LATTICES IN PRINCIPAL SERIES TYPES IN THE COHOMOLOGY

LATTICES IN PRINCIPAL SERIES TYPES IN THE
COHOMOLOGY OF U(3) SHIMURA VARIETIES
DANIEL LE
Abstract. Under hypotheses required for the Taylor-Wiles method, we prove
for forms of U (3) which are compact at infinity that the lattice structure on
principal series types given by the λ-isotypic part of completed cohomology is
a local invariant of the Galois representation associated to λ when this Galois
representation is residually irreducible locally at places dividing p. A crucial
ingredient that we establish is a mod p multipliciity one result for principal
series types. To prove this result, we combine Hecke theory and weight cycling
with the Taylor-Wiles method.
1. Introduction
Our goal is to establish a local-global compatibility result in the p-adic Langlands
program for GL3 (Qp ). Recall classical local-global compatibility for GLn : if Π =
⊗0v πv is an irreducible cohomological automorphic representation associated to a padic n-dimensional Galois representation ρ via the cohomology of a unitary Shimura
variety over F + , then πv is obtained from the Weil-Deligne representation attached
to ρ|Dv via the local Langlands correspondence, and is independent of the Hodge
filtration attached to ρ|Dv by Fontaine ([Fon94]). Suppose that ρ|Dv has HodgeTate weights (0, 1, . . . n − 1). The following na¨ıve question extrapolates from the
p-adic Langlands correspondence for GL2 (Qp ) ([Col10], [Paˇs13]).
Question 1.0.1. Is there a natural GLn (Fv+ )-invariant norm on πv which corresponds to the Hodge filtration for ρ|Dv ?
Completed cohomology gives πv an integral structure, and hence also a GLn (Fv+ )invariant norm, which is of an a priori global nature. The following na¨ıve question
extrapolates from p-adic local-global compatibility for GL2 (Q) ([Eme11]).
Question 1.0.2. Does this integral structure depend only on ρ|Dv ?
If ρ|Dv is potentially crystalline of principal series type τ , then τ appears in πv
with multiplicity one. The following conjecture is the analogue in the GL3 (Qp )-case
of Breuil’s conjecture made in the GL2 -case (Conjecture 1.2 of [Bre]). Suppose that
n = 3 and p splits completely in F , the CM field over which U splits.
Conjecture 1.0.3. The lattice structure on τ given by completed cohomology depends only on ρ|Dv .
The following is the main theorem of the paper (see Theorem 6.2.3 for more details).
Theorem 1.0.4. Suppose that ρ|Dv is irreducible for all places v|p of F + . Under
mild hypotheses necessary for the Taylor-Wiles method, Conjecture 1.0.3 is true.
In fact, the lattice can be described explicitly in terms of ρ|Dv .
Date: 9.30.14.
1
2
DANIEL LE
1.1. Detailed description of results. Let F be a CM field with totally real subfield F + and assume that F/F + is unramified at all finite places. Suppose that
p 6= 2 splits completely in F . Let E be an extension of Qp . Let ρ : GF → GL3 (OE )
be an essentially conjugate self-dual Galois representation that is potentially crystalline at places dividing p with Hodge-Tate weights (0, 1, 2) and ramified only at
places that split in F/F + . Suppose that ρ|Dw has tame inertial type η aw ⊗η bw ⊗η cw
for each place w|p of F where η is the Teichmuller character and aw > bw > cw are
integers with a − c < p − 1.
Let λ ⊂ T be the Hecke eigensystem attached to ρ. Here T is the spherical Hecke
algebra at places of F + that split in F and where ρ is unramified. For each place v
of F + that splits in F , choose a place ve|v of F and an isomorphism Fve ∼
= Fv+ . Fix a
+
0
+
0
∼
place v|p of F . For each place v 6= v of F , let τv0 ⊂ τv0 = τve0 be lattices in types
corresponding to ρ by results towards the inertial local Langlands correspondence
(see Proposition 6.5.3 of [BC09]). Note that τv0 is trivial for all but finitely many
places v 0 , and in particular at all inert places.
Consider a form of U (3) over F + that splits over F , is quasisplit at all finite
b 0 be the completed cohomology with Zp
places, and is compact at infinity. Let H
b 0 [λ] 6= 0 (or in
coefficients of the associated Shimura variety and assume that H
other words ρ is modular). Let
b 0 [λ]))
π = HomK v (⊗0 0 τ 00 , H
v 6=v v
v
GL3 (Ap,∞
F )
where K ⊂
is a hyperspecial compact open subgroup away from
v∞. Then HomGL3 (Ov ) (τv , π[p−1 ]) is one-dimensional (see Theorems 5.4 and 5.9 of
[Lab11]).
Theorem 1.1.1. Assume that ρ is modular. Assume that ρ is strongly generic,
satisfies the Taylor-Wiles conditions, the local deformation rings with the types τve0
for places v 6 |p of F + are regular, and that ρ|GFve is irreducible for places v|p. Then
the lattice (τv ⊗Zp Qp ) ∩ π ⊂ τv ⊗Zp Qp depends only on ρ|GFve in an explicit way.
This is an analogue for GL3 (Qp ) of Conjecture 1.2 of [Bre], which was established
in [EGS13]. See also [Le14] for results on algebraic vectors rather than for types.
Modifications of the Taylor-Wiles patching method [TW95] are well suited for this
result and play a large role in its proof. Kisin’s local-global modification ([Kis09])
provides a natural setting to compare the local and global Langlands correspondences integrally. The modification in [CEG+ 14] allows one to use Hecke theory
and weight cycling as in [EGH13] to study lattices. We use Diamond ([Dia97])
and Fujiwara’s ([Fuj06]) modifications, the Hecke theoretic methods of [Le14], and
the strategy of Section 10 of [EGS13] to show the following mod p multiplicity one
theorem, which plays a crucial role in the proof of Theorem 1.1.1. Let π be the
reduction of π modulo the maximal ideal of Zp .
Theorem 1.1.2. The space HomGL3 (Ov ) (τ v , π)[Ui,v ] is one-dimensional, where
Ui,v are Hecke operators at v.
1.2. Overview of sections. We give a quick overview of the paper. In Section
2, we define certain lattices of interest in principal series types and some of the
maps between them. In Section 3, we make computations in Hecke theory that
relate the Hecke algebras for algebraic vectors and principal series types. This is
the automorphic analogue of relating the special fibers of crystalline and potentially crystalline of Hodge-Tate weights (0, 1, 2) Galois deformation rings. These
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
3
play an important role in Section 5. In Section 4, we describe the patching process
that plays a key role in Section 6. Section 5 is the technical heart of the paper,
where we prove cyclicity of patched modules for principal series types, a mod p
multiplicity one statement (see Theorem 5.4.1). In Section 6, we prove the main
lemma (Lemma 6.1.1) establishing part of the lattice structure in types in families.
We then use classical local-global compatibility to prove the main theorem (Theorem 6.2.3) about describing the lattice in principal series types given by integral
completed cohomology.
1.3. Acknowledgements. We thank Florian Herzig for correspondence regarding
Proposition 2.1.2. We thank Christophe Breuil for sharing a note about lattice
conjectures for GL3 (Qp ) which in addition to [Bre] and [BD12] inspired this paper.
