COMPATIBLE SYSTEMS OF GALOIS REPRESENTATIONSASSOCIATED TO THE EXCEPTIONAL GROUP E6GEORGE BOXER, FRANK CALEGARI, MATTHEW EMERTON, BRANDON LEVIN,
KEERTHI MADAPUSI PERA, AND STEFAN PATRIKIS

1. Introduction
In [Ser94], Serre raised the question of whether G2 or E8 was the motivic Galois
group of a motive M over a number field, and one can evidently ask the same question for the other exceptional simple Lie groups. A slightly weaker version of thisquestion asks for a motive M such that the associated p-adic Galois representations have algebraic monodromy group equal to the exceptional group in question.
In this form, Serre’s question was answered in the affirmative by Yun in [Yun14],
who also dealt with the case of the exceptional group E7 . (A stronger version of
the question for the group G2 had previously been answered by Dettweiler andReiter [DR10].) This left open the cases of E6 or F4 . In [Pat16], the last authorof this paper succeeded in constructing geometric Galois representations for theremaining exceptional groups using arguments inspired by Ramakrishna’s lifting
theorems [Ram02], at least for a set of primes p of density one (improved to all
but finitely many in Theorem 1.2 of [Pat17]). While this answered a weak formof (the E6 analogue of) Serre’s question, the Galois representations constructedin [Pat16, Pat17] did not obviously come from motives M or from compatible systems of Galois representations (although that would certainly be a consequence of
the Fontaine–Mazur conjectures [FM95]). The main goal of this paper is to remedy
this lacuna for the group E6 .
Theorem 1.1. Let F/F + be a totally imaginary CM field with maximal totally realsubfield F + . Let G denote the simply connected form of E6 , and fix a minusculerepresentation G → GL27 . Then there exists a strongly compatible system of Galois
representations with coefficients in a number field M such that the representations
rλ : GF → G(M λ ) ,→ GL27 (M λ )have images with Zariski closure G(M λ ) for all primes λ. Moreover, this compatible
system is potentially automorphic and motivic in the sense that there is a CM
extension H/F such that:• There a cuspidal automorphic representation π for GL27 /H such that rλ |GHis the compatible system of Galois representations associated to π.
• The compatible system rλ satisfies the conclusion of the Fontaine–Mazur
conjecture: there is a smooth projective variety X/F and integers i and jsuch that rλ is a GF -sub-representation of H i (XF , Ql (j)).
G.B. was supported in part by NSF postdoctoral fellowship DMS-1503047, F.C. was supported
in part by NSF Grant DMS-1701703, M.E. was supported in part by NSF Grant DMS-1601871,
K.M.P. was supported in part by NSF Grant DMS-1803623, S.P. was supported in part by NSF
Grant DMS-1700759.
1

The main idea of the paper is to follow the strategy of [Pat16], but to replace thelifting theorems inspired by Ramakrishna with those inspired by the work of Khareand Wintenberger [KW09a, KW09b], exploiting the modularity lifting theoremsof [BLGGT14]. In order to do this, one must link G-representations with GLn representations by choosing some (faithful) representation r : G → GLn . The methodsof [BLGGT14] require that the corresponding Galois representations have distinct
Hodge–Tate weights. This imposes a strong restriction on the representation r,
namely, that its formal character should be multiplicity-free. In particular, themethod of this paper only applies to the exceptional groups G2 , E6 , and E7 , in their
quasi-minuscule (G2 ) or minuscule (E6 and E7 ) representations (we concentrateon E6 because other methods are available in the other cases). As in [Pat16, Pat17],
we require a seed representation ρ : GF → G(Fp ) from which to construct geometric
lifts. In [Pat16], suitable representations ρ came from composing representations
associated to modular forms with the principal SL2 . These representations are notsuitable for our purposes, because their composition with the minuscule representation is reducible. Instead, we construct a representation related to the action of
the Weyl group of E6 on the weight space of our representation. The fact that the
representation we consider is irreducible in GLn relies on the assumption that r
is minuscule. The reason we succeed in controlling the monodromy groups at allprimes is a consequence of elementary combinatorial properties of the formal character of E6 (using ideas of Larsen and Pink [LP92]) together with our ability to
exploit independence of p results in compatible systems of Galois representations
associated to automorphic forms ([TY07, Shi11, Car12]).
We end the introduction with some remarks on what the methods of this paper
cannot do.
Remark 1.2. Galois representations for Q versus imaginary quadratic
fields. Our construction gives compatible systems of E6 -representations over any
imaginary quadratic field F ; these extend to representations of GQ whose image is
Zariski-dense in the L-group G o Out(G) of an outer form of E6 . We leave open thequestion as to whether actual E6 -systems exist over Q, noting that the methods ofthis paper will not succeed in constructing them. Indeed, our methods require thatthe corresponding Galois representations have regular weight, and there do not exist
any such compatible systems of Galois representations over Q (see Remark 5.1).
Remark 1.3. Motives versus motives with coefficients. We ultimately construct compatible systems of 27-dimensional Galois representations for a coefficientfield L over which we have little control. One can ask the more refined questionof whether there exists a motive M with coefficients over Q of type E6 (Yun’s result [Yun14] answer this question in the affirmative for E8 ). One reason that thisrefinement is interesting is that it would have applications to the to the inverse
Galois problem (see Corollary 6.1).
It seems to us that the Galois theoretic methods of either this paper or of [Pat16,
Pat17] are unsuited to answering such a question. It is illustrative to consider
the simpler case of GL2 . By constructing a geometric Galois representation ρ :GQ → GL2 (Qp ), one can hope to prove it is automorphic and hence associated to amodular form f , and thus to construct a corresponding motive Mf ([Sch90]). But it
seems very hard to impose conditions on ρ to ensure that the form f has coefficientsin Q. For example, in weight 2 (on the modular form side) one would want to put
conditions on ρ to ensure that it actually come from an elliptic curve rather than2

an abelian variety of GL2 -type. In practice, we actually work in highly regularweight, and there is a certain amount of numerical evidence [Rob17] pointing to
the fact that there may not exist any motives M at all with coefficients in Q andmonodromy group GL2 with Hodge–Tate weights [0, k − 1] when k > 50.
2. The mod p representation
In this section, we construct the mod p representations that we will lift in thenext section using potential automorphy theorems.
Definition 2.1. Let G be the split simply-connected reductive group scheme over
Z of type E6 . Fix a pinned based root datum of G, and let L G = G o Out(G)with the non-trivial element τ of Out(G) = Z/2Z acting through the corresponding
pinned automorphism. The induced action of τ on WG is conjugation by the longest
element w0 , since τ acts on T , the maximal torus of the pinning, by the opposition
involution −w0 . Let W = WG o Out(G), and continue to write τ for the non-trivial
element of Out(G) ⊂ W . We will write B for the Borel subgroup of G associatedto the based root datum.
Remark 2.2. There is an isomorphism W ' WG × Z/2Z given by the identity on
WG and sending τ to (w0 , 1).
We fix, once and for all, a choice of minuscule representation rmin : G → GL27 ,writing Λmin for the weights of T in rmin . Now we explain how to extend rmin toL
G. Let G27 = (GL27 × GL1 ) o Z/2Z, where the non-trivial element  ∈ Z/2Z acts
via  (g, µ)−1 = (µ · t g −1 , µ). The representation g 7→ rmin (τ gτ −1 ) is isomorphic
to the dual minuscule representation of G, so there exists A ∈ GL27 such thatrmin (τ gτ −1 ) = A · t rmin (g)−1 A−1 . Iterating, we see that A · t A−1 commutes withrmin , so must be a scalar: A = t A · ε. Clearly ε ∈ {±1}, and since 27 is odd, wemust in fact have ε = 1 by considering determinants.
We can now extend rmin tormin : L G → G27
by rmin (τ ) = (A, ε, ) = (A, 1, ). The fact that ε = 1 has the following consequence,
which we will need later:
Lemma 2.3. Let ν : G27 → Gm be the character given by ν(g, a, 0) = a, ν() = −1.
Let x ∈ L G be any element with non-trivial projection to Out(G) (the case ofinterest will be x such that conjugation by x induces a Chevalley involution of G).
Then ν ◦ rmin (x) = −1.
Lemma 2.4. Let E + /F + be a Galois extension of totally real fields whose Galois
group is identified with a subgroup P of WG , and let F/F + be a quadratic totally
imaginary extension. Let E = F.E + . Then the composite∼∼∼

Gal(E/F + ) −→ Gal(E + /F + ) × Gal(F/F + ) −→ P × Z/2Z ⊂ WG × Z/2Z −→W
sends complex conjugation to (w0 , τ ) ∈ W = WG o Out(G). In this setting we will
write L P for the image of P × Z/2Z in W .
∼Proof. This follows from the definition of the isomorphism WG × Z/2Z −→ W(Remark 2.2).

