IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL
FIELDS AND MULTIPLICITY TWO
CHUANGXUN CHENG
Abstract. In this paper, under some technical conditions on the coefficients, we prove
Ihara’s lemma for Shimura curves over totally real fields. We then construct examples
where the multiplicities of Galois representations in the cohomology groups of Shimura
curves are two. These examples provide some evidence for the BDJ conjecture.
1. Introduction
Ihara’s lemma is a result about the kernel of a certain map between cohomology groups
(or Jacobians) of modular curves with different level structures. In the Taylor-Wiles system
method, Ihara’s lemma is a key ingredient to extend an R = T theorem in the minimal
case to the nonminimal case.
In [9], the authors proved an analogue of Ihara’s lemma for Shimura curves over Q. The
argument cannot apply to general totally real fields case directly because in the later case
the Shimura curves are in general not moduli spaces of abelian varieties and we do not
have a universal object to construct the crystals as in [9]. Nevertheless, in [23], the author
showed how we could construct crystals over Shimura curves from the crystals over unitary
Shimura curves. In this paper, we formulate an analogue of Ihara’s lemma for Shimura
curves over totally real fields and prove it in certain cases by combining the methods in
[9] and [23]. The result is Theorem 1.2 below.
In [5], by the Diamond-Taylor-Wiles method, the author proved an R = T theorem
and a multiplicity one result for Shimura curves over totally real fields (see Theorem 1.3
below.) The results in [5] deal primary with the minimal case. For those coefficients to
which we may apply Theorem 1.2, we can extend the results in [5] from minimal case to
nonminimal case. In particular, we have Theorem 1.5 below. Starting with this result, we
then construct examples where the multiplicity one fails. More precisely, by adapting the
method in [22], we construct examples where the multiplicities of Galois representations
in the cohomology groups of Shimura curves are two (see Theorem 1.6 below.)
To state our main results, we first fix some notation. Let F be a totally real field of
degree n = [F : Q] > 1, HomF (F, R) = {τ1 , · · · , τn }. Let SD be a finite set of finite
places of F such that |SD | ≡ n − 1 (mod 2). Let D be a quaternion algebra over F
which is ramified at {τ2 , · · · , τn } ∪ SD . (There is a unique one up to isomorphism.) Let
G = ResF/Q D× be the algebraic group over Q attached to D× . Let L be a finite extension
of F which splits D. We also assume that L is sufficiently large. In particular, we assume
that L contains the image of each embedding F → Q̄.
Let p > 3 be a rational prime number unramified in F . Assume
that p factors as
`
p = p1 p2 · · · pr . We also fix a bijection between Hom(F, R) and p|p Hom(Fp , Q̄p ), denote
1
2
CHUANGXUN CHENG
this set by S. We fix a prime λ | p of L. Let O be the ring of integers of Lλ and κ the
residue field.
Fix a maximal order OD of D and isomorphisms OD ⊗OF OFv = M2 (OFv ) for finite
×
primes v - SD . Let K0 ⊂ (D ⊗F A∞
F ) be the open compact subset such that the v×
component (K0 )v = (OD ⊗OF OFv ) . In particular, (K0 )v = GL2 (OFv ) for finite primes
v - SD . Define K0 (N ), for an ideal Nprime
to SD , to be the subgroup of K0 consisting
∗ ∗
of those u for which uv congruent to
mod(v ordv (N ) ) for every v | N .
0 ∗
1.1. The curves. Let X be the G(R)-conjugacy class of the map
h : C× → G(R) ' GL2 (R) × H× × · · · × H× ,
a −b
which maps a+ib to (
, 1, ..., 1). The conjugacy class X is naturally identified with
b a
the union of the upper and lower half plane by the map ghg −1 7→ g(i), where g(i) = a+ib
c+id
a b
if g = (
, · · · ).
c d
Let M = M (G, X) = (MH )H be the canonical model defined over F of the Shimura
variety defined by G and X. (Here H runs through the open compact subgroups of
G(A∞ ).) Each MH is proper and smooth but not necessarily geometrically connected
over F , and
MH (C) ' G(Q)\X × G(A∞ )/H.
For each H and H 0 and g ∈ G(A∞ ) with g −1 H 0 g ⊂ H and H sufficiently small (see [3,
Lemma 1.4.1.1],) there is an etale map %g : MH 0 → MH which on complex points coincides
with the one induced by right multiplication by g in G(A). For a normal subgroup H 0 of H,
the etale cover %1 : MH 0 → MH is Galois, and mapping g −1 7→ %g defines an isomorphism
of H/H 0 with a group of covering maps.
The weight homomorphism w : Gm,R → GR attached to h is given by
w(r) = (r−1 , 1, · · · , 1) r ∈ R.
This is not defined over Q if n > 1. Then we cannot expect a description of MH (C) as a
moduli space of abelian varieties. Indeed, if MH (C) is a moduli space, we could describe
h as given by the Hodge structure of an abelian variety, which is rational.
0
Suppose that H and H are sufficiently small open compact subgroups of G(A∞ ), and
∞
g ∈ G(A ). Let F be a sheaf over the Shimura curves attached to a certain module (see [7, Section 6.1].) There is a natural identification of sheaves on MH∩gH 0 g−1 :
0 −1
F gH g |M
= F H∩gH
0
H∩gH g −1
curve MR . Then define
0
0 −1
g
. Here F R means that we consider F as a sheaf over the
0
0
[HgH ] : H j (MH 0 ⊗ F̄ , F H ) → H j (MH 0 ∩g−1 Hg ⊗ F̄ , F H |M
(1.1)
→ H j (MgH 0 g−1 ∩H ⊗ F̄ , F
= H j (MgH 0 g−1 ∩H ⊗ F̄ , F
→ H j (MH ⊗ F̄ , F H ),
0
H ∩g −1 Hg
0 −1
gH g
|M
0
H∩gH g −1
)
0
gH g −1 ∩H
)
)
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
3
where the first arrow is the restriction map, the second arrow is induced from %g :
MH 0 ∩g−1 Hg → MgH 0 g−1 ∩H , and the last arrow is the trace map. See [15, Section 15]
0
for more details. In this paper, we are
Q interested in the case where Hv = Hv = GL2 (OFv )
for v | p and F is associated with a v|p GL2 (OFv )-module.
0
Let H = H . If q is a prime of OF which is unramified in D and does not divide p, let
ωq ∈ A ∞
F be such that ωq is a uniformizer at q and is 1 at every other place. Then write
ωq 0
Tq = [H
H].
0 1
If also Hq = GL2 (Oq ), define
ωq 0
Sq = [H
H].
0 ωq
If H = K0 (N ), denote by TO (H, F) the O-algebra generated by Tq for q - N SD and Sq
for q with Hq = GL2 (Oq ). Write TA (H, F) = TO (H, F) ⊗ A for any O-algebra A. Write
Uωq = Tq if q | N .
A maximal ideal of TA (H, F) is Eisenstein if it contains Tv − 2 and Sv − 1 for all but
finitely many primes v of F which split completely in some finite abelian extension of F .