We thank Matthew Emerton for many conversations about patching and the p-adic
Langlands program. This paper also owes a debt to ideas (some unpublished and
some in [CEG+ 14]) of Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty,
Vytautas Paskunas, and Sug Woo Shin, and the ideas and results of Matthew
Emerton, Toby Gee, and Florian Herzig, and it is a pleasure to acknowledge this.
1.4. Notation. For a field k, Gk denotes the absolute Galois group of k. HodgeTate weights are normalized so that the cyclotomic character has weight −1. ComG
pact induction is denoted indG
H while usual induction is denoted IndH . The symbols
∨
d
· and · are used to denote the Pontriagin dual and Schikhov dual, respectively
(see Section 1.5 of [CEG+ 14]). The letter v is used to denote places of F + while
the letter w is used to denote places of F .
For λ ∈ X∗ (T), let W (λ) be the Zp -points of the algebraic GL3 (Zp )-representation
IndG
B s0 λ where s0 is the longest Weyl element. Let W (λ) be the reduction of W (λ)
modulo p, and let F (λ) be the (irreducible) socle of W (λ). We label the other elements of S3 as e = (), s1 = (12), s2 = (23), r1 = (132), r2 = (123), and s0 = (13).
The letters a, b, and c are used to describe weights of representations. We assume
throughout that a > b > c are integers with a − c < p − 1. In Section 3, we assume
that a − c < p − 2 in order to calculate Hecke algebras. In Section 5.1, we assume
that a − c < p − 3 in order to apply Fontaine-Laffaille theory. In Sections 5 and 6,
we make the assumption of strong genericity as in [EGH13] in order to eliminate
Serre weights.
A (Serre) weight (of GL3 (Fp )) is an (absolutely) irreducible GL3 (Fp )-representation
over Fp . By Theorem 2.8 and Section 4 of [And87], we have the following results.
Note that [And87] describes SL3 (Fp )-representations, but the proofs carry over verbatim to the GL3 (Fp )-setting. If F and F 0 are weights, then dimF Ext1GL3 (Fp ) (σ 0 , σ) =
dimF Ext1GL3 (Fp ) (σ, σ 0 ) ≤ 1. If F (a, b, c) is a weight, let E(a, b, c) is the set of triples
(x, y, z) such that Ext1GL3 (Fp ) (F (a, b, c), F (x, y, z)) 6= 0. If a > b > c are integers
such that a − c < p − 1, then
E(a, b, c) := {(b + p − 1, a, c), (a, c, b − p + 1), (c + p − 2, b, a − p + 2),
(a + 1, c − 1, b − p + 1), (b + p − 1, a + 1, c − 1), (a, c − 1, b − p + 2),
(b + p − 2, a + 1, c)}.
Moreover, Ext1GL3 (Fp ) (σ 0 , σ) = 0 if both σ and σ 0 are upper alcove weights. If
(x, y, z) ∈ E(a, b, c), let E((a, b, c), (x, y, z)) be the unique up to isomorphism nontrivial extension of F (a, b, c) by F (x, y, z).
4
DANIEL LE
2. Lattices in principal series types
In this section, we discuss the different lattices in principal series types. Let
G = GL3 (Qp ), Z the center of G, K = GL3 (Zp ), I ⊂ K be the subgroup of matrices
that are upper triangular mod p, and I1 ⊂ I be the subgroup of matrices that are
unipotent mod p.
2.1. Integral structure on principal series types. Let a > b > c ∈ Z with
a − c < p − 1. These inequalities imply that the principal series types to be defined
×
will be absolutely irreducible and residually multiplicity free. Let η : F×
p → Zp be
a
b
c
the Teichmuller character. Let χ = η ⊗η ⊗η be a character of I by inflation
via the
∼
3
map I → I/I1 → (F×
p ) . Extend χ to a character of IZ by setting χ
p 0 0
0 p 0
0 0 p
= 1.
Identify S3 with the group of permutation matrices in GL3 . Let ` be the length
function on S3 . Let χw be the character with factors permuted by w ∈ S3 . Let e
be the identity, s0 = (13), r1 = (132), r2 = (123), s1 = (12), and s2 = (23). Then
for example χr1 = η b ⊗ η c ⊗ η a and χr2 = η c ⊗ η a ⊗ η b .
KZ s
s
For each s ∈ S3 , consider the KZ-representation τ s = indK
I χ = indIZ χ over
e
s
Zp . Let τ denote τ . The representations τ are lattices in the principal series
s
s0 s
types τ s ⊗Zp Qp . For s, s0 ∈ S3 , the intertwiner ιss0 s : indKZ
,→ indKZ
IZ χ
IZ χ
from the classical representation theory of GL3 (Fp ) is nonzero modulo p and gives
isomorphisms of principal series types τ ⊗Zp Qp ∼
= τ s ⊗Zp Qp for all s ∈ S3 . By
0
Frobenius reciprocity, this map is determined by the image of v s s . The intertwiner
is given by
X
0
(1)
ιss0 s : v s s 7→
gs0 v s .
g∈I1 /(s0 I1 s0−1 ∩I1 )
0
00 0
00 0
KZ s
s
Proposition 2.1.1. The composition ιss s ◦ ιss0 ss s : indKZ
IZ χ ,→ indIZ χ
00
0
00 0
00 0
1
p 2 (`(s )+`(s )−`(s s )) ιss s s .
Proof. See Lemme 2.2 of [Bre].
ss
is
Florian Herzig provided the argument for the following proposition, which describes the submodule structure of τ .
Proposition 2.1.2. The socle and cosocle filtrations of τ have associated graded
pieces
F (a, b, c)
F (a, c, b − p + 1) ⊕ F (b + p − 1, a, c)
F (c+p−1, a, b)⊕F (b−1, c, a−p+2)⊕F (a−1, b, c+1)⊕F (c+p−2, a, b+1)⊕F (b, c, a−p+1)
F (c + p − 1, b, a − p + 1),
where any number of bottom rows are the graded pieces of a submodule. Furthermore, all nontrivial extensions that can occur do occur (see Section 1.4). The same
statement is true for τ s0 except that the order of the rows is reversed.
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
5
Proof. Satz 4.4 of [Jan84] gives a filtration with associated graded pieces as described above (see also the last lemma of Section 2 and Section 5.2 of [Jan84]).
Recall that a, b, and c are distinct mod p − 1. Using Frobenius reciprocity and
Lemma 2.3 of [Her11], the socle and cosocle of τ are F (c + p − 1, b, a − p + 1) and
F (a, b, c), respectively. Hence, F (c + p − 1, b, a − p + 1) and F (a, b, c) are pieces of
the associated graded for both the socle and cosocle filtrations. It remains to show
that the socle and cosocle filtrations have length 4, to describe the other associated
graded pieces, and show that all nontrivial extensions that can occur between the
inner layers n
do occur.
o
n ∗ ∗ ∗ o
∗∗∗
GL (F )
∗∗∗
0 ∗∗
Let P1 =
, P2 =
⊂ GL3 (Fp ). Consider IndP1 3 p F (b+p−
0 0 ∗
0 ∗∗
1, a)⊗F (c) ⊂ τ , which has Jordan Holder factors F (b+p−1, a, c), F (b−1, c, a−p+
2), F (a−1, b, c+1), F (c+p−2, a, b+1), F (b, c, a−p+1), and F (c+p−1, b, a−p+1)
by Lemma 6.1.1 of [EGH13]. Again using Frobenius reciprocity and Lemma 2.3 of
GL (F )
[Her11], IndP1 3 p (F (b+p−1, a)⊗F (c)) has socle and cosocle F (c+p−1, a, b−p+1)
and F (b + p − 1, a, c), respectively. This shows that the socle and cosocle filtrations
must have length 4 and further that F (b + p − 1, a, c) is in the second layer and
F (b − 1, c, a − p + 2), F (a − 1, b, c + 1), F (c + p − 2, a, b + 1), and F (b, c, a − p + 1)
are in the third with all possible nontrivial extensions occuring.
