_{V_{l}}^{sc}). We rewrite the sequence (2.2.1) as

1 → Ker(φ) → [f_{G_{m}}^{−1}(T_{v})]^{0}⟶^{φ}T_{v} → 1.  (2.2.2)

2.3. Recall that S_{V_{l}} ⊗ \overline{ℚ}_{l} is a \overline{ℚ}_{l}-almost simple algebraic group ⇔ S_{V_{l}} ⊗ \overline{ℚ}_{l} is semisimple and does not have infinite normal algebraic \overline{ℚ}_{l}-subgroups different from S_{V_{l}} ⊗ \overline{ℚ}_{l}.

If S_{V_{l}} ⊗\overline{ℚ}_{l} is a \overline{ℚ}_{l}-almost simple algebraic group, then by Corollary 1.5 the l-adic representation ρ_{l} is defined by the minuscule weights of the classical Lie algebras.

2.4. Assume that S_{V_{l}} ⊗ \overline{ℚ}_{l} is not a \overline{ℚ}_{l}-almost simple algebraic group. Since S_{V_{l}}^{sc} ⊗ \overline{ℚ}_{l} is a simply connected algebraic \overline{ℚ}_{l}-group, it is a product of two proper semisimple \overline{ℚ}_{l}-subgroups:

S_{V_{l}}^{sc} ⊗ \overline{ℚ}_{l} = S_{1} × S_{2}.

Consider the canonical projections pr_{i} : G_{m} × S_{1} × S_{2} → S_{i} (i = 1, 2) and pr_{0} : G_{m} × S_{1} × S_{2} → G_{m}. Since [f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0} is a maximal \overline{ℚ}_{l}-subtorus of G_{m} × S_{1} × S_{2}, we have:

[f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0} =

pr_{0}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) × pr_{1}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) × pr_{2}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}),

and pr_{i}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) is a maximal \overline{ℚ}_{l}-subtorus of S_{i} (i = 1, 2), pr_{0}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) = G_{m}. We get the decomposition

[f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0} = G_{m} × pr_{1}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) × pr_{2}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}). (2.4.1)

We can find an element y_{v} ∈ ([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})(\overline{ℚ}_{l}) such that φ(y_{v}) = ρ_{l}(F −1 ; \overline{v}) (recall that ρ_{l}(F −1 ; \overline{v}) ∈ T_{v}(ℚ) ⊂ T_{v}(\overline{ℚ}_{l})).

So the decomposition (2.4.1) gives the decomposition y_{v} = λ × y_{v1} × y_{v2}, where

λ ∈ G_{m}(\overline{ℚ}_{l}) = \overline{ℚ}_{l}^{×},
y_{vi} ∈ pr_{i}([f_{G_{m}}^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})(\overline{ℚ}_{l}) ⊂ S_{i}(\overline{ℚ}_{l}).

On the other hand,

U_{l} ⊗ \overline{ℚ}_{l} = \overline{ℚ}_{l} ⊗ W_{1} ⊗ W_{2},

where W_{1} is an irreducible S_{1}-module, W_{2} is an irreducible S_{2}-module. Let

ρ_{0} : G_{m} ⊗ \overline{ℚ}_{l} → GL(\overline{ℚ}_{l}) = \overline{ℚ}_{l}^{×},
ρ_{i} : S_{i} → GL(W_{i}) (i = 1, 2)

are the corresponding representations. We have a commutative diagram

\begin{matrix}(G_{m} ⊗ \overline{ℚ}_{l}) × S_{1} × S_{2} & ⟶^{ρ_{0}⊗ρ_{1}⊗ρ_{2}} & GL(\overline{ℚ}_{l} ⊗ W_{1} ⊗ W_{2}) \\ ↓ f_{G_{m}} & & || \\ G_{V_{l}} ⊗ \overline{ℚ}_{l} & ⊂ & GL(\overline{ℚ}_{l} ⊗ W_{1} ⊗ W_{2}). \end{matrix}  (2.4.2)

Since f_{G_{m}}(y_{vi}) ∈ T_{v}(\overline{ℚ}_{l}) ⊂ GL(U_{l})(\overline{ℚ}_{l}), we can diagonalize the linear operator f_{G_{m}}(y_{vi}) (over \overline{ℚ}_{l}). Let Δ_{i} be the set of its eigenvalues. The commutative diagram (2.4.2) shows that the set Δ of all eigenvalues of the linear operator f_{G_{m}}(y_{v}) = φ(y_{v}) = ρ_{l}(F −1 ; \overline{v}) is equal to

Δ = λ · Δ_{1} · Δ_{2}  (2.4.3)

with the natural action of Gal(\overline{ℚ}/ℚ) on Δ (recall that the characteristic polynomial of ρ_{l}(F −1 ; \overline{v}) belongs to ℤ[t]).

