ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADICREPRESENTATIONS OF Gal(Q/Q)

SHIANG TANG

Abstract. A connected reductive algebraic group G is said to be an l-adic algebraic mon-odromy group for ΓQ if there is a continuous homomorphism

Gal(Q/Q)→ G(Ql)

with Zariski-dense image. In this paper, we give a classification of connected l-adic alge-braic monodromy groups for Gal(Q/Q), in particular producing the first such examples forSLn,Sp2n,Spinn and Esc

6 . To do this, we start with a mod-l representation of Gal(Q/Q)coming from the Weyl group of G and use a variation of Stefan Patrikis’ generalization of amethod of Ravi Ramakrishna to produce its desired l-adic lifts.

1. Introduction

For a split connected reductive group G and a prime number l, it is natural to study twotypes of continuous representations of ΓQ = Gal(Q/Q): the mod l representations

ρ : ΓQ → G(Fl)and the l-adic representations

ρ : ΓQ → G(Ql)

where we use the l-adic topology for G(Ql). Mod l representations of ΓQ are closely relatedto the inverse Galois problem for finite groups of Lie type, which asks for the existence ofsurjective homomorphisms ρ : ΓQ � G(k) for k a finite extension of Fl. It is still wide open,even for small groups such as SL2. This is difficult since we stick with Q as the base field. Ifwe replace ΓQ by ΓF for some number field F , it is not hard to show that every finite groupis a Galois group over some number field, but if we insist on ΓQ then the problem becomesvery difficult. On the other hand, we can ask for its analogues in the l-adic world:

Question 1: Are there continuous homomorphisms ρ : ΓQ → G(Ql) with Zariski-denseimages?

We also ask a refined question which takes geometric Galois representations (in the sense of[FM93]) into account:

Question 2: Are there continuous geometric Galois representations ρ : ΓQ → G(Ql) withZariski-dense images?

This paper gives an almost complete answer to Question 1. Before explaining our result,let us make some remarks on Question 2. We shall call a reductive group G an l-adicalgebraic monodromy group, or simply an l-adic monodromy group for ΓQ if thehomomorphisms in Question 1 exist, and a geometric l-adic monodromy group for ΓQif the homomorphisms in Question 2 exist. Many classical groups are known as geometric

1

2 SHIANG TANG

l-adic monodromy groups for ΓQ: for example, the Tate module of a suitable n-dimensionalabelian variety A over Q realizes GSp2n as a geometric l-adic monodromy group for ΓQ. Butwe expect it to be impossible to realize some algebraic groups as geometric l-adic monodromygroups for ΓQ.

Example 1.1. (A heuristic argument) There is no homomorphism ρ : ΓQ → SL2(Ql) thatis unramified almost everywhere, potentially semi-stable at l, and has Zariski-dense image.

In fact, assuming the Fontaine-Mazur and the Langlands conjectures (see [7] and [3]), ifsuch ρ exists, then ρ = ρπ for some cuspidal automorphic representation π on GL2(AQ).But ρ is even, i.e., det ρ(c) = 1, so π∞ (the archimedean component of π) is a principalseries representation, and π is associated to a Maass form. Therefore by the Fontaine-Mazurconjecture, ρπ has finite image, a contradiction. �

In contrast, 1.3 shows in particular that SL2 is an l-adic monodromy group for ΓQ. On theother hand, SL2 can be a geometric l-adic monodromy group for ΓF for some finite extensionF/Q:

Example 1.2. Let f be a non-CM new eigenform of weight 3, level N , with a nontrivialnebentypus character ε. Such f exist for suitable N , see [9]. We write E for the field ofcoefficients of f . Then for all l and λ|l, there is a continuous representation rf,λ : ΓQ →GL2(Eλ) which is unramified outside {v : v|Nl} and Tr(rf,λ(Frp)) = ap for p not dividingNl, with ap the p-th Hecke eigenvalue of f . We have det(rf,λ) = κ2ε where κ is the l-adiccyclotomic character. By a theorem of Ribet [14], for almost all l, rf,λ(ΓQ) contains SL2(k)for a subfield k of kλ (the residue field of Eλ). It follows that rf,λ has Zariski dense image.Let F be a finite extension of Q that trivializes ε, then the image of r′ := κ−1 · rf,λ|ΓF landsin SL2(Eλ) and is Zariski-dense.

Most of the exceptional algebraic groups are known as geometric l-adic monodromy groups,see [21], [12], and [2]. For example, in [12], Patrikis constructs geometric Galois represen-tations (in the sense of [7]) for ΓQ with full algebraic monodromy groups for essentially allexceptional groups of adjoint types. Patrikis has obtained an extension to general reductivegroups of Ravi Ramakrishna’s techniques for lifting odd two-dimensional Galois representa-tions to geometric l-adic representations. In [2], a better answer is given for type E6: theyshow that E6 (to be precise, the L-group of an outer form of E6) is the algebraic monodromygroup of a Galois representation appearing in the cohomology of a Shimura variety over Q.

As 1.1 indicates, the question of determining whether a simple algebraic group G is a geo-metric l-adic monodromy group for ΓQ is sensitive to the form G takes: PGL2 comes fromgeometry (Tate module of elliptic curves), whereas its simply-connected form SL2 shouldnot. Thus, it is a interesting (and seemingly very difficult) question to determine what sim-ple algebraic groups (adjoint, simply-connected, or others) are geometric l-adic monodromygroups for ΓQ. Moreover, for those that are not, we ask if they are l-adic monodromy groupsat least. We have the following:

Theorem 1.3. For a simple algebraic group G and for a prime number l, there are infinitelymany non-conjugate continuous homomorphisms

ρl : ΓQ → G(Ql)

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 3

with Zariski-dense images, where we take l to be large enough for all G not equal to Bscn , C

scn , E

sc7 ,

l ≡ 1(4) for G = Bscn , C

scn , and l ≡ 1(3) for G = Esc

7 .

Remark 1.4. Patrikis has shown that Ead7 , E8, F4, G2 and the L-group of an outer form of

Ead6 are geometric l-adic monodromy groups for ΓQ. The congruence conditions on l for

G = Bscn , C

scn , E

sc7 can be removed using the methods in an upcoming paper by Fakhruddin,

Khare and Patrikis ([8]).

I believe Theorem 1.3 contains the first sighting of SLn, Sp2n, Spinn, Esc6 , E

ad6 , E

sc7 as any sort

of arithmetic monodromy groups for ΓQ.

1.3 and 1.4 imply the following theorem (see Section 5):

Theorem 1.5. For a connected reductive algebraic group G, there are continuous homomor-phisms

ρl : ΓQ → G(Ql)

with Zariski-dense image for large enough primes l if and only if the dimension of Z(G) isat most one.

In the rest of this introduction, we will sketch the strategy of Theorem 1.3, which makesuse of Patrikis’ generalization of Ramakrishna’s techniques but is very different from hisarguments in many ways. For the rest of this section, we will assume that G is a simplealgebraic group split over Zl with a split maximal torus T . Let Φ = Φ(G, T ) be the associatedroot system. Let O be the ring of integers of an extension of Ql whose reduction moduloits maximal ideal is isomorphic to k, a finite extension of Fl. We start with a well-chosenmod l representation and then use a variant of Ramakrishna’s method to deform it to Owith big image. Achieving this is a balancing act between two difficulties: the Inverse GaloisProblem for G(k) is difficult, so we want the residual image to be relatively ‘small’. On theother hand, Ramakrishna’s method works when the residual image is ‘big’.

1. Building the residual representations: Weyl groups. Before explaining our con-struction, let us first recall a different construction used in [12]. Patrikis uses the principalGL2 homomorphism to construct the residual representation

ρ : ΓQr−→ GL2(k)

ϕ−→ G(k)

for r a surjective homomorphism constructed from modular forms and ϕ a principal GL2

homomorphism (for the definition, see [15] and [12], Section 7.1). But the principal GL2 isdefined only when ρ∨ (the half-sum of coroots) is in the cocharacter lattice X∗(T ), which isnot the case for G = SL2n, Sp2n, E

sc7 , etc. Nonetheless, the principal SL2 is always defined;

but it is not known whether SL2(k) is a Galois group over Q and the surjectivity of r iscrucial in applying Ramakrishna’s method.

For this reason, we use a different construction which is similar to the construction of resid-ual representations in [2]. By abuse of notation, we still use G, T to denote G(k), T (k),respectively. We consider the following exact sequence finite groups, to which we shall referas the N-T sequence:

1→ T → NG(T )π−→ W → 1

where W = NG(T )/T is the Weyl group of G. We want to take N = NG(T ) as the image ofthe residual representation ρ. It turns out that the adjoint action of N on the Lie algebra

4 SHIANG TANG

g over k decomposes into at most three irreducible pieces (2.2), which is a very pleasantproperty for making our variant of Ramakrishna’s technique work. It has been known fora long time that W is a Galois group over Q, but what we need is realizing N as a Galoisgroup over Q. A natural approach would be solving the embedding problem posed by the N-T sequence, i.e., to suppose there is a Galois extension K/Q realizing W , and then to find anfinite Galois extension K ′/Q containing K such that the natural surjective homomorphismGal(K ′/Q) � Gal(K/Q) realizes π : N � W .