GL (F )
Similarly examining IndP2 3 p F (a)⊗F (c, b−p+1) shows that F (a, c, b−p+1) is
in the second layer and F (c+p−1, a, b) is in the third. Furthermore, all extensions of
F (a, c, b−p+1) occur. The statement for τ s0 is obtained similarly or by duality. For s ∈ S3 , let s(a, b, c) be an ordered triple mod p − 1 congruent to the spermutation of (a, b, c).
0
Proposition 2.1.3. The image of ιss is the unique lattice up to homothety with
cosocle F (s(a, b, c)).
0
Proof. By Proposition 2.1.2, the image of ιss is a lattice with cosocle F (s(a, b, c)).
0
Uniqueness follows from Lemma 4.1.1 of [EGS13] and the fact that τ s is residually
multiplicity free.
2.2. Shadow lattices. Let τ1 , τ2 , and τ3 ⊂ τ be the unique lattices up to homothety with cosocle F (b − 1, c, a − p + 2), F (a − 1, b, c + 1), and F (c + p − 2, a, b + 1),
respectively. It remains to describe the submodule structure of τ 1 , τ 2 , and τ 3 .
Proposition 2.2.1.
graded pieces
(1) The socle and cosocle filtrations of τ 1 have associated
F (b − 1, c, a − p + 2)
F (b + p − 1, a, c) ⊕ F (a, c, b − p + 1) ⊕ F (c + p − 1, b, a − p + 1)
F (a, b, c)⊕F (b, c, a−p+1)⊕F (c+p−1, a, b)⊕F (c+p−2, a, b+1)⊕F (a−1, b, c+1).
(2) The socle and cosocle filtrations of τ 2 have associated graded pieces
6
DANIEL LE
F (a − 1, b, c + 1)
F (b + p − 1, a, c) ⊕ F (a, c, b − p + 1) ⊕ F (c + p − 1, b, a − p + 1)
F (a, b, c)⊕F (b, c, a−p+1)⊕F (c+p−1, a, b)⊕F (c+p−2, a, b+1)⊕F (b−1, c, a−p+2).
(3) The socle and cosocle filtrations of τ 3 have associated graded pieces
F (c + p − 2, a, b + 1)
F (b + p − 1, a, c) ⊕ F (a, c, b − p + 1) ⊕ F (c + p − 1, b, a − p + 1)
F (a, b, c)⊕F (b, c, a−p+1)⊕F (c+p−1, a, b)⊕F (b−1, c, a−p+2)⊕F (a−1, b, c+1).
Again, any number of bottom rows are the graded pieces of a submodule. Moreover,
all nontrivial extensions that can occur do occur (see Section 1.4).
Proof. We will prove the proposition for τ 1 , the other cases being similar. The
weight F (b − 1, c, a − p + 2) is in the first layer for both filtrations by construction.
Recall that in the proof of Lemma 4.1.1 of [EGS13], τ1 ⊂ τ s0 can be taken to be
the inverse image of the minimal submodule of τ s0 containing F (b − 1, c, a − p + 2)
as a Jordan-Holder factor. Proposition 2.1.2 shows that F (a, c, b − p + 1) and
F (b + p − 1, a, c) are in the second layer of the cosocle filtration of τ 1 . Similarly,
looking at the inclusion τ1 ⊂ τ s1 , we see that F (c + p − 1, b, a − p + 1) is also in the
second layer of the cosocle filtration of τ 1 . In fact, F (a, c, b − p + 1) ⊕ F (b + p −
1, a, c)⊕F (c+p−1, b, a−p+1) is the second layer of the cosocle filtration since these
weights are the only Jordan-Holder factors of τ that extend F (b − 1, c, a − p + 2)
(see Section 1.4).
The other five weights are all in the third layer of the cosocle filtration since
there are no extensions between them (see Section 1.4). It remains to show that all
possible nontrivial extensions between the bottom two rows occur. Since the map
τ 1 → τ s1 has image with Jordan-Holder factors F (b − 1, c, a − p + 2), F (a, c, b − p +
1), F (c + p − 1, b, a − p + 1), and F (c + p − 1, a, b) by Proposition 2.1.2, these are
the Jordan-Holder factors of the cokernel of τ s1 → τ 1 . Hence all possible nontrivial
extensions of F (b + p − 1, a, c) occur in τ 1 since they occur in im(τ s1 → τ 1 ) by
Proposition 2.1.2. All other nontrivial extensions are established analogously. From
this we see that the cosocle and socle filtrations agree.
3. Hecke algebras of principal series types and extensions
In this section, we describe Hecke operators for principal series types and extensions of two weights, and relate them in Proposition 3.2.2.
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
7
3.1. Hecke operators for principal series types. Keep the notation of Section
2. Note that by Frobenius reciprocity, we have
∼ HomK (indKZ χ, indKZ χ) ∼
HomIZ (χ, indKZ χ) =
= Zp
IZ
IZ
IZ
s
χ. For each s ∈ S3 , let v ∈ τ ⊂ indG
IZχ be a (unique
p
00
up to scaling) generator of the χs -isotypic submodule. Let t1 = 0 1 0 and t2 =
0
0
1
p 0 0
0 p 0 . Let (i, j) be a permutation of (1, 2).
by irreducibility of
indKZ
IZ
s
s
0 0 1
s
Proposition 3.1.1. The element ri ti v s spans an I1 Z-invariant subspace of indG
IZ χ
ri s
on which IZ acts by χ . Moreover, by Frobenius reciprocity, we define maps
ri s
s
ri s
νrsi s : indG
→ indG
7→ ri ti v s , which are isomorphisms.
IZ χ
IZ χ by v
Proof. The first statement follows from two easy computations. First, one checks
that ri ti normalizes I1 and so ri ti v s spans an I1 Z-invariant subspace. Second,
we have that tri ti v s = ri (ri−1 tri )ti v s = ri ti (ri−1 tri )v s = χri s (t)ri ti v s for any t ∈
T(Zp ).
For the second statement, note that the composition
G
G
ri
νrsi s νsri s : indG
IZ χ → indIZ χ → indIZ χ
p 0 0
maps v s to ri ti rj tj v s = 0 p 0 v = v and is hence the identity map.
0 0 p
s
Let Ujs be the operator on indG
IZ χ given by
P
g∈I1 /(tj I1 t−1
j ∩I)
gtj .
Proposition 3.1.2. The composition
∼
G
s
ri s
ri s
,→ indG
νsri s ◦ ιsri s : indG
IZ χ → indIZ χ
IZ χ
is the operator Ujri s .