3

We now construct the mod p Galois representation. Let P be a Sylow 3-subgroupof WG ; we note that |WG | = 27 · 34 · 5. Let F/F + denote a fixed totally imaginary
quadratic extension of our fixed totally real field F + . Our starting point will be
to construct P as a Galois group over F + , with some local restrictions whose
significance will become apparent when we apply potential automorphy theorems.
Lemma 2.5. There exists a totally real Galois extension E + /F + together with anequality Gal(E + /F + ) = P (we write “=” to denote a fixed isomorphism) and all
the primes of F + which ramify in E + are split in F/F + .
Proof. The Scholz–Reichardt theorem ([Ser08, Thm 2.1.1]) guarantees the existenceof a number field L with Gal(L/Q) = P , and, because |P | is odd, such an extension
will automatically be totally real. It suffices to show that we can construct such an
extension ramified only at primes which are totally split in F (equivalently, in the
Galois closure of F ). This forces the intersection of L with the Galois closure of F
to be unramified everywhere over Q and hence trivial, and thus E + = L.F + will
have Galois group P over F and produce the desired extension. The result follows
immediately by induction and from the following lemma, which is extremely close
to [Ser08, Thm 2.1.3]:
e → A be a central extension of a finite group A by Z/pZ,
Sublemma 2.6. Let A
e divides pn . Let L/Q be Galois with Galois
and assume that the exponent of A
group A, and assume that every prime l which ramifies in L has the following
properties:(1) l ≡ 1 mod pn .
(2) The inertia group(s) of l in A coincides with the decomposition group(s)at l.
(3) l splits completely in F .
eeThen there exists an extension L/Lwhich is Galois over Q with Galois group Ae satisfy the same conditions as above.
and such that the primes l which ramify in L
The only difference between this statement and Theorem 2.1.3 of [Ser08] is theextra requirement that the primes l splits completely in F . There are two inductive
steps in the proof of Theorem 2.1.3 of [Ser08], and we indicate the required argument
to show that the new auxiliary prime q may be chosen to split completely in F .
e = A × Z/pZ is a split extension. Pick any prime q ≡ 1
Suppose first that A
mod pn which is totally split in the Galois closure of F and also totally split in thefield L(ζpn , {l1/p }l∈Ram(L/Q) ) given by adjoining the pth roots of primes l whiche to be the composite of L and the sub-extension of
ramify in L. Then take L
Gal(Q(ζq )/Q) with Galois group Z/pZ.
e is a non-split extension. The argument in [Ser08] proceeds
Now suppose that A
e and then modifying L
e so that it is ramified at theby first finding an extension L
same places as L. Hence the ramified primes l automatically satisfy the required
splitting condition in F . The final step is to modify the field further so that it hasproperty (2). This is achieved by choosing an auxiliary prime q ≡ 1 mod p alongwith a character χ : (Z/qZ)× → Z/pZ satisfying the following properties:(1) The prime q ≡ 1 mod pn .
(2) For every prime l which ramifies in L, there is an equality χ(l) = cl where clis determined from the extension L.
4(3) The prime q splits completely in L.
(4) The prime q splits completely in F .
Only the last condition is new. The first three conditions are Čebotarev conditionsin the field L(ζpn , {l1/p }l∈Ram(L/Q) ), whereas the fourth condition is a Čebotarev
condition in the Galois closure of F . By construction, the primes l are totally
split in the Galois closure of F . Hence the intersection of the Galois closure of F
with L(ζpn , {l1/p }l ) must be contained inside Q(ζpn ). Since the first condition
implies that q splits completely in Q(ζpn ), there is no obstruction to finding suchprimes q satisfying all four conditions using the Čebotarev density theorem provided
there is no obstruction without the last hypothesis. But this is exactly what follows
from the proof of Theorem 2.1.3 of [Ser08].

As in Lemma 2.4, let L P be the image of P × Z/2Z in W ; explicitly, it is the
product P × h(w0 , τ )i inside W . Our reason for working with the group P is the
combination of the following two properties:
Lemma 2.7.
(1) The restriction of the extensionπ

1 → T (Z) → NG (T )(Z) o Out(G) −→ WG o Out(G) → 1
to L P splits.
(2) P acts transitively on the set Λmin of weights of the minuscule representation rmin .
Proof. Since P is a 3-group while T (Z) is an F2 -module, the Hochschild–Serre
spectral sequence
H i (P, H j (h(w0 , τ )i, T (Z))) =⇒ H i+j (L P, T (Z))
degenerates at the E2 -page, and restriction induces an isomorphism∼

H 2 (L P, T (Z)) −→ H 2 (h(w0 , τ )i, T (Z))P .
The image of our extension under this restriction isomorphism is trivial, because
the element (w0 , τ ) lifts to an order two element of N (T )(Z) o Out(G) (see for
instance [AH16, Lemma 3.1], noting that ZG has order 3).
For the second part of the lemma, note that WG acts transitively on Λmin , so the
3-part of the stabilizer of any element λ ∈ Λmin has order 3 (recall |WG | = 27 · 34 · 5
and |Λmin | = 33 ). It follows that the orbit of P on λ has order at least |P |/3 = 33 ,and thus (equality holds and) P acts transitively. (More generally, a finite group G
acts transitively on a set X of p-power order if and only if a p-Sylow subgroup P
acts transitively on X.)
The work of Shafarevich on the inverse Galois problem for solvable groups implies
that every split embedding problem with nilpotent kernel has a proper solution;
we need some precise local control so will not be able to invoke this theorem,
and unfortunately the following construction, guided by the demands of the local
deformation theory as in [Pat16] and of automorphy lifting as in [BLGGT14], issomewhat technical.
Applying Lemma 2.7, let us fix a splitting of the extensionπ

1 → T (Z) → π −1 (L P ) −→ LP → 15and write s : Gal(E/F + ) → π −1 (L P ) for the resulting lift. This representation is
not yet suitable for potential automorphy theorems, so we modify it in the following
Proposition.
We first establish some notation. For a fixed prime p, we write  for the padic cyclotomic character and ¯ for its mod p reduction. For a prime l exactly
dividing p − 1, let pr(l) be the projection from F×p onto the l-torsion subgroup×F×[l]
restricting
totheidentityonF[l].
Finally,let¯[l] = pr(l) ◦ ¯: GF + → F×p [l].
pp
For any homomorphism ρ̄ : GF + → G(Fp ) and any place v of F + , let ρ̄v := ρ̄|GF + .
v
Finally, for any homomorphism of groups ρ : Γ → Γ0 , and any ρ(Γ)-module M , letρ(M ) denote M regarded as a Γ-module.
Proposition 2.8. Consider pairs of primes (l, p) such that:• All primes above l split in F/F + .
• p splits in E/Q and p − 1 is divisible by l but not by l2 .
Then there exist infinitely many primes l such that there exist infinitely many
pairs (l, p) such that there exists a homomorphismρ̄ : GF + → T (Fp )[l] · π −1 (L P ) ⊂ NG (T )(Fp ) o Out(G)lifting our fixed identification Gal(E/F + ) = L P and satisfying the following:(1) The restriction ρ|GF is ramified only at places split in F/F + .
(2) For any choice of complex conjugation c ∈ GF + , ρ(c) is a Chevalley involution of G, i.e. dim(gAd(ρ(c))=1 ) = dim(G/B) = 36.1
(3) For all places v|p of F + , fix any choice of integers nv,α indexed by simple
roots α ∈ ∆ = ∆(G, B, T ). Then ρ|GF + is equal to
v
Y
∨
nv,α
α ◦ ¯[l]
.
α∈∆

