A family of Calabi-Yau varieties and
potential automorphy
Michael Harris
1
Department of Mathematics,
University of Paris 7,
Paris, France.
Nick Shepherd-Barron
DPMMS,
Cambridge
University,
Cambridge, CB3
0WB, England.
Department of Mathematics, Harvard University,
Cambridge, MA 02138,
U.S.A.
May 25, 2006
1
Richard Taylor 2
Institut de Mathґematiques de Jussieu, U.M.R. 7586 du CNRS, member Institut
Universitaire de France
2Partially supported by NSF Grant DMS-0100090
Introduction
This paper is an attempt to generalise the methods of [T1] and [T2] to Galois
representations of dimension greater than 2. Recall that these papers showed
that some quite general two dimensional Galois representations of Gal( Q/Q)
became modular after restriction to some Galois totally real field. This has
proved a surprisingly powerful result. To work in higher dimensions one faces
two problems: to generalise the modularity theorems of Wiles [W] and
TaylorWiles [TW] from GL2to GSpn(or some similar group) and to find some
replacement for the families of abelian varieties used in [T1] and [T2].
Clozel and two of us (M.H. and R.T.) have tackled the first of these
problems in [CHT] and [T3], on which papers this work depends in an
essential way. We work with unitary groups as this seems to make life easier,
but, using base change, one can immediately deduce results for (for example)
GSpn(if n is even) over a totally real field. When this paper was submitted only
[CHT] was available. In that paper we had succeeded in generalising the
arguments of [TW] to prove modularity of ‘minimal’ lifts but had been only able
to generalise the results [W] conditionally under the assumption of a
generalisation of Ihara’s lemma (lemma 3.2 of [I], see conjecture A in the
introduction of [CHT] for our conjectured generalisation). Thus at that time the
main results of this paper were all conditional on conjecture A of the
introduction of [CHT]. However, while this paper was being refereed, one of us
(R.T.) found a way to apply generalisations of the arguments of [TW] directly in
the non-minimal case thus avoiding the level raising arguments of [W] and the
appeal to conjecture A of [CHT]. This means that the results of this paper also
became unconditional.
The modularity theorems in [CHT] and [T3] are of the form rmod
lautomorphic implies rautomorphic. Of course rhas to be unramified at all but
finitely many places and potentially semi-stable at l. However there are a
number of further restrictions and all the precise statements we know are
unfortunately complicated- see [T3] for details. Essential to the method seems
to be that ris self dual up to a twist; that it is odd (e.g. if it is symplectic, then
the multiplier of complex conjugation should be -1); and that rshould have
distinct Hodge-Tate numbers. At present we are also forced to assume that
(the Frobenius semi-simplification) of rrestricted to the decomposition group at
some prime not dividing lis indecomposable. Stabilisation of trace formulae
related to unitary groups may allow one to forgo this last assumption, but we
have not worked out the details. We also assume that ris infact crystalline and
that rmod lhas ‘large’ image.
Using these results, the aim of this paper is to prove that certain l-adic
1
rare modular over some (Galois) totally real field without
assuming that rmod lis modular. One will of course need to keep the other
assumptions mentioned in the last paragraph. To copy the arguments of [T1]
and [T2] one needs families of ‘motives’ whose cohomology has h= 1 for all
p,qandwhich has largemonodromy. In [T1] and [T2] familiesofelliptic curves
and Hilbert-Blumenthal abelian varieties were used. In the case dimr>2
families of abelian varieties will not work. The key insight of this paper is to
work instead with (part of the cohomology) of the projective family:
Y(s:t) : s(Xn+1 0 +Xn+1 1 +...+Xn+1 n) = (n+1)tX0X1...Xn
. (More precisely we look at the H' = ker(µn+1 Q-→ µn+1
n+1
n-1H1
over P1
(s:t)
2
n-1) invariants in Hof a fibre in this family.) Note that in the case n= 2 this is just
a family of elliptic curves, so our theory is in a sense a natural generalisation
of the n= 2 case.
There is one further wrinkle in the method. We can only handle
representations rsuch that the restriction of rmod lto inertia at lis isomorphic to
that obtained from some element of this, or some similar family. For n= 2 this
is no real restriction. For n>2 it seems to be a serious restriction. Hence our
main theorems only apply to elements of such families. We consider two
cases: the above family (see theorem 3.5) and the family of Symm(E) as
Evaries over a family of elliptic curves (see theorem 3.3). (To study the latter
one needs to also study the former as the monodromy of the latter has rather
small image.)