Proof. From the respective formulas in (1) and Proposition 3.1.1, we see that the
composition is defined by
X
X
v ri s 7→
gri rj tj v ri s =
gtj v ri s
g∈I1 /(ri I1 ri−1 ∩I1 )
g∈I1 /(ri I1 ri−1 ∩I1 )
=
X
gtj v ri s = Uj v ri s .
g∈I1 /(tj I1 t−1
j ∩I1 )
3.2. Hecke operators for extensions. Let a > b > c be integers with a − c <
p − 2. These inequalities, which are stronger than those in Section 2, are used
in Proposition 3.2.1 to split certain exact sequences. Let E be one of E((b + p −
1, a, c), (a, b, c)), E((a, c, b − p + 1), (a, b, c)), E((c + p − 2, a, b + 1), (b + p − 1, a, c)),
or E((c + p − 2, a, b + 1), (a, c, b − p + 1)), and let E 0 be the unique up to isomorphism nonsplit extension with the reverse socle filtration of E. Thus E 0 is one of
E((a, b, c), (b + p − 1, a, c)), E((a, b, c), (a, c, b − p + 1)), E((b + p − 1, a, c), (c + p −
2, a, b + 1)), or E((a, c, b − p + 1), (c + p − 2, a, b + 1)). Let σ be the socle of E and
σ 0 the cosocle.
σ is the cosocle
E 0 and σ 0 the socle.
nThus
o
n ∗ ∗ ∗ofo
∗∗∗
∗∗∗
0 ∗∗
Let P1 =
, P2 =
⊂ GL3 (Fp ). Let P = P2 (resp. P1 , P1 ,
0 0 ∗
0 ∗∗
and P2 ). Let P be the opposite parabolic. Let N and N be the unipotent radicals
8
DANIEL LE
of P and P , respectively. Let M = P/N ∼
= P /N . The representations σ, σ 0 , E,
0
and E are K-modules by inflation via the natural map K GL2 (Fp ).
0
Proposition 3.2.1.
(1) σ N ∼
.
= σN and σ 0N ∼
= σN
(2) If E = E((b + p − 1, a, c), (a, b, c)) or E((a, c, b − p + 1), (a, b, c)), then
0 ∼ N
EN ∼
= EN
= (E 0 )N ∼
= σ ⊕ σ 0N .
= EN ∼
(3) If E = E((c+p−2, a, b+1), (b+p−1, a, c)) or E((c+p−2, a, b+1), (a, c, b−
0 ∼ N
0 ∼
0
p + 1)), then E N ∼
.
= EN
= σ and EN ∼
= EN
= σN ⊕ σN
Proof. We have the exact sequences
0 →σ N → E N → σ 0N
0 →σ 0N → (E 0 )N → σ N
0
σN → EN → σN
→0
0
0
σN
→ EN
→ σN → 0
By Lemma 2.3 of [Her11] and the inequalities satisfied by a, b, and c, σ N ∼
= σN
0
and σ 0N ∼
have different actions of the GL1 factor of M . We conclude that
= σN
the corresponding short exact sequences split. One now uses Proposition 2.1.4 of
[Le14] and argues as in Proposition 3.2.2 of [Le14].
By Proposition 2.1.2 of [Le14], there is a Hecke operator T ∈ H(E 0 )µ defined
∼
up to scalar corresponding to a fixed nonzero isomorphism σN → σ N where µ =
(0, 1, 1) (resp. (0, 0, 1), (0, 0, 1), or (0, 1, 1)). Note that in [Le14], we consider indG
K
rather than indG
KZ , but this makes no difference in the proofs. Proposition 3.2.2
now fixes this scalar. The following proposition, which uses notation of Section 2,
relates Hecke operators for principal series types and the extensions in Proposition
3.2.1.
Proposition 3.2.2. By Frobenius reciprocity, the map T corresponds to a map
Kt KZ
E 0 → indKZµ
E0.
(1) If E = E((b + p − 1, a, c), (a, b, c)), then we have the commutative diagram
Kt KZ
τ s2
Kt KZ
τ s2
Kt KZ
τ s2
Kt KZ
E0
τ s2
τ s0
indKZµ
τ s2
τ s0
indKZµ
τ r2
τ s0
indKZµ
E0
E
indKZµ
where all maps are nonzero mod p, the composition of the top row corresponds to U1s2 , the composition of the bottom row corresponds to T .
(2) If E = E((a, c, b − p + 1), (a, b, c)), then we have the commutative diagram
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
Kt KZ
τ s1
Kt KZ
τ s1
Kt KZ
τ s1
Kt KZ
E0
τ s1
τ s0
indKZµ
τ s1
τ s0
indKZµ
τ r1
τ s0
indKZµ
E0
E
indKZµ
9
where all maps are nonzero mod p, the composition of the top row corresponds to U2s1 , the composition of the bottom row corresponds to T .
(3) If E = E((c + p − 2, a, b + 1), (b + p − 1, a, c)), then we have the commutative
diagram
Kt KZ
τe
Kt KZ
τe
Kt KZ
τe
Kt KZ
E0
τe
τ r2
indKZµ
τe
τ r2
indKZµ
τ0
τ r2
indKZµ
E0
E
indKZµ
where all maps are nonzero mod p, the composition of the top row corresponds to U2e , the composition of the bottom row corresponds to T . Here, τ 0
is the unique lattice such that τ 0 has socle F (c + p − 2, a, b + 1) (by Lemma
4.1.1 of [EGH13]).
(4) If E = E((c + p − 2, a, b + 1), (a, c, b − p + 1)), then we have the commutative
diagram
Kt KZ
τe
Kt KZ
τe
Kt KZ
τe
Kt KZ
E0
τe
τ r1
indKZµ
τe
τ r1
indKZµ
τ0
τ r1
indKZµ
E0
E
indKZµ
10
DANIEL LE
where all maps are nonzero mod p, the composition of the top row corresponds to U1e , the composition of the bottom row corresponds to T . Again,
τ 0 is the unique lattice such that τ 0 has socle F (c+p−2, a, b+1) (by Lemma
4.1.1 of [EGH13]).
Proof. We prove this in the case E = E((b + p − 1, a, c), (a, b, c)). The proofs in
the other cases are similar. The commutative diagram in the proposition mod p is
obtained from Proposition 2.1.2 of [Le14] and the commutative diagram
τ sN2
GL (F )
2
p
IndB2 (F
(F (b) ⊗ F (a)) ⊗ F (c)
p)
(τ s2 )N
∼
=
GL (F )
τ rN2
2
p
IndB2 (F
(F (b) ⊗ F (a)) ⊗ F (c)
p)
(τ r2 )N
0
EN
F (a, b) ⊗ F (c)
(E 0 )N ,
where B2 ⊂ GL2 is the subgroup of upper triangular matrices. The maps in the first
two horizontal rows are obtained by Proposition 2.1.4 of [Le14] and maps between
lattices in principal series given by Lemma 4.1.1 of [EGS13] (see Section 2). In
the first two cases, these maps are given by classical intertwiners (see Proposition
2.1.3).
By Proposition 3.1.2, U1s2 factors through τ s0 . By Proposition 2.1.2 of [Le14],
the reduction of U1s2 corresponds to a map τ sN2 → (τ s2 )N that factors through
GL (F )
2
p
IndB2 (F
(F (b) ⊗ F (a)) ⊗ F (c). Again using Proposition 2.1.4 of [Le14], we see
p)
that the space of such maps is one dimensional. We conclude that the reduction of
U1s2 must fit into the commutative diagram in the statement of the proposition. 4. Patching
Following [EGS13], we use Kisin’s local-global modification of the Taylor-Wiles
method to study lattices in completed cohomology in families on local deformation rings. In this section, we introduce the Taylor-Wiles patching method mostly
following [CHT08] and [CEG+ 14].