Let Γ_{Δ_{i}} be the multiplicative subgroup of \overline{ℚ}^{×} generated by Δ_{i}, Γ_{\{λ\}} = \{λ^{m} | m ∈ ℤ\}. We denote by X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})) the group of characters of \overline{ℚ}_{l}-torus pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}). It is evident that

Γ_{v} ⊂ Γ_{{λ}} · Γ_{Δ_{1}} · Γ_{Δ_{2}},

so

rank Γ_{v} ≤ ∑ rank X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}))(pr_{i}(y_{v})) ≤
∑ rank X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}) = rank(G_{m} × S_{1} × S_{2}) = rank(G_{V_{l}}). (2.4.4)

Condition (2.1.6) implies that

rank X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}))(pr_{i}(y_{v})) = rank X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})). (2.4.5)

Since there exists the canonical exact sequence

X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})) → X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{L})]^{0}))(pr_{i}(y_{v})) → 0,

and X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0})) is torsion-free, we deduce from (2.4.5) that

Γ_{Δ_{i}} = X^{∗}(pr_{i}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}))(pr_{i}((y_{v}))

is torsion-free. So (2.4.4) implies that

Γ_{{λ}} · Γ_{Δ_{1}} · Γ_{Δ_{2}} = Γ_{{λ}} × Γ_{Δ_{1}} × Γ_{Δ_{2}}.  (2.4.6)

On the other hand, W_{i} is a symplectic or orthogonal S_{i}-module, so W_{i}^{∗} = W_{i}. Hence Δ_{i}^{−1} = Δ_{i}.

The Galois group Gal(\overline{ℚ}/ℚ) acts in a natural way on Δ and on Δ · Δ. For each element η ∈ Δ · Δ we define a map T_{η} : Δ → \overline{ℚ}^{×} by the formula T_{η}(δ) = ηδ^{−1}. It is easy to see that for each element σ ∈ Gal(\overline{ℚ}/ℚ) we have (counting multiplicities)

Card(T_{η}(Δ) ∩ Δ) = Card(T_{σ(η)}(σ(Δ)) ∩ σ(Δ)) = Card(T_{σ(η)}(Δ) ∩ Δ),

so for any constant c the set

{η ∈ Δ · Δ | Card(T_{η}(Δ) ∩ Δ) = c}

is Gal(\overline{ℚ}/ℚ)-invariant. It is easy to see that

η = λ^{2} ⇔ Card(T_{η}(Δ) ∩ Δ) = Card(Δ).

Consequently λ^{2} ∈ ℚ, and λ^{2} is of absolute value p_{v} (by the condition (2.1.3) and the well known A.Weil theorem [6, §21, Theorem 4]), so we get the equality λ^{2} = ±p_{v}.

Let ρ be a complex conjugation on \overline{ℚ} induced by some embedding \overline{ℚ} → ℂ. According to (2.4.6) each element δ ∈ Δ has the unique decomposition δ = λ·δ_{1} ·δ_{2}, where δ_{i} ∈ Δ_{i}. We claim that

ρ(λδ_{1}δ_{2}) = λδ_{1}^{−1}δ_{2}^{−1}.  (2.4.7)

Indeed, ρ(λδ_{1}δ_{2}) = λδ′_{1}δ′_{2} for some δ′_{i} ∈ Δ_{i}. Then by the A.Weil theorem and by relation (2.1.3) we have λδ_{1}δ_{2}ρ(λδ_{1}δ_{2}) = p_{v}, so

λ^{2}δ_{1}δ_{2}δ′_{1}δ′_{2} = p_{v}, λ^{4}(δ_{1}δ′_{1}δ_{2}δ′_{2})^{2} = p_{v}^{2}, p_{v}^{2}(δ_{1}δ′_{1}δ_{2}δ′_{2})^{2} = p_{v}^{2}, (δ_{1}δ′_{1}δ_{2}δ′_{2})^{2} = 1.

Note that λ^{2} = λδ_{1}δ_{2} · λδ_{1}^{−1}δ_{2}^{−1} ∈ Γ_{v}, so δ_{1}δ′_{1}δ_{2}δ′_{2} = λδ_{1}δ_{2} · λδ′_{1}δ′_{2}/λ^{2} ∈ Γ_{v}. Since Γ_{v} is torsion free (by the condition (2.1.5)) the relation (δ_{1}δ′_{1}δ_{2}δ′_{2})^{2} = 1 implies δ′_{1}δ′_{2} = (δ_{1}δ_{2})^{−1}. So the claim (2.4.7) is proved.

The A.Weil theorem and (2.4.7) imply that

λ^{2} = λδ_{1}δ_{2} · ρ(λδ_{1}δ_{2}) = p_{v}.  (2.4.8)

2.5. Theorem . Assume that the decomposition Δ = λ · Δ_{1} · Δ_{2} corresponds to the decomposition S_{V_{l}}^{sc} ⊗ \overline{ℚ}_{l} = S_{1} × S_{2}, where S_{1} and S_{2} are semisimple algebraic \overline{ℚ}_{l}-groups. Then for each i Δ_{i} ·Δ_{i} does not contain a Gal(\overline{ℚ}/ℚ)-invariant subset ≠ {1}(without counting multiplicities).

Proof . Assume that Δ_{1} · Δ_{1} containes a Gal(\overline{ℚ}/ℚ)-invariant subset B ≠ \{1\} (without counting multiplicities). It is evident that

Card(B) < \frac{1}{2} · Card(Δ) · Card(Δ) ≤ \frac{1}{2} · (2 · dim_{k} J)^{2} < \frac{1}{2} · l^{n}.  (2.5.1)

Since Δ_{2}^{−1} = Δ_{2}, we have 1 ∈ Δ_{2} · Δ_{2}, so

λ^{2} · B ⊂ λ^{2} · Δ_{1} · Δ_{1} ⊂ λ^{2} · Δ_{1} · Δ_{1} · Δ_{2} · Δ_{2} = Δ · Δ.