In the first section we study the family Y. Most of the results we state seem
to be well known, but, when we can’t find an easily accessible reference,
wegivetheproof. Inthesecondsectionwerecallsomesimplealgebraicnumber
theory results that we will need. The main substance of the paper is
contained in section three where we prove various potential modularity
theorems. In the final section we give some example applications in the style
of [T2]. Others are surely also possible. Here are two examples of what we
prove:
p,qrepresentations
Theorem A Let E/Q be an elliptic curve with multiplicative reduction at a prime
q.
For any odd integer mthere is a finite Galois totally real field F/Q such
that Symm(E) becomes automorphic over F. (One can choose an F that will
work simultaneously for any finite set of odd positive integers.)
mH11.
2. For any positiveintegermthe L-functionL(SymmmH1(E),s) hasmeromorphic
continuation to the whole complex plane and satisfies the expected
functional equation. It does not vanish in Res= 1+m/2.
3. The Sato-Tate conjecture is true for E.
Theorem B Suppose that nis an even, positive integer, and that t∈ Q- Z. Then
the L-function L(Vt,s) of Hn-1(Y(1:t)Ч Q,Ql)H'
is independent of l, has meromorphic continuation to the whole complex plane
and satisfies the expected functional equaltion
L(Vt,s) = o(Vt,s)L(Vt,n- s).
(See theorem 4.4 for details.) The authors wish to thank the following
institutions for their hospitality,
which have made this collaboration possible: the Centre Emile Borel, for
organizing the special semester on automorphic forms (R.T.); Cambridge
University, and especially John Coates, for a visit in July 2003 (M.H. and
R.T.); and HarvardUniversity, for anextendedvisit during the spring
of2004(M.H.). We also thank Michael Larsen for help with the proof of
theorem 4.4; and Ahmed Abbes, Christophe Breuil, Johan de Jong and
Takeshi Saito for helping us prove proposition 1.15, as well as for helping us
try to prove stronger related results which at one stage we thought would be
necessary. Finally we thank Nick Katz for telling us, at an early stage of our
work, that corollary 1.10 was true (an important realisation for us) and
providing a reference.
roots of 1. We will use ζm
l
for the group scheme of mth
th
3
Notation
e will write µm
to denoteaprimitivemrootof1. Wewillalsodenotebyothel-adiccyclotomic
character.
cwill denote complex conjugation.
W
If T
is a
vari
ety
and
ta
poi
nt
of T
we
will
writ
e
O
T,t
∧
T,tfo
r
the
loc
al
ring
of T
at t.
We
will
use
k(t)
to
den
ote
its
resi
due
fiel
d
and
Oto
den
ote
its
co
mpl
etio
n.
If
ri
s
a
re
pr
e
s
e
nt
at
io
n
w
e
wi
ll
w
rit
e
rss
Чf
or
its
s
e
m
isi
m
pl
ifi
c
at
io
n.
L
et
K
b
e
a
pa
di
c
fi
el
d
a
n
d
v:
K
→
→
Z
its
v
al
u
at
io
n.
W
e
wi
ll
w
rit
e
O
KK
fo
r
its
ri
n
g
of
in
te
g
er
s
a
n
d
k(
K)
or
k(
v)
fo
r
its
re
si
d
u
e
fi
el
d.
W
e
wi
ll
d
e
n
ot
e
b
y
|
|t
h
e
a
b
s
ol
ut
e
v
al
u
e
o
n
K
d
e
fi
n
e
d
b
y
|a
|K
=
(#
k(
K)
)-v
(a)
.
W
e
wi
ll
al
s
o
d
e
n
ot
e
b
y
W
Kt
h
e
W
ei
l
gr
o
u
p
of
K
a
n
d
b
y
It
h
e
in
er
ti
a
s
u
b
gr
o
u
p
of
W
K.
W
e
wi
ll
w
rit
e
Fr
o
b
K
or
Fr
o
bv
Kf
or
th
e
g
e
o
m
et
ri
c
Fr
o
b
e
ni
u
s
el
e
m
e
nt
in
W
K/
IK
.