4.1. Compact open subgroups and Taylor-Wiles primes. Let F be a CM
field with F + its totally real subfield such that F/F + is unramified at all finite
places and p splits completely in F . Following [CHT08], let G/Z be the group
scheme defined to be the semidirect product of GL3 ×GL1 by the group {1, j}, which
acts on GL3 × GL1 by j(g, s) = (s · (g −1 )t , s). Let ρ : GF + → G(OE ) be a Galois
representation with reduction ρ : GF + → G(kE ) such that ρ−1 (GL3 (E)×GL1 (E)) =
GF . Suppose that ρ is potentially crystalline at places w|p of F and ramifies only at
places that are split in F/F + . Suppose further that ρ satisfies the following axiom.
Axiom 4.1.1. We say that ρ satisfies the Taylor-Wiles conditions (TW) if
• ρ|GF has image containing GL3 (F) for some F ⊂ kE with #F ≥ 7, and
• F
ker ad ρ|GF
does not contain F (ζp ).
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
11
Note that the first condition, which is stronger than the usual condition of adequacy
(see Definition 2.3 of [Tho12]), allows us to later choose a place v1 .
As in Section 2.3 of [CEG+ 14], we let U/OF + be a model for a definite unitary
group that is quasisplit at all finite places of F + and splits over F . For each place v
∼
of F + that splits as v = wwc in F , fix an isomorphism ιw : U(OFv+ ) → GL3 (OFw ).
Let λ ⊂ T be the Hecke eigensystem corresponding to ρ where T = TSp ,univ is
defined in Section 2.3 of [CEG+ 14].
0
+
Axiom 4.1.2. We say that ρ|GF is modular
)\U(A∞
F + )/U, OE )[λ] 6=
Q (M) if H (U(F
∞
0 for some compact open subgroup U = v Uv ⊂ U(AF + ) for which Uv is ramified
(not hyperspecial ) at only only finitely many v, all of which split over F .
Assume (M) and let Σ be the set of places v 6 |p in F + where U is ramified, or
in other words where Uv is not a hyperspecial compact open subgroup. We choose
a place (which exists by (TW)) v1 ∈
/ Σ of F + which is prime to p such that
c
• v1 splits in F as v1 = w1 w1
• v1 does not split completely in F (ζp ).
• ρ(Frobw1 ) has distinct F-rational eigenvalues, no two of which have ratio
(N v1 )±1 .
Q
For m ∈ N, let Um = v Um,v ⊂ U(A∞
F + ) be the compact open subgroup where
• Um,v = U(Ov ) for all places v which split in F other than v1 and those
dividing p;
• Um,v1 is the preimage of the upper triangular matrices under
∼
U(Ov1 ) → U(kv1 ) → GL3 (kw1 );
ιw1
• Um,v is the kernel of the map U(Ov ) → U(Ov /pm ) for v|p;
• Um,v is a hyperspecial maximal compact open subgroup of U(Fv ) if v is
inert in F .
−1
U(F + )t ∩ U does not
Note that v1 was chosen so that for all t ∈ U(A∞
F + ), t
contain an element of order p, necessary for Proposition 4.2.1.
We now choose Taylor-Wiles primes. As in Section 2.5 of [CEG+ 14], for each
e N of F + and F , respectively. Let q
N ≥ 1 we choose finite sets of places QN and Q
e N . For v ∈ QN , we let U1 (QN )v ⊂ U(O + ) be the
be the size of the sets QN and Q
Fv
corresponding parahoric
compact
open
subgroup
defined
in
Section
5.5
of [EG14].
Q
Let U1 (QN )m = v U1 (QN )m,v ⊂ U(A∞
F + ) be the compact open subgroup where
U1 (QN )m,v = Um,v for v 6∈ QN and U1 (QN )m,v = U1 (QN )v for v ∈ QN .
4.2. Patching algebraic modular forms. Let Z be the center of U. Let ξ be
the character of GF + such that ξχ3 = det ρ. Normalizing class field theory to
take uniformizers to arithmetic Frobenii, we consider ξ = ⊗v ξv as a character of
∼
∞ ×
∼
Z(A∞
F + ) = (AF + ) . For v ∈ Σ, let τv be the U(Ov ) → GL3 (Ow )-representation
ιw
over E which is the type corresponding to ρ|GF + ∼
=GFw by results towards inertial
v
local Langlands (see Proposition 6.5.3 of [BC09]). Let τv0 be a GL3 (Ow )-invariant
OE -lattice in τv . Let Sτ (U1 (QN )m ) be the functions
0
f : U(F + )\U(A∞
F + ) → ⊗v∈Σ τv
such that f (gzu) = u−1 ξ(z)f (g) for all g ∈ U(A∞
Z(A∞
F + ), z ∈ Q
F + ) and u ∈
0
U1 (QN )m where U1 (QN )m acts on ⊗v∈Σ τv by its projection to v∈Σ Um,v .
12
DANIEL LE
Let S be the set of places v1 , those in Σ, and those dividing p in F + . For each
place v in S let ve be a place in F lying over v and let Se be the set of these places
,ξw
ve. For places w in F , let Rw
be the universal OE -lifting ring of ρ|GFw with
fixed determinant ξw . For ve with v ∈ Σ, let Rve,τv ,ξv be the reduced p-torsion free
quotient of Rv
e corresponding to lifts of inertial type τv and determinant ξv . Let
e OE , ρ, 1−n δ n + , {R,ξv1 } ∪ {R,ξv }v|p ∪ {R,τv ,ξv }v∈Σ
S = F/F + , S, S,
F/F
v
e
v
e
v
e1
be the deformation problem in the terminology of [CHT08]. There is a universal
deformation ring RSuniv and a universal S-framed deformation ring RSS in the sense
of Definition 1.2.1 of [CHT08].
ψ
For each v ∈ QN , let Rve ve be the quotient of Rv
e defined in Section 5.5 of [EG14].
We let SQN be the deformation problem
e N , OE , ρ, 1−n δ n + ,
SQN = F/F + , S ∪ QN , Se ∪ Q
F/F
,ξ
ψ
v ,ξv
}v∈Σ ∪ {Rve ve }v∈Qn .
{Rve1 v1 } ∪ {Rve,ξv }v|p ∪ {Rv,τ
e
Again, we have the universal deformation ring RSuniv,ξ
and the universal S-framed
QN
∨
S ,ξ
deformation ring RSQ . Let M1,QN = pr Sτ (U1 (QN )2N , OE /$N )mQN
where
N
·∨ denotes the Pontrjagin dual, pr is defined in Section 2.5 of [CEG+ 14], and mQN
is the maximal ideal of the spherical Hecke algebra at places away from S ∪ QN cor
= M1,QN ⊗Runiv,ξ RSQS ,ξ .
responding to ρ (see Section 2.3 of [CEG+ 14]). Let M1,Q
N
SQ
N
N
We now patch these spaces of algebraic automorphic forms. Let
v1
v b
v ,ξv
b v|p Rv,ξ
b v∈Σ Rv,τ
b v,ξ
⊗R
Rloc = ⊗
⊗ ⊗
e
e
e
1
where all completed tensor products are taken over OE . Let R∞ = Rloc [[x1 , . . . , xq−3[F + :Q] ]]
and S∞ = OE [[z1 , . . . , z9#S−1 , y1 , . . . , yq ]]. Let Gv denote U(Fv ) and Kv ⊂ Gv the
maximal compact subgroup. As in Section
2.6 of [CEG+ 14], we patch the spaces
Q
M1,QN together to get an R∞ × S∞ [[ v|p Kv ]]-module M∞ .