Consequently ∑_{z∈λ^{2}·B} z ∈ ℤ.

Assume that p_{v} does not divide ∑_{z∈λ^{2}·B} z. Then for each place w of the field \overline{ℚ} lying over p_{v} we have

w ( ∑_{z∈λ^{2}·B} z) = 0.

It follows that there exists z_{w} ∈ λ^{2} ·B such that w(z_{w}) = 0. Hence for each element δ_{2} ∈ Δ_{2}

0 = w(z_{w}) = \frac{1}{2}\{w(z_{w}δ_{2}^{2}) + w(z_{w}δ_{2}^{−2})\}.

Since both summands in the last brackets are nonnegative in virtue of the relations z_{w}δ_{2}^{±2} ∈ Δ · Δ, we have the equalities

w(z_{w}δ_{2}^{2}) = w(z_{w}δ_{2}^{−2}) = 0.

So w(δ_{2}) = 0 for all w|p_{v}. It follows that δ_{2} is a root of 1 [14, Sublemma 3.4.0]. Since δ_{2} ∈ Δ_{2} is an arbitrary element, we conclude that the multiplicative group Γ_{Δ_{2}} generated by Δ_{2} is a finite subgroup of \overline{ℚ}^{×}. We know that Γ_{Δ_{2}} is torsion-free. So Γ_{Δ_{2}} = {1}, and

rank Γ_{v} ≤ 1 + rank Γ_{Δ_{1}} = 1 + rank X^{∗}(pr_{1}([φ^{−1}(T_{v} ⊗ \overline{ℚ}_{l})]^{0}))(pr_{1}(y_{v})) ≤

1+rank X^{∗}(pr_{1}([φ^{−1}(T_{v}⊗\overline{ℚ}_{l})]^{0})) = 1+rank(S_{1}) < rank (G_{m}×S_{1}×S_{2}) = rank(G_{V_{l}})

contrary to the condition (2.1.6).

Consequently p_{v} divides ∑_{z∈λ^{2}·B} z. Since λ^{2} = p_{v} by (2.4.8) and each b ∈ B is of absolute value 1 in virtue of the A.Weil theorem [6, §21, Theorem 4], we get the relation

∑_{z∈λ^{2}·B} z = p_{v} · b_{v},

where b_{v} ∈ ℤ is of absolute value ≤ Card(B) < \frac{1}{2} · l^{n} in virtue of (2.5.1).

By the condition (2.1.3) p_{v} is unramified in k, and Norm_{k/ℚ}(v) = p_{v}. Consequently p_{v} splits completely in k, and we have

k ⊗_{ℚ} ℚ_{p_{v}} ≃ ℚ_{p_{v}} × ... × ℚ_{p_{v}}.  (2.5.2)

By the condition (2.1.1) k containes all the (l^{n})^{th} roots of unity. On the other hand it is well known that (ℚ_{p_{v}}^{×})_{tors} ≃ ℤ/(p_{v} − 1)ℤ. It follows from (2.5.2) that l^{n}|(p_{v} − 1) and hence

p_{v} = 1 ( mod l^{n}).  (2.5.3)

By the condition (2.1.1) all the l^{n}-torsion points of J(\overline{k}) are rational over k. It follows that ρ_{l}|_{Gal(\overline{k}/k)} is trivial mod l^{n} and hence for each δ ∈ Δ we have the relation δ = 1 (mod l^{n}). The same relation holds for each element of Δ · Δ.

The known relation λ^{2} · B ⊂ Δ · Δ implies

∑_{z∈λ^{2}·B} z = Card(B) ( mod l^{n}).  (2.5.4)

In virtue of (2.5.3) we deduce from (2.5.4) that b_{v} = Card(B) ( mod l^{n}). It is clear that the inequalities |b_{v}| ≤ Card(B) < \frac{1}{2} · l^{n} imply the relation

b_{v} = Card(B).

Consequently all elements of the set λ^{2} · B are equal to p_{v} = λ^{2} contradicting to the condition B ≠ {1}. Theorem 2.5 is proved.

2.6. In virtue of the condition (2.1.5) Γ_{v} is torsion-free. So we may consider the canonical embeddings

Δ ⊂ Γ_{v} ⊂ Γ_{v} ⊗ ℚ.

In virtue of Corollary 1.5 there exists a simple simply connected algebraic \overline{ℚ}_{l}-group S_{2} of classical type such that the highest weight of each irreducible S_{2}-submodule of H_{et}^{1}(J ⊗ \overline{k}, ℚ_{l}) ⊗ \overline{ℚ}_{l} is a minuscule weight.

2.7. Lemma . The set Δ_{2} contains no three distinct points which are lying on one affine line of the space Γ_{Δ_{2}} ⊗ ℚ.

Proof . Assume that S_{2} is a simple group of type A_{m} (m ≥ 1). Then in N.Bourbaki’s notations

ω_{i} = ε_{1} + ... + ε_{i} − \frac{i}{m + 1}(ε_{1} + ... + ε_{m+1}),

Weyl’s group W (R) is the group of all permutations of {ε_{1}, ..., ε_{m+1}} [2, Chapter 6, Section 4.7], dim E(ω_{r}) = \binom{m+1}{r} [2, Chapter 8, Table 2], E(ω_{r}) is symplectic or orthogonal ⇔ r = \frac{m+1}{2} [2, Chapter 8, Table 1], and in this case

ω_{r} = \frac{1}{2}(ε_{1} + ... + ε_{(m+1)/2} − ε_{(m+3)/2} − ... − ε_{m+1}),

ch E(ω_{r}) = ch E(ω_{(m+1)/2}) = ∑_{{a_{i}∈\{±1\} \atop a_{1}+...+a_{m+1}=0}} e^{a_{1}ε_{1}/2+...+a_{m+1}ε_{m+1}/2} =

= ∑_{{a_{i}∈\{±1\} \atop a_{1}+...+am∈\{±1\}}} e^{a_{1}(ε_{1}−ε_{m+1})/2+...+a_{m}(ε_{m}−ε_{m+1})/2}.