W
e
wi
ll
w
rit
e
Ar
tK
fo
r
th
e
Ar
ti
n
is
o
m
or
p
hi
s
m
Ar
tK
:
K
Ч
~a
b
K
denote a surjective homomorphism tK,l : IK . If l = p, we will let t→→
Weil-Deligne represena
normalised to send uniformisers to lifts of FrobK
GL(V) is a homomorphi
r(s)Nr(s)= |Artfor the Fro
(r,N)F-ss,N) of (r,N). We
irreducible smooth repre
semi-simple Weil-Delign
continuous finite dimens
WD(W) for the associat
instance [TY]).
1 A family of hyp
Let nbe an even positive
Ч
Y ⊂ P n P1
over Z[1 n+1] defined by
s(Xn+1 +Xn+1 1 +· · · +Xn+1 n) = (n+1)tX0
0
We
will
con
sid
er
Y
as
a
fam
ily
of
sch
em
es
ove
r
P11
by
· X1 · · · · ·
Xn.
proj
ecti
on
p to
the
sec
ond
fact
or.
We
will
lab
el
poi
nts
of
P1w
ith
refe
ren
ce
to
the
affi
ne
pie
ce
s=
1. If
tis
a
poi
nt
of
Pw
e
sha
ll
writ
e
Yfo
r
the
fibr
e of
Y
abo
ve
t.
Let
T0=
P1({
8}
∪
µn+1
t)/Z[
1/(n
+1)]
.
The
ma
ppi
ng
Y|T0
→
T8is
reg
ular
.
Ifζn+
1=
1th
enY
ζ0is
sm
oot
h.
The
tota
lsp
ace
YYha
son
lyis
olat
eds
ing
ular
ities
atp
oint
s
wh
ere
all
the
Xith’
s
are(
n+1
)roo
tsof
unit
y
with
pro
duc
t
ζ-1.
The
se
sin
gul
ariti
es
are
ordi
nar
y
qua
drat
ic
sin
gul
ariti
es.
4
ζ is a primitive (n+ 1)root of unity then over Z[1/(n+ 1),ζ] the scheme Y gets
a natural action of the group H = µn+1 n+1/µwith the subµn+1embedded
diagonally: (ζ0,...,ζn)(X0: ...: Xn) = (ζ0X0: ...: ζnXnn+1). We will let H0denote the
subgroup of elements (ζi) ∈ Hwith ζ0ζ1...ζ= 1. Then H0acts on every fibre Ytn+1.
If t= 1 then H0npermutes transitively the singularities of Yt. The whole group
Hacts on Y0. For Ncoprime to n+1 set
Vn[N] = V[N] = (Rn-1p*Z/NZ)H, a lisse sheaf on T00Ч SpecZ[1/N(n+ 1)].
(Although the action of His only defined over a cyclotomic extension, the
H00invariants make sense over Z[1/N(n+1)].) If l |n+1 is prime set
Vn,l = Vl =
V[lm])⊗Zl Ql.
(lim←
thIf
m
Similarly define V = (Rn-1p*Z)H0
a locally constant sheaf on T0(C)
- ({ 8} ∪ µn+1))H0
and VDRn-1 DR= H(Y/P1
iVDR
0. The locally constant
l
sheaf on T0
Ч Vl -→ Ql
l
5
and Vl
Lemma 1.1 V[N], Vl
(and aa connection) over T(C) corresp
locally free coherent sheaf with a decreasing filtration F are natural perfect alternating pairing
and
VЧ V coming from Poincare duality.
The following facts seem to be we
Nick Katz has told us that many of the
he only wrote up the case n= 3.
and V⊗ Q are all locally free of rank
Proof: We need only check the fibre at 0. In the case V⊗ C this is shown to
be locally free of rank nin proposition I.7.4 of [DMOS]. The same argument
works in the other cases. ✷
Corollary 1.2 If (N,n+1) = 1 then V/NV is the locally constant sheaf on T 0(C)
corresponding to V[N]. Lemma 1.3 Under the action of H/H0~ = µn+1the fibres
(V ⊗ C)and (Vl⊗QlQl)00split up as none dimensional eigenspaces, one for
each non-trivial character of µn+1. Proof: This is just proposition I.7.4 of
[DMOS]. ✷
Lemma 1.4 The monodromy of V ⊗ Q around a point in ζ ∈ µn+1has
1eigenspace of dimension at least n- 1.