Q
Proposition 4.2.1. The S∞ [[ v|p Kv ]]-module M∞ is projective. Moreover, M∞
Q
Q
admits a v|p Gv -action that extends the action of v|p Kv .
Proof. See Proposition 2.8 of [CEG+ 14].
Q
For a v Kv -representation ⊗v τv over OE , we denote by M∞ (⊗v τv ) the R∞ ∨ ∨
module HomK (⊗v τv , M∞
) , where ·∨ denotes the Pontrjagin dual. By Lemma 4.13
+
d d
of [CEG 14], if ⊗v τv has no OE -torsion, then M∞ (⊗v τv ) ∼
)
= HomK (⊗v τv , M∞
d
+
where · denotes the Schikhof dual (see Section 1.5 of [CEG 14]). Note that in
[CEG+ 14], M∞ (⊗v τv ) is only defined for OE -torsion-free objects and the definition
with Schikhof duality is used instead.
Q
Proposition 4.2.2. If ⊗v τv is a locally algebraic v Kv -representation over OE ,
M∞ (⊗v τv ) is maximal Cohen-Macaulay over its support as an R∞ -module.
Proof. The S∞ -module M∞ (⊗v τv ) is free by Proposition 4.2.1. Hence,
depthR∞ M∞ (⊗v τv ) ≥ depthS∞ M∞ (⊗v τv ) = dim S∞
≥ dim SuppR∞ M∞ (⊗v τv ),
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
13
where the first inequality comes from the fact that the S∞ -action factors through
R∞ and the last inequality follows from the local algebraicity of ⊗v τv and Theorem
3.3.4 of [Kis08].
Recall that π is defined in Section 1.1. We define π in the same way, but with
kE -coefficients instead of OE -coefficients.
Proposition 4.2.3. Let ℘ ⊂ m ⊂ R∞ be ideals corresponding to ρ and ρ, respecd
∨
tively. Then π = M∞
[℘] and π = M∞
[m].
Proof. This follows from the construction of M∞ (see Section 2.6 of [CEG+ 14]).
5. Cyclicity of patched modules
In this section we prove that the patched modules for principal series types are
cyclic over R∞ with certain Hecke operators adjoined (see Theorem 5.4.1) using
an induction argument from [EGS13] with base case provided by the method of
Diamond and Fujiwara ([Fuj06] and [Dia97]).
5.1. Isomorphisms of patched lattices. Fix a place v|p of F + . Recall the
definition of π from Section 1.1. We say that ρ is modular of (Serre) weight F
if F appears in the Kv ∼
= GL3 (Qp )-socle of π. We use Theorem 7.5.6 of [EGH13]
to prove some isomorphisms of patched modules. This theorem states that the set
of Serre weights for which ρ is modular is
W (ρ) ={F (a, b, c), F (c + p − 2, a, b + 1), F (b, c − 1, a − p + 2),
F (c + p − 2, b + 1, a − p + 1), F (b + p − 1, a + 1, c − 1), F (a, c, b − p + 1),
F (c + p − 2, b, a − p + 2), F (b + p − 1, a, c), F (a, c − 1, b − p + 2)}.
∼ GL3 (Zp )-representation V , we write M∞ (V ) to denote
From now on, for a Kv =
for M∞ (Vv ⊗v0 6=v Vv0 ) where Vv0 is a fixed Kv0 -representation. The representation
Vv0 is specialized to certain representations in Section 5.3.
Proposition 5.1.1. The map ιrsii : M∞ (τ si ) ,→ M∞ (τ ri ) is an isomorphism.
Proof. By Lemma 2.1.3 and Proposition 2.1.2, the cokernel of ιrsii : τ si ,→ τ ri
contains no Serre weights of ρ and so the cokernel of ιrsii : M∞ (τ si ) ,→ M∞ (τ ri ) is 0
by Proposition 4.2.3 and Nakayama’s lemma. By Nakayama’s lemma, the cokernel
of ιrsii : M∞ (τ si ) ,→ M∞ (τ ri ) is 0.
Denote by ιi : τi ,→ τ s0 the unique inclusions which are nonzero mod p for
i = 1, 2, 3.
Proposition 5.1.2. The induced map M∞ (τ3 ) ,→ M∞ (τ s0 ) is an isomorphism.
Proof. By the proof of Lemma 4.1.1 of [EGS13], the image of ι3 is the minimal
submodule containing F (c+p−2, a, b+1) as a Jordan-Holder factor. By Proposition
2.1.2, we see that the cokernel of τ 3 ,→ τ s0 contains no Jordan-Holder factors in
W (ρ). We conclude that the cokernel of M∞ (τ 3 ) → M∞ (τ s0 ) is 0. By Nakayama’s
lemma, the cokernel of M∞ (τ3 ) ,→ M∞ (τ s0 ) is 0.
14
DANIEL LE
5.2. The base cases: The method of Diamond and Fujiwara. Let
Y
s
End(M∞ (⊗v0 |p τv0v0 ))
R⊂
sv0 ∈S3
s
be the subring generated by the image of R∞ and the Hecke operators Ui,vv00 for
i = 1, 2 and sv0 ∈ S3 of Section 2.
,τw
b w∈Σ Rw
Axiom 5.2.1. We say ρ has regular type (RT) if ⊗
is a regular local ring.
e
Note that this is a hypothesis at places not dividing p. Assume that ρ satisfies
(RT). We make this assumption at places not dividing p to apply the method of
Diamond and Fujiwara. Assume the axioms (TW) and (M) of Section 4. The
following proposition establishes the base case of cyclicity of patched modules, and
follows directly from the method of Diamond and Fujiwara ([Fuj06] and [Dia97]).
Recall from Section 1.1 the definition of aw , bw , and cw . Let av = ave, bv = bve, and
cv = cve. Assume that av −cv < p−3 so that the Hodge-Tate weight (av +2, bv +1, cv )
is Fontaine-Laffaille.
Proposition 5.2.2. The R-module M∞ (⊗v|p F (av , bv , cv )) is cyclic.
Proof. See Theorem 6.2.2 of [Le14].
5.3. The induction step: weight cycling. Assume that ⊗v|p F (av , bv , cv ) is a
strongly generic modular lower alcove weight for ρ where strongly generic means
that av − bv > 1, bv − cv > 3, and av − cv < p − 7 for all v|p. These inequalities are
hypotheses of Theorem 5.2.5 of [EGH13], which is used in the proof of Lemma 5.3.1
to eliminate weights. Finally, assume that ρ|GFve is irreducible for each place v|p in
order to apply weight cycling as in [EGH13]. Fix a place v|p of F + , as we continue
to use the notation M∞ (V ) of Section 5.1.
Let E, E 0 , F , F 0 , N , P , N , P , and T be as in Section 3. Note that the operator
T is given by certain Hecke operators determined in Proposition 3.2.2. Note that
we use here Proposition 5.3.1 in the first two cases of Proposition 3.2.2 and an
analogous fact in the latter two cases to show that the left-hand columns of the
diagrams in Proposition 3.2.2 induce maps M∞ (E 0 ) ,→ M∞ (τ s ) for s = s2 (resp.