(4)

Let ρ∨
G denote the half-sum of the positive coroots of (G, B, T ); it lies inX• (T ) since #ZG = 3. There is a set Sreg of two primes q split in E/Q andof order l modulo p such that for some place v|q of F + , ρ|GF + is unramifiedvwith Frobenius mapping to ρ∨G (q) (which lands inside the group T (Fp )[l]because q l ≡ 1 (mod p)).
Moreover, in addition to satisfying the above conditions, we can choose p > 56 =
2 · (27 + 1), l > hG = 12 (the Coxeter number of G), {nv,α }, and ρ such that
(1) For all v|p, the composite rmin ◦ ρ|GF + is a direct sum of distinct powers ofv
¯[l].
(2) For all v|p and for any Borel subgroup B (with Lie algebra b) containing T ,the cohomology groups H 0 (GFv+ , ρv (g/b)) and H 0 (GFv+ , ρv (g/b)(1)) bothvanish. Moreover, for any Borel subgroup B27 containing the maximal
torus of SL27 that stabilizes the weight spaces of rmin , H 0 (GFv+ , (rmin ◦ρv )(sl27 /b27 )) = 0 and H 0 (GFv+ , (rmin ◦ ρv )(sl27 /b27 )(1)) = 0.
(3) The composite rmin ◦ ρ|GF (ζp ) is absolutely irreducible.
Proof. For now let l be any odd prime such that the places of F + above l are
all split in F/F + , and E and Q(µl2 ) are linearly disjoint over Q (eg, take l split
in E/Q); later in the argument we will require l to be larger than some absolute
1S.P. apologizes for the non-standard term “split Cartan involution” used in [Pat16], [Pat17].
6

bound depending only on the group G. Let p be any prime split in E/Q suchthat l, but not l2 , divides p − 1 (such p exist by Čebotarev). Fix a set Sreg of two
rational primes q that are split in E/Q and have order l modulo p: such q exist since
E and Q(µp ) are linearly disjoint over Q (p splits in E/Q). We will modify theoriginal lift s : Gal(E/F + ) → π −1 (L P ) by an element of H 1 (GF + , T (Fp [l])) so as tosatisfy the local conditions (here GF + continues to act via the quotient Gal(E/F + );note that T (Fp )[l] is τ -stable). Let T (Fp )[l] = ⊕Wi be the decomposition into
irreducible Fl [P ]-modules. Recall that (w0 , c) ∈ L P acts on T by −1, so this is also
a decomposition as L P -module, and the splitting field F + (Wi ) of the Gal(E/F + )module Wi contains F for each i (since l 6= 2). Let Σ be the set of places of F +that are either split or ramified in F/F + (implicitly including the infinite places in
the former condition); in particular, the action of GF + on T (Fp )[l] factors throughGF + ,Σ . For any finite subset T of Σ, [NSW08, Theorem 9.2.3(v)] implies that the
restriction map
M
H 1 (GF + ,Σ , Wi ) →H 1 (GFv+ , Wi )T

is surjective for all i. Assembling the different i (with a common set T ), the restriction map
M
H 1 (GF + ,Σ , T (Fp )[l]) →H 1 (GFv+ , T (Fp )[l])T

is also surjective. We apply this observation to the following set T and the following
local cohomology classes: let T be the union of the following sets of places of F + :• places which are ramified in F/F + ;• places dividing p;• places above the rational primes q ∈ Sreg ;• an auxiliary place w lying above a rational prime r that splits completelyin E(ζp )/Q.
Consider the following local classes in H 1 (GFv+ , T (Fp )[l]) for v ∈ T :• trivial at places which ramify inQF/F + ;• the prescribed homomorphism α∈∆ α∨ ◦ ¯[l]nv,α for v|p,
• the unramified homomorphism Frv 7→ ρ∨G (q) for v|q and q ∈ Sreg , whereρ∨
denotesthehalf-sumofthepositive
coroots of G, which is in fact aG
cocharacter of G (we will only need this construction for one of the places
above q).
• the unramified homomorphism Frw 7→ t, where t is any element of T (Fp )[l]such that the values λ(t) for λ ∈ Λmin are all distinct (for l sufficiently
large, such t exist).
Let φ ∈ H 1 (GF + ,Σ , T (Fp )[l]) be a class with these local restrictions, and set ρ = φ·s.
We claim the conclusions of the proposition hold for this ρ. The conditions at finite
places are all evident from the construction, and the condition on complex conjugation is satisfied because any order two element (we0 , τ ) ∈ NG (T )(Fp ) lifting (w0 , τ ) ∈WG o Out(G) gives a Chevalley involution of g: it acts by −1 on Lie(T ), and it
sends a root space gα to g−α , so we get precisely dim(G/B) = dim(gAd((we0 ,τ ))=1 ).
For the second list of assertions, note that for v|p, rmin ◦ ρ|GF + is equal to
v
M
P
∨
n
hλ,α
i
¯[l] α∈∆ v,α,λ∈Λmin7where Λmin is the set of weights of rmin . For varying λ, we want these exponents
to be distinct modulo l. We simply choose the (nv,α )α so that the exponents are
distinct in Z, and then any l sufficiently large will do. To evaluate the cohomology
groups appearing in the conclusion of the proposition, note that as a GFv+ -module,ρv (g/b) is a direct sum of characters of the form
¯[l]

P

α∈∆ nv,α hβ,α

∨

i

,

where β is a negative root of G. Clearly the choice of l can be modified if necessary
to ensure that these exponents (which don’t depend on p) are all (distinct, non-zero)
integers in (−l, l), and so regardless of how p is chosen the group H 0 (GFv+ , ρv (g/b))
will vanish. (Since ρv factors through T , this argument in fact shows that ρv (g/t)
has no invariants; in particular, it applies to all Borel subgroups B containing T .)The vanishing of H 0 (GFv+ , ρv (g/b)(1)) is even more straightforward: the order of¯ is divisible by some prime other than l, whereas all powers of ¯[l] have order 1
or l. We can similarly deduce the vanishing of H 0 (GFv+ , (rmin ◦ ρv )(sl27 /b27 )) andH 0 (GFv+ , (rmin ◦ρv )(sl27 /b27 )(1)), where b27 is the Lie algebra of any Borel subgroup
B27 ⊂ SL27 containing the maximal torus of SL27 characterized by the property
that it stabilizes each of the weight spaces of rmin . Then (rmin ◦ ρv )(sl27 /b27 ) is adirect sum of charactersP∨¯[l] α∈∆ nv,α hλ1 −λ2 ,α i ,for distinct weights λ1 , λ2 ∈ Λmin . These exponents are by construction non-zero
modulo l, so we win, and the same argument as for ρv (g/b)(1) applies to show
H 0 (GFv+ , (rmin ◦ ρv )(sl27 /b27 )(1)) is zero as well.
The condition at the auxiliary prime w ensures absolute irreducibility of rmin ◦ρ|GF (ζp ) : P acts transitively on Λmin by Lemma 2.7(2), so any non-zero submoduleof rmin ◦ ρ|GF (ζp ) has non-zero projection to each weight space (recall that F (ζp ) is
linearly disjoint from E over F ); but for a place w0 |w of F (ζp ), the image of GF (ζp )w0
in T (Fp ) acts via distinct characters on the different weight spaces of rmin .