Since the representation of S_{2} in W_{2} is symplectic or orthogonal, we have the following relation:

Δ_{2} = {α_{1}^{a_{1}}...α_{m}^{a_{m}} | a_{j} ∈ {±1}, a_{1} + ... + a_{m} ∈ {±1}},

where α_{1}, α_{2}, ..., α_{m} are multiplicatively independent algebraic numbers. Hence elements of Δ_{2} are vertices of some cube in Γ_{Δ_{2}} ⊗ ℚ. It is evident that no three such vertices are lying on one affine line of Γ_{Δ_{2}} ⊗ ℚ.

Assume that S_{2} is a simple group of type B_{m} (m ≥ 2). Then in N.Bourbaki’s notations dim E(ω_{m}) = 2^{m} [2, Chapter 8, Table 2],

ω_{m} = \frac{1}{2}(ε_{1} + ... + ε_{m}),

ch E(ω_{m}) = ∑_{a_{i}∈{±1}} e^{a_{1}ε_{1}/2+...+a_{m}ε_{m}/2}

[2, Chapter 6, Section 4.5], so we may assume that

Δ_{2} = {α_{1}^{a_{1}}...α_{m}^{a_{m}} | a_{j} ∈ {±1}},

where α_{1}, α_{2}, ..., α_{m} are multiplicatively independent algebraic numbers. Hence elements of Δ_{2} are vertices of some cube in Γ_{Δ_{2}} ⊗ ℚ, and no three such vertices are lying on one affine line of Γ_{Δ_{2}} ⊗ ℚ.

Assume that S_{2} is a simple group of type C_{m} (m ≥ 2). Then dim E(ω_{1}) = 2m [2, Chapter 8, Table 2], ω_{1} = ε_{1},

ch E(ω_{1}) = ∑_{{a_{i}∈\{±1\} \atop i∈\{1,...,m\}}} e^{a_{i}ε_{i}}

[2, Chapter 6, Section 4.6]. So we may assume that Δ_{2} = {α_{j}^{a_{j}} | a_{j} ∈ {±1}, j ∈ {1, ..., m}},

where α_{1}, α_{2}, ..., α_{m} are multiplicatively independent algebraic numbers. Hence elements of Δ_{2} are vertices of some cube in Γ_{Δ_{2}} ⊗ ℚ, and no three such vertices are lying on one affine line of Γ_{Δ_{2}} ⊗ ℚ.

Assume that S_{2} is a simple group of type D_{m} (m ≥ 3). Then dim E(ω_{1}) = 2m [2, Chapter 8, Table 2], ω_{1} = ε_{1},

ch E(ω_{1}) = ∑_{{a_{i}∈\{±1\} \atop i∈\{1,...,m\}}} e^{a_{i}ε_{i}},

dim E(ω_{m−1}) = dim E(ω_{m}) = 2^{m−1},

ch E(ω_{m−1}) = ∑_{{a_{i}∈\{±1\} \atop Card(\{i|a_{i}=−1\})=1( mod 2 )}} e^{a_{1}ε_{1}/2+...+a_{m}ε_{m}/2},

ch E(ω_{m}) = ∑_{{a_{i}∈\{±1\} \atop Card(\{i|a_{i}=−1\})=0( mod 2 )}} e^{a_{1}ε_{1}/2+...+a_{m}ε_{m}/2}

[2, Chapter 6, Section 4.8]. So we may assume that

Δ_{2} = {α_{j}^{a_{j}} | a_{j} ∈ {±1}, j ∈ {1, ..., m}},

or

Δ_{2} = {α_{1}^{a_{1}}...α_{m}^{a_{m}} | a_{j} ∈ {±1}, Card({j|a_{j} = −1}) = 1(mod 2)},

or

Δ_{2} = {α_{1}^{a_{1}}...α_{m}^{a_{m}}(a_{j} ∈ {±1}, Card({j | a_{j} = −1}) = 0(mod 2)},

where α_{1}, α_{2}, ..., α_{m} are multiplicatively independent algebraic numbers. Hence in any case elements of Δ_{2} are vertices of some cube in Γ_{Δ_{2}} ⊗ ℚ, and no three such vertices are lying on one affine line of Γ_{Δ_{2}} ⊗ ℚ. Lemma 2.7 is proved.

2.8. Lemma . If x, y, z ∈ Δ are three distinct points lying on some affine line L ⊂ Γ_{v} ⊗ ℚ, then x = λ · δ′_{1} · δ_{2}, y = λ · δ_{1} · δ_{2}, z = λ · δ′′_{1} · δ_{2}, where δ′_{1}, δ_{1}, δ′′_{1} ∈ Δ_{1}, and δ_{2} ∈ Δ_{2}.