GL (Q )
0 N
N
⊂ IndK 3 p E 0 induces a
⊂ IndG
s1 , e, and e). The map E → IndG
P (E )
P σ
0
map M∞ (E) → M∞ (E ). The following lemma is a reformulation of weight cycling
in terms of patched modules for extensions.
Lemma 5.3.1. The map M∞ (E) → M∞ (E 0 ) is an isomorphism.
Proof. Corollary 2.2.2 of [Le14] with W taken to be E 0 and W 0 taken to be E,
Proposition 4.2.3, and duality implies that the map M∞ (E) → M∞ (E 0 ) is surjective by Nakayama’s lemma if the hypotheses of Corollary 2.2.2 of [Le14] are
satisfied. Since 0 6= σ N 6⊂ σ 0 ⊂ E 0 and σ 0 is the unique nontrivial submodule
N
of E 0 , σ N generates E 0 . To see that HomGv (IndG
P σv /Ev , π) = 0, see Theorem
5.2.5, Proposition 6.1.3, and the proof of Theorem 6.2.3 of [EGH13]. Note that
in the application of Theorem 5.2.5 of [EGH13], we use strong genericity and the
assumption that ρ|GFw is irreducible for each place w|p.
Let J be the kernel of the map. Suppose that M∞ (E) and M∞ (E 0 ) have support
of dimension d as R-modules (see Theorem 3.3.4 of [Kis08] for a formula for d).
Let Zd be the functor that assigns to an R-module the associated d-dimensional
cycle (see Definition 2.2.5 of [EG14]). We have Zd (M∞ (E)) = Zd (M∞ (E 0 )) by
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
15
exactness of M∞ (·) from Proposition 4.2.1, the fact that E and E 0 have the same
Jordan-Holder factors, and additivity of Zd in exact sequences. We conclude that
Zd (J) = 0. Since M∞ (Ev ⊗ ⊗w Vw ) is maximal Cohen-Macaulay over its support
in R∞ by Proposition 4.2.2, there are no embedded associated primes by Theorem
17.3(i) of [Mat89], and so J = 0.
The following proposition is the induction step to prove cyclicity of patched
modules. The idea to use Hecke operators to show cyclicity of patched modules
originates in Theorem 6.2.4 of [Le14]. Let Vv0 be one of σv0 , σv0 0 , Ev0 , or Ev00 .
Proposition 5.3.2. The R-modules M∞ (⊗v0 Vv0 ) where at most one Vv0 is reducible
are cyclic.
Proof. We proceed by induction on the number of Vv0 that are not lower alcove
weights. If none are, we are done by Proposition 5.2.2. Suppose that Vv is not
∼
a lower alcove weight. The isomorphism M∞ (E) → M∞ (E 0 ) from Lemma 5.3.1
∼
0
gives an isomorphism M∞ (E/σ) → M∞ (E )/T (see Section 3).
∼
Suppose that σ 0 is a lower alcove weight. Then since M∞ (E/σ) → M∞ (σ 0 ) is
cyclic by the induction hypothesis, we conclude that M∞ (E 0 ) is cyclic by Nakayama’s
lemma. By Lemma 5.3.1, M∞ (E) is cyclic as well. Since M∞ (E 0 ) M∞ (σ), we
conclude that M∞ (σ) is cyclic as well.
Now if σ 0 is an upper alcove weight, then σ 0 is either F (b+p−1, a, c) or F (a, c, b−
p + 1). The previous paragraph with E = E((c + p − 2, a, b + 1), (b + p − 1, a, c))
or E = E((c + p − 2, a, b + 1), (a, c, b − p + 1)) shows that M∞ (F (b + p − 1, a, c))
and M∞ (F (a, c, b − p + 1)) are cyclic. We now apply the same argument as in the
previous paragraph.
5.4. Cyclicity and an application. We use Proposition 5.3.2 and Lemma 10.1.13
from [EGS13] to prove that patched modules for principal series types are cyclic.
Theorem 5.4.1. The R-module M∞ (⊗v|p τvsv ) for sv ∈ S3 is free of rank 1.
Proof. To show that M∞ (⊗v τvsv ) is cyclic over R, it suffices by Nakayama’s lemma
to show that M∞ (⊗v τ svv ) is cyclic over R. By Theorem 7.5.6 of [EGH13] and
Proposition 2.1.2, the weights in W (ρ) that appear in the Jordan-Holder factors of
τ are F (av , bv , cv ), F (av , cv , bv −p+1), F (bv +p−1, av , cv ), and F (cv +p−2, av , bv +1).
By Proposition 2.1.2, there is a subquotient Dv of τ svv with precisely these four
weights. We conclude that M∞ (⊗v τ svv ) = M∞ (⊗v Dv ), and it suffices to show that
M∞ (⊗v Dv ) is cyclic.
By Proposition 2.1.2, the Hasse diagram describing the nonsplit extensions of
the Jordan-Holder factors of Dv is connected with irreducible socle with each edge
corresponding to one of the extensions in Section 1.4. Using Proposition 5.3.2
and Lemma 10.1.13 of [EGS13], we see that M∞ (⊗v Dv ) is cyclic. Freeness over
R follows by the definition of R and the fact that M∞ (⊗v τvsv ) are isomorphic for
different sv ∈ S3 (see Section 6.1).
We apply Theorem 5.4.1 to the following lemma which will be used later to
compute the reduction of the principal series type. Again fix a place v|p of F + , as
we use the notation of Section 5.1.
Lemma 5.4.2. Let τ 1 ⊂ τ be a lattice with irreducible cosocle σ ∈ W (ρ). Suppose
that M∞ (τ 1 ) is cyclic and τ 2 ( τ 1 . Then M∞ (τ 2 ) ⊂ mM∞ (τ 1 ).
16
DANIEL LE
Proof. The inclusion τ 2 ,→ τ 1 and surjection τ 1 σ whose composition is 0 induce
the diagram
M∞ (τ 2 )
M∞ (τ 1 )
M∞ (F )
M∞ (τ 1 )/m
M∞ (σ)/m
where the composition of the top row is 0. The map M∞ (τ 1 ) → M∞ (σ) is surjective by exactness of M∞ (·), and so the bottom row is surjective. By assumption,
M∞ (τ 1 ) is one dimensional, and so the bottom row is an isomorphism. We conclude
that the composition M∞ (τ 2 ) → M∞ (τ 1 ) → M∞ (τ 1 )/m is 0.
6. Lattices in patched modules
In this section, we use the patched modules of Section 4, the results of Section
5, and the Hecke theory of Section 3 to study lattices in principal series in families.
6.1. Lattices via Hecke theory. We keep the notation of Section 2 such as τvs
for s ∈ S3 and τi,v for i = 1, 2, 3. Fix a place v|p of F + . Fix lattices τv00 ⊂ τv0
for places v 0 |p of F + with v 0 6= v. From now on we denote M∞ (Vv ⊗ ⊗v0 6=v τv00 )
by M∞ (V ) where V is a K ∼
= GL3 (Ov )-representation. Using Theorem 5.4.1,
∼
we can and do choose an isomorphism ψ s0 : R → M∞ (τ s0 ). The compositions
∼
∼
ψ si = νss0i ◦ ψ s0 : R → M∞ (τ s0 ) → M∞ (τ si ) are also isomorphisms.