This embedding problem is solvable when the sequence splits, by an elementary case ofa famous theorem of Igor Shafarevich, see [16], Claim 2.2.5. In [1], the splitting/non-splitting the N-T sequence is determined completely: for instance, it does not split forG = SLn, Sp2n, Spinn, E7. We find our way out by replacing N with a suitable subgroup N ′

for which the decomposition of the adjoint representation remains the same, then realizingN ′ as a Galois group over Q with certain properties, see Section 2.1.3-2.1.6. Finally, wedefine our residual representation ρ to be the composite

ΓQ � N ′ → G = G(k)

where the first arrow comes from the realization of N ′ as a Galois group over Q and thesecond arrow is the inclusion map. We write ρ(g) for the Lie algebra g/k equipped with aΓQ-action induced by the homomorphism

ΓQρ−→ G

Ad−→ GL(g)

2. A variant of Ramakrishna’s method. For a residual representation

ρ : ΓQ → G(k)

unramified outside a finite set of places S containing the archimedean place and a globaldeformation condition for ρ (which consists of a local deformation condition for each v ∈ S),a typical question in Galois deformation theory is to try to find continuous l-adic lifts

ρ : ΓQ → G(O)

of ρ with O the ring of integers of a finite extension of Ql whose residue field is k, such thatfor all v, ρ|ΓQv

(we fix an embedding Q → Qv) satisfies the prescribed local deformationcondition at v. Ravi Ramakrishna has an ingenious method [13] for obtaining the desiredlifts which has been generalized and axiomatized in [18], [6], [12], etc. By the Poitou-Tateexact sequence, if the dual Selmer group

H1L⊥(ΓQ,S, ρ(g)(1))

associated to the global deformation condition for ρ vanishes, then such lifts exist. Here L(resp. L⊥) is the Selmer system (resp. dual Selmer system) of tangent spaces (resp. annihila-tors of the tangent spaces under the local duality) of the given local deformation conditions.Ramakrishna discovered that if one imposes finitely many additional local deformation con-ditions of ‘Ramakrishna type’ in place of the unramified ones at a finite set of well-chosenplaces of Q disjoint from S, then the new dual Selmer group will vanish. However, thistechnique is very sensitive to the image of ρ, which has to be ‘big’ to make things work: ifρ(g) is irreducible then all is good, but finding such a ρ can be very difficult. In practice, wewould prefer those ρ for which ρ(g) does not decompose too much. Unfortunately, the formof Ramakrishna’s method in [12] (see 3.4) does not work for our ρ.

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 5

Inspired by the use of Ramakrishna’s method in [6], we hurdle this by making two obser-vations. For our ρ, ρ(g) decomposes as ρ(t) (the Lie algebra of T over k equipped with anirreducible action of ρ(ΓQ)) and a complement (see 2.2). Our first observation is that if

H1L(ΓQ,S, ρ(t))

(see 3.6 for the definition) vanishes, then we can kill the full dual Selmer group using Ramakr-ishna’s method; moreover, we cannot find an auxiliary prime w /∈ S with a Ramakrishnadeformation LRamw (see 3.1) such that h1

L⊥∪LRam,⊥w(ΓQ,S∪w, ρ(g)(1)) < h1

L⊥(ΓQ,S, ρ(g)(1)) when

H1L(ΓQ,S, ρ(t)) 6= {0}. But it is hard to achieve H1

L(ΓQ,S, ρ(t)) = {0} in general. Our secondobservation is as follows: suppose that 0 6= h1

L(ΓQ,S, ρ(t)) ≤ h1L⊥(ΓQ,S, ρ(t)(1)) (the inequality

is easy to guarantee), and let φ be a non-trivial class in H1L⊥(ΓQ,S, ρ(t)(1)). We can then find

an auxiliary prime w /∈ S with a Ramakrishna deformation LRamw such that φ|ΓQw/∈ LRam,⊥w ,

which impliesh1

L⊥∪LRam,⊥w(ΓQ,S∪w, ρ(t)(1)) < h1

L⊥∪L⊥w

(ΓQ,S∪w, ρ(t)(1)),

where Lw is the intersection of LRamw and the unramified condition at w. It turns out that(see the proof of 3.11) the right side of the inequality equals h1

L⊥(ΓQ,S, ρ(t)(1)); then a doubleinvocation of Wiles’ formula gives

h1L∪LRamw

(ΓQ,S∪w, ρ(t)) < h1L(ΓQ,S, ρ(t)).

By induction, we can enlarge L finitely many times to make H1L(ΓQ,S, ρ(t)) vanish, which in

turn allows us (see the proof of 3.14) to enlarge L even further to make H1L⊥(ΓQ,S, ρ(g)(1))

vanish, as remarked in the first observation. Thus we obtain an l-adic lift ρ : ΓQ → G(O)satisfying the prescribed local deformation conditions. By specifying carefully the localdeformation conditions, we make ρ have Zariski-dense image in G(Ql).

In the end, let us remark that the process described in the previous paragraph is very delicate,depending heavily on the algebraic structure of the summand of ρ(g) we want to annihilate.Fortunately, it works well for ρ(t).

Remark 1.6. An intricate generalization of Ramakrishna’s techniques is developed in theupcoming paper [8]. Their method applies to the case when ρ(g) is semi-simple, multiplicity-free, and each summand is absolutely irreducible. In particular, we can annihilate the abovedual Selmer group using their method. But it is much more complicated than ours. We willonly use it to remove the congruence conditions on l in 1.3. A simpler method for removingthese conditions might exist, but the author does not know at this point.

Acknowledgement: I would like to thank my adviser Stefan Patrikis for introducing thisinteresting project to me, for his constant guidance and encouragement, without which noneof the above work would be possible. I would also like to thank Jeffrey Adams for helpfulconversations on Weyl groups.

Notation: For a field F (typically Q or Qp), we let ΓF denote Gal(F/F ) for some fixedchoice of algebraic closure of F of F . When F is a number field, for each place v of F we fixonce and for all embeddings F → F v, giving rise to inclusions ΓFv → ΓF . If S is a finite setof places of F . we let ΓF,S denote Gal(FS/F ), where FS is the maximal extension of F in Funramified outside of S. If v is a place of F outside S, we write Frv for the correspondingarithmetic frobenius element in ΓF,S. When F = Q, we will sometimes write Γv for ΓFv andΓS for ΓQ,S. For a representation ρ of ΓF , we let F (ρ) denote the fixed field of Ker(ρ).

6 SHIANG TANG

Consider a group Γ, a ring A, an algebraic group G over Spec(A), and a homomorphismρ : Γ → G(A). We write g for both the Lie algebra of G and the A[G]-module induced bythe adjoint action. We let ρ(g) denote the A[Γ]-module with underlying A-module g inducedby ρ. Similarly, for a A[G]-submodule M of g, we write ρ(M) for the A[Γ]-module withunderlying A-module M induced by ρ.

We call an algebraic group simple if it is connected, nonabelian and has no proper normalalgebraic subgroups except for finite subgroups. It is sometimes called an almost simplegroup in the literature. Consider a simple algebraic group G, we write Gsc (resp. Gad) forthe simply-connected form (resp. adjoint form) of G.

Let O be the ring of integers of a finite extension of Ql. We let CNLO denote the categoryof complete noetherian local O-algebras for which the structure map O → R induces anisomorphism on residue fields.

All the Galois cohomology groups we consider will be k-vector spaces for k a finite extensionof Fl. We abbreviate dimHn(−) by hn(−).

We write κ for the l-adic cyclotomic character, and κ for its mod l reduction.

2. Constructions of residual representations

In this section, we construct residual representations

ρ : ΓQ → G(Fl)

for G a simple, simply-connected algebraic group.

2.1. Constructions based on the Weyl groups. I would like to thank Stefan Patrikisfor suggesting me to take variants of Weyl groups as the residual images, which turns out towork remarkably well.

2.1.1. Some group-theoretic results. We recall a property of the Weyl group of an irreducibleroot system Φ:

Lemma 2.1. W acts irreducibly on the C-vector space spanned by Φ and transitively onroots of the same length.

Let k be a finite extension of Fl. Let G = G(k) and T = T (k) be a maximal split torus ofG. Let Φ = Φ(G, T ) and N = NG(T ).

Corollary 2.2. For any α, β ∈ Φ of the same length, there exists w ∈ N such thatAd(w)gα = gβ. The adjoint action Ad(N) on g decomposes into submodules t and

gΦ :=∑α∈Φ

gα

when Φ is simply-laced; and is the direct sum of t,

gl :=∑

α∈Φ,α is long

gα

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 7

and

gs :=∑

α∈Φ,α is short

gα

otherwise.

Moreover, t is irreducible, and gΦ, gl, gs are irreducible if l is sufficiently large.

Proof. It suffices to show that gΦ, gl, gs are irreducible. We will only show that gΦ is ir-reducible, for the other two cases are similar. Take a nonzero vector X ∈ gΦ, writeX =

∑1≤i≤kXi where 0 6= Xi ∈ gαi for some distinct roots α1, α2, · · · , αk ∈ Φ. Since l

is sufficiently large, we can choose t ∈ T such that α(t), α ∈ Φ are all distinct. We have

ad(tj)X =∑i

αi(t)jXi

where 0 ≤ j ≤ k − 1. It follows that X, ad(t)X, ad(t2)X, · · · , ad(tk−1)X are linearly inde-pendent and hence they span the same subspace as X1, · · · , Xk do. In particular, Xi ∈ gαibelongs to the N -submodule of gΦ generated by X. But N acts transitively on the set ofroot spaces, it follows that X generates gΦ. Therefore, gΦ is irreducible. �

Remark 2.3. The above discussion can be carried out verbatim for a subgroup N ′ of N thatmaps onto a subgroup W ′ of W acting transitively on roots of the same length.