1.2. Ihara’s lemma. Let U ⊂ K0 be a sufficiently small open compact subgroup such
that Uv = (OD ⊗ OFv )× for almost all v and for all v | pSD . Let q be a finite prime
of F such that q - pSD and Uq = GL2 (Oq ). Let U0 (q) be the subgroup of U defined by
U0 (q)v = Uv if v 6= q and
a b
a b
∗ ∗
U0 (q)q = {
∈ Uq |
≡
(mod q)}.
c d
c d
0 ∗
Q
Let V be an irreducible mod p representation of GL2 (OF /p) = v|p GL2 (kv ) given by
V = ⊗v|p ⊗τ :kv →F̄p Symmbτ −2 with 2 ≤ bτ ≤ p + 1. Let H = U or U0 (q). We attach to V
a locally constant sheaf
FV = G(Q)\(X × G(A∞ )) × V /H
(1.2)
π
MH
over MH . Let ηq ∈
×
(D ⊗F A∞
F )
be the element
1 0
. Then we have ηq−1 U0 (q)ηq ⊂ U
0 ωq
and two maps %1 , %ηq : MU0 (q) → MU .
Conjecture 1.1. Assume that the genus of MU is greater than 1, then the following map
is injective:
1
1
1
Het
(MU ⊗ F̄ , FV )m ⊕ Het
(MU ⊗ F̄ , FV )m → Het
(MU0 (q) ⊗ F̄ , FV )m
1 0
(f1 , f2 ) 7→ 1∗ f1 +
f .
0 ωq ∗ 2
Here m is a non-Eisenstein maximal ideal of the Hecke algebra.
One of the main results we prove in this paper is the following theorem.
4
CHUANGXUN CHENG
Theorem 1.2. If
P
τ ∈S bτ
− 1 ≤ p − 2, then Conjecture 1.1 holds.
1.3. Multiplicity two. We first recall the statement of the multiplicity one result in [5].
Let N be a squarefree ideal of F such that (N, pSD ) = 1.
Let MK0 (N ) be the Shimura curve over F attached to (G, X) with level K0 (N ). Let
σ : GL2 (OF /p) → Aut(W/F̄p ) be a regular Serre weight, FV be the sheaf attached to
V = σ op = ⊗v|p ⊗τ :kv →F̄p detaτ Symmbτ −2 (regularity means that 2 ≤ bτ − 1 ≤ p − 2), m be
a maximal non-Eisenstein ideal of the corresponding Hecke algebra. The following result
is [5, Theorem 5.3].
Theorem 1.3. Let ρ̄ : GF → GL2 (F̄p ) be a modular Galois representation of weight σ
with conductor dividing pN SD . Assume that ρ̄ satisfies the following conditions
p
(1) the restriction ρ̄|GF 0 is irreducible, where F 0 = F ( (−1)(p−1)/2 p);
(2) if v | p, then EndF̄p [GFv ] (ρ̄|GFv ) = F̄p ;
(3) if v | N , then ρ̄ is ramified at v;
(4) if v | SD , and Norm(v)2 ≡ 1 (mod p), then ρ̄ is ramified at v.
1 (M
Then dimF̄p HomGF (ρ̄, Het
K0 (N ) ⊗ F̄ , FV )m ) ≤ 1.
1 (M
1
Remark 1.4. (a) We have the obvious embedding Het
K0 (N ) ⊗F̄ , FV )[m] ,→ Het (MK0 (N ) ⊗
F̄ , FV )m . It induces an isomorphism (see [2, Lemma 4.10])
∼
1
1
HomGF (ρ̄, Het
(MK0 (N ) ⊗ F̄ , FV )[m]) −
→ HomGF (ρ̄, Het
(MK0 (N ) ⊗ F̄ , FV )m ).
1 (M
∼
On the other hand, by Eichler-Shimura relation, we know that Het
K0 (N ) ⊗ F̄ , FV )[m] =
a
r̄ for some Galois representation r̄ : GF → GL2 (F̄p ) and some integer a (see for example
[1].) Then the above theorem says that if a > 0 and ρ̄ ∼
= r̄, then a = 1.
(b) The theorem is proved by using Diamond’s refined Taylor-Wiles system [8]. The
condition (1) is necessary for this argument and the corresponding Galois deformation
problem. The condition (3) is here because we do not have a general Ihara’s lemma
for Shimura curves over totally real fields. The condition (4) is essential for having a
multiplicity one result, as we shall see in Theorem 1.6.
Theorem 1.5. Let ρ̄ : GF → GL2 (F̄p ) P
be a modular Galois representation of weight σ
with conductor dividing pN SD such that τ ∈S bτ − 1 ≤ p − 2. Assume that ρ̄ satisfies the
following conditions
p
(1) the restriction ρ̄|GF 0 is irreducible, where F 0 = F ( (−1)(p−1)/2 p);
(2) if v | p, then EndF̄p [GFv ] (ρ̄|v ) = F̄p ;
(3) if v | SD , and Norm(v)2 ≡ 1 (mod p), then ρ̄ is ramified at v.
1 (M
Then dimF̄p HomGF (ρ̄, Het
K0 (N ) ⊗ F̄ , FV )m ) ≤ 1.
Proof. By [2, Lemma 2.3], we may assume that all the aτ are 0. Let T(K0 (N ), F) be the
1 (M
Hecke algebra attached to Het
K0 (N ) ⊗ F̄ , FV ). Then
1
dimF̄p HomGF (ρ̄, Het
(MK0 (N ) ⊗ F̄ , FV )m ) = 1
1 (M
if and only if Het
K0 (N ) ⊗ F̄ , FV )m is free of rank two over T(K0 (N ), F)m . Combining
Theorem 1.2 and Theorem 1.3, by a standard argument in Taylor-Wiles method (see for
example [24, Section 3] and [14, Section 5.3],) we can remove condition (3) in Theorem
1.3.
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
5
Let p and q be two primes of F such that (pq, SD ) = 1 and pq | N . Let MK0 (N ) be
the Shimura curve attached to D with level K0 (N ) as in Theorem 1.5. Let D0 be anther
quaternion algebra over F such that D0 is ramified at p, q and at the primes where D is
ramified. Fix a maximal order OD0 of D0 and isomorphisms OD0 ⊗OF OFv = M2 (OFv ) for
finite primes v - pqSD . Let MO0 be the Shimura curve attached to D0 with level O, where
Ov = K0 (N )v if v - pq, Ov = (OD0 ⊗OF OFv )× if v | pq.
Let ρ̄ : GF → GL2 (F̄p ) be a modular Galois representation which is ramified at q. Then
we have the following theorem.
1 (M
Theorem 1.6. With the notation as above, if dimF̄p HomGF (ρ̄, Het
K0 (N ) ⊗ F̄ , FV )m ) =
1
0
1, then dimF̄p HomGF (ρ̄, Het (MO ⊗ F̄ , FV )m ) ≤ 2. The dimension is two if and only if ρ̄
is unramified at p and ρ̄(Frobp ) is ±1.
Remark 1.7. We prove this theorem by adapting the idea of Ribet [22], where the author
proved a multiplicity two result for the Jocobians of Shimura curves over Q. The idea is
used in [20] to prove a level-lowering result for Galois representations attached to Hilbert
modular forms. In fact, we use some arguments in [20]. These ideas also appeared in for
example [3] and [15].
1.4. Notation. If L is a perfect field we will let L̄ denote the algebraic closure of L and
GL its absolute Galois group Gal(L̄/L). If L is a number field, we let AL denote the ring
of adeles over L, and A∞
L denote the ring of finite adeles over L. If L = Q, we write A
and A∞ for AQ and A∞
Q , respectively.
Let F , p be as above. For any prime v of F , let Fv be the completion of F at v,
OFv the ring of integers of Fv , kv the residue field of OFv , $v a uniformizer of OFv , and
Frobv ∈ Gal(F̄v /Fv ) an arithmetic Frobenius element. Write IFv ⊂ GFv for the inertia
group at prime v.