3. Lifting Galois Representations
Let ρ : GF + → NG (T )(Fp )oOut(G) ⊂ L G(Fp ) be a homomorphism constructed
as in Proposition 2.8. We would like to lift ρ to a homomorphismρ : GF + → L G(Zp )that belongs to a compatible system of representations, all having Zariski-dense
image and appearing in the cohomology of an algebraic variety. To achieve Zariskidense monodromy for a lift ρ, we follow the approach of [Pat16]: ensuring local
Steinberg-type ramification at one auxiliary prime and sufficiently general Hodge–
Tate cocharacter suffices. To produce the lift, and to put it in a compatible system,
we compare deformation rings for ρ and rmin ◦ ρ : GF + → G27 (Fp ). We control asuitable deformation ring for rmin ◦ ρ using the method of Khare–Wintenberger andthe automorphy lifting results of [BLGGT14].
Let S be a finite set of primes of F + which split in F containing all those whereρ is ramified (by the construction of ρ, the primes of F + at which ρ is ramified split
in F ). We will enlarge the set S as necessary. Let FS be the maximal extension+(in F ) of F unramified outside (places above) S, and set GS = Gal(FS /F + ). We
will be deforming GS -representations. For each v ∈ S, fix an extension ṽ of v to F ,and fix a member of the GF,S -conjugacy class of homomorphisms GFṽ → GF,S .
8

Via the inclusion GF,S ⊂ GS , these choices specify what we mean by restricting ρ(or its lifts) to GFṽ . We do not review in detail the mechanics of the deformation
theory in this setting, but instead refer the reader to [Pat16, §9.2] (for L G) and[CHT08, §2] (for G27 ). We now specify local deformation conditions to define two
global deformation functors, one for ρ and one for rmin ◦ ρ.
For v ∈ S, define the following local deformation conditions Pv on lifts of ρ|GFṽ :(1)• For v above the auxiliary primes q ∈ Sreg (see Proposition 2.8), we impose
the Steinberg deformation condition as in [Pat16, §4.3], with respect to theBorel subgroup B of G specified by our based root datum. Note that byconstruction the order of ¯: GFṽ → F×
p is greater than hG − 1 (since l > hG
is the order of q modulo p).
• For v|p, we take an ordinary deformation condition as in [Pat16, §4.1]. To
be precise, fix the following lift χT of ρ|IFṽ to T (Zp ):YχT =α∨ ◦ (ñv,α · χ−nv,α ),α∈∆where the ñv,α are sufficiently general positive (positive ensures, as in[Pat16, Lemma 4.8], that our characteristic zero lifts are de Rham) integers
congruent to nv,α modulo l − 1, and χ is the Teichmüller lift of ¯ · ¯[l]−1 .
• For all other primes v ∈ S, the inertial image ρ(IFṽ ) has order prime top (indeed, ρ lands in the prime-to-p group NG (T )(Fp ) o Out(G)), and wetake the minimal deformation condition of [Pat16, §4.4].
Lemma 3.1. For all v ∈ S, let Pv be the local condition just defined. Let P ={Pv }v∈S , and let LiftρP be the associated global lifting functor associated to thiscollection of local conditions (see [Pat16, §9.2]).
(1) For all v ∈ S not above p, the local lifting ring associated to the condition Pv
just defined is formally smooth, and the associated deformation functor hastangent space Lv of dimension dim H 0 (GFṽ , ρ(g)). For v|p, the same holds,
except the tangent space has dimension dim H 0 (GFṽ , ρ(g)) + dim(G/B).
P(2) The associated deformation functor Def P
ρ is representable. Let Rρ be theP
representing object. For some integer δ, Rρ has a presentation as thequotient of a power series ring over Zp in δ variables by an ideal generatedby (at most) δ relations.
Proof. The local claims follow from [Pat16, §4.1, 4.3, 4.4]. The global claims follow
from [Pat16, Proposition 9.2]), using that:• the centralizer of ρ in g is trivial;• for all complex conjugations c, ρ(c) is a Chevalley involution of G;• the local lifting rings have dimensions as computed in the first part of thelemma.
(We remark that the integer δ is the common dimension of the Selmer and dual
Selmer groups associated to the global deformation functor.)
Next we define an analogous deformation ring for rmin ◦ρ : GS → G27 (Fp ). Recall thecharacter ν : G27 → Gm . The composition ν ◦(rmin ◦ρ) is the non-trivial character ofGal(F/F + ), and we fix µ : GF + → Z×p equal to its Teichmüller lift (we will consider
lifts with this fixed character). To define a global deformation problem in the senseof [BLGGT14, §1.5] (see [CHT08] for more details), we must, for each v ∈ S, choose9an irreducible component Cv of the (generic fiber) lifting ring (in the case v not abovep) Rrmin ◦ρ|G [1/p] or (in the case v|p) limK Rrmin ◦ρ|G ,{H},K−ss [1/p], where here
Fṽ

Fṽ

we follow the notation of [BLGGT14]: H is a collection (indexed by embeddings
Fṽ → Qp , but for us Fṽ = Qp ) of multi-sets of Hodge numbers, K varies over finite
extensions of Fṽ , and the lifting ring in question is the one constructed by Kisin([Kis08]), whose characteristic zero points parametrize potentially semistable deformations, semistable over K, with the prescribed Hodge numbers; to be precise, it is
the maximal reduced p-torsion-free quotient of Rrmin ◦ρ|G whose Qp -points satisfy
Fṽ

these properties (see [BLGGT14, §1.4] for an overview). In both cases v|p andv
given by the maximal reducedv - p, we then associate the lifting ring RrCmin
◦ρ|G
Fṽ

p-torsion-free quotient of (v - p) Rrmin ◦ρ|GFṽor (v|p) limK Rrmin ◦ρ|G

Fṽ,{H},K−ss that

is, after inverting p, supported on the component Cv . Namely, we take:• For v above the auxiliary primes q ∈ Sreg , recall that ρ|GFṽ is unramified
with (arithmetic) Frobenius Frṽ mapping to ρ∨G (q). Let ϕ : PGL2 → G be
the principal homomorphism associated to our fixed pinning of G, so ρ(Frv )equals ϕ(diag(q, 1)). The composite rmin ◦ ϕ decomposes as S16 ⊕ S8 ⊕ S0
(see [Gro00, §7]), where we write S i for the ith symmetric power of thestandard representation of SL2 . Consider the lift ρṽ of rmin ◦ ρ|GFṽ given
by r16 ⊕ r8 ⊕ r0 where ri is the tame unipotent representation on the Sicomponent given by the matrices(
q i−2a+2 if a = b,(ri (Frv ))a,b =0
if a 6= b,
and on a topological generator τv of tame inertia by(
p if b = a + 1,
(ri (τv ))a,b =0 if b 6= a + 1.
A quick calculation shows that, for all finite extensions K/Fṽ , we havean equality H 0 (GK , ad(ρṽ )(1)) = 0, so ρṽ is a robustly smooth point ofRrmin ◦ρ|G [1/p], in the sense of [BLGGT14, §1.3]. In particular, ρṽ lies onFṽ

a unique irreducible component of Rrmin ◦ρ|GFṽ[1/p], and we take Cv to be

this component.
• For v|p, let the set H of Hodge–Tate weights beX{hλ =ñv,α hλ, α∨ i}λ∈Λmin ,α∈∆where Λmin is the set of weights of rmin as before. Borel subgroups of SL27
containing the maximal torus T27 (the unique torus stabilizing the weightspaces in rmin ) are in bijection with orderings of the set Λmin ; let B27 be
the Borel defined by the ordering λ > λ0 ⇐⇒ hλ > hλ0 . The ordinary
deformation ring (again following the notation of [Pat16, §4.1]) associatedto the Borel B27 and the lift χT27 : IFṽ → T27 (Zp ) of rmin ◦ ρ|GFṽ given by!
YχT27 = rminα∨ (ñv,α χ−nv,α )α∈∆10

is formally smooth and receives by the universal property a canonical surjection from Rrmin ◦ρ|G ,H,K−ss for K = Fṽ (χ) (recall that χ is the Teichmüller
Fṽ

lift of ¯ · (¯[l])−1 ), and after inverting p it induces an isomorphism from aunique irreducible component of the source (the equality of dimensions follows from [Kis08] and [Pat16, §4.1]). We take Cv to be this component.
• For all other v ∈ S, we take the irreducible component of Rrmin ◦ρ|GFṽparametrizing minimal deformations in the sense of [Pat16, §4.4]; using
the standard argument, this is a power series ring and induces a unique
irreducible component Cv of Rrmin ◦ρ|G [1/p].
Fṽ