2.1.2. Some basic results in the inverse Galois theory. We list some elementary results withproofs in the inverse Galois theory, some of which are modified in order to suit our purposes.See Serre’s beautiful lecture notes [16] for details.

The first result dates back to David Hilbert:

Theorem 2.4. For n ≥ 2 (resp. n ≥ 3), there are infinitely many polynomials with rationalcoefficients f(T ) which realize the symmetric group of n letters Sn (resp. the alternatinggroup of n letters An) as Galois groups over Q. Moreover, for Sn (resp. for An with n ≥ 4),f can be chosen to have at least a pair of non-real roots.

The next result is an elementary case of a theorem due to Igor Shafarevich [11], which willbe used frequently in our constructions:

Theorem 2.5. Let G be a finite group. Suppose that there is a finite Galois extension K/Qsuch that Gal(K/Q) ∼= G. Let H be a finite abelian group with exponent m. Suppose thatthere is a split exact sequence of finite groups

1→ H → S → G→ 1

Then there is a finite Galois extension M/Q containing K such that the natural surjectivehomomorphism Gal(M/Q) � Gal(K/Q) realizes the surjective homomorphism S � G. Inother words, the split embedding problem has a proper solution.

Moreover, for any prime l that is outside the ramification locus of K/Q and prime to m, wecan choose M so that l is unramified in M .

8 SHIANG TANG

Proof. The argument is a minor modification of the proof of [16], Claim 2.2.5. Put L =K(µm). H can be regarded as a m-torsion module on which Gal(L/Q) acts. So there isa finite free (Z/mZ)[Gal(L/Q)]-module F of which H is a quotient. Suppose that r is thenumber of copies of (Z/mZ)[Gal(L/Q)] in F . Let S ′ be the semi-direct product of Gal(L/Q)and F . To solve the embedding problem posed by 1→ H → S → G→ 1, it suffices to solvethe embedding problem posed by

1→ F → S ′ → Gal(L/Q)→ 1

Claim: There is a Galois extension M ′/Q that solves the above embedding problem. More-over, for any prime l that does not divide m and is outside of the ramification locus of K/Q,we can choose M ′ so that l is unramified in M ′.

To see this, we choose places v1, · · · , vr of Q away from l such that vi splits completely in L.Let wi be a place of L extending vi, 1 ≤ i ≤ r. Any place of L extending vi can be writtenuniquely as σwi for some σ ∈ Gal(L/Q). Let w0 be a place of L extending l. For 1 ≤ i ≤ r,choose θi ∈ OL such that for 0 ≤ j ≤ r, (σwj)(θi) = 1 if σ = 1 and i = j and 0 if not. Theexistence of θi follows from the Weak Approximation Theorem. Let

M ′ = L( m√σθi|σ ∈ Gal(L/Q), 1 ≤ i ≤ r)

M ′ is Galois over Q, being the composite of L and the splitting field of the polynomial∏i

∏σ(Tm − σθi) ∈ Q[T ]. It is easy to see that Gal(M ′/L) ∼= F as Gal(L/Q)-modules. In

fact, fix i, there is an isomorphism

φi : Gal(L( m√σθi|σ ∈ Gal(L/Q))/L

) ∼= (Z/mZ)[Gal(L/Q)]

For an element g on the left side and any σ ∈ Gal(L/Q), g( m√σθi) = ζσ · m

√σθi for some

ζσ ∈ µm ∼= Z/mZ. We then define

φi(g) =∑σ

ζσ · σ ∈ (Z/mZ)[Gal(L/Q)]

It is clear that φi is an isomorphism by our choice of θi. It follows that Gal(M ′/L) ∼= F bylinear disjointness.

Therefore we obtain an exact sequence

1→ F → Gal(M ′/Q)→ Gal(L/Q)→ 1

Since F ∼= IndGal(L/Q){1} Z/mZ, by Shapiro’s lemma, H2(Gal(L/Q), F ) = H2({1},Z/mZ) = 0

and hence the sequence splits. Thus, Gal(M ′/Q) ∼= S ′.

It remains to show that l is unramified in M ′. We have ∀σ ∈ Gal(L/Q), ∀i, w0 is unramifiedin L( m

√σθi), because w0(σθi) = 0 and l does not divide m. So w0 is unramified in their

composite M ′. On the other hand, l is unramified in K by assumption and is unramified inQ(µm) since l does not divide m, so l is unramified in L. It follows that l is unramified inM ′. We have proved the claim.

Finally, let M be the fixed field of the kernel of the natural surjective homomorphism S ′ � S,we obtain a solution to the original embedding problem. �

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 9

2.1.3. SLn. We use the notation in 2.1.1. Let G = SLn(k), so W ∼= Sn. By [1], the N-Tsequence only splits when n is odd. We consider the subgroup W ′ = An of W . Let T be themaximal torus consisting of diagonal elements in SLn(k) and Φ = Φ(G, T ).

Lemma 2.6. Suppose n ≥ 4. Then An, as a subgroup of W , acts transitively on Φ.

Proof. This follows from the fact that An acts doubly transitively on {1, 2, · · · , n} if andonly if n ≥ 4. �

Let N ′ = π−1(W ′).

Lemma 2.7. The following exact sequence of finite groups splits:

1→ T → N ′ → W ′ → 1

Proof. We think of An as a subgroup of N = NG(T ) by realizing it as the group of n×n evenpermutation matrices. Then An normalizes T and N ′ is a semi-direct product of them. �

Let l be large enough. Since T is abelian of exponent |k| − 1 which is prime to l, by 2.5, N ′

is a Galois group over Q that is unramified at l. In other words, there is a surjection

ΓQ � N ′

which is unramified at l. Define ρ to be the composite

ΓQ � N ′i−→ SLn(k)

which is unramified at l.

Remark 2.8. It is tricker to realize N as a Galois group over Q. This can be reduced torealizing the Tits group T of SLn as a Galois group. T can be identified with the group ofn×n signed permutation matrices with determinant one, which is an index two subgroup ofthe group of n×n signed permutation matrices. The latter is isomorphic to the Weyl groupof type Bn, hence a Galois group over Q. As the Tits group sequence splits if and only if nis odd, so T can be realized over Q for n odd. When n is even, this problem is open exceptfor small n, as far as I know.

Let g = sln(k).

Proposition 2.9. For l sufficiently large and n ≥ 4, ρ(g) decomposes into irreducible ΓQ-modules ρ(t) and ρ(gΦ).

Proof. This follows from 2.2 and 2.3. �

There remains the case when G is SL2 or SL3. For SL3, see Section 2.2. For SL2, see Section4.2.3.

10 SHIANG TANG

2.1.4. Sp2n. We use the notation in 2.1.1. Let G = Sp2n(k), then W is isomorphic to asemi-direct product of Sn and D := (Z/2Z)n, see [4], for example. We fix a maximal splittorus T in Sp2n(k). The N-T sequence does not split by [1].

Lemma 2.10. Consider the N-T sequence for Sp2n. Sn ⊂ W has a section to N , D ⊂ Wdoes not have a section to N but there is a subgroup D of N such that π(D) = D andD ∼= (Z/4Z)n. Moreover, as subgroups of N , Sn normalizes D and Sn ∩ D = {1}. We letW1 be the subgroup of N generated by Sn and D.

Therefore, by 2.5, W1 can be realized as a Galois group over Q. N is generated by W1 and T .W1 normalizes T and W1∩T = TZ ∼= (Z/2Z)n. Let S be the (abstract) semidirect product ofW1 and T . By 2.5, for l large enough, S can be realized as a Galois group over Q unramifiedat l. But there is a natural quotient map S � N , so N is a Galois group over Q unramifiedat l as well. In other words, there is a surjection

ΓQ � N

unramified at l. Define ρ to be the composite

ΓQ � Ni−→ Sp2n(k)

which is unramified at l.

Remark 2.11. Note that W1∼= T and hence the Tits group for Sp2n is easily realized as a

Galois group over Q.

Let g = sp2n(k). The root system Φ of g is not simply laced.

Proposition 2.12. For l sufficiently large, ρ(g) decomposes into irreducible ΓQ-modules ρ(t),ρ(gl) and ρ(gs).

Proof. This follows from 2.2. �

2.1.5. Spin2n and Spin2n+1. For spin groups, the N-T sequence does not split by [1]. Weconsider a subgroup of W . For G = Spin2n, W is isomorphic to a semi-direct product ofSn and D := (Z/2Z)n−1 and we take W ′ be the subgroup generated by An and D. ForG = Spin2n+1, W is isomorphic to a semi-direct product of Sn and D := (Z/2Z)n and wetake W ′ be the subgroup generated by An and D. Similar to the symplectic case, we willshow that N ′ = π−1(W ′) is a Galois group over Q.

Lemma 2.13. Consider the N-T sequence for G = Spin2n(k) or Spin2n+1(k). An ⊂ W has

a section to N , and there is an abelian subgroup D of N such that π(D) = D. Moreover, assubgroups of N , An normalizes D and let W1 be their product.

Therefore, by 2.5, W1 can be realized as a Galois group over Q. N ′ is generated by W1 andT and W1 normalizes T . Let S be the (abstract) semidirect product of W1 and T . By 2.5,for l large enough, S can be realized as a Galois group over Q unramified at l. But there isa natural quotient map S � N ′, so N ′ is a Galois group over Q unramified at l as well. Inother words, there is a surjection

ΓQ � N ′

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 11

unramified at l. For G = Spin2n or Spin2n+1, define ρ to be the composite

ΓQ � N ′i−→ G(k)

which is unramified at l.

Let g be the Lie algebra of G(k). The corresponding root system Φ is simply laced ifG = Spin2n and not if G = Spin2n+1.