Q
Let Σ be a set of primes of F . If a group U has the form U = v∈Σ Uv , and J is an
ideal which is aQproduct of some elements in Σ, we will write U J for theQsubgroup of U
given by U J = v∈Σ,v-J Uv and UJ for the subgroup of U given by UJ = v∈Σ,v|J Uv .
This paper is organized as follows. In section 2, we recall some basic properties of
Shimura curves and prove some lemmas we need. In section 3, we prove Theorem 1.2 following the strategy of [9]. Then in section 4, we study the Galois structure on cohomology
groups of Shimura curves and prove Theorem 1.6.
2. Construction of certain sheaves on Shimura curves
In this section, we recall some facts about Shimura curves over totally real fields, the
construction of unitary Shimura curves, and the construction of certain sheaves on them.
The details can be found in [4] and [23]. We also review facts about reductions of Shimura
curves, which are proved in [4], [15], and [26].
√
2.1. Unitary Shimura curves. Let E0 = Q( −a) be an imaginary quadratic extension
of Q splits p. Fix embedding E0 ,→ C and E0 ,→ Qp . Let E = E0 F = F ⊗Q E0 , B =
D ⊗F E = D ⊗Q E0 . Let G00 be the algebraic group over Q of D× ×F × E × ∼
= D× ·E × ⊂ B × .
00
×
Let ν := NrdD/F × NormE/F : G → F be the product of the reduced norm and norm.
Let G0 be the algebraic group over Q of the inverse image of Q× ⊂ F × of the surjective
6
CHUANGXUN CHENG
map ν. Fix a prime P | p of F . Then P splits in E and we fix a prime P0 of E which
divides P. Therefore FP = EP0 , OFP = OEP0 .
We consider the G0 (R)-conjugacy class X 0 (G00 (R)-conjugacy class X 00 ) of
h0 :
C× → G0 (R) ⊂ G00 (R) = GL2 (R) · C× × H× · C× × · · · × H× · C×
−1
x y
−1
−1
z = x + iy 7→
,1 ⊗ z ,··· ,1 ⊗ z
−y x
The conjugacy classes X 0 , X 00 have natural structures of complex manifolds and are isomorphic to the upper half plane H+ and to the union of upper and lower half planes
H, respectively. Let M 0 = M (G0 , X 0 ), M 00 = M (G00 , X 00 ) be the canonical models of the
Shimura varieties defined over the reflex field E. The inclusions G → G00 and G0 → G00
induce morphisms of Shimura varieties
M → M 00 ← M 0 .
2.2. Moduli interpretation of M 0 . We recall the moduli interpretation of M 0 following
[4] and [23]. Let the involution ∗ on B = D ⊗F E be the tensor product of the main
involution of D and the conjugation of E and let ψ be the non-degenerate alternating
form on B defined by
√
ψ(x, y) = TrE/Q ( −a TrdB/E xy ∗ ).
Let OB be a maximal order in B stable under the involution ∗. An abelian scheme
A over a scheme S is called an OB -abelian scheme over S when a ring homomorphism
m : OB → EndS (A) is given. Let Lie(A) denote the locally free OS -module Lie(A/S) =
HomOS (0∗ Ω1A/S , OS ), where 0 : S → A denotes the 0-section. When S is a scheme over
Spec E, for an OB -abelian scheme A over S, we define direct summands Lie2 A ⊃ Lie1,2 A
of the OB ⊗Z OS = B ⊗Q OS -module Lie(A) as follows. The submodule Lie2 A is defined
to be the submodule on which the action of E0 ⊂ B and that of E0 ⊂ OS are conjugate
to each other over Q. Similarly, Lie1,2 A is the submodule where the action of E ⊂ B and
that of E ⊂ OS are conjugate to each other over F . They are the same as the tensor
products Lie2 A = Lie(A) ⊗E0 ⊗E0 E0 , Lie1,2 A = Lie(A) ⊗E⊗E E and hence are direct
summands. If A is an OB -abelian scheme, the dual A∗ is considered as an OB -abelian
scheme by the composite map
∗
m
∗
opp
m∗ : OB →
− OB
−→ End(A)opp →
− End(A∗ )
where opp denotes the opposite ring. A polarization θ ∈ Hom(A, A∗ )sym of an OB -abelian
scheme A is called an OB -polarization if it is OB -linear.
Let K 0 ⊂ G0 (A∞ ) be a sufficiently small open compact subgroup. Assume that K 0 ⊂
(OB ⊗ Ẑ)× . Let T̂ ⊂ B ⊗ A∞ be an OB ⊗ Ẑ-lattice satisfying ψ(T̂ , T̂ ) ⊂ Ẑ. We define
0
a functor MK
on the category of schemes over E as follows. For a scheme S over E,
0 ,T̂
0
let MK 0 ,T̂ (S) be the set of isomorphism classes of the triples (A, θ, ζ) consisting of the
following data
• An OB -abelian scheme A over S of dimension 4n such that Lie2 (A) = Lie1,2 A and
that it is a locally free OS -module of rank two.
• An OB -polarization θ ∈ Hom(A, A∗ )sym of A.
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
7
• A K 0 -equivalent class ζ of an OB ⊗ Ẑ-linear isomorphism ι : T̂ (A) → T̂ such that
there exists a Ẑ-linear isomorphism ι0 making the following commutative diagram
T̂ (A) × T̂ (A)
T̂ × T̂
By [4], the functor
/ T̂ (A) × T̂ (A∗ )
Tr ψ
/ Ẑ x
ι×ι
0
MK
0 ,T̂
(1,θ∗ )
/ Ẑ(1)
ι0
0 . This functor is independent
is represented by the scheme MK
0
of the choice of T̂ up to unique canonical isomorphism. Moreover, there is an integral
0 for the moduli
version of this moduli description (see [4, Section 2.6].) We still write MK
0
0 . Let π 0 :
space over OE . Let AK 0 over OE be the universal abelian scheme over MK
0
K
0
1
1
AK 0 → MK 0 be the structure map. Then (πK 0 )∗ ΩA 0 /M 0 and R (πK 0 )∗ OAK 0 are OB ⊗
K
K0
OM 0 0 -modules locally free of rank one. Note that we have a decomposition
K
OB ⊗ O L ∼
= O 1 ⊕ O 1 ⊕ · · · ⊕ OB 1 ⊕ O 2 ⊕ O 2 ⊕ · · · ⊕ OB 2
B2
B1
⊂B ⊗ L ∼
=
j
where
Bi
B11
⊕
B21
⊕ ··· ⊕
⊕
B12
B2
B1
n
Bn1
⊕
B22
⊕ ··· ⊕
n
Bn2
is isomorphic to M2 (L). Let ei denote the idempotent whose Bi2 component is
1 0
and the other components are zero. Then ei (πK 0 )∗ Ω1A 0 /M 0 and ei R1 (πK 0 )∗ OAK 0
K
0 0
K0
are locally free OS -modules of rank one.
Let ωi0 = ei (πK 0 )∗ Ω1A 0 /M 0 ⊗ O and Hi0 = ei R1 (πK 0 )∗ OAK 0 ⊗ O.
K
K0
Lemma 2.1. Assume that K 0 has no level structure at the prime P0 . Then
(1) Hi0 ∼
= (ωi0 )−1 .
(2) The exact sequence
0 → (πK 0 )∗ Ω1M 0
K0
induces an isomorphism
/OE
P0
→ Ω1A
K 0 /OEP0
∼
ωi0 −
→ Ω1M 0
K0
/OE
P0
→ Ω1A
0
K 0 /MK 0
→0
⊗ Hi0 .