In each case, we writev
RrCmin◦ρ|GF

for the associated local lifting ring. Recallṽ

v([BLGGT14, §1.3]) that for v not above p, RrCmin◦ρ|Gis the maximal quotientFṽ

of Rrmin ◦ρ|Gthat is reduced, p-torsion-free, and after inverting p is supported on

Fṽv
the component Cv . For v|p, RrCmin◦ρ|G

is in general constructed similarly, but forFṽ

us it is simply the formally smooth ordinary deformation ring produced by [Pat16,§4.1].
v
Lemma 3.2. For all v ∈ S, the representation rmin induces a map RrCmin
◦ρ|G

Pv
Rρ|
GF

→
Fṽ

.
ṽ

Proof. For v|p, this follows directly
P from the definitions once we check that rmin (B) ⊂B27 . Let λ be the co-character α∈∆ ñv,α α∨ . Since all ñv,α are positive, B is thelocus of g ∈ G where limt→0 Ad(λ(t))g exists. By construction, B27 is the locus
where limt→0 Ad(rmin ◦ λ(t))g exists. The claim follows.
Pv
For v|q, we write Rρ|for the Steinberg lifting ring; recall that under our hyGFṽ

potheses it is a power series ring over Zp (in dim(g) variables). We have surjections
Rrmin ◦ρ|G

Fṽv
Recall that RrCmin
◦ρ|G

Fṽ


→ Rρ|
G

Fṽ

Fṽ

.
Fṽ

is the maximal reduced p-torsion-free quotient of Rrmin ◦ρ|G

Pv
whose Qp -points lie on Cv . Since Rρ|G

Rrmin ◦ρ|G

Pv
→ Rρ|
G

Fṽ

is a reduced, p-torsion-free quotient ofFṽ

Pv, it suffices to show that every Qp -point of Rρ|Glies on the same ir-

Fṽ

reducible component of Rrmin ◦ρ|GFṽ[1/p] as the representation ρṽ (defined above)

Pv
that characterizes the component Cv . As Rρ|G[1/p] is itself irreducible, it suf-

Fṽ

fices to check this for a single Qp -point, which we will do for the unramified lift∨ρunṽ (Frṽ ) = ρG (q). This claim follows from applying [BLGGT14, Lemma 1.3.5] toρṽ , whose associated graded for the obvious filtration is canonically identified withrmin ◦ ρunṽ .
For the other ramified primes v ∈ S, the lemma is evident.
Now consider the global deformation problem (in the sense of [BLGGT14, §1.5];
see [CHT08] for details)
S = (F/F + , S, {ṽ}v∈S , Zp , rmin ◦ ρ, µ, {Cv }v∈S ).
11

Since rmin ◦ρ|GF is absolutely irreducible, this deformation functor is pro-represented
by some RrSmin ◦ρ .
Lemma 3.3. The representation rmin induces a surjection RrSmin ◦ρ → RρP .
Proof. There is an induced map RrSmin ◦ρ → RρP by Lemma 3.2. It is a surjection
because the GF + -module ρ(g) is a direct summand of ad(rmin ◦ ρ) (indeed, these
representations factor through representations of a finite prime-to-p group), so theassociated map on tangent spaces is injective; dually, the map on co-tangent spaces
is surjective, and we conclude by Nakayama’s lemma.
Finally, we can invoke the main results of [BLGGT14] to deduce that RρP has a
Qp -point ρ such that rmin ◦ ρ is potentially automorphic:
Theorem 3.4. For sufficiently general choice of lifts ñv,α as in Equation 1, therepresentation ρ : GF + → L G(Fp ) constructed in Proposition 2.8 admits a geometriclift ρ : GF + → L G(Zp ) such that:(1) The Zariski closure of the image of ρ is L G.
(2) The composite rmin ◦ρ|GF is potentially automorphic in the sense of [BLGGT14].
(3) The composite rmin ◦ ρ|GF belongs to a compatible system of l-adic representations: there exist a number field M and a strictly pure — in the senseof [BLGGT14, §5.1] — compatible systemrλ : GF → GL27 (M λ )indexed over all finite places λ of M . In particular, the restriction of rλto Iv for v above the auxiliary primes q ∈ Sreg is unipotent with Jordan
blocks of size 1, 9, and 17 as long as v has residue characteristic different
from λ.
Proof. By the proof of [BLGGT14, Theorem 4.3.1] (see especially the last paragraph), RrSmin ◦ρ is a finite Zp -module. Lemma 3.3 then implies that RρP is a finite
Zp -module. We have already seen in Lemma 3.1 that it has dimension at least one,so we conclude that RρP (Zp ) is non-empty. Let ρ be an element of RρP (Zp ). Then:• By [BLGGT14, Theorem 4.5.1], the composite rmin ◦ ρ|GF is potentiallyautomorphic (in the sense of [BLGGT14]).
• The Zariski closure Gρ of the image of ρ|GF is G: by [Pat16, Lemma 7.8],
it suffices to show
– Gρ is reductive;– Gρ contains a regular unipotent element of G; and– for some v|p, ρ|GFṽ is B-ordinary, and, for all simple roots α, α◦ρ|IK =rα for some finite extension K/Fṽ and for distinct integers rα .
Reductivity is immediate since ρ is irreducible. The third condition follows
by taking the integers {ñv,α }α∈∆ in the definition of the local condition Pv(see the discussion preceding Lemma 3.1) to be sufficiently general. Finally,
to show that the image of ρ contains a regular unipotent element, we checkthat for v|q, the tame inertia in ρ|GFṽ acts by a regular unipotent. Lemma
3.5 below shows that this would follow from the corresponding claim thatrmin ◦ ρ(IFṽ ) contains a unipotent element with Jordan blocks of dimension
17, 9, and 1. Let π be the automorphic representation of GL27 (AF 0 ), for12

a suitable finite extension F 0 /F , witnessing the potential automorphy ofrmin ◦ ρ, and let v 0 be a place of F 0 above v. By local-global compatibility
at l 6= p (Proved in general by [Car12], but known for odd dimensional
representations by previous work of [HT01, TY07, Shi11]), the (Frobenius
semi-simple) Weil–Deligne representation associated to rmin ◦ ρ|GF 0 is isov0

morphic to the image of πv0 under the local Langlands correspondence. It
follows (eg, using [BLGGT14, Lemma 1.3.2(1)]) that rmin ◦ ρ|GFṽ lies on aunique irreducible component of Rrmin ◦ρ|G . By construction, rmin ◦ ρ|GFṽ
Fṽ

and rmin ◦ ρṽ lie on the same irreducible component Cv of Rrmin ◦ρ|G

, and

Fṽ

since they both lie on a unique component, [BLGGT14, Lemma 1.3.4(2)]implies their inertial restrictions are isomorphic. The result follows.
The claim that rmin ◦ ρ can be put in a compatible system, follows from
[BLGGT14, Theorem 5.5.1], and the claim concerning the restriction to Iv
for v|q follows as above from local-global compatibility at l 6= p.
In the next section, we will show that in fact all members of the compatible
system {rλ }λ have algebraic monodromy group equal to G.
Finally, here is the Lie-theoretic lemma promised in the proof of Theorem 3.4.
Lemma 3.5. Let G be an almost-simple algebraic group over an algebraicallyclosed field of characteristic zero, and let T be a maximal torus in G. Fix a systemof simple roots ∆ = {α1 , . . . , αr } of g with respect to T . Let λ ∈ X • (T ) be adominant (with respect to ∆) minuscule weight, with corresponding highest weight
representation rλ : G → Aut(Vλ ). Let N ∈ g be a regular nilpotent element. Thenfor any nilpotent element X ∈ g, rλ (X) is conjugate to rλ (N ) if and only if X isregular nilpotent.
Proof. It is well-known that the regular nilpotent elements of g form a single Gorbit, so one direction is clear. By the theorems of Jacobson-Morosov, Kostant, andMal’cev ([CM93, Theorem 3.3.1, 3.4.10, 3.4.12]), there is an sl2 -triple {X, Y, H} in
g containing X, this triple is unique up to conjugacy, and its conjugacy class isdetermined by that of H. Replacing {X, Y, H} by a conjugate, we may assume
that H ∈ Lie(T ) and α(H) ∈ {0, 1, 2} for all α ∈ ∆ (see [CM93, §3.5]). When we
likewise extend N to an sl2 -triple with semi-simple element HN , and assume after
conjugating that HN ∈ Lie(T ) is dominant, we have α(HN ) = 2 for all α ∈ ∆.
Since the conjugacy class of H determines that of X, X is regular if and only ifα(H) = 2 for all α ∈ ∆. We claim that the highest eigenvalue of H on Vλ is enoughto determine whether X is regular. For each weight µ of Vλ , writeµ=Xbα (µ)αα∈∆

for some bα (µ) ∈ Q. Now we use the assumption that λ is minuscule: for allα ∈ ∆, we have bα (λ) ≥ bα (µ) P([Bou68, Ch. VI, Ex. 1.23-1.24]). The maximal
eigenvalue of H on Vλ is thus
α (λ)α(H), and it follows easily that thisα∈∆ bPmaximal eigenvalue is strictly less than α∈∆ bα (λ) · 2 if X is not regular.
13