Remark 2.14. The above construction can also realize the Tits groups for Spin2n and Spin2n+1

as Galois groups over Q. For, Sn has a section to the Tits group for the orthogonal groupand its pullback in the spin group is a two-fold central extension of Sn which is a Galoisgroup over Q by a result of Jack Sonn [17]. But we will not need this.

Proposition 2.15. For l sufficiently large and n ≥ 4, ρ(g) decomposes into irreducible ΓQ-modules ρ(t) and ρ(gΦ) when G = Spin2n; and it decomposes into irreducible ΓQ-modulesρ(t), ρ(gl) and ρ(gs) when G = Spin2n+1.

Proof. Note that the action of W ′ on Φ is transitive if and only if n ≥ 4. Then the propositionfollows from 2.2 and 2.3. �

Remark 2.16. There remains the case when G is one of Spin4, Spin5, Spin6, Spin7. ButSpin4(C) ∼= SL2(C)× SL2(C), Spin5(C) ∼= Sp4(C), Spin6(C) ∼= SL4(C) which are included inother cases. For Spin7, the half sum of coroots ρ∨ = 3α∨1 + 5α∨2 + 3α∨3 has integer coefficients(see [4], for example), so the principal GL2 map is well-defined, see Section 2.2.

2.1.6. Esc7 . We refer the reader to [4] for detailed information on the root system of type E7.

Let G = Esc7 (k). The Weyl group W is isomorphic to the direct product of [W,W ] and Z/2Z.

By [1], the N-T sequence does not split. We choose a subgroup W ′ of W which lifts to N asfollows. Consider the extended Dynkin diagram of type E7, there is a sub-root system Φ′ of Φwhich is of type A7. The alternating group A8 is a subgroup of S8

∼= W (A7) ≤ W = W (E7).

Lemma 2.17. A8 ≤ W lifts to N .

Lemma 2.18. The action of A8 on Φ has an orbit of size 56 and an orbit of size 70.

Proof. We first consider the action of S8∼= W (A7) on Φ. By 2.1, S8 acts transitively on

Φ′, which has 56 roots. A straightforward calculation shows that for some α ∈ Φ − Φ′,S8 · α has exactly 70 roots and the stablizer of α in S8 is isomorphic to S4 × S4 ⊂ S8. Since56 + 70 = 126 is the number of roots in Φ, the lemma is true for S8. Now we consider thealternating group A8. 2.6 implies that A8 still acts transitively on Φ′. As (S4×S4)∩A8 (thestablizer of α in A8) has order 1

2|S4 × S4| = 288, A8 · α has exactly |A8|/288 = 70 roots. �

It is clear that A8, considered as a subgroup of N , normalizes T and A8 ∩ T = {1}. Let N ′

be the subgroup of G = Esc7 (k) generated by A8 and T . By 2.4 and 2.5, for l large enough,

N ′ can be realized as a Galois group over Q unramified at l. In other words, there is asurjection

ΓQ � N ′

that is unramified at l. Define ρ to be the composite

ΓQ � N ′i−→ Esc

7 (k)

12 SHIANG TANG

which is unramified at l.

Let ga (resp. gb) be the direct sum of the root spaces corresponding to the orbit of size 56(resp. size 70) in 2.18.

Proposition 2.19. For l sufficiently large, ρ(g) decomposes into irreducible ΓQ-modules ρ(t),ρ(ga) and ρ(gb).

Proof. The proof is very similar to the proof of 2.2. �

2.2. Constructions based on the principal GL2. We record some facts on the principalSL2. For more details, see [15] and [12]. Let G/k be a simple algebraic group with a BorelB containing a split maximal torus T . Let Φ = Φ(G, T ) be the root system of G with theset of simple roots ∆ corresponding to B. We fix a pinning {Xα}α∈∆ and let

X =∑α∈∆

Xα

which can be extended to an sl2-triple (X,H, Y ) with

H =∑α>0

Hα

for Hα the coroot vector corresponding to α.

A principal SL2 homomorphism is a homomorphism

ϕ : SL2 → G

such that

ϕ(

(1 t0 1

)) = exp(tX)

and

ϕ(

(t 00 t−1

)) = exp(tH) = 2ρ∨(t)

with ρ∨ the half-sum of coroots. ρ∨ is always defined when G is adjoint. Suppose thatρ∨ : Gm → Gad lifts to G and fix a lift which is again denoted ρ∨. A principal GL2

homomorphism is a homomorphism

ϕ : GL2 → G

such that

ϕ(

(1 t0 1

)) = exp(tX)

and

ϕ(

(t 00 1

)) = ρ∨(t)

By definition, a principal GL2 factors through PGL2.

By examining the list in [4], we get

Lemma 2.20. For G a simple algebraic group, ρ∨ : Gm → Gad lifts to Gsc if and only if Gis one of the following types: A2n, B4n, B4n+3, D4n, D4n+1, E6, E8, F4, G2.

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 13

The operator ad(H) preserves gX (the centralizer of X in g) and

gX =∑m>0

V2m

where V2m is the eigenspace of H corresponding to the eigenvalue 2m. The following propo-sition is due to Kostant.

Proposition 2.21. The dimension of gX is equal to the rank of g. V2m is nonzero if andonly if m is an exponent of g. Letting GL2 act on g via ϕ, there is an isomorphism ofGL2-representations

g ∼=⊕m>0

Sym2m(k2)⊗ det−m⊗V2m

In [12], the following crucial lemma is verified using Magma for exceptional Lie algebras.

Lemma 2.22. Assume l is large enough for G. For g of type An, Bn, Cn and exceptionaltypes, there is a root α ∈ Φ such that every irreducible submodule of g in 2.21 has a vectorwith nonzero lα component and a vector with nonzero g−α component.

Remark 2.23. Note that we have excluded the Dn case in the above lemma since the calcu-lation is very messy for Dn and I do not know a conceptual proof.

Suppose that ρ∨ : Gm → Gad lifts to G. We take f to be as in 1.2. By Ribet’s theorem, theprojective image of rf,λ is either PGL2(k) or PSL2(k) for a subfield k of kλ. Define

ρ : ΓQrf,λ−−→ GL2(k)

ϕ−→ G(k)

Remark 2.24. This construction works for all exceptional groups but Esc7 for which ρ∨ :

Gm → Ead7 does not lift to Esc

7 . We will only use this construction for G of type E6, A2 andB3.

3. Annihilating the dual Selmer group: Ramakrishna’s method and itsvariant

Given ρ : ΓQ → G(k) defined in the previous section, we want to obtain its l-adic liftρ : ΓQ → G(O) with O the ring of integers of a finite extension of Ql whose residue fieldis k satisfying a given global deformation condition. Just as in [12], we use Ramakrishna’smethod to annihilate the associated dual Selmer group. The new feature is a double useof Patrikis’ extension of Ramakrishna’s method when the original form fails to work, seeSection 3.2.

3.1. Review on Ramakrishna’s method.

3.1.1. Ramakrishna deformations. In this section, we review Patrikis’ extension of Ravi Ra-makrishna’s techniques for lifting odd two-dimensional Galois representations to geometricl-adic representations. We will only list the key points and results here. For proofs, see [12],Section 4.2. For a review on deformation theory of Galois representations, see [12], Section3.

14 SHIANG TANG

We begin by defining a type of local deformation condition called Ramakrishna conditionwhich will be used at the auxiliary primes of ramification in Ramakrishna’s global argument.Let F be a finite extension of Qp for p 6= l, and let ρ : ΓF → G(k) be an unramifiedhomomorphism such that ρ(FrF ) is a regular semi-simple element. Let T be the connectedcomponent of the centralizer of ρ(FrF ); this is a maximal k-torus of G, but we can lift it toan O-torus uniquely up to isomorphism, which we also denote by T , and then we can liftthe embedding over k to an embedding over O which is unique up to G(O)-conjugation. Bypassing to an etale extension of O, we may assume that T is split.

Definition 3.1. ([12], Definition 4.9) Let ρ, T be as above. For α ∈ Φ(G, T ), ρ is said to beof Ramakrishna type α if

α(ρ(FrF )) = κ(FrF )

Let Hα = T · Uα be the subgroup generated by T and the root subgroup Uα correspondingto α. Ramakrishna deformation is a functor

LiftRamρ : CNLO → Sets

such that for a complete local noetherian O-algebra R, LiftRamρ (R) consists of all lifts

ρ : ΓF → G(R)

of ρ such that ρ is G(R)-conjugate to a homomorphism ΓFρ′−→ Hα(R) with the resulting

composite

ΓFρ′−→ Hα(R)

Ad−→ GL(gα ⊗R) = R×

equal to κ.

We shall call such a ρ to be of Ramakrishna type α as well.

We denote by DefRamρ the corresponding deformation functor.

Lemma 3.2. ([12], Lemma 4.10) LiftRamρ is well-defined and smooth.

Consider the sub-torus Tα = Ker(α)0 of T , and denote by tα its Lie algebra. There is acanonical decomposition tα ⊕ lα = t with lα the one-dimensional torus generated by thecoroot α∨.

The next lemma is crucial to the global argument.

Lemma 3.3. ([12], Lemma 4.11) Assume ρ is of Ramakrishna type α. Let W = tα⊕gα, W⊥

the annihilator of W under the Killing form on g. Let LRamρ (resp. LRam,⊥ρ ) be the tangent

space of DefRamρ (resp. the annihilator of LRamρ under the local duality pairing). Then

(1) LRamρ∼= H1(ΓF , ρ(W )).

(2) dimLRamρ = h0(ΓF , ρ(g)).