Proof. (1) It suffices to prove that (πK 0 )∗ Ω1A 0 /M 0 and R1 (πK 0 )∗ OAK 0 are dual to each
K
K0
other. Let A∗K 0 be the dual abelian scheme of AK 0 , the principal polarization gives
an isomorphism Lie(AK 0 ) = Lie(A∗K 0 ). Since we have an isomorphism of OB -sheaves
R1 (πK 0 )∗ OAK 0 ∼
= Lie(A∗K 0 ), the lemma follows.
0
0 and A 0 are smooth over O
(2) Since K has no level at P0 , both MK
0
EP0 . Therefore
K
the sequence is exact. Taking the long exact sequence of this short exact sequence, we
obtain the following map
(πK 0 )∗ Ω1A
0
K 0 /MK 0
→ R1 (πK 0 )∗ (πK 0 )∗ Ω1M 0
K0
∼
= Ω1M 0
K0
/OE
P0
1
/OE
P0
⊗ R (πK 0 )∗ OAK 0 .
By the same argument as in [9, Lemma 7], this map is an isomorphism. Thus the result
follows.
Remark 2.2. This is also proved in [18, Proposition 3.1].
8
CHUANGXUN CHENG
2.3. Sheaves on MK . By the construction in [23, Section 6.2], the universalP
OB -abelian
0
0
00
00
scheme A over M extends to an OB -abelian scheme A over M . Let m = i∈S bi − 1.
Let A00m be the m-fold self fibre product of A00 → M 00 . Now we construct our sheaves as
at the beginning of [23, Section 6.3]. Let A (resp. Y ) be the base change of A00 (resp.
A00m ) by M → M 00 . Let πK be the structure map πK : AK → MK . Then we may define
an algebraic correspondence e on YNsuch that eRq (πK )∗ κ = FV if q = m. In fact, e is
defined by ei where we have FV ∼
= i∈S Symmbi −2 ⊗ detai (ei R1 (πK )∗ κ).
Similarly, ei (πK )∗ Ω1AK /MK and ei R1 (πK )∗ OAK are locally free OS -modules of rank one.
Let ωi = ei (πK )∗ Ω1AK /MK ⊗ O and Hi = ei R1 (πK )∗ OAK ⊗ O.
Lemma 2.3. Assume that K has no P-level structure, then
(1) Hi ∼
= ωi−1 .
(2) The exact sequence
0 → (πK )∗ Ω1MK /OF → Ω1AK /OF → Ω1AK /MK → 0
P
P
induces an isomorphism
∼
ωi −
→ Ω1MK /OF ⊗ Hi .
P
Proof. Those properties in Lemma 2.1 extend to the objects over M 00 . The lemma then
follows by pulling back.
By Riemann-Roch, we have the following corollary.
Corollary 2.4. Assume that K has no P-level structure, then the degree of ωi is (gK − 1).
Here gK is the genus of MK .
2.4. Supersingular and ordinary points on the special fibre of MK . Before we
prove the main theorem, we review some properties of reductions of Shimura curves.
Let ℘ be a finite prime of F , K be an open compact subgroup of G(A∞ ). We suppose
that K factors as K℘ H. Then we have the following results. The first one is proved in [4].
The second one is indicated in [4] and the detail proof is given in [15].
Theorem 2.5. (1) Suppose K℘ is the subgroup K℘0 = GL2 (OF℘ ). Then, if H is sufficiently
small, there exists a model M0,H of MK defined over OF,(℘) . This model is proper and
smooth.
(2) Suppose K℘ is the subgroup K℘n of matrices congruent to I modular ℘n . Then, if H
is sufficiently small, there exists a regular model Mn,H of MK with a map to M0,H . The
morphism Mn,H → M0,H is finite and flat.
Theorem 2.6. (1) Suppose K℘ is the group
a b
Γ0 (℘) = {
∈ GL2 (OF℘ ) | c ∈ ℘}.
c d
Then, if H is sufficiently small, there exists a regular model MΓ0 (℘),H of MK defined over
M0,H . The morphism MΓ0 (℘),H → M0,H is finite and flat.
(2) The special fibre MΓ0 (℘),H × k̄℘ is isomorphic to a union of two copies of M0,H × k̄℘
intersecting transversely above a finite set of points ΣH .
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
9
The points in ΣH are the supersingular points of M0,H × k̄℘ . We can describe this
set in another way. Let D̄ be another quaternion algebra over F ramified at primes in
SD ∪ {℘} ∪ {τ | τ |∞}. So it is totally definite. Let Ḡ = ResF/Q D̄× be the corresponding
algebraic group over Q, then there is a bijection
×
ΣH ∼
= Ḡ(Q)\Ḡ(A∞ )/H × OD̄
℘
(2.1)
∞ ℘
×
∼
= Ḡ(Q)\Ḡ(A ) × F℘ /H × OF×℘
where the second isomorphism is induced by the reduced norm D̄℘× → F℘× . Since D̄ is
totally definite, ΣH is a finite set.
According to [4] and [15], the action of GL2 (F℘ ) on the inverse system of Shimura curves
descends to an action on the set ΣH of supersingular points, and the action factors through
det : GL2 (F℘ ) → F℘× . Further, if we normalize the reciprocity map of class field theory
so that the arithmetic Frobenius elements correspond to uniformizers, then an element
σ ∈ W (F℘ab /F℘ ) acts on the set ΣH in exactly the same way as the element [σ] ∈ F℘×
corresponding to σ by class field theory.
Let K = GL2 (O℘ )H and ](ΣH ) be the number of supersingular points on M0,H × k̄℘ .
Let gR be the genus of any geometric fibre of MR . Then we have the following lemma.
Lemma 2.7. ](ΣH ) = (Norm(℘) − 1)(gGL2 (O℘ )H − 1).
Proof. The argument is similar to the proof of [9, Lemma 6]. We compute gΓ0 (℘)H in two
ways. On one hand, in characteristic 0, we have a map MΓ0 (℘)H → MGL2 (O℘ )H which is
of degree Norm(℘) + 1. Therefore, gΓ0 (℘)H − 1 = (Norm(℘) + 1)(gGL2 (O℘ )H − 1). On the
other hand, modulo ℘, MΓ0 (℘)H is singular. One can still define its arithmetic genus and
it is equal to gΓ0 (℘)H since MΓ0 (℘)H is flat. Applying Riemann-Hurwitz formula to the
map (MΓ0 (℘)H )/℘ → (MGL2 (O℘ )H )/℘ , we obtain the following equation
2gΓ0 (℘)H − 2 = 2(2gGL2 (O℘ )H − 2) + 2](ΣH ).
Comparing the two equations, the lemma follows.
2.5. p-adic uniformization of Shimura curves. The discussion in this subsection is
needed in section 4. Let v be a finite place of F at which D0 is ramified. Here D0 is
the quaternion algebra defined in section 1.3. Let P be an open compact subgroup of
(D0 ⊗F AF )× such that Pv = (OD0 ⊗OF OFv )× . Then we have a Shimura curve MP0 which
is defined over F .
Let Fvur be the maximal unramified extension of Fv , let ΩFv be Drinfeld’s upper half
plane over Fv and Ω = ΩFv ⊗Fv Fv nr . We let g ∈ GL2 (Fv ) act on Ω via the natural
val(det g)
(left) action on ΩFv and the action of Frobv
on Fvur . We also let n ∈ Z act on Ω
−n
nr
through the action of Frobv on Fv . This gives an Fv -rational action of GL2 (Fv ) × Z on
Ω. Moreover, the Fv -analytic space GL2 (Fv )\(Ω × (P v \(D0 ⊗F AvF )× /G0 (Q))) algebraizes
canonically to a scheme XS over Fv . Let M 0 and X be the inverse limits of MP0 and XP
over all P . Then a special case of [26, Theorem 5.3] gives the following theorem. (See also
[17, Section 1].)