Proof: Let t∈ T0(C). Picard-Lefschetz theory (see [SGA7]) gives an Hn-1orbit ∆
of elements of H(Yn-1(Yζt(C),Z) and an exact sequence (0) -→ H(C),Z) -→
Hn-1(Yt(C),Z) -→ Z∆n-1(Yt(C),Z) maps to (xd. If x∈ H) ∈ Z∆0then the monodromy
operator sends xto x±d∈∆xdd. Taking H0invariants we get an exact sequence
(0) -→ Hn-1(Yζ(C),Z)H0
ζ d-→ V
ζ
0 1 d∈∆
l ...X
=
or V[l] over T0
(
X
n
+
1
0
0
Write Qt
-→ Z and the monodromy
operator sends x∈ Vto x± d(x) d. ✷ We remark that this argument works
equally well for VЧ Z[1/l(n+1)].
We also want to analyse the monodromy at infinity. For simplicity we will
argue analytically as in [M] and [LSW], which in turn is based on Griffith’s
method [G] for calculating the cohomology of a hypersurface. (Indeed the
argument below is sketched in [LSW].) One of us (N.I.S-B.) has found an
H-equivariant blow up of Y which is semistable at 8, and it seems possible
that combining this with the Rapoport-Zink spectral sequence would give an
algebraic argument, which might give more precise information.
n
+...+Xn+1 n)/(n+1)- tX
X
n
∧ ...∧ dXi-1 ∧ dXi+1 ∧ ...∧ dX .
Ω
6
=
i=0(-1)iXidX0
n
, and
Then for i= 1,...,n+1
= (i- 1)!(X0X1...Xn)i-1Ω/Qi t
ω' i
isameromorphicdifferentialonPn(C)withapoleoforderialo
ngY. Moreover dω' i/dt= ω' i+1. Also set ωii= tω' iso that
ωiis H-invariant andt
tdωi/dt= iωi +ωi+1.
(C). We claim that for i= 1,...,nwe h
Suppose that t∈ { 8} ∪ µn+1
∈ Hi(Yt)- Hi-1(Yt)
ω notation of section 5 of [G]. If this were not the case then pro
in the
4.6 ' of [G] would tell us that (X0X1...Xn)i-1lies in the ideal generated b
Xn j-i tX0...Xj-1Xj+1...Xn. Hence (X0X1...Xn)iwould lie in the ideal gener
the Xn+1 j- tXandusingthefactthatC[X00X1...X,...,Xn]nH0. Symmetrising
the action of H= C[Z,Y0,...,Yn]/(Zn+1- Y0...Yn0)(with Yj= Xn+1 jandZ= X
wewouldhavethatZilies intheidealgenerated by the Yj- tZ and Zn+1C[Z,Y0,...,Yni]. Taking the degree i homogeneous part and using th
that i<n+ 1 we would have that Z
lies in the ideal generated by the - tZin C[Z,Y0,...,Yn
Yj
0 = Y1
n
' i n-iVDR)t n-1gives a linear form Hn(Pn(C) Y(Yt(C),Z) → Hn(Pnt(C)'
gives a class R(ω' i
Yn-1(Yt(C),C)H0t
i
⊗ C- (Fn+1-iVDR)t
R(ω' i) ∈ (F
FjV
].
Setting Z= 1 and Y= ...= Y= twould then give a
contradiction, proving the claim. Integration against
ω(C),Z) → C. Composing this with the map H(C),Z)
shows that ω) in H. According to theorem 8.3 of [G]
⊗ C. Thus the R(ω' i) for i= 1,...,nare a basis of
Hn-1(Yt(C),C)H0. Moreover we have the following lemma
(due to Deligne, see proposition I.7.6 of [DMOS]).
= 1.
Lemma 1.5 For j= 0,...,n- 1 we have
dimFjVDR/Fj+1VDR
Moreover if ζis a primitive (n+1)th
root of unity then Hacts on
DR,0/Fj+1V DR,0 ⊗ Z[1/(n+1),ζ]
) -→ (ζ0...ζn)n-j.