(3) LRam,⊥ρ∼= H1(ΓF , ρ(W⊥)(1)).

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 15

(4) Let LRam,�ρ (resp. LRam,⊥,�ρ ) be the preimage in Z1(ΓF , ρ(g)) of LRamρ (resp. LRam,⊥ρ ).Under the canonical decomposition

g =⊕γ

gγ ⊕ tα ⊕ lα

all cocycles in LRam,�ρ (resp. LRam,⊥,�ρ ) have lα (resp. g−α) component equal to zero.

3.1.2. The global argument. In this section, we assume G is semi-simple. Let ρ : ΓQ,S → G(k)be a continuous homomorphism for which h0(ΓQ, ρ(g)) = h0(ΓQ, ρ(g)(1)) = 0. In particular,the deformation functor is representable.

Proposition 3.4. ([12], Proposition 5.2) Suppose that there is a global deformation conditionL = {Lv}v∈S consisting of smooth local deformation conditions for each place v ∈ S. LetK = Q(ρ(g), µl). We assume the following:

(1) ∑v∈S

(dimLv) ≥∑v∈S

h0(ΓQv , ρ(g))

(2) H1(Gal(K/Q), ρ(g)) and H1(Gal(K/Q), ρ(g)(1)) vanish.

(3) Assume Item 2 holds. For any pair of non-zero Selmer classes φ ∈ H1L⊥(ΓQ,S, ρ(g)(1))

and ψ ∈ H1L(ΓQ,S, ρ(g)), we can restrict them to ΓK where they become homomor-

phisms, which are non-zero by Item 2. Letting Kφ/K and Kψ/K be their fixed fields,we assume that Kφ and Kψ are linearly disjoint over K.

(4) Consider any φ and ψ as in the hypothesis of Item 3. There exists an element σ ∈ ΓQsuch that ρ(σ) is a regular semi-simple element of G, the connected component ofwhose centralizer we denote T , and such that there exists a root α ∈ Φ(G, T ) satisfying(a) α(ρ(σ)) = κ(σ).

(b) k[ψ(ΓK)] has an element with non-zero lα component; and

(c) k[φ(ΓK)] has an element with non-zero g−α component.

Then there exists a finite set of primes Q disjoint from S, and a lift ρ : ΓQ,S∪Q →G(O) of ρ such that ρ is of type Lv at all v ∈ S and of Ramakrishna type at allv ∈ Q.

3.2. A variant of Ramakrishna’s method in the Weyl group case. In this section,we let G be a simple algebraic group of classical type or type E7 and let ρ : ΓQ → G(k) beas in Section 2.1. In particular, the adjoint Galois module ρ(g) decomposes into the directsum of irreducibles: ρ(t), ρ(gΦ) when Φ is simply-laced, ρ(t), ρ(gl), ρ(gs) otherwise. Let usremark that the method in this section would also work for other exceptional groups if onecan realize the normalizer of a split maximal torus in G(k) (or an suitable variant of which)as a Galois group over Q.

Let M be a finite dimensional k-vector space with a continuous ΓQ-action. Define its Tatedual M∨ = Hom(M,µ∞) to be a space with the following ΓQ-action:

(σf)(m) := σ(f(σ−1m))

16 SHIANG TANG

Proposition 3.5. For any continuous homomorphism ρ : ΓQ → G(k), ρ(g)∨ ∼= ρ(g)(1). Forρ as in Section 2.1, ρ(t)∨ ∼= ρ(t)(1).

Proof. The Killing form is a non-degenerate G-invariant symmetric bilinear form on g, whichidentifies the contragredient representation g∗ with g, and hence identifies ρ(g)∨ with ρ(g)(1)as Galois modules. If ρ is as in Section 2.1, then the Galois action on t factors through W . Itis easy to see that the standard bilinear form on t is non-degenerate and W -invariant. Justas above, we deduce that ρ(t)∨ ∼= ρ(t)(1) as Galois modules. �

Definition 3.6. Let L = {Lv}v∈S be the Selmer system corresponding to a global deforma-tion condition for ρ that is unramified outside a finite set of places S, and let L⊥ = {L⊥v }v∈Sbe the associated dual Selmer system. Suppose that ρ(g) is semi-simple. Then for anyΓQ-submodule M of ρ(g), M∨ is isomorphic to M(1) ⊂ ρ(g)(1). Moreover, the isomor-phism is canonical if ρ(g) decomposes into non-isomorphic irreducible submodules. Definethe M-Selmer as follows:

H1L(ΓS,M) = Ker

(H1(ΓS,M)→

⊕v∈S

H1(Γv,M)/(Lv ∩H1(Γv,M))

)and define the M-dual Selmer as follows:

H1L⊥(ΓS,M

∨) = Ker

(H1(ΓS,M

∨)→⊕v∈S

H1(Γv,M∨)/(L⊥v ∩H1(Γv,M

∨))

)

The motivation of the arguments in this section is the difficulty to fulfill Item 4 of 3.4 forour choice of the residual representation. Let us explain it. In practice, we need to find aregular semi-simple element Σ in ρ(ΓQ), the connected component of whose centralizer wedenote T ′, for which there exists a root α ∈ Φ′ := Φ(G, T ′) such that

(1) α(Σ) ∈ (Z/lZ)×.

(2) Every irreducible summand of ρ(g) has an element with non-zero lα component.

(3) Every irreducible summand of ρ(g) has an element with non-zero g−α component.

Since ρ is unramified at l and Q(µl) is totally ramified at l, Q(ρ) and Q(µl) are linearlydisjoint over Q. Thus, there exists an element σ ∈ ΓQ such that ρ(σ) = Σ and κ(σ) = α(Σ).It follows that α(ρ(σ)) = κ(σ).

If we take Σ to be a regular semi-simple element in T , then there is no α ∈ Φ fulfilling both(2) and (3). Therefore, we need to look for Σ ∈ N = NG(T ) for which π(Σ) 6= 1 in W .

Let t′ be a regular semi-simple element in G and t ∈ T be an element conjugate to t′.Then t and t′ determines a unique bijection between Φ = Φ(G, T ) and Φ′ = Φ(G, T ′) withT ′ = ZG(t′)0: for any α ∈ Φ, define α′ ∈ Φ′ such that

α′(h) = α(g−1hg)

for any h ∈ T ′, where g is an element in G such that g−1t′g = t. Since t is regular semi-simple,α′ is independent of g.

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 17

Lemma 3.7. For a long root α ∈ Φ, let Σ be a regular semi-simple element in N (whichplays the role of t′) such that π(Σ) = sα. We fix an element t ∈ T conjugate to Σ and letT ′ = ZG(Σ)0. Σ and t determine a bijection between Φ and Φ′ as above. If Φ is of type An orDn, then the root α′ ∈ Φ′ corresponding to α fulfills (1) and (3). For (2), gΦ has an elementwith non-zero lα′ component, but t does not. If Φ is of type Bn, Cn or E7, then α′ fulfills (1)and t has a vector with non-zero g−α′ component. Moreover, t′ ∩ t = t′α′ ∩ t = W ∩ t withW = t′α′ ⊕ gα′ in either case.

Lemma 3.8. Assume that Φ is of type Bn, Cn or E7, so ρ(g) = ρ(t)⊕ ρ(gl)⊕ ρ(gs) for Bn

and Cn, ρ(g) = ρ(t)⊕ ρ(ga)⊕ ρ(gb) for E7.

(1) (Type Bn and Cn) For a pair of non-perpendicular β, γ ∈ Φ with β long and γ short,let Σ be a regular semi-simple element in N such that π(Σ) = sβ · sγ. We fix anelement t ∈ T conjugate to Σ and let T ′ = ZG(Σ)0. Σ and t determine a bijectionbetween Φ and Φ′ = Φ(G, T ′). Then for all long root α′ in the span of β′ and γ′

(which is a sub-system of Φ′ of type C2), (3) is satisfied. For (2), gl (resp. gs) hasan element with non-zero lα′ component, but t does not. α′(Σ) has order 4 in k×,hence (1) is satisfied only for l ≡ 1(4).

(2) (Type E7) For a pair of non-perpendicular β, γ ∈ Φ with gβ ⊂ ga and gγ ⊂ gb, let Σbe a regular semi-simple element in N such that π(Σ) = sβ · sγ. We fix an elementt ∈ T conjugate to Σ and let T ′ = ZG(Σ)0. Σ and t determine a bijection betweenΦ and Φ′ = Φ(G, T ′). Then for all root α′ in the span of β′ and γ′ (which is asub-system of Φ′ of type A2), (3) is satisfied. For (2), ga (resp. gb) has an elementwith non-zero lα′ component, but t does not. α′(Σ) has order 3 in k×, hence (1) issatisfied only for l ≡ 1(3).

Moreover, W ∩ t ⊂ t′ with W = t′α′ ⊕ gα′.

Remark 3.9. In general, no matter what element of N we pick for Σ and what root α′ we pickfrom the associated root system Φ′, t does not have an element with non-zero lα component.This means the irreducible summand t is ‘too small’ in some sense. But the next lemmaindicates how we may find out way out despite of this.

Lemma 3.10. Let σ be an element in ΓQ such that α(ρ(σ)) = κ(σ) and Σ := ρ(σ) is aregular semi-simple element in NG(T ) satisfying 3.7 for Φ of type An or Dn, and 3.8 for Φof type Bn, Cn or E7. Assume there is a Selmer system L = {Lv}v∈S for which the t-SelmerH1L(ΓS, ρ(t)) is trivial. Then Item 4 of 3.4 is satisfied.