4. Controlling the image in the compatible system
Theorem 3.4 provides the existence of a compatible system {rλ } of GF representations with the property that the geometric monodromy group at one prime
is precisely E6 . Our goal in this section is to use known properties of compatible
systems ([LP92]) together with the additional properties our compatible system
satisfies at the auxiliary primes Sreg to ensure that the monodomy group is E6
at all primes λ.
Let M denote the coefficient field of our compatible system. For a pair of reductive groups Gi over M λ together with faithful irreducible representations ρi , recall
that (G1 , ρ1 ) is similar to (G2 , ρ2 ) if and only if, for some maximal tori Ti ⊂ Gi ,the pairs (Ti , ρi |Ti ) are isomorphic (cf. §5.1 of [LP92]). We would like to compare
the following two faithful representations of reductive groups over M λ :(1) The monodromy group Gλ of rλ over M λ (that is, the Zariski closure of
the image of rλ over M λ ) with its tautological representation rλ ,(2) The group G|M λ , where G is (as above) the split simply-connected reductive
group scheme over Z of type E6 , and rmin is the minuscule representation.
Lemma 4.1. The monodromy group Gλ for each prime λ of M has the following
properties:(1) The component group of Gλ is is trivial.
(2) The rank of Gλ is 6.
(3) If Tλ is a maximal torus in Gλ , and T |M λ is a maximal torus in G|M λ , thenthere is an isomorphism(Tλ , rλ |Tλ ) ' (T |M λ , rmin |T |M ).
λ

(4)
(5)

If Gλ = G◦λ acts irreducibly, then (Gλ , rλ ) is similar to (G|M λ , rmin ).
If Gλ = G◦λ acts irreducibly, then Gλ is isomorphic to G|M λ and rλisisomorphic to rmin .
(6) There exists a unipotent element in the image with Jordan blocks of size 1,
9, and 17.

Proof. The first three properties (1), (2), (3) involve quantities which are constantin a compatible system, c.f. Propositions 6.12 and 6.14 of [LP92]. We choose λ sothat rλ = rmin ◦ ρF as in Theorem 3.4(1). Then, by Theorem 3.4(1), we deducethat (Gλ , rλ ) ' (G|M λ , rmin ) in this case. Claim (4) follows immediately from (3)and the definition of similarity recalled above. Claim (5) follows from Theorem 5.6of [LP92], noting that E6 does not occur in the explicit list of groups which gives
rise to any of the basic similarity relations of §5 of ibid. The nilpotent operatorof the Weil–Deligne representation of rλ at the auxiliary prime v|q for q ∈ Sreg
decomposes (by Theorem 3.4 (3)) into Jordan blocks of size 1, 9, and 17, assumingthat the residual characteristic of q is different from that of λ. Yet Sreg was chosen(for this purpose!) to consist of two primes, so this holds for at least one prime v,
which gives part (6).

We now show that Conditions (2), (3), and (6) of Lemma 4.1 are sufficient toimply — purely by representation theoretic methods — that Gλ acts irreducibly,
which by Lemma 4.1(5) will prove the claims in Theorem 1.1 concerning the monodromy groups of {rλ }. In light of the existence of the unipotent element whose
14

existence is guaranteed by Lemma 4.1 (6), it suffices to show that Gλ cannot act
faithfully on a direct sum of representations of dimension
27 = 26 + 1 = 18 + 9 = 17 + 10 = 17 + 9 + 1
unless Gλ = E6 and the representation is irreducible.
4.1. The Formal Character of e6 . The root lattice Φ of E6 consists of 72 roots; it
6
may begiven as Φ+ ∪ Φ− , where the positive roots Φ+ are given
explicitly    inRby
5
5
5
5
the 2
= 20 vectors ei ±ej for 2 ≤ i < j ≤ 6, and the 24 =
+
+
=
2
0
2
4
16 vectors!
√
3 ±1 ±1 ±1 ±1 ±1
,
,
,
,
,
2 2 2 2 2 2P
where there are an even number of minus signs. If 2ρ =α∈Φ+ α, then ρ =√(4 3, 4, 3, 2, 1, 0). The root lattice is not self-dual, but has discriminant 3. A
weight µ corresponding to a choice of minuscule representation is given by
!
√
2 3
µ=
, 0, 0, 0, 0, 0 .
3
The 27 weights Σ of the corresponding minuscule representation may be obtained
from µ from the orbit of the Weyl group; all 27 such weights may be obtainedby applying at most 2 reflections in the roots of Φ to µ. The 27 roots√ can also
be described explicitly in 
thisbasis;
there
are
10
roots
of
the
form
−
3/3e1 ± ei    
55
54
for i = 2, . . . , 5, and 2 =++= 16 roots of the form
1
3
5
!
√
3 ±1 ±1 ±1 ±1 ±1
,
,
,
,
,
6 2 2 2 2 2
where there are an odd number of minus signs, and finally µ itself. These three sets(of orders 10 + 16 + 1 = 27) are precisely the orbits under the stabilizer WG,µ of µin the Weyl group WG .

Lemma 4.2. Of the 27= 2925 collections of 3 vectors in Σ, exactly 45 such3
triples generate a subspace of dimension 2, and they all consist of a triple ofweights (µ, µ0 , µ00 ) with µ + µ0 + µ00 = 0.
Proof. This can be checked explicitly using magma given the explicit descriptionof Σ given above.

We also recall the following facts (cf. [Lur01]) which are also elementary toverify by direct computation. (These facts are included purely for the reader’senjoyment and are not used in the sequel, except perhaps tangentially in the proof
of Lemma 5.2.) If Λ is the weight lattice, then Σ injects into V = Λ/2Λ, which
acquires the structure of a quadratic space via the map q(µ) = 21 hµ, µi (note that Λ
is an even lattice). The pairing hx, yi = q(x + y) − q(x) − q(y) is preserved by
the Weyl group WG , which may be identified with the corresponding orthogonalgroup O(V, q) ([Lur01, Remark 4.3.4]). The triples {µ, µ0 , µ00 } of Lemma 4.2 lie
inside a maximal isotropic subspace U ⊂ V of dimension 2. The stabilizer of U
(and of a triple) is a subgroup of WG of index 45, which correspondingly acts
15