Theorem 2.8. There exists a (D0 ⊗F AvF )× × Z-equivariant, Fv -rational isomorphism
M 0 ⊗F Fv ∼
= X.
10
CHUANGXUN CHENG
In particular, we have an Fv -rational isomorphism
M 0 ⊗F Fv ∼
= XP .
P
Furthermore, there exists an integral model M0P for MP0 over OFv , and the above isomorphism can be extended to schemes over OFv .
We apply the above theorem to the case P = O and v = q to study the singular points
of the mod q reduction of M0O .
The special fibre of M0O has non-degenerate quadratic singular points. The dual graph
G attached to the special fibre of M0O ⊗ kq is the quotient
GL2 (Fq )+ \(∆ × (Oq \G0 (A∞ )q /G0 (Q)),
where ∆ is the well-known tree attached to SL2 (Fq ), GL2 (Fq )+ is the kernel of the map
µ : GL2 (Fq ) → Z/2Z
defined by the formula
µ(γ) = ordq (det γ)
(mod 2).
0
M0O ⊗kq
The singular points of
correspond to the edges of G. Let D̄ be a definite quaternion
algebra over F ramified at places in SD ∪ {p} ∪ {v | v|∞}. (Note that D0 is ramified at q
and unramified at τ1 , and D̄0 is unramified at q and ramified at τ1 .) Let G¯0 = ResFQ (D̄0 )×
be the associated algebraic group. Let S be an open compact subgroup of Ḡ0 (A∞ ) such
that Sv = Ov if v 6= q, and Sq = Γ0 (q) where
a b
Γ0 (q) = {
∈ GL2 (OFq ) | c ∈ q}.
c d
Then the edges of ∆ are in one-to-one correspondence with GL2 (Fq )+ /(Sq F × ). Therefore,
the edges of G are bijective to G¯0 (Q)\G¯0 (A)/S.
Remark 2.9. Let us consider the curves MK0 (N ) and MO0 and their reductions MK0 (N )
(mod p) and MO0 (mod q). By the definition of the quaternion algebras D and D0 , we see
that there is a bijection between the set of singular points of MK0 (N ) (mod p) and the set
of singular points of MO0 (mod q).
3. Ihara’s lemma
In this section, we prove Theorem 1.2 following the strategy of [9]. Some of the arguments are exactly the same as in [9]. Recall that U has no level structure at pq. Fix P | p.
From the two natural maps α1 , α2 : MU0 (q) → MU we get a map
α := α1∗ + α2∗ : H 0 (MU × kP , ωi⊗bi )2 → H 0 (MU0 (q) × kP , ωi⊗bi ).
Lemma 3.1. If 1 ≤ bi ≤ p − 2, then α is injective.
Proof. Assume that (f1 , f2 ) is in thekernel. The map is Uωq -equivariant if we let Uωq act
Tq
−1
on the left hand side by the matrix
. Here, Tq and Sq are q-th Hecke
Sq Norm(q) 0
operators on H 0 (MU × kP , ωi⊗bi ), Uωq is the q-th Hecke operator on H 0 (MU0 (q) × kP , ωi⊗bi ).
We may assume that(f1 , f2 ) is an eigenvector of Uωq . Similarly, we may assume that f1 is
an eigenform of Sq . This implies that f1 is a multiple of f2 . Therefore α2∗ (f2 ) = −α1∗ (f1 ) is
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
11
a multiple of α1∗ (f2 ). Now if x is apoint such that f2 (x) = 0, then by the above analysis,
1 0
f2 (ηq (x)) = 0 where ηq =
. By the following lemma and Lemma 2.7, the degree
0 ωq
of ωi⊗bi is either 0 or ≥ (Norm(p) − 1)(gU − 1). This is impossible because of Corollary
2.4. Therefore f2 = 0 and f1 = f2 = 0.
Definition 3.2. Let x be a point of MU × kP . The q-orbit of x is defined to be Hq (x) =
{ηqn (x)|n ∈ Z}.
Lemma 3.3. (1) If x is ordinary, then Hq (x) has infinitely many elements.
(2) If x is supersingular, then Hq (x) is the set of all supersingular points.
Proof. (1) By [4, Section 1.3], the set of connected components of MU is bijective to the
set
T (Q)\T (A∞ )/ Nrd(U )
where T = ResF/Q Gm,F and ηq acts on this set as the multiplication by det ηq . Assume
that ηqm (for some integer m) acts trivially on this set, it suffices to show that the set
{ηqnm (x)|n ∈ Z}, which is a subset of the intersection of Hq (x) and the connected component containing x, is infinite. By [4, Proposition 4.5.4], we know that every irreducible
component of MU × F 0 is isomorphic to an irreducible component of MU0 0 × F 0 for some
open compact subgroup U 0 and some unramified extension F 0 /FP . Let z ∈ MU0 0 × kP
corresponds to x under this identification. It is ordinary, which means that the corresponding abelian variety Az is ordinary. The set {ηqnm (x)|n ∈ Z} is bijective to the set of
abelian varieties which are qm -isogenous to Az . From [4, Section 11.4], we know that A
is ordinary if and only if EndD A = E 0 where E 0 is a quadratic extension of E. (Also, A
is supersingular if and only if EndD A ∼
= D̄ for some definite quaternion algebra over F .)
Therefore, we may apply the argument in [9, Lemma 8] to prove the statement.
(2) This part is now clear since we know the set of supersingular points and the action
of ηq from subsection 2.4.
As in [9, Section 4], we interpret the above results in the content of crystalline cohomology. For a smooth proper scheme M over OFP and a non-negative integer a,
let MF ∇
[0,a] (M ) be the category of torsion F -crystals as defined in [12]. By [12, The1
•
orem 6.2], we have an object in MF ∇
[0,1] (MU × OFP ) defined by R π∗ (ΩAU /MU ×OF ) ⊗ κ.
P
This has a natural action of OB , so we can define ei R1 π∗ (Ω•AU /MU ×OF ) ⊗ κ. Let Ei deP
1
•
note the object in MF ∇
[0,1] (MU × OFP ) corresponding to ei R π∗ (ΩAU /MU ×OF ) ⊗ κ and
P
bi −2
Ei . Finally, let
Ebi denote the object in MF ∇
[0,bi −1] (MU × OFP ) defined by Symm
∇
E = ⊗τ ∈S Ebτ . By [12, Theorem 2.1], it is an object in MF [0,m] (MU × OFP ). Note that
P
we assumed that m = (bi − 1) ≤ p − 2. Write E(U ) = H 1 (MU × OFP , E ⊗ Ω•MU /OF ) =
P
H 1 (MU × kP , E ⊗ Ω•MU /kP ). By [12, Theorem 4.1], it is an object in MF ∇
[0,m] (OE ) with
∼
coefficient κ ⊗Fp OF /p = ⊕τ ∈S κ. We may decompose E(U ) as
E(U ) = ⊕τ ∈S E(U )τ
where each E(U )τ is a free κ-module.
12
CHUANGXUN CHENG
Lemma 3.4.
gr E(U )τ ∼
=
j

⊗(2−bτ )
1

)
H (MU × kP , ωτ
H 0 (M
U
×
j=0
⊗(b )
kP , ωτ τ )
j = bτ − 1
otherwise


0
Proof. By Faltings’ comparison theorem in [12], we have
grj E(U )τ ∼
= H 1 (MU × kP , grj (E ⊗ Ω•
τ
MU /kP ))
= H 1 (MU × kP , grj (Ebτ ⊗ Ω•MU /kP ))
By the same argument as in [9, Lemma 10], we have that
 ⊗(2−bτ )

ωτ
j
•
∼
gr (Ebτ ⊗ ΩMU /kP ) = ωτ⊗(bτ −2) ⊗ Ω1M /k
U
P


0
j = bτ − 1
j=0
otherwise
The lemma follows.