The next proposition achieves the vanishing assumption of the t-Selmer in 3.10 by using ofa variant of the cohomological arguments in Ramakrishna’s method.

Proposition 3.11. Suppose that

h1L(ΓS, ρ(t)) ≤ h1

L⊥(ΓS, ρ(t)(1))

Then there is a finite set of places Q disjoint from S and a Ramakrishna deformation con-dition for each w ∈ Q with tangent space LRamw such that

H1L∪{LRamw }w∈Q

(ΓS∪Q, ρ(t)) = 0

18 SHIANG TANG

We may assume that H1L(ΓS, ρ(t)) is nontrivial, for otherwise we are done. The inequality

in 3.11 then implies that H1L⊥(ΓS, ρ(t)(1)) is nontrivial. Let 0 6= φ ∈ H1

L⊥(ΓS, ρ(t)(1)).

Lemma 3.12. There exists σ ∈ ΓQ with the following properties:

(1) ρ(σ) is a regular semisimple element of G(k), the connected component of whosecentralizer we denote T ′.

(2) There exists α ∈ Φ(G, T ′), such that α(ρ(σ)) = κ(σ).

(3) k[φ(ΓK)] has an element with nonzero g−α-component.

Proof. This follows immediately from 3.7. �

Lemma 3.13. There exist infinitely many places w /∈ S such that ρ|Γw is of Ramakrishnatype α and φ|Γw /∈ LRam,⊥w .

Proof. The proof is identical to the proof of [12], Lemma 5.3, which uses 3.3, 3.12 andChebotarev’s density theorem. �

Proof of 3.11: Let w be chosen as in 3.13. We will show that

h1

L⊥∪LRam,⊥w(ΓS∪w, ρ(t)(1)) < h1

L⊥(ΓS, ρ(t)(1)) (1)

and

h1L∪LRamw

(ΓS∪w, ρ(t))− h1

L⊥∪LRam,⊥w(ΓS∪w, ρ(t)(1)) = h1

L(ΓS, ρ(t))− h1L⊥(ΓS, ρ(t)(1)) (2)

which will imply thath1L∪LRamw

(ΓS∪w, ρ(t)) < h1L(ΓS, ρ(t))

from which 3.11 follows by induction.

We first show (2): A double invocation of Wiles’ formula gives

[LHS of (2)−RHS of (2)]=dim(LRamw ∩ H1(Γw, ρ(t))) − h0(Γw, ρ(t)) = h1(Γw, ρ(W ∩ t)) −h0(Γw, ρ(t)) = dim(W ∩ t)− dim(t′ ∩ t) = 0 (3.7).

It remains to prove (1). Let Lw = Lunrw ∩ LRamw , so L⊥w = Lunr,⊥w + LRam,⊥w . We have thefollowing obvious inclusions

H1L∪Lw(ΓS∪w, ρ(t)) ⊂ H1

L∪LRamw(ΓS∪w, ρ(t)) (A)

H1

L⊥∪LRam,⊥w(ΓS∪w, ρ(t)(1)) ⊂ H1

L⊥∪L⊥w

(ΓS∪w, ρ(t)(1)) (A′)

H1L∪Lw(ΓS∪w, ρ(t)) ⊂ H1

L(ΓS, ρ(t)) = H1L∪Lunrw

(ΓS∪w, ρ(t)) (B)

H1L⊥(ΓS, ρ(t)(1)) = H1

L⊥∪Lunr,⊥w(ΓS∪w, ρ(t)(1)) ⊂ H1

L⊥∪L⊥w

(ΓS∪w, ρ(t)(1)) (B′)

As φ|Γw /∈ LRam,⊥w , (A′) is a strict inclusion. We claim that (B′) is an isomorphism whichimplies (1). To prove our claim, we consider (B) first. There is an exact sequence

0→ H1L∪Lw(ΓS∪w, ρ(t))→ H1

L(ΓS, ρ(t))→ (Lunrw ∩H1(Γw, ρ(t)))/(Lw ∩H1(Γw, ρ(t)))

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 19

The top of its last term has dimension dim(t′ ∩ t), the bottom of its last term has dimensiondim(t′α ∩ t). By 3.7, these dimensions are equal. So the last term is zero, and hence (B) isan isomorphism. A double invocation of Wiles’ formula gives

h1L⊥∪L⊥

w(ΓS∪w, ρ(t)(1))− h1

L⊥(ΓS, ρ(t)(1))

= h1L∪Lw(ΓS∪w, ρ(t))− h1

L(ΓS, ρ(t)) + h0(Γw, ρ(t))− dim(Lw ∩H1(Γw, ρ(t)))

Because (B) is an isomorphism and dim(Lw ∩ H1(Γw, ρ(t))) = h1(Γw, ρ(W ∩ t′ ∩ t)) =dim(W ∩ t′ ∩ t) = dim(t′ ∩ t) = h0(Γw, ρ(t)) (3.7), the right hand side of the above identityis zero. Therefore (B′) is an isomorphism which completes the proof of the proposition. �

Theorem 3.14. Let L = {Lv}v∈S be a family of smooth local deformation conditions forρ unramified outside a finite set of places S containing the real place and places where ρ isramified. Suppose that ∑

v∈S

dimLv ≥∑v∈S

h0(Γv, ρ(g)).

and ∑v∈S

dim(Lv ∩H1(Γv, ρ(t))) ≤∑v∈S

h0(Γv, ρ(t)).

Assume l is large enough, and when Φ is doubly-laced, assume l ≡ 1(4).

Then there is a finite set of places Q disjoint from S and a continuous lift

ρ : ΓS∪Q → G(O)

of ρ such that ρ is of type Lv for v ∈ S and of Ramakrishna type for v ∈ Q.

Proof. The second inequality and Wiles’ formula imply that h1L(ΓS, ρ(t)) ≤ h1

L⊥(ΓS, ρ(t)(1)).By 3.11, we can enlarge L by adding finitely many Ramakrishna deformation conditions toget a new Selmer system L′ = {Lv}v∈S′ with S ′ ⊃ S such that H1

L′(ΓS′ , ρ(t)) = {0}. By3.3,(2), replacing L by L′ preserves the first inequality.

If Φ is of type An or Dn, we take Σ, α as in 3.7. There exists σ ∈ ΓQ such that ρ(σ) = Σand κ(σ) = α(Σ) which imply that α(ρ(σ)) = κ(σ). By Chebotarev’s density theorem, thereare infinitely many places w /∈ S ′ such that ρ|Γw is of Ramakrishna type α. If Φ is of typeBn, Cn or E7, we take Σ, α as in 3.8. For the same reason, there are infinitely many placesw /∈ S ′ such that ρ|Γw is of Ramakrishna type α if l ≡ 1(4).

For the Ramakrishna deformation LRamw associated to ρ|Γw ,

H1L′∪LRamw

(ΓS′∪w, ρ(t)) = 0.

Indeed, LRamw ∩H1(Γw, ρ(t)) = H1(Γw, ρ(W ∩ t)) ⊂ H1(Γw, ρ(t′ ∩ t)) = H1(Γw/Iw, ρ(t)). So

H1L′∪LRamw

(ΓS′∪w, ρ(t)) ⊂ H1L′(Γ

′S, ρ(t)) = {0}.

Let us check the assumptions of 3.4. Item (1) is satisfied by assumption. For Item (2), since|Gal(K/Q)| divides (l − 1)|ρ(ΓQ)| which is prime to l by the definition of ρ (see Section2.1). Thus the cohomology groups vanish in positive degrees. For Item (3), it sufficesto show that for any pair of irreducible summands M,N of ρ(g), distinct or not, M andN(1) are non-isomorphic. This can be proved using the argument of [12], Lemma 6.8. As

20 SHIANG TANG

H1L′(Γ

′S, ρ(t)) = {0}, Item (4) is satisfied by 3.10. Therefore, 3.4 applies and we get desired

l-adic lifts. �

3.3. Ramakrishna’s method in the principal GL2 case. We use the notation in Section2.2. Patrikis has shown that all simple algebraic groups of exceptional types are geomet-ric monodromy groups for ΓQ except for Ead

6 , Esc6 , E

sc7 . In this paper, the principal GL2

construction is mainly used for proving that Ead6 , E

sc6 , SL3, Spin7 are algebraic monodromy

groups for ΓQ. Unlike what happens in Section 3.2, there is no issue with the original formof Patrikis’ extension of Ramakrishna’s method (3.4) for this construction. The proof of thefollowing theorem is identical to that of [12], Theorem 7.4.

Theorem 3.15. Let L = {Lv}v∈S be a family of smooth local deformation conditions forρ unramified outside a finite set of places S containing the real place and places where ρ isramified. Suppose that ∑

v∈S

dimLv ≥∑v∈S

h0(Γv, ρ(g)).

Let l be large enough.

Then there is a finite set of places Q disjoint from S and a continuous lift

ρ : ΓS∪Q → G(O)

of ρ such that ρ is of type Lv for v ∈ S and of Ramakrishna type for v ∈ Q.

4. Simple, simply-connected algebraic groups as monodromy groups for ΓQ

In this section, we prove 1.3.

4.1. Local deformation conditions.

4.1.1. The archimedean place. Recall that in Section 2.1.3 through Section 2.1.5, we con-struct the residual representation by first realizing Sn or An as a Galois group over Q andthen repeatedly applying 2.5 to build the Galois extension realizing N or a subgroup of Nover Q. Let f be the polynomial in 2.4. We write c for the nontrivial element in ΓR, thecomplex conjugation.