transitively on the set of 45 triples. The stabilizer is also isomorphic to the Weylgroup of F4 .
Lemma 4.3. The quotient of Tλ which acts faithfully on any sub-representationof dimension ≥ 4 has rank at least 3, while Tλ itself acts faithfully on any subrepresentation of dimension 26. In particular, if Gλ has a sub-representation ofdimension ≥ 4 or 26, then the quotient of Gλ which acts faithfully on this subrepresentation has rank at least 3 or 6 respectively.
Proof. Recall (in the notation of Lemma 4.1) that we have an isomorphism (Tλ , rλ |Tλ ) '(T |M λ , rmin |T |M ). Hence it suffices to prove the corresponding claims for the forλmal character of the minuscule representation of E6 . For the first claim, it suffices toshow that any four weights appearing in the sub-representation must generate a sublattice of the weight lattice of rank at least 3. Assume otherwise. Then certainly any
two of these weights must generate a sublattice of rank at most 2. By Lemma 4.2,
such triples of weights are all of the form (µ, µ0 , µ00 ) where µ + µ0 + µ00 = 0. But(among four distinct weights) such a relation can hold for at most one such triple.
Hence we obtain a contradiction. The second claim follows from the fact that thesum of all 27 weights in Σ is zero, while none of the weights in Σ is zero.
We now note the following:
Lemma 4.4. Suppose that g is a reductive Lie algebra with a faithful irreducible
representation of dimension d for d ∈ {17, 26, 18}. Then g = h or h ⊕ t, where his semi-simple and t is a rank one torus. Furthermore, assuming that h is simple
when d = 18, then h is one of the following:hd rank(h)
sl2
17
1
so17
17
8
sl17
17
16
sl2
26
1
f4
26
4
so13 × sl2 26
7sl13 × sl2 26
13
sp26
26
13
so26
26
13
sl26
26
25
sl2
18
1
sp18
18
9
so18
18
9
sl18
18
17
Proof. It suffices to classify all small (of dimension at most 27) representations of
the simple Lie groups; one may (and we did) compute these using the Weyl character
formula. One can also deduce this result immediately for groups of rank ≤ 8 by
consulting [MP81, Table 1, p.25–83]. Groups of rank > 8 can also be handled
directly by hand, but they also play no role in the argument below because we arealways able to immediately dismiss any groups of rank > 6 = rank(E6 ).
Let us now return to the possible cases in which our 27 dimensional Galois
representation is reducible, and consider the corresponding monodromy groups.
16Suppose there is a constituent of dimension 17. The rank must be bounded by 6.
From Lemma 4.4, it follows that the Lie algebra of the monodromy group on thissummand must be sl2 or sl2 × t. But the rank of these algebras is at most 2,
which violates Lemma 4.3. Suppose there is a constituent of dimension 26. Thenby Lemma 4.3, the rank of the Lie algebra of this representation is exactly 6, andhence the rank of h with g = h or h×t is 6 or 5. Since there are no such groups of thisrank with irreducible representations of dimension 26 by Lemma 4.4, we once morederive a contradiction. Hence the only remaining possibility is that the 27 dimensional representation decomposes into two irreducible pieces of dimensions 9 and 18,
corresponding to a decomposition of weights Σ = Σ9 ∪ Σ18 . On the other hand, weknow that the 18 dimensional representation must have a unipotent element with
Jordan blocks of size 1 + 17. The tensor product of two Jordan blocks of size mand n with n ≤ m decomposes into blocks of size n+m−1, n+m−3, . . . , n−m+1.
In particular, it must be the case that the 18-dimensional representation does notfactor into a product of smaller dimensional representations, since otherwise there
could not be a Jordan block of a unipotent element of size as larger as 17. Hencethe monodromy group on this representation must have a simple Lie algebra (upto a torus). Again, we deduce from Lemma 4.4 and using rank considerations
that g = sl2 or sl2 × t, once more contradicting Lemma 4.3.
We conclude:
Corollary 4.5. Let {rλ : GF → GL27 (M λ )}λ be the compatible system producedin Theorem 3.4. Then for all λ, the Zariski closure of the image of rλ is isomorphic
to G|M λ under the minuscule representation rmin .
5. E6 motives over CM fields
We conclude by showing that rλ is a sub-representation of the cohomology ofsome smooth projective variety over F , thus completing the proof of Theorem
1.1. Note that, according to the Tate conjecture, we expect that such a compatible
family should be cut out by correspondences over F , and thus arise from a motive M
over F . We do not have any idea how to prove this. On the other hand, we do knowthat the compatible family rλ becomes automorphic over a CM extension H/F/Q,
and (since we are in highly regular weight), using standard methods combined withthe work of Shin [Shi11], one can associate a motive M over H whose associated padic Galois representations {rλ |GH } have monodromy group E6 . More precisely,
we should say that one expects to be able to associate such a motive where thecorrespondences cutting out M arise from Hecke operators. In practice, we take ashortcut and deduce from [Shi11] the weaker claim that the Galois representations
over GH (and thus over GF by restriction of scalars) came from cohomology. We
apologize for the omission and leave it as an exercise to the more responsible reader
to write down the correct argument.
To set up all the required notation would be quite cumbersome, so we will simply
use the notation of [Shi11], giving precise references to where the relevant terms
are defined. We hope that a reader with a copy of [Shi11] at hand can easily∼
follow this argument. Fix an isomorphism ιl : Ql −→ C; it is implicit in all of theconstructions of [Shi11]. By Theorem 3.4, there is a CM extension H/F and acuspidal automorphic representation Π0 of GL27 (AH ) such that• (Π0 )∨ ∼= (Π0 )c .
17

1−n
• Rl (Π0 ) ∼= rλ |GH · 2 , in the notation of [Shi11, Theorem 7.5]. (Note that n
is odd, and we have taken into account a standard twist in the association of
Galois representations to cohomological automorphic representations, as in[Shi11]; the relation between the conjugate self-dualities on the automorphic
and Galois sides is explicitly stated in [BLGGT14, Theorem 2.1.1]. Of
course it will suffice to show this Tate twist of rλ appears in cohomology.)• [H + : Q] ≥ 2, and H contains a quadratic imaginary field (we can simply
enlarge an initial choice of H to ensure these conditions).

Set n = 27, for ease of reference to [Shi11]; we will recall the construction of Rl (Π0 )and see as a result that after some further base-change that there is an explicit
description of this Galois representation in the cohomology of a unitary similitudegroup Shimura variety. We begin with two reductions. Let E be an imaginary
quadratic field not contained in H satisfying the four bulleted conditions in Step (II)of the proof of [Shi11, Theorem 7.5]. Replace H by HE and Π0 by BCHE/H (Π0 ).
Then having made this replacement the triple (E, H, Π0 ) satisfies the six bulleted
conditions at the beginning of Step (I) of the proof of [Shi11, Theorem 7.5]. Let H 0
be an imaginary quadratic extension of H + satisfying the three bulleted conditions(defining the set denoted F(H) — but note our H is Shin’s F ) in Step (I) of[Shi11, Theorem 7.5]. Then replace H by HH 0 and Π0 by BCHH 0 /H (Π0 ). Again
having made this replacement, Proposition 7.4 of [Shi11] now applies to the triple
×0(E, H, Π0 ). There is a Hecke character ψ of A×
E /E such that, setting Π = ψ ⊗ Π∼(an automorphic representation of the group Gn (A) = GL1 (AE ) × GLn (AH ) of
[Shi11, §3.1]), we have (in the notation of [Shi11, Corollary 6.8])
el0 (Π) ⊗ recl,ι (ψ c )|G ,Rl (Π0 ) := Rl0 (Π) := RHlwhere (see [Shi11, 5.5, 6.23], and note that the group G no longer denotes E6 , butrather the unitary similitude group defined in [Shi11, §5.1]!)X
n−1 ∞ ssel0 (Π) =CG · RRξ,l(π ) .
π ∞ ∈Rl (Π)