The two natural maps α1 and α2 induce morphisms
MF [0,p−2] (OE ). We let αDR denote the map
∗
αi,DR
: E(U ) → E(U0 (q)) in
∗
∗
α1,DR
+ α2,DR
: E(U )2 → E(U0 (q)).
Lemma 3.5. If 2 ≤ bτ ≤ p − 2, then grj (Ker(αDR )) = 0 for j 6= 0.
Proof. It suffices to prove that grj (Ker(αDR )τ ) = 0 for j 6= 0. This follows from Lemma
3.1 and Lemma 3.4.
We can now prove Theorem 1.2.
Proof of Theorem 1.2. The proof is the same as that of [9, Theorem 4]. We sketch it here.
Let Ξ be the following map
1
1
1
Ξ : Het
(MU ⊗ F̄ , FV ) ⊕ Het
(MU ⊗ F̄ , FV ) → Het
(MU0 (q) ⊗ F̄ , FV ).
Let V be the functor defined below [9, Corollary 6]. Then we have V(αDR ) = Ξ,
and therefore V(Ker αDR ) = Ker Ξ. Any Jordan-Holder constituent of Ker Ξ induces
1 (M ⊗ F̄ , F ). Suppose we have a Jordan-Holder cona Jordan-Holder constituent of Het
U
V
1
stituent W of Het (MU ⊗ F̄ , FV ), then it must be a Jordan-Holder constituent of some
1 (M ⊗ F̄ , F ) annihilated by a maximal idea m. If m is
T(U, V )-subquotient of Het
U
V
non-Eisenstein, then W is contained in some irreducible rank two Galois representation
rm : GF → GL2 (T(U, V )/m). Therefore W must be the whole rm . Since rm |GFP is of the
form V(W) where W has a two step filtration, the theorem follows.
One immediate corollary is Ihara’s lemma for some p-adic sheaves. If W is a free
O-module, we know that the natural map
1
1
Het
(MH ⊗ F̄ , FW )m ⊗O κ → Het
(MH ⊗ F̄ , FW ⊗O κ )m
is an isomorphism. (See for example [5, Lemma 1.1].) Assume that (kτ )τ ∈S ∈ Zn with
2 ≤ kτ ≤ p + 1 and all kτ are of the same parity. Let W be the GL2 (Op ) representation
defined by
W = ⊗v|p ⊗τ :Fv →Q̄p (Symmkτ −2 OF2 v ) ⊗ O.
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
13
Then we have the following result.
Theorem 3.6. If the genus of MU is greater than 1, and
following map is injective:
P
τ ∈S (kτ
− 1) ≤ p − 2, then the
1
1
1
Het
(MU , FW )m ⊕ Het
(MU , FW )m → Het
(MU0 (q) , FW )m
1 0
(f1 , f2 ) 7→ 1∗ f1 +
f .
0 ωq ∗ 2
Here m is a non-Eisenstein maximal ideal of the Hecke algebra.
4. Examples with multiplicity two
In this section, we study certain exact sequences following Carayol [3] and Jarvis [15].
For simplicity, we will use MK to denote the integral model MK (as stated in subsections
2.4 and 2.5) if MK exists.
4.1. Exact sequences of MK0 (N ) . First, we consider the Shimura curve MK0 (N ) and the
sheaf F = FV . Let H = K0 (N )p , then K0 (N ) = Γ0 (p)H. The exact sequence of vanishing
cycles for the proper morphism MΓ0 (p),H ⊗ OFp → Spec OFp and the sheaf F is
(4.1)
1
1
0 → Het
(MΓ0 (p),H ⊗ k̄p , F) → Het
(MΓ0 (p),H ⊗ F¯p , F)
M
2
→
(R1 Φη̄ F)x → Het
(MΓ0 (p),H ⊗ k̄p , F)
→
x∈ΣH
2
Het
(MΓ0 (p),H
⊗ F¯p , F) → · · · ,
where ΣH denotes the set of singular points of the k̄p -scheme MΓ0 (p),H ⊗ k̄p . Note that
ΣH consists of a finite number of non-degenerate quadratic points.
Write
2 (M
2
¯
L(H) = Ker(Het
Γ0 (p),H ⊗ k̄p , F) → Het (MΓ0 (p),H ⊗ Fp , F)),
1
Z(H) = Het (MΓ0 (p),H ⊗ k̄p , F),
1 (M
¯
M (H) = Het
Γ0 (p),H ⊗ Fp , F),
L
1
X(H) = x∈ΣH (R Φη̄ F)x ,
X̃(H) = Ker(X(H) → L(H)).
Then we have a short exact sequence
(4.2)
0 → Z(H) → M (H) → X̃(H) → 0.
We construct a second exact sequence, based on the comparison between the cohomology
of the special fibre and the cohomology of the normalization of the special fibre. Recall
we have a map r : M0,H ⊗ k̄p t M0,H ⊗ k̄p → MΓ0 (p),H ⊗ k̄p , and MΓ0 (p),H ⊗ k̄p can be
regarded as two copies of M0,H ⊗ k̄p glued together transversally above each supersingular
point of M0,H ⊗ k̄p . As r is an isomorphism away from supersingular points, there is an
exact sequence of sheaves
0 → F → r∗ r∗ F → G → 0,
14
CHUANGXUN CHENG
where G is a skyscraper sheaf supported on ΣH . Taking the long exact sequence, we have
0
0
0 → Het
(MΓ0 (p),H ⊗ k̄p , F) → Het
(MΓ0 (p),H ⊗ k̄p , r∗ r∗ F)
(4.3)
0
1
→ Het
(MΓ0 (p),H ⊗ k̄p , G) → Het
(MΓ0 (p),H ⊗ k̄p , F)
1
→ Het
(MΓ0 (p),H ⊗ k̄p , r∗ r∗ F) → 0.
Write
0 (M
∗ F) → H 0 (M
L̃(H) = Im(Het
Γ0 (p),H ⊗ k̄p , r∗ r
Γ0 (p),H ⊗ k̄p , G)),
et
L
0
Y (H) = Het (MΓ0 (p),H ⊗ k̄p , G) = x∈ΣH Gx ,
1 (M
∗
R(H) = Het
Γ0 (p),H ⊗ k̄p , r∗ r F),
Ỹ (H) = Y (H)/L̃(H).
Then we have another short exact sequence
(4.4)
0 → Ỹ (H) → Z(H) → R(H) → 0.
The following lemma collects several properties of the exact sequences (4.2) and (4.4).
Lemma 4.1. (1) The sequences (4.2) and (4.4) are equivariant for the Hecke action and
the Galois action.
i (M
∗
2
∼ i
(2) Het
Γ0 (p),H ⊗ k̄p , r∗ r F) = Het (M0,H ⊗ k̄p , F) .
(3) There is an isomorphism
M
HΣ1 H (MΓ0 (p),H ⊗ k̄p , F)→
˜
Hx1 (MΓ0 (p),H ⊗ k̄p , RΨη̄ F).
x∈ΣH
(4) There is an isomorphism
0
(MΓ0 (p),H ⊗ k̄p , G) ∼
Het
= HΣ1 H (MΓ0 (p),H ⊗ k̄p , F).
(5) There is an isomorphism
N : X(H)(1) → Y (H).