Lemma 4.1. Let G be one of SLn, Sp2n, Spin2n+1, Spin2n and ρ : ΓQ → G(k) be as inSection 2.1. We assume that n ≥ 4 for SLn, Spin2n+1, Spin2n, and n ≥ 2 for Sp2n. Let kbe the sum of the number of real roots of f and the number of pairs of non-real roots off . Let ε1, · · · , εn be a sequence of numbers such that k of the εi’s are 1, and n − k of theεi’s are −1. Let π be the covering homomorphism from the spin group to the orthogonalgroup. Then ρ(c) ∼ diag(ε1, · · · , εn) for SLn, ρ(c) ∼ diag(ε1, · · · , εn, εn, · · · , ε1) for Sp2n,π(ρ(c)) ∼ diag(ε1, · · · , εn, εn, · · · , ε1) for Spin2n, π(ρ(c)) ∼ diag(ε1, · · · , εn, 1, εn, · · · , ε1) forSpin2n+1.

Corollary 4.2. Let G be of classical type and ρ : ΓQ → G(k) be as in Section 2.1. Thenh0(ΓR, ρ(g)) is rk(g) + f where

f = 2((k

2

)+

(n− k

2

))

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 21

for SLn,

f = 4((k

2

)+

(n− k

2

))+ 2n

for Sp2n,

f = 4((k

2

)+

(n− k

2

))+ 2k

for Spin2n+1,

f = 4((k

2

)+

(n− k

2

))for Spin2n.

Lemma 4.3.dimk

(Sym2n(k2)⊗ det−n

)diag(1,−1)

equals n when n is odd, and n+ 1 when n is even.

Corollary 4.4. Let ρ : ΓQ → G(k) be as in Section 2.2. Then h0(ΓR, ρ(g)) = 4 for G = SL3,h0(ΓR, ρ(g)) = 9 for G = Spin7, h0(ΓR, ρ(g)) = 38 for G = Esc

6 .

4.1.2. The place l. As we are not looking for geometric l-adic Galois representations in thispaper, we impose no condition at the place l. So the tangent space is equal to H1(Γl, ρ(g)).By the local Euler characteristic formula,

h1(Γl, ρ(g)) = h0(Γl, ρ(g)) + h2(Γl, ρ(g)) + dimk g

Lemma 4.5. Let ρ be as in Section 2.1 or Section 2.2. Then h2(Γl, ρ(g)) = 0 for largeenough primes l.

Proof. By local duality, it suffices to show that h0(Γl, ρ(g)(1)) = 0. For ρ in Section 2.1,ρ(IQl) is trivial by construction but κ(IQl) is nontrivial, so ρ(g)(1)IQv is trivial. In particular,h0(Γl, ρ(g)(1)) = 0. For ρ in Section 2.2, a similar argument to that in [20], Proposition 4.4shows h0(Γl, Sym2n(k2)⊗ det−n⊗κ) = 0 for n ≥ 1 and large enough primes l which impliesh0(Γl, ρ(g)(1)) = 0. �

Corollary 4.6. h1(Γl, ρ(g)) = h0(Γl, ρ(g)) + dimk g.

4.1.3. A zero-dimensional deformation. In order to enlarge the image of the l-adic lift of theresidual representation, we need to use some simple local deformation conditions at someunramified places.

Suppose that p 6= l and ρ : Γp → G(k) is an unramified representation and let g ∈ G(O) bea lift of ρ(Frp).

Definition 4.7. DefineLiftgρ : CNLO → Sets

such that for a complete local noetherian O-algebra R, Liftgρ(R) consists of all lifts

ρ : ΓF → G(R)

of ρ such that ρ is unramified and ρ(Frp) is G(R)-conjugate to g. So the tangent space iszero-dimensional and the deformation condition is smooth.

22 SHIANG TANG

4.1.4. Steinberg deformations. In [12], a local deformation condition of ‘Steinberg type’ istaken at a place in order to obtain a regular unipotent element in the image of the l-adiclift. We will only need this to deform ρ constructed from the principal GL2. We refer thereader to [12], Section 4.3 for the definition and properties of the Steinberg deformation.

4.1.5. Minimal prime to l deformations. This deformation condition is well-known, see [12],Section 4.4 for the definition. We will use this deformation condition at places v 6= l forwhich ρ(IQv) is nontrivial and ρ(ΓQv) has order prime to l.

4.2. Algebraic monodromy groups of the l-adic lifts of ρ. In this section, we specifythe global deformation condition and compute the Wiles formula, then use the results inSection 3 to prove 1.3. Let us recall Wiles’ formula, for a proof, see [12], Proposition 9.2.

Proposition 4.8. Let M be a finite-dimensional k-vector space with a continuous ΓQ actionunramified outside a finite set of places S. Let L = {Lv}v∈S (resp. L⊥ = {L⊥v }v∈S) be aSelmer system (resp. dual Selmer system) for M . Then

h1L(ΓS,M)− h1

L⊥(ΓS,M∨) = h0(ΓS,M)− h0(ΓS,M

∨) +∑v∈S

(dimk Lv − h0(Γv,M)).

We will compute the right hand side of the identity for the global deformation condition to bespecified in below. For ρ(g) from either Section 2.1 or Section 2.2, note that h0(ΓS, ρ(g)) =h0(ΓS, ρ(g)(1)) = 0.

4.2.1. Weyl group case. For G a simple, simply connected group of classical type or type E7,let ρ : ΓQ → G(k) be as in Section 2.1. Here we exclude the A1, A2, B3 cases. We imposeno condition at v = l, minimal prime to l condition at v ∈ S − {∞, l} which is legitimatesince ρ(ΓQ) has order prime to l by our construction. Moreover, take a semi-simple elementg in T = T (k) ⊂ Im(ρ) such that α(g) = 1, β(g) 6= 1 for all β ∈ Φ − {±α} (which ispossible since l is large enough for G). Let g = s · u ∈ G(O) with s ∈ T (O) lifting g and u anontrivial element in the root subgroup Uα(O) whose reduction modulo the maximal idealof O is trivial. Let σ ∈ ΓQ such that ρ(σ) = g. By Chebotarev’s density theorem, there is aprime p /∈ S such that Frp = σ. We take Liftgρ|Γp

at p. Let L be the Selmer system associated

to the above local deformations.

Lemma 4.9. Let f be as in Section 4.1.1 which is chosen to have at least a pair of non-realroots. Then

∑v∈S dimk Lv ≥

∑v∈S h

0(Γv, ρ(g)).

Proof. This follows directly from the previous computations. �

Lemma 4.10.∑

v∈S dimk(Lv ∩H1(Γv, ρ(t))) ≤∑

v∈S h0(Γv, ρ(t)).

Let us make the following observation which is from [12], Lemma 7.7. It will be usedfrequently in the proof of 4.11, 4.15, 4.16: suppose that ρ : ΓQ → G(k) is a continuousrepresentation with a continuous lift ρ : ΓQ → G(O). Let Gρ be the Zariski closure of G(O)

in G(Ql). Then Lie(Gρ), Lie(Gρ)∩ gO, and (Lie(Gρ)∩ gO)⊗O k are ΓQ-modules. Moreover,the last one is a submodule of ρ(g) and thus is a direct sum of some irreducible summands ofρ(g). If Lie(Gρ) = g(Ql) or (Lie(Gρ)∩gO)⊗Ok = g, then Gρ = G(Ql) (since G is connected).

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 23

Proposition 4.11. For G a simple, simply-connected group of classical type or type E7,excluding type A1, A2 and B3, and for almost all primes l when Φ is of type An or Dn, foralmost all primes l ≡ 1(4) when Φ is of type Bn or Cn, for almost all primes l ≡ 1(3) whenΦ is of type E7, there are infinitely many non-conjugate l-adic lifts

ρ : ΓQ → G(O)

of ρ : ΓQ → G(k) defined as in Section 2.1 with Zariski-dense images in G(Ql).

Proof. By 4.9 and 4.10, we can apply 3.14 to obtain a lift ρ : ΓQ → G(O) satisfying theprescribed local conditions. Suppose first that G is of classical type. The condition at pimplies that Gρ has infinitely many elements which implies that Lie(Gρ) is nontrivial. By2.9, 2.12 and 2.15, (Lie(Gρ) ∩ gO)⊗O k is then either t or g (since it is a Lie subalgebra ofg). But the condition at p implies that there is a non-trivial unipotent element in Gρ, so(Lie(Gρ)∩gO)⊗O k cannot be t. Thus (Lie(Gρ)∩gO)⊗O k = g. Now suppose that G = Esc

7 .By 2.19, (Lie(Gρ) ∩ gO) ⊗O k is either t, t ⊕ ga or g (since it is a Lie subalgebra of g). Wetake α for which gα ⊂ gb, the condition at p implies that there is a non-trivial unipotentelement in Gρ which is an element of Uα(O). It follows that (Lie(Gρ) ∩ gO)⊗O k cannot bet or t⊕ ga, hence (Lie(Gρ) ∩ gO)⊗O k = g.

Finally, by choosing infinitely many non-conjugate g ∈ G(O) for Liftgρ|Γp, we obtain infinitely

many non-conjugate l-adic lifts of ρ. �

4.2.2. Principal GL2 case. For G a simply connected group of one of the following types:A2, B3, E6, let ρ : ΓQ → G(k) be as in Section 2.2.