∼
For each i, Proposition 5.1.1 gives an isomorphism ψ ri = ιrsii ◦ ψ si : R →
∼
M∞ (τ si ) → M∞ (τ ri ). This also gives an isomorphism
∼
ψ e = νre1 ◦ ψ r1 : R → M∞ (τ e ).
When appropriate, for each s ∈ S3 , we consider the operators Uis as elements of R
∼
via the isomorphisms R → EndR (M∞ (τ s )).
Lemma 6.1.1.
(1) The map (ψ s0 )−1 ◦ ιss0i ◦ ψ si ∈ EndR (R) is given by multiplication by Ujsi .
(2) The map (ψ s0 )−1 ◦ ιsri0 ◦ ψ ri ∈ EndR (R) is given by multiplication by Ujsi .
(3) The map (ψ s0 )−1 ◦ ιse0 ◦ ψ e ∈ EndR (R) is given by multiplication by U2s1 U1e .
Proof. For (1), we note that
(ψ s0 )−1 ◦ ιss0i ◦ ψ si = (ψ s0 )−1 ◦ (νss0i )−1 ◦ νss0i ◦ ιss0i ◦ ψ si = (ψ si )−1 Ujsi ψ si = Ujsi
using Proposition 3.1.2. For (2),
(ψ s0 )−1 ιsri0 ψ ri = (ψ s0 )−1 ◦ ιsri0 ◦ ιrsii ◦ (ιrsii )−1 ◦ ψ ri = (ψ s0 )−1 ◦ ιss0i ◦ ψ si = Ujsi
where we use Proposition 2.1.1 and (1). For (3),
(ψ s0 )−1 ◦ ιse0 ◦ ψ e = (ψ s0 )−1 ◦ ιsr10 ◦ ψ r1 ◦ (ψ r1 )−1 ◦ ιre1 ◦ ψ e = U2s1 ◦ U1e
where we use (2) and the fact that (ψ r1 )−1 ◦ ιre1 ◦ ψ e = U1e by an argument similar
to the one for (1).
LATTICES IN TYPES FOR U(3) SHIMURA VARIETIES
17
6.2. Local-global compatibility and the main theorem. This subsection deduces the main theorem, Theorem 6.2.3 (cf. Theorem 1.1.1) from the local-global
compatibility result Proposition 6.2.1 and Theorem 6.1.1.
Proposition 6.2.1. Suppose that eigenvalues of ϕ acting on the crystalline Dieudonn´e
module Dcris (ρ|GFv ) are α1 , α2 , and α3 on the η av , η bv , and η cv eigenspaces, respectively. Then U1s and U2s act on HomGL3 (Ov ) (τvs , π) by αs(1) and αs(1) αs(2) /p,
respectively.
Proof. See Theorem 1.2 of [BLGGT12].
Proposition 6.2.2. Given a Zp -lattice L ⊂ τ ⊗Zp Qp , L is the sum of the saturations of each lattice with irreducible cosocle as described in Lemma 4.1.1 of [EGS13].
Proof. The image of this sum in L modulo the maximal ideal of Zp is L since every
Jordan-Holder factor is in the image. By Nakayama’s lemma, the image of the sum
is L.
Theorem 6.2.3. Let ρ : GF → GL3 (Zp ) be an essentially conjugate self-dual
Galois representation such that
• ρ satisfies (RT);
• for all places w|p of F , ρ|Fw is a lattice in a potentially crystalline representation with Hodge-Tate weights (0, 1, 2) with tame type η aw ⊗ η bw ⊗ η cw
where F (aw , bw , cw ) is strongly generic i.e. aw − bw > 1, bw − cw > 3, and
aw − cw < p − 7;
• for all places w|p of F , ρ|Fw is irreducible;
• its reduction ρ : GF + → G3 (kE ) satisfies (TW);
• and ρ satisfies (M).
Then the lattice τv ⊗Zp Qp ∩π ⊂ τv ⊗Zp Qp is isomorphic to τ s0 +(α1 α2 /p)−1 ιsr10 (τ r1 )+
α1−1 ιsr20 (τ r2 )+(α12 α2 /p)−1 ιse0 (τ e )+α0 ι1 (τ1 )+α0 ι2 (τ2 ) where α0 is the one of (α1 α2 /p)−1
and α1−1 which has larger valuation.
Proof. The lattice τv ⊗Zp Qp ∩π takes this form by Proposition 6.2.2. Let θ : R → Zp
be the map corresponding to ρ. The map from R∞ factors through the natural
∼
map R∞ ⊗S∞ OE → RSuniv,ξ → OE . The images of the Hecke operators are given
by Proposition 6.2.1. The result follows from Lemma 6.1.1 and Proposition 5.1.2
except for the coefficients of ι1 (τ1 ) and ι2 (τ2 ). Lemma 6.1.1 and Proposition 5.1.2
give coefficients in terms of θ(Uis ) for i = 1, 2 and s ∈ S3 (see the proof of Theorem
7.2.4 of [Le14]). Note that ι3 (τ3 ) ⊂ τ s0 and ιss0i (τ si ) ⊂ ιsri0 (τ ri ) induce isomorphisms
after applying M∞ (·) by Propositions 5.1.1 and 5.1.2 and so these terms can be
omitted from the sum.
We cannot study the shadow lattices τ1 and τ2 as we did the other lattices
because M∞ (τ1 ) and M∞ (τ2 ) are not cyclic. However, we can use Lemma 5.4.2
to study the images of each summand in π. We will calculate the Jordan-Holder
factors of each summand aside from the saturations of τ1 and τ2 and find that the
factors F (b − 1, c, a − p + 2) and F (a − 1, b, c + 1) do not appear. We conclude
that the saturations of τ1 and τ2 are nonzero mod p, allowing us to calculate the
coefficients of ι1 (τ1 ) and ι2 (τ2 ).
The image of ι3 (τ3 ) has image F (c + p − 2, a, b + 1) by Propositions 5.4.2 and
5.1.2, and so the image of τ s0 has Jordan-Holder factors F (c + p − 1, b, a − p + 1)
18
DANIEL LE
and F (c + p − 2, a, b + 1). Similarly, the image of (α1 α2 /p)−1 ιss01 (τ s1 ) has JordanHolder factors F (b + p − 1, a, c) and F (b, c, a − p + 1); the image of α1−1 ιss02 (τ s2 )
has Jordan-Holder factors F (a, c, b − p + 1) and F (c + p − 1, a, b); and the image
of (α12 α2 /p)−1 ιse0 (τ e ) is F (a, b, c). Since the image of the lattice in π must contain
every Jordan-Holder factor of τ , we conclude that the images of αι1 (τ1 ) and αι2 (τ2 )
in π must be nonzero.
The image of the lattice must have all Serre weights in the socle of π by Lemma
5.4.2 and nowhere else since τ is residually multiplicity free. Hence F (a, b, c) and
F (c + p − 2, a, b + 1) cannot be in the image by Proposition 2.2.1. Therefore the
images of αι1 (τ1 ) and αι2 (τ2 ) in π must contain F (b + p − 1, a, c) or F (a, c, b − p + 1)
(or both). By Propositions 2.1.2 and 2.1.3, ιss0i factors through both τ1 and τ2 . We
conclude that α can be (α1 α2 /p)−1 or α1−1 . It must be the one with larger valuation
because the images of (α1 α2 /p)−1 ιss01 (τ s1 ) and α1−1 ιss02 (τ s2 ) in π are nonzero by
Lemma 5.4.2.
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