(6) L̃(H)m = 0, L(H)m = 0.
Proof. These results are proved in [15, Sections 16-18].
Remark 4.2. (1) We have an isomorphism
1
1
Het
(M0,H ⊗ k̄p , F) ∼
(M0,H ⊗ F¯p , F),
= Het
because M0,H has good reduction at p. Therefore, we also have
1
R(H) ∼
(M0,H ⊗ F¯p , F)2 .
= Het
(2) Jarvis [15] proved more than those listed in the lemma. In particular, [15, Proposition 17.4] gives the following diagram which describes the action of the inertia group and
is very important in our application:
τ −1
(4.5)
M (H) −−−−→ M (H)
x


α
βy

X(H) −−−−→ Y (H)
V ar(τ )
Here τ is an element in the inertia group, and V ar(τ ) is the variation map.
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
15
By the lemma, X̃(H)m = X(H)m and Ỹ (H)m = Y (H)m . Then combining exact sequences (4.2) and (4.4), we have the following diagram
(4.6)
0
Y (H)m
/ Z(H)m
0
/ M (H)m
/ X(H)m
/0
R(H)m
0
together with an isomorphism N : (X(H)m )(1)→Y
˜ (H)m .
Fix an embedding F = T(K, F)m /m → F̄p . Tensoring the above diagram with F̄p , we
obtain
(4.7)
Y (H) ⊗ F̄p
Z(H) ⊗ F̄p
/ M (H) ⊗ F̄p
/ X(H) ⊗ F̄p
/0
R(H) ⊗ F̄p
0
together with an isomorphism N : X(H)(1) ⊗ F̄p →Y
˜ (H) ⊗ F̄p .
Lemma 4.3. If M (H) ⊗ F̄p →X(H)
˜
⊗ F̄p is an isomorphism, then ρ̄ is unramified at p
Proof. We can compute the action of the inertia group IFp . An element τ ∈ IFp acts by
τ −1
(4.8)
M (H) ⊗ F̄p −−−−→ M (H) ⊗ F̄p

x

α
βy

X(H) ⊗ F̄p −−−−→ Y (H) ⊗ F̄p
V ar(τ )
If β is an isomorphism, then by the above diagram we see that α is the zero map. Therefore,
τ − 1 = 0, i.e., the action is unramified.
We have the following lemma, which corresponds to [22, Proposition 1].
Lemma 4.4. dimF̄p X(H) ⊗ F̄p ≤ 2. The dimension is 2 if and only if ρ̄ is unramified at
p and Frobp acts as ±1.
16
CHUANGXUN CHENG
0
1
Proof. Let Wp be the Atkin-Lehner operator on MK0 (N ) defined by the matrix
−$p 0
(see [16, Section 3]), Uωp the p-th Hecke operator on MK0 (N ) . By [15, Theorem 10.2], the
involution Wp permutes the two components of MK0 (N ) ⊗ kp . Note that X(H) ⊗ F̄p is an
unramified GFp -module and Frobp acts on X(H) ⊗ F̄p as ± Norm(p) by the discussion in
subsection 2.4.
We have a surjection M (H)⊗ F̄p X(H)⊗ F̄p with dim M (H)⊗ F̄p = 2 and dim X(H)⊗
F̄p ≥ 1. If dim X(H) ⊗ F̄p = 2, then M (H) ⊗ F̄p ∼
= X(H) ⊗ F̄p . The GFp -action on
M (H) ⊗ F̄p is unramified. Taking the determinant of Frobp , we see Norm(p)2 ≡ Norm(p)
(mod m) and therefore Frobp acts as ±1.
Assume now that ρ̄ is unramified at p and Frobp acts as ±1. Taking determinant, we
see that Norm(p) ≡ 1 (mod m). Define map A by
1
1
A : Het
(MK0 (N/p) ⊗ k̄p , F)2m → Het
(MK0 (N ) ⊗ k̄p , F)m
(f, g) 7→ 1∗ f + (ηp )∗ g
and map B induced by the normalization map
1
1
B : Het
(MK0 (N ) ⊗ k̄p , F)m → Het
(MK0 (N/p) ⊗ k̄p , F)2m .
Id V er
An easy computation shows that B ◦ A =
where V er is the transpose of
V er Id
the Frobenius. Since each component of MK0 (N ) ⊗ kp is rational over kp , Uωp acts on
ΣH as the same way as Frobp . By [3, Theorem 0.7A] or [20, Proposition 7], the action
of Frobp on Z(H)m is the same as that of Norm(p)Uωp . Thus Uωp ≡ ±1 (mod m) on
1 (M
1
Het
K0 (N ) ⊗ k̄p , F)m ⊗ F̄p . Moreover, Uωp + Wp factors through Het (MK0 (N ) ⊗ k̄p , F)m ⊗
1 (M
F̄p → Het
K0 (N/p) ⊗ k̄p , F)m ⊗ F̄p (see [16, Proposition 3.1],) hence Uωp + Wp acts as 0
on Y (H) ⊗ F̄p . Thus Wp ≡ ∓1 (mod m).
Let C be the diagonal map (resp. anti-diagonal map)
1
1
C : Het
(MK0 (N/p) ⊗ k̄p , F)m → Het
(MK0 (N/p) ⊗ k̄p , F)2m
2
1
1 (M
if Frobp acts as −1 (resp. as +1.) Note that Het
K0 (N/p) ⊗ F̄ , F)m ⊗ F̄p → Het (MK0 (N ) ⊗
F̄ , F)m ⊗ F̄p is surjective because ρ̄ is irreducible. In particular, A ⊗ F̄p is surjective. Since
Wp (A(f, g)) = A(g, f ), we see that the following map is a surjection on to Im((B ⊗ F̄p ) ◦
(A ⊗ F̄p )) = Im(B ⊗ F̄p ) = R(H) ⊗ F̄p .
1
1
C 0 : Het
(MK0 (N/p) ⊗ k̄p , F)m → Het
(MK0 (N/p) ⊗ k̄p , F)2m ⊗ F̄p .
The composition (A ⊗ F̄p ) ◦ C 0 is surjective. Also V er ≡ ±1 (mod m) since Norm(p) ≡ 1
1 (M
(mod m), it is easy to check that (B ⊗ F̄p ) ◦ (A ⊗ F̄p ) ◦ C 0 (Het
K0 (N/p) ⊗ k̄p , F)m ) = 0.
Therefore (A⊗ F̄p )(R(H)⊗ F̄p ) ⊂ Ker(B ⊗ F̄p ) = Y (H)⊗ F̄p . Hence Z(H)⊗ F̄p ⊂ Y (H)⊗ F̄p
and maps to 0 in M (H) ⊗ F̄p . The lemma follows.
4.2. Exact sequences of MO0 . Next, we describe the corresponding picture for the curve
MO0 . Note that now we consider the reduction mod q. The exact sequence of vanishing
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
17
cycles for the proper morphism MO0 ⊗ OFq → Spec OFq and the sheaf F is
(4.9)
1
1
0 → Het
(MO0 ⊗ k̄q , F) → Het
(MO0 ⊗ F¯q , F)
M
2
2
→
(R1 Φη̄ F)x → Het
(MO0 ⊗ k̄q , F) → Het
(MO0 ⊗ F¯q , F) → · · · ,
x∈ΣO
where ΣO denotes the set of singular points for the k̄q -scheme MO0 ⊗ k̄q . Note that ΣO
consists of a finite number of non-degenerate quadratic points.