Proof . According to (2.4.6) we may consider the canonical projection

pr_{Δ_{2}} : Γ_{v} ⊗ ℚ = (Γ_{{λ}} ⊗ ℚ) × (Γ_{Δ_{1}} ⊗ ℚ) × (Γ_{Δ_{2}} ⊗ ℚ) → Γ_{Δ_{2}} ⊗ ℚ. If x, y, z are distinct elements of Δ lying on an affine line L, then pr_{Δ_{2}}(x), pr_{Δ_{2}}(y) , pr_{Δ_{2}}(z) are lying on pr_{Δ_{2}}(L). Note that if y is situated between x and z, then pr_{Δ_{2}}(y) is situated between pr_{Δ_{2}}(x) and pr_{Δ_{2}}(z). If pr_{Δ_{2}}(L) is a line, then pr_{Δ_{2}}(x), pr_{Δ_{2}}(y), pr_{Δ_{2}}(z) are distinct elements of pr_{Δ_{2}}(Δ) = Δ_{2} lying on an affine line pr_{Δ_{2}}(L) contrary to Lemma 2.7. Consequently pr_{Δ_{2}}(L) is a single point of Δ_{2}. Lemma 2.8 is proved.

§3. Roots, weights and Frobenius eigenvalues

3.1. Now we start to prove Theorem 0.6.

Let R be the root system of Lie Im(ρ_{l}) ⊗ \overline{ℚ}_{l} relative to a Cartan subalgebra h. We denote by ( , ) : h^{∗} × h → \overline{ℚ}_{l} the tautological pairing. Let R^{V} be the dual root system. The sets R and R^{V} are invariant under the natural action of the Weyl group W. We define the lattice of weights P (R) in h^{∗}:

P (R) = {λ ∈ h^{∗} | (λ, α^{V}) ∈ ℤ ∀α ∈ R}.

3.2. Let Ω(V_{l} ⊗ \overline{ℚ}_{l}) be the collection of weights of h ⊂ Lie Im(ρ_{l}) ⊗ \overline{ℚ}_{l}. We have

Ω(V_{l} ⊗ \overline{ℚ}_{l}) ⊂ P (R) ⊂ h^{∗}.

The collection Ω(V_{l} ⊗ \overline{ℚ}_{l}) is W -invariant and R-saturated [2, Chapter 8, §7, Propositions 3, 4]. This means that, for a weight λ ∈ Ω(V_{l} ⊗ \overline{ℚ}_{l}) and a root α ∈ R such that m = (λ, α^{V}) > 0, the collection Ω(V_{l} ⊗ \overline{ℚ}_{l}) containes the arithmetic progression λ, λ − α, ..., λ − mα of length m. If m = (λ, α^{V}) < 0, then Ω(V_{l} ⊗ \overline{ℚ}_{l}) contains the arithmetic progression λ, λ + α,...,λ + (−m)α of length (−m).

3.3. Recall that a nonzero weight λ ∈ P (R) is called a minuscule weight (microweight)⇔(λ, α^{V}) ∈ {0, 1, −1} for all α ∈ R [2, Chapter 8, §7.3].

If the canonical l-adic representation ρ_{l} is not defined by the minuscule weights, then Ω(V_{l} ⊗ \overline{ℚ}_{l}) containes the arithmetic progression of type λ, λ − α, λ − 2α or λ, λ + α, λ + 2α. In this case there exist distinct elements δ ≠ δ′ ≠ δ′′ ∈ Δ such that δ^{2} = δ′ ·δ′′ (indeed, if λ, λ−α, λ−2α ∈ Ω(V_{l} ⊗\overline{ℚ}_{l}), then 2·(λ−α) = λ+(λ−2α)).

3.4. Remark . It is evident that δ is the middle point of the segment [δ′, δ′′] ⊂ Γ_{v} ⊗ ℚ. Hence δ′, δ and δ′′ are lying on one affine line L ⊂ Γ_{v} ⊗ ℚ.

3.5. Consider the set

Δ^{line} = {(δ′, δ, δ′′) | δ ≠ δ′ ≠ δ′′ ∈ Δ, δ^{2} = δ′ · δ′′}.

It is evident that Δ^{line} ⊂ Δ×Δ×Δ is invariant under the action of the Galois group Gal(\overline{ℚ}/ℚ) because for each σ ∈ Gal(\overline{ℚ}/ℚ) we have (σ(δ))^{2} = σ(δ^{2}) = σ(δ′) · σ(δ′′).

The last relation shows that the set

B = {δ′/δ, δ′′/δ = (δ′/δ)^{−1} | (δ′, δ, δ′′) ∈ Δ^{line}} ⊂ Γ_{v}

is invariant under the natural action of Gal(\overline{ℚ}/ℚ).

3.6. Let (δ′, δ, δ′′) ∈ Δ^{line}. According to Lemma 2.8 we have the following decompositions:

δ′ = λ · δ′_{1} · δ_{2}, δ = λ · δ_{1} · δ_{2}, δ′′ = λ · δ′′_{1} · δ_{2},  (3.6.1)

where δ′_{1}, δ_{1}, δ′′_{1} ∈ Δ_{1}, δ_{2} ∈ Δ_{2}.

Let b = δ′/δ ∈ B. Then b = δ′_{1}/δ_{1}, and we get from (3.6.1) the following decompositions:

δ′ = λ · b · δ_{1} · δ_{2}, δ = λ · δ_{1} · δ_{2}, δ′′ = λ · b^{−1} · δ_{1} · δ_{2}.