In our case, the integer CG = τ (G) ker1 (Q, G) (defined in [Shi11, Theorem 6.1]) is
e0 (Π) is irreducible (since
2: this is explained in [Tay12, p. 411–412]. Moreover, Rlthe image of rλ |ΓH is Zariski-dense in E6 ), and Rl (Π) in fact contains at least twoelements: following [Tay12, p. 413], there will be two automorphic representationsof G(AQ ), differing by a twist but having isomorphic base-changes to GLn (AH ),that contribute to Rl (Π) (these are denoted πe and πe ⊗ (δA/Q ◦ ν) in [Tay12, p.
n−1 ∞∞e0 (Π) ∼413]). We fix one such πe ∈ Rl (Π). It follows that R(eπ ) (no= Rξ,llsemisimplification necessary because these are irreducible representations).
Now recall the decomposition ([Shi11, 5.5])
Mn−1 ∞H n−1 (Sh, Lξ ) =π ∞ ⊗ Rξ,l
(π ).
π∞

e0 (Π) is contained in H n−1 (ShU ×H H, Lξ ) (itAt some finite level U , we deduce that Rlis even a direct summand cut out by Hecke operators). Finally, letting AU → ShU
denote the universal abelian scheme (arising from the PEL moduli problem), and(m)letting AU denote its m-fold fiber product over ShU (for any integer m ≥ 1), then18

there are integers mξ and tξ (see [HT01, p. 98]) such that(m )H n−1 (ShU ×H H; Lξ ) ∼= ε · H n−1+mξ (AU ξ ×H H, Ql )(tξ ),where ε is a suitable idempotent projector. We recall that our assumption [H + :(m )Q] ≥ 2 implies that ShU , and therefore AU ξ , are smooth projective varieties
1−nover H. Thus (rλ ·  2 )|GH ⊗ recl,ιl (ψ c )−1 |GH is a sub-representation of the co(m )homology of the smooth projective variety AU ξ over H. Possibly replacing H
by a finite extension, we can find a product of CM abelian varieties A/H suchthat recl,ιl (ψ c )|GH is a sub-representation of H i (A ×H H, Ql )(j) for some integers i and j ([DMOS82, IV. Proposition D.1]). We conclude that rλ |GH is a subrepresentation of H r (X ×H H, Ql )(s) for some smooth projective variety X/H(m )(namely, X = A ×H AU ξ ). By Frobenius reciprocity and irreducibility of rλ ,rF
rλ (as GF -representation) is a sub-representation of IndG
GH H (X ×H H, Ql )(s) .
This induction is just the cohomology of X regarded as a variety over F (i.e., via
X → Spec(H) → Spec(F )), so the proof of Theorem 1.1 is complete.
Remark 5.1. One may reasonably ask whether there exist E6 motives (or strongly
compatible systems) over Q. We do not know the answer. There is, however, atechnical obstruction for applying the methods of this paper. To use automorphic
methods, the Hodge–Tate weights of rmin ◦ ρλ must be distinct. However, there
cannot exist such a compatible system over Q (or any totally real field), since, by
Corollary 5.4.3 of [BLGGT14], this would imply that the trace of complex conjugation representation must be ±1, and, as the following lemma shows, there are nosuch involutions in E6 (C) ⊂ GL27 (C).
Lemma 5.2. There are two conjugacy classes of non-trivial involutions in E6 (C)with traces −5 and 3 respectively.
Proof. This lemma is “well known” in the sense that at least one of the authors of
this paper claims to know it. In lieu of a reference, we give the following computational proof, which we explain in as conceptual terms as possible. Let X• = X• (T )denote the cocharacter group of G, and X • = X • (T ) the character group. We
may assume any involution lies in T and thus can be considered as an elementof X• ⊗ µ2 ' (X • /2X • )∨ . We may identify the latter group with the F2 -dual V ∨of V = Λ/2Λ. This is a quadratic space, thus self-dual, and so there is a WG equivariant isomorphism V ∨ ' V . There are 63 = 26 − 1 non-zero elements of V .
These are divide into two sets depending on whether the vectors are isotropic ornot. The action of WG correspondingly has two orbits of size 27 and 36 respectively.
This is almost true by pure thought, since the orbits must have order greater than 2(V is an irreducible WG -representation) and the only non-trivial permutation representations of WG of degree less than 40 and greater than 2 are of orders 27 and 36
respectively ([CCN+ 85]). In fact, one can identify these orbits of size 27 and 36
are precisely the images of the set of weights and positive roots, respectively. If µ
is the highest weight vector corresponding to rmin , let WG,µ the stabilizer of µ
in WG . Since rmin is minuscule, the corresponding weights are precisely those of
the form wµ for w ∈ WG /WG,µ . Hence the trace of an involution corresponding
to v ∨ turns out to be equal toX∨(−1)hv ,wµi
w∈WG /WG,µ19

By computing this sum explicitly over all v ∨ , we see that the traces of the two nontrivial involutions in the minuscule representation are −5 and 3 for precisely 27
and 36 of the v ∨ respectively. For example, for the first sum, we have alreadynoted that the stabilizer Wµ acts on the roots with orbits of size 1, 10, and 16
respectively given explicitly before Lemma 4.2; the corresponding signs in the sum
are +1, +1, and −1 respectively, and hence the sum is 1+10−16 = −5. The second
sum can be computed directly from the first by averaging over the WG action andthen using the fact that, for any fixed v ∨ , one has
X∨(−1)hv ,vi = 0.
v∈V

Alternatively, if WG,α is the stabilizer of the image of a root α ∈ V (and hence ofindex 36 in WG ), then the orbits of WG,α on the weights are of size 15 and 12, andthe corresponding signs are +1 and −1 and the trace is 15 − 12 = +3.
As a confirmation of these numbers, we can appeal to the following variafG of WG
tion. There exists a cover (the extended Weyl group of Tits [Tit66]) W
6by V = (Z/2Z) which lives as a subgroup of E6 (C). The group has an explicit
description as the Z-points of the normalizer N of a (split) maximal torus T of G.
fG admits a 27 dimensional representation via the minuscule repreThe group W
sentation of E6 which turns out to be absolutely irreducible. Then one can check(using magma) from the character table that the two conjugacy classes of involution
fG → WG ) (of orders 27 and 36 respectively) have traces −5 and 3
in V = ker(WfG .
under both irreducible 27 dimensional representations of W

6. Complements
We note the following application of Theorem 1.1 to the inverse Galois problem.
Let us write E6sc (Fl ) for the Fl -points of the (simply connected) form of E6 that
we have been considering. The group E6sc (Fl ) is the Schur cover of the simple
Chevalley group of type E6 (with Schur multiplier of order (3, l − 1)), which wedenote below by E6 (Fl ). The groups E6sc (Fl ) and E6sc (Fl ).2 are known to occur as
Galois groups over Q for p ≡ 4, 5, 6, 9, 16, 17 mod 19 (these are primes of order 9
in F×19 ) by [Mal88, Thm 2.3] as a consequence of the rigidity method. In contrast,
we can prove that E6sc (Fl ).2 is a Galois group over Q for a positive density ofprimes ≡ 1 mod 19 (by taking E = Q(ζ19 ) below).
Corollary 6.1. Let F be an imaginary quadratic field. Then, for a set S ofprimes l of positive density, there exists a number field L/Q containing F with
Gal(L/Q) = L G(Fl ) = E6sc (Fl ).2 and Gal(L/F ) = E6sc (Fl ). Moreover, one may
assume that all the primes in S split completely in any finite extension E/Q.
Proof. We combine the previous result with Theorem 3.17 of [Lar95], to deducethat E6sc (Fl ) is the Galois group of the kernel of ρλ over F for a relative density one
set of primes l which split completely in the coefficient field M of the compatible
system. In particular, the intersection of S with the set of primes split completely
in E/Q has positive density. Since the representation rλ : GF → GL27 (Fl ) is, byconstruction, conjugate self-dual, it extends to a representation of GQ to G27 (Fl )whose kernel therefore has Galois group L G(Fl ).

20

Remark 6.2. Note that Corollary 6.1 remains true if one replaces E6sc (Fl ).2
and E6sc (Fl ) by E6 (Fl ).2 and E6 (Fl ), for the obvious reason that the latter groups
are quotients of the former groups.
6.1. Acknowledgments. This work traces its origins to a discussion following a
number theory seminar given by one of us (S.P.) at the University of Chicago in
February 2016, where all the authors of this paper were in attendance. We would
like to thank Sug Woo Shin for answering questions related to his paper [Shi11]
relevant for the discussion in §5.

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22Department of Mathematics, University of Chicago, 5734 S University Ave Chicago,
IL 60637
E-mail address: gboxer@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S University Ave Chicago,
IL 60637
E-mail address: fcale@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S University Ave Chicago,
IL 60637
E-mail address: emerton@math.uchicago.edu
Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
E-mail address: bwlevin@math.arizona.edu
Department of Mathematics, Boston College, Chestnut Hill, MA 02467
E-mail address: keerthi.madapusipera@bc.edu
Department of Mathematics, The University of Utah, 155 S 1400 E, Salt Lake City,UT 84112
E-mail address: patrikis@math.utah.edu

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