We begin with the following proposition due to Tom Weston ([20], Proposition 5.3):

Proposition 4.12. Let π = πf be a cuspidal automorphic representation correspondingto a holomorphic eigenform f of weight at least 2. Assume that for some prime p, πp isisomorphic to a twist of the Steinberg representation of GL2(Qp). Then for almost all λ, thelocal Galois representation rf,λ|Γp (in the notation of 1.2) has the form

rf,λ|Γp ∼(χκ ∗0 χ

)where the extension * in H1(Γp, kλ(κ)) is non-zero.

Suppose that G admits a principal GL2. Let f be a non-CM weight 3 cuspidal eigenformthat is a newform of level Γ1(p) ∩ Γ0(q) for some primes p and q; the nebentypus of f is acharacter ε : (Z/pqZ)× → (Z/pZ)× → C×. Such a form f exists, for example, [9], 15.3.7.aand 15.3.11.a. At p we take the Steinberg deformation condition. At q we use the minimalprime to l deformation condition, as ρ(Γq) has order prime to l. At l we impose no condition.Moreover, choose an element σ ∈ ΓQ such that ρ(σ) is regular semi-simple in T (k) togetherwith a lift g ∈ T (O) such that α(g), α ∈ ∆ are distinct. By Chebotarev’s density theorem,there is a prime r /∈ {∞, l, p, q} such that Frr = σ. We take Liftgρ|Γr

at r. Let L be the

Selmer system associated to the above local deformations.

Lemma 4.13. Under the above global deformation condition, the right hand side of Wiles’formula is 2 for A2, 9 for B3, and 34 for E6.

24 SHIANG TANG

Proposition 4.14. For G = SL3, Spin7 or Esc6 and for almost all primes l, there are infinitely

many non-conjugate l-adic liftsρ : ΓQ → G(O)

of ρ : ΓQ → G(k) defined as in Section 2.2 with Zariski-dense images in G(Ql).

Proof. The proof is very similar to the proof of [12], Theorem 7.4, so we skip a few detailshere.

We first show that 3.4 applies to ρ. Item 1 is satisfied by 4.13; Item 2 and 3 are satisfied bythe proof of [12], Theorem 7.4; For Item 4, take σ ∈ ΓQ such that ρ(σ) = 2ρ∨(a) is regularwith 1 6= a ∈ (Z/lZ)× and κ(σ) = a2 (which is possible, again, see the proof of [12], Theorem7.4). It follows that (a) is satisfied for any simple root α. (b) and (c) are also satisfied by2.22.

Therefore, we can deform ρ to a continuous representation ρ : ΓQ → G(O) satisfying theprescribed local conditions on S and the Ramakrishna condition on a set of auxiliary primesdisjoint from S. We write Gρ for the Zariski closure of the image of ρ in G(Ql). By [12],Lemma 7.7, Gρ is reductive. By 4.12, the Steinberg condition at p ensures that Gρ containsa regular unipotent element (see the proof of [12], Theorem 8.4). By a theorem of Dynkin,Gρ is then of type A1 or A2 for G = SL3; type A1, G2 or B3 for G = Spin7; and type A1, F4,

or E6 for G = Esc6 . But α(ρ(Frr)), α ∈ ∆ are distinct, so Gρ = G(Ql) in all three cases (see

the proof of [12], Lemma 7.8).

Finally, by choosing infinitely many non-conjugate g ∈ T (O) for Liftgρ|Γr, we obtain infinitely

many non-conjugate l-adic lifts of ρ. �

Remark 4.15. For any simple, adjoint algebraic group G, the principal GL2 is well-definedand the proof of the above proposition carries over to G, modulo some local calculations.Therefore, all simple, adjoint algebraic groups are l-adic monodromy groups for ΓQ.

4.2.3. SL2. The alternating group An admits a unique nontrivial extension An by Z/2Z forn 6= 6, 7. By a result of N. Vila and J-F. Mestre [16], An can be realized as a Galois group overQ. In particular, we get a surjection r : ΓQ � A5. On the other hand, A5 can be describedas follows: the symmetries of an icosahedron induce a 3-dimensional irreducible faithfulrepresentation of A5, i.e., there is an injective homomorphism A5 → SO(3). The pullback ofA5 along the two-fold covering map SU(2) � SO(3) is a nontrivial central extension of A5 byZ/2Z, hence is isomorphic to A5. In particular, we get an embedding A5 → SL2(C). As thematrix entries of the image lie in a finite extension of Q, we can choose a finite extension kof Fl for which there is a well-defined embedding A5 → SL2(k). Precomposing it with r, weobtain a representation ΓQ � SL2(k) which we denote by ρ. It is easy to see that the adjointmodule ρ(sl2(k)) is irreducible. Let S be a finite set of places containing the archimedeanplace and ramified places. We impose no condition at l, and take the minimal prime to ldeformation condition at all other places in S, which is legitimate since the residual imagehas order prime to l for l > 120. Let Σ ∈ A5 be an element of order 4, whose image in SL2(k)is conjugate to diag(

√−1,−

√−1). As A5 has trivial abelian quotient, Q(µl) and Q(ρ) are

linearly disjoint over Q. So there is an element σ ∈ ΓQ such that ρ(σ) = Σ and κ(σ) = −1.By Chebotarev’s density theorem, there is a prime p /∈ S for which Frp = σ. Therefore, ρ|Γpis of Steinberg type and we take the Steinberg deformation condition at p.

ALGEBRAIC MONODROMY GROUPS OF G-VALUED l-ADIC REPRESENTATIONS OF Gal(Q/Q) 25

Proposition 4.16. For ρ : ΓQ → SL2(k) defined as above and for almost all primes l, thereis an l-adic lift

ρ : ΓQ → SL2(O)

of ρ with Zariski-dense image in SL2(Ql). Moreover, there are infinitely many non-conjugaterepresentations as such.

Proof. We first show that 3.4 applies to ρ. Item 1 is satisfied: the left hand side of theinequality equals the right hand side; Item 2 is satisfied since |Gal(K/Q)| has order primeto l by the definition of ρ; Item 3 is satisfied since ρ(g) and ρ(g)(1) are non-isomorphic; ForItem 4, we take σ to be as above, the connected component of whose centralizer is denotedT , and α be a root of Φ(G, T ) which has rank one, so (a) is satisfied. As ρ(g) is irreducible,(b) and (c) are satisfied.

Therefore, we can deform ρ to a continuous representation ρ : ΓQ → SL2(O) satisfying theprescribed local conditions on S and the Ramakrishna condition on a set of auxiliary primesdisjoint from S. We write Gρ for the Zariski closure of the image of ρ in SL2(Ql). As ρ(g) isirreducible, Lie(Gρ) is either trivial or gQl . If the former is true, then Gρ is finite. But ρ|Γpis Steinberg, so in particular the image of ρ is infinite, a contradiction.

Finally, by choosing infinitely many linearly disjoint extension of Q realizing A5, we obtaininfinitely many non-conjugate lifts. �

Remark 4.17. For G a simple but non-simply connected group, suppose there is a homo-morphism ρl : ΓQ → Gsc(Ql) with Zariski-dense image. We compose ρl with the projectionGsc(Ql) � G(Ql), the resulting map has Zariski-dense image in G(Ql). Therefore, 4.11, 4.14and 4.16 imply 1.3.

5. Connected reductive groups as algebraic monodromy groups for ΓQ

In this section, we explain how the hoped-for classification of l-adic monodromy groups (1.5)can be deduced from the case when G is a simple algebraic group.

Following [10], a connected algebraic group G is said to be the almost-direct product of itsalgebraic subgroups G1, ..., Gn if the map

(g1, · · · , gn) 7→ g1 · · · gn : G1 × · · · ×Gn → G

is a surjective homomorphism with finite kernel; in particular, this means that the Gi com-mute with each other and each Gi is normal in G.

Proposition 5.1. [10] An algebraic group is semi-simple if and only if it is an almost directproduct of simple algebraic groups. (Here a simple algebraic group is called almost simple in[10])

Proposition 5.2. (Goursat’s Lemma) Let G1, G2 be groups, let H be a subgroup of G1×G2

such that the two projections p1 : H → G1, p2 : H → G2 are surjective. Let N1 (resp. N2)be the kernel of p2 (resp. p1). Then the image of H in G1/N1 × G2/N2 is the graph of anisomorphism G1/N1

∼= G2/N2.

5.1, 5.2, 4.15 together imply

26 SHIANG TANG

Corollary 5.3. Let G be a connected semi-simple group with trivial center, then there arecontinuous homomorphisms

ΓQ → G(Ql)

with Zariski-dense images for almost all l.

Proposition 5.4. (Tate)[5] Let F be a number field, let G′ � G be a surjection of algebraicgroups over Ql with kernel a central torus, then any continuous homomorphism ΓF → G(Ql)lifts to a continuous homomorphism ΓF → G′(Ql).

5.3 together with 5.4 imply

Corollary 5.5. Let G be a connected reductive group such that Z(G) is a one-dimensionaltorus, then there are continuous homomorphisms

ΓQ → G(Ql)

with Zariski-dense images for almost all l.

As Q only has one Zl-extension by class field theory, we have

Lemma 5.6. Let G be a connected reductive group such that dimZ(G) > 1, then for anyprime l, there is no continuous homomorphism

ΓQ → G(Ql)

with Zariski-dense image.

Therefore, to prove 1.5, it remains to show the analogue of 5.3 and 5.5 in the case when Ghas non-trivial finite center. By 1.3, 1.4, 5.1, 5.2, we have

Theorem 5.7. Let G be a connected semi-simple group with non-trivial finite center, thenthere are continuous homomorphisms

ΓQ → G(Ql)

with Zariski-dense images for almost all primes l.

Therefore, the proof of 1.5 is complete!

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Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090

E-mail address: tang@math.utah.edu