Since b · δ_{1} = δ′_{1} ∈ Δ_{1} and δ_{1}^{−1} ∈ Δ_{1}, we have the relation b ∈ Δ_{1} · Δ_{1}. So B ⊂ Δ_{1} · Δ_{1}. On the other hand, B ≠ \{1\}, so we have a nontrivial Gal(\overline{ℚ}/ℚ)- invariant subset B ⊂ Δ_{1} · Δ_{1} contrary to Theorem 2.5.

Hence ρ_{l} is defined by the minuscule weights. A simple factor of type E_{6} can’t appear because E(ω_{1,6}) is not a symplectic or orthogonal representation [2, Chapter 8, Table 1]. We exclude the case of a simple factor of type E_{7} in the next paragraph.

§4. Exclusion of type E_{7}

4.1. In Bourbaki’s notations [2, Chapter 6, §4] a simple Lie algebra of type E_{7} has the following system of roots:

R = {±ε_{i} ± ε_{j} (1 ≤ i < j ≤ 6), ±(ε_{7} − ε_{8}),

±\frac{1}{2} · (ε_{7} − ε_{8} + ∑_{i=1}^{6}(−1)^{ν(i)}ε_{i}) (∑_{i=1}^{6} ν(i) = 1 ( mod 2))\}.

Base of the system of roots is the following set:

\{α_{1} = \frac{1}{2}(ε_{1} + ε_{8}) − \frac{1}{2}(ε_{2} + ε_{3} + ε_{4} + ε_{5} + ε_{6} + ε_{7});

α_{2} = ε_{1} + ε_{2}; α_{3} = ε_{2} − ε_{1}; α_{4} = ε_{3} − ε_{2};

α_{5} = ε_{4} − ε_{3}; α_{6} = ε_{5} − ε_{4}; α_{7} = ε_{6} − ε_{5}}.

For a given α ∈ R consider the reflection

S_{α}(x) = x − 2 · \frac{(x, α)}{(α, α)} · α.

Since (α, α) = 2 for all α ∈ R, we get the following formula:

S_{α}(x) = x − (x, α) · α.

These reflections S_{α} (α ∈ R) generate the Weyl group W (R) of order 2^{10} · 3^{4} · 5 · 7.

4.2. We want to find the set of all weights of irreducible symplectic representation E(ω_{7}) of degree 56 = 2^{3} · 7 [2, Chapter 8, Tables 1, 2]. Since

ω_{7} = ε_{6} + \frac{1}{2}(ε_{8} − ε_{7}) = \frac{1}{2}(2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + 3α_{7})

is a munuscule weight (microweight), the set of all weights of E(ω_{7}) coincides with W (R) · ω_{7} [2, Chapter 8, §7, Proposition 6].

It is easy to find the matrix A = (a_{ij}) = (S_{α_{i}}(α_{j})):

A = ( \begin{matrix}−α_{1} & α_{2} & α_{1} + α_{3} & α_{4} & α_{5} & α_{6} & α_{7} \\ α_{1} & −α_{2} & α_{3} & α_{2} + α_{4} & α_{5} & α_{6} & α_{7} \\ α_{1} + α_{3} & α_{2} & −α_{3} & α_{3} + α_{4} & α_{5} & α_{6} & α_{7} \\ α_{1} & α_{2} + α_{4} & α_{3} + α_{4} & −α_{4} & α_{4} + α_{5} & α_{6} & α_{7} \\ α_{1} & α_{2} & α_{3} & α_{4} + α_{5} & −α_{5} & α_{5} + α_{6} & α_{7} \\ α_{1} & α_{2} & α_{3} & α_{4} & α_{5} + α_{6} & −α_{6} & α_{6} + α_{7} \\ α_{1} & α_{2} & α_{3} & α_{4} & α_{5} & α_{6} + α_{7} & −α_{7} \end{matrix} ) .

We consider a positive root

ε_{3} + ε_{6} = α_{2} + α_{3} + 2α_{4} + α_{5} + α_{6} + α_{7}

and the corresponding reflection S_{ε_{3}+ε_{6}}. It is easy to see that S_{ε_{3}+ε_{6}}(α_{1}) = α_{1} + α_{2} + α_{3} + 2α_{4} + α_{5} + α_{6} + α_{7}, S_{ε_{3}+ε_{6}}(α_{2}) = α_{2}, S_{ε_{3}+ε_{6}}(α_{3}) = α_{3}, S_{ε_{3}+ε_{6}}(α_{4}) = −α_{2} − α_{3} − α_{4} − α_{5} − α_{6} − α_{7}, S_{ε_{3}+ε_{6}}(α_{5}) = α_{2} + α_{3} + 2α_{4} + 2α_{5} + α_{6} + α_{7}, S_{ε_{3}+ε_{6}}(α_{6}) = α_{6}, S_{ε_{3}+ε_{6}}(α_{7}) = −α_{2} − α_{3} − 2α_{4} − α_{5} − α_{6}.