Write
2 (M 0 ⊗ k̄ , F) → H 2 (M 0 ⊗ F
¯q , F)),
L(O) = Ker(Het
q
et
O
O
1
0
Z(O) = Het (MO ⊗ k̄q , F),
1 (M 0 ⊗ F
¯q , F),
M (O) = L
Het
O
1
X(O) = x∈ΣO (R Φη̄ F)x ,
X̃(O) = Ker(X(O) → L(O)).
Then we have a short exact sequence
(4.10)
0 → Z(O) → M (O) → X̃(O) → 0.
We construct a second exact sequence. Let r be the normalization map of the special
fibre of MO0 . We define G 0 via the following exact sequence of sheaves
0 → F → r∗ r∗ F → G 0 → 0.
Then G 0 is a skyscraper sheaf supported on ΣO . Taking the long exact sequence, we obtain
(4.11)
0
0
0
0 → Het
(MO0 ⊗ k̄q , F) → Het
(MO0 ⊗ k̄q , r∗ r∗ F) → Het
(MO0 ⊗ k̄q , G 0 )
1
1
→ Het
(MO0 ⊗ k̄q , F) → Het
(MS0 ⊗ k̄q , r∗ r∗ F) → 0.
Write
0 (M 0 ⊗ k̄ , r r ∗ F) → H 0 (M 0 ⊗ k̄ , G 0 )),
L̃(O) = Im(α : Het
q L∗
q
et
O
O
0
0
Y (O) = Het (MO ⊗ k̄q , G 0 ) = x∈ΣO Gx0 ,
1 (M 0 ⊗ k̄ , r r ∗ F),
R(O) = Het
q ∗
O
Ỹ (O) = Y (O)/L̃(O).
Then we have another short exact sequence
(4.12)
0 → Ỹ (O) → Z(O) → R(O) → 0.
i (M 0 ⊗ k̄ , r r ∗ F) ∼ H i (P1 ⊗ k̄ , F)b , for some positive integer b.
Lemma 4.5. (1) Het
= et
q ∗
q
O
(2) There is an isomorphism
M
HΣ1 O (MO0 ⊗ k̄q , F)→
˜
Hx1 (MO0 ⊗ k̄q , RΨη̄ F).
x∈ΣO
(3) There is an isomorphism
0
Het
(MO0 ⊗ k̄q , G 0 ) ∼
= HΣ1 O (MO0 ⊗ k̄q , F).
(4) There is an isomorphism
N : X(O)(1) → Y (O).
(5) L̃(O)m = 0, L(O)m = 0, R(O)m = 0.
18
CHUANGXUN CHENG
Proof. (1) We know that reduction Ω (mod p) is a union of P1 . By Theorem 2.8, the
special fibre MO0 ⊗ k̄q is several copies of P1 glued together transversally above the singular
points. The normalization of the special fibre is a disjoint union of finitely many P1 . The
statement follows.
(2)(3)(4) The arguments are the same as the arguments in Lemma 4.1.
(5) The first two terms vanish because they come from the 0th and 2nd cohomology
groups and m is a non-Eisenstein ideal. The third term vanishes because the 1st cohomology group of P1 vanishes.
Combining exact sequences (4.10) and (4.12), we obtain the following short exact sequence
0 → Y (O)m → M (O)m → X(O)m → 0,
together with an isomorphism N : (X(O)m )(1)→Y
˜ (O)m . Then we have the following exact
sequence
Y (O) ⊗ F̄p → M (O) ⊗ F̄p → X(O) ⊗ F̄p → 0,
together with an isomorphism N : X(O)(1) ⊗ F̄p →Y
˜ (O) ⊗ F̄p .
Lemma 4.6. There is an isomorphism of Hecke modules
Y (H)m ∼
= Y (O)m .
Proof. We deduce this from the exact sequence (3.14) of [20]. Rewriting that sequence in
our setting, we have
0 → Y (H 0 )2m → Y (H)m → Y (O)m → 0.
Here Y (H 0 ) corresponds to the group Y (H) for the curve MK 0 where K 0 = K0 (N/q).
Note that ρ̄ is ramified at q, hence m is not q-old. On the other hand, MK 0 has no level
structure at q and Y (H 0 ) is q-old. Therefore, Y (H 0 )m = 0 and the claim follows.
4.3. Proof of Theorem 1.6. We combine the results in the above sections to prove a
general multiplicity two result.
Lemma 4.7. ρ̄ is unramified at q if and only if there is an isomorphism
(4.13)
M (O) ⊗ F̄p →X(O)
˜
⊗ F̄p .
Proof. By the diagram in Remark 4.2, we can compute the action of the inertia group IFp .
An element τ ∈ IFp acts by
τ −1
(4.14)
M (O) ⊗ F̄p −−−−→ M (O) ⊗ F̄p

x

α
βy

X(O) ⊗ F̄p −−−−→ Y (O) ⊗ F̄p
V ar(τ )
If ρ̄ is unramified at q, then M (O) ⊗ F̄p is unramified at q. Thus αV ar(τ )β = τ − 1 = 0 for
all τ ∈ IFp . This is the same as αN β = 0 since V ar(τ ) = −(τ )N . Now β is a surjection
and N is an isomorphism, so α = 0. Thus by the exact sequence, β is an isomorphism.
On the other hand, if β is an isomorphism, then by the exact sequence we see that α is
the zero map. Therefore, τ − 1 = 0, i.e., the action is unramified.
We have assumed that ρ̄ is ramified at q, then we have the following lemma.
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
19
Lemma 4.8. dimF̄p M (O) ⊗ F̄p = 2 dimF̄p X(O) ⊗ F̄p .
Proof. By the exact sequences constructed above, we have
dimF̄p M (O) ⊗ F̄p ≤ dimF̄p X(O) ⊗ F̄p + dimF̄p Y (O) ⊗ F̄p
(4.15)
= 2 dimF̄p Y (O) ⊗ F̄p = 2 dimF̄p Y (H) ⊗ F̄p
= 2 dimF̄p X(H) ⊗ F̄p ≤ 2 dimF̄p M (H) ⊗ F̄p ≤ 4
By the above lemma, we see that dimF̄p M (O) ⊗ F̄p = dimF̄p X(O) ⊗ F̄p = 2 cannot happen
since ρ̄ is ramified at q. The lemma follows.
1 (M 0 ⊗ F̄ , F) ) ≤ 2 follows from equation (4.15).
Proof of Theorem 1.6. dimF̄p Hom(ρ̄, Het
m
O
Since dimF̄p Y (H) ⊗ F̄p = dimF̄p Y (O) ⊗ F̄p , we have dimF̄p X(H) ⊗ F̄p = dimF̄p X(O) ⊗ F̄p .
In particular, dimF̄p X(O) ⊗ F̄p ≤ 2. The dimension is 2 if and only if Gal(F¯p /Fp )-action
is unramified and Frobp acts as ±1. Theorem 1.6 is now clear from Lemmas 4.8.
Remark 4.9. Now we have a multiplicity two result. If we start with a Shimura curve such
that the cohomology group has multiplicity two, we can do the same thing as above (as
long as we have enough primes in the level): construct a new Shimura curve by changing
local invariants, compare the corresponding cohomology groups, etc.. Then we may get a
multiplicity 4 result similar to Theorem 1.6. In this way we may construct Shimura curves
with multiplicity 2a . This is closely related to the conjectures in [2, Section 4].
Acknowledgements The author would like to thank Matthew Emerton, for his support, encouragement, and many enlightening discussions. He would like to thank Brian
Conrad, Fred Diamond, and Mark Kisin for helpful conversations and correspondences,
Panagiotis Tsaknias for pointing out a mistake in an early version of this paper.
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