So we get the following chains:

2 · ω_{7} = 2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + 3α_{7} ⟶^{S_{ε_{3}+ε_{6}}}

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{1}}}

α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{3}}}

α_{2} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{5}}}

α_{2} + 2α_{4} + α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{4}}}

α_{2} + α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{2}}}

−α_{2} + α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{6}}}

−α_{2} + α_{5} + α_{7};

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{4}}}

2α_{1} + α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{2}}}

2α_{1} + 3α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{3}}}

2α_{1} + 3α_{2} + 4α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{4}}}

2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{5}}}

2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{6}}}

2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + α_{7};

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{5}}}

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{6}}}

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} + α_{7}⟶^{S_{α_{7}}}

2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} − α_{7};

α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{4}}}

α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{2}}}

3α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7};

α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{5}}}

α_{2} + 2α_{3} + 2α_{4} + α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{6}}}

α_{2} + 2α_{3} + 2α_{4} + α_{5} + α_{7}⟶^{S_{α_{3}}}

α_{2} + 2α_{4} + α_{5} + α_{7}⟶^{S_{α_{4}}}

α_{2} + α_{5} + α_{7}⟶^{S_{α_{5}}}

α_{2} − α_{5} + α_{7};

2α_{1} + α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}⟶^{S_{α_{3}}}

2α_{1} + α_{2} + 4α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7};

α_{2} + 2α_{3} + 2α_{4} + α_{5} + α_{7}⟶^{S_{α_{7}}}

α_{2} + 2α_{3} + 2α_{4} + α_{5} − α_{7}⟶^{S_{α_{3}}}

α_{2} + 2α_{4} + α_{5} − α_{7};

α_{2} + α_{5} + α_{7}⟶^{S_{α_{7}}}

α_{2} + α_{5} − α_{7}. The continuation of this procedure leads to old weights or to opposite weights.

Consequently the set Ω(E(ω_{7})) of all weights of E(ω_{7}) is equal to \frac{1}{2}Ω ∪ (−\frac{1}{2}Ω), where Ω is the set of all elements in the chains above. As a result, we obtain the following list of elements of Ω:

2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + 3α_{7}; 2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + α_{7}; 2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 2α_{6} + α_{7}; 2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + 3α_{2} + 4α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + 3α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + α_{2} + 4α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}; 2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} + 2α_{6} + α_{7}; 2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} + α_{7}; 2α_{1} + α_{2} + 2α_{3} + 2α_{4} + α_{5} − α_{7};

3α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7};

α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7};
α_{2} + 2α_{3} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7};
α_{2} + 2α_{3} + 2α_{4} + α_{5} + 2α_{6} + α_{7};
α_{2} + 2α_{3} + 2α_{4} + α_{5} + α_{7};

α_{2} + 2α_{3} + 2α_{4} + α_{5} − α_{7}; α_{2} + 2α_{4} + 3α_{5} + 2α_{6} + α_{7}; α_{2} + 2α_{4} + α_{5} + 2α_{6} + α_{7};

α_{2}  + 2α_{4} + α_{5}  + α_{7};
α_{2}  + 2α_{4} + α_{5}  − α_{7};

α_{2} + α_{5} + 2α_{6} + α_{7};

α_{2}  + α_{5}  + α_{7};
α_{2}  + α_{5}  − α_{7};
α_{2}  − α_{5}  + α_{7};

α_{2} − α_{5} − 2α_{6} − α_{7}; α_{2} − α_{5} − α_{7}.

4.3. Let (x_{1}, ..., x_{7}) be the coordinates on ℚα_{1} + ... + ℚα_{7} with respect to {α_{1}, ..., α_{7}}. It is evident that the affine hyperplane {x | x_{7} = 1} contains 27 points of Ω∪(−Ω), and the affine hyperplane {x | x_{7} = −1} containes 27 points of Ω∪(−Ω). Consequently there is a partition

Ω ∪ (−Ω) = {±(2α_{1} + 3α_{2} + 4α_{3} + 6α_{4} + 5α_{5} + 4α_{6} + 3α_{7})}∪
{x ∈ Ω ∪ (−Ω) | x_{7} = −1} ∪ {x ∈ Ω ∪ (−Ω) | x_{7} = 1}.

Since W (R) acts transitively on Ω(E(ω_{7})) = \frac{1}{2}(Ω ∪ (−Ω)), we see that for each weight ω ∈ Ω(E(ω_{7})) there exist parallel affine hyperplanes L_{−ω}, L_{ω} of dimension 6, which give a partition

Ω(E(ω_{7})) = {±ω} ∪ {L_{−ω} ∩ Ω(E(ω_{7}))} ∪ {L_{ω} ∩ Ω(E(ω_{7}))},

where

Card(L_{±ω} ∩ Ω(E(ω_{7}))) = 27.

These hyperplanes separate −ω and ω like on the following picture (where stars denote elements of Ω(E(ω_{7}))):

\begin{aligned}& & ∗ & | & & ∗ & | \\ & | & ∗...∗ & | & | & ∗...∗ & | \\ ∗ & | & L_{−ω} & | & | & L_{ω} & | & ∗ \\ −ω & | & ∗...∗ & | & | & ∗...∗ & | & ω \\ & | & ∗ & & | & ∗ \end{aligned}  (4.3.1)

4.4. Definition . We say that a subset D ⊂ Δ has a configuration of type E_{7}(ω_{7}) ⇔ there exist a 7-dimensional affine subspace L ⊂ Γ_{v} ⊗ ℚ and an affine isomorphism f : L ≃ ℚα_{1} + ... + ℚα_{7} such that the following conditions hold:

1) D = L ∩ Δ;

2) f(D) = Ω(E(ω_{7})) is the system of all weights of the irreducible representation E(ω_{7}) of a simple Lie algebra of type E_{7}.

4.5. Remark . If S_{V_{l}}^{sc} ⊗ \overline{ℚ}_{l} has a simple factor of type E_{7}, then there exists a subset D ⊂ Δ with a configuration of type E_{7}(ω_{7}).

Indeed, if Δ′_{1} corresponds to a simple factor of type E_{7}, then due to 2.4 we get the decomposition

Δ = λ · Δ′_{1} · Δ′′_{1} · Δ_{2}.

It is evident that for each δ′′_{1} ∈ Δ′′_{1}, δ_{2} ∈ Δ_{2} the affine space L = (Γ_{Δ′_{1}} ⊗ ℚ) · δ′′_{1} · δ_{2} contains the subset D = Δ′_{1} · δ′′_{1} · δ_{2} with a configuration of type E_{7}(ω_{7}).

4.6. Lemma . If D ⊂ Δ has a configuration of type E_{7}(ω_{7}), then pr_{Δ_{2}}(D) is a single point.

Proof . We denote by Φ the composition of the following maps:

ℚα_{1} + ... + ℚα_{7} ⟶^{f^{−1}} L ⟶^{pr_{Δ_{2}}} Γ_{Δ_{2}} ⊗ ℚ.

The map Φ is an affine map (possibly, degenerated). According to the results of Section 2.7

Φ(Ω(E(ω_{7}))) = pr_{Δ_{2}}(f^{−1}(Ω(E(ω_{7}))) = pr_{Δ_{2}}(D)

is a subset of the set of all vertices of a cube in Γ_{Δ_{2}} ⊗ ℚ.

Assume that this subset contains at least two distinct points. Then we get from (4.3.1) the following picture (where affine subspaces Φ(L_{±ω}) are parallel and separate vertices Φ(−ω) and Φ(ω)):

\begin{aligned}& & ∗ & | & & ∗ & | \\ & | & ∗...∗ & | & | & ∗...∗ & | \\ ∗ & | & Φ(L_{−ω}) & | & | & Φ(L_{ω}) & | & ∗ \\ Φ(−ω) & | & ∗...∗ & | & | & ∗...∗ & | & Φ(ω) \\ & | & ∗ & & | & ∗ \end{aligned}  (4.6.1)

Since all stars are vertices of a cube, we can find an edge of a cube such that the following conditions hold: 1) Φ(−ω) is lying on this edge; 2) the image of Φ(ω) under the canonical projection of a cube onto this edge is not equal to Φ(−ω).

Let Φ′ be the composition of Φ and the projection of a cube onto the edge under consideration. We get from (4.6.1) the following picture:

\begin{matrix}∗ & ∗ & ∗ & ∗ \\ Φ′(−ω) & Φ′(L_{−ω}) & Φ′(L_{ω}) & Φ′(ω) \end{matrix}

Consequently we arrive to a contradiction, because there are no “intermediate” vertices between two distinct vertices of a cube lying on one edge. Lemma 4.6 is proved.

4.7. Remark . Assume that D ⊂ Δ has a configuration of type E_{7}(ω_{7}). Then for each element σ ∈ Gal(\overline{ℚ}/ℚ) the set σ(D) ⊂ Δ has a configuration of type E_{7}(ω_{7}).

It is evident because Gal(\overline{ℚ}/ℚ) acts on Γ_{v} ⊗ ℚ as a group of affine transformations.

4.8. Consider the set

Δ^{E_{7}} = {(δ′, δ′′, ..., δ^{(56)}) ∈ Δ × ... × Δ = Δ^{56} | δ′, δ′′, ...δ^{(56)} are distinct points of Δ and the set {δ′, δ′′, ..., δ^{(56)}} has a configuration of type E_{7}(ω_{7})}. In virtue of Remark 4.7 this set is invariant under the action of the Galois group

Gal(\overline{ℚ}/ℚ). Consequenly the set

B = {1, δ′′/δ′, ..., δ^{(56)}/δ′, δ′/δ′′, ..., δ^{(56)}/δ′′, ...} | (δ′, ..., δ^{(56)}) ∈ Δ^{E_{7}}}

is invariant under the natural action of Gal(\overline{ℚ}/ℚ). According to Lemma 4.6 we have the following decompositions:

δ′ = λ · δ′_{1} · δ_{2}, δ′′ = λ · δ′′_{1} · δ_{2}, ... δ^{(56)} = λ · δ_{1}^{(56)} · δ_{2},  (4.8.1)

where δ′_{1}, δ′′_{1}, ..., δ_{1}^{(56)} ∈ Δ_{1}, δ_{2} ∈ Δ_{2}.

Hence B = {1, δ′′_{1}/δ′_{1}, ..., δ_{1}^{(56)}/δ′_{1}, δ′_{1}/δ′′_{1}, ..., δ_{1}^{(56)}/δ′′_{1}, ...} |(δ′, ..., δ^{(56)}) ∈ Δ^{E_{7}}}

is a Gal(\overline{ℚ}/ℚ)-invariant subset of Δ_{1} · Δ_{1}^{−1} = Δ_{1} · Δ_{1}, and B ≠ \{1\} contrary to Theorem 2.5. Thus a simple factor of type E_{7} can’t appear. Theorem 0.6 is proved.

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Vladimir State University, ul. Gorkogo, 87, Vladimir, 600026 , Russia

E-mail address: tankeev%rtf@vpti.vladimir.su