7
by (ζ0,...,ζn
Now assume in addition that t = 0. Then the class [ω] is in the span of the
classes [ω1],...,[ωnn+1]. In section 4 (particularly equation (4.5)) of [G] a method
is described for calculating its coefficients. To carry it out we will need certain
integers A0,ji,jdefined recursively for j>i= 0 by • Ai+1,j= Ai,i+1+2Ai,i+2= 1 for all j>0
and • A+...+(j- i- 1)A. Note that these also satisfy Ai,i+1i,j-1= 1 for all iand Ai,j=
Ai,j-1+(j- i)Ai-1,j-1
for j- 1 >i>0. We claim that for all non-negative integers iand nwe have
= min(n,i)
An-j,n+1 i!/(i- j)!.
n(i+1)
j=0
This can be proved by induction on n. The case n= 0 is clear. For general i we
see that
min(n,i) j=0i!/(i- j)! =min(n,i) j=1An-j,n+1An-j,ni!/(i- j)!+ min(n-1,i) j=0i!/(i- j)! =min(n-1,i-1)
j=0An-j-1,n(j+1)An-j-1,ni!(i- j+j+1)/(i- j)!+A(i+1)i! = (i+1)min(i,n-1) j=0ni!/(i- j)! = (i+1),
where we set An-i-1,nAn-1-j,nn-i-1,n= 0 if i= n. Thus we see that, as polynomials in
T
n
Tn = j=0 Aj,n+1(T- 1)(T- 2)...(T+j- n).
A1,n+1
z-1
0 1 n+
8
Write
A(z) =
100
... 0 0
120
... 0 0
013
... 0 0
A
n
,
n
+
1
.
z
n- 2 1
An0 1,n+
1
.1 z
n- 1
. 1 An-
2,n+1
z-1
A3,n+1
z-1
A2,n+1
z-1
.
Then expanding along the last column we see that A(0) has characteristic polynomial
n.
n+1Aj,n+1 (T- 1)(T- 2)...(T+j- n) = T
j=0
It also has rank n- 1 and so has minimal polynomial T
n
0
zdv(z)/dz= -A(z)v(z)/(n+1). In a
neighbourhood of zero its solutions are of the form
S(z)exp(-A(0)log(z)/(n+1))v0
is a constant vector. (See section 1 of [M].) We will prove
by induction on ithat
n+1]- t(A1,n+1[ωn]+A2,n+1[ωn-1]+...+Ai,n+1[ωn+1-i
i,j(X0...Xj)j-i-1(Xj+1...Xn)n+j-i Ω/Qn-i t .
n j=i+1
0...Xn)n
n+1
0...Xj-1Xj+1...Xn)(X0...Xj-1)j-1Xj j(Xj+1...Xn)n+j
n j=0(Xn j
the formula n t
j=i-A
(X0 [ t
ω (X
(j+1- i)An-i
n
(X0...Xj )j-i(Xj+1...X
. )n+k-i
. [ωn]+...+A
.
X
)k-i-1Xk-ikk(Xk+1 1
)n+1+j-i
[ ]).
ω
. Consider the differential
equation
where S(z) is a single matrix valued function in a neighbourhood of 0 and v
]) = (n- 1- i)!ttn+1j-i(j- i)A
)(XTo prove the case i= 0 combine formula (4.5) of [G] with the formula (1t- X= . To prove the case i>0 combine the case i- 1 andn formula (4.5) of
[G] with
+1(1- t)[ωn+1
j+1-i
(X0...
X
i-1,j
A
)
n+1-i
)
t
n
..
.
X
= . The special case i=
- 1 (A1,n+1
nthen tells us that
]=1
i,n+1
n k=i+1
...X
n+1
γ
k
S
up
po ],...,[ωn
se
th
at
-(n+1)t
X
0.
..
X
k1
X
k+1
n
n,n+1
1
G.
n
..
k-i
Gt
1
i,k
t
t∈
H
n(
P
n(
C
)
Y
(
C
),
C
)H
.
0
∈
Hn-1(Yt
(C),Z)
t
1
H0
ωn
m
a
p
s
n-1(Y
t
t
o
G
t
t
ω
v(γt) =
9
(C),Z). Then the coefficients of γwith respect to the basis of Hdual to [ω] is given by
As explained in [M] if γis a locally constant section of the local system of the
Hn-1(Yt(C),Z)H0tthen the Gtcan be taken locally constant and so tdv(γt-(n+1))/dt=
A(t)v(γ). Let z0be close to zero in P1tand let Pbe a loop in a small
neighbourhood of 0 based at z0
-(n+1) 0 n-1and going mtimes around 0. Let Pbe a lifting of this path
the map P1
under→ P1-(n+1)under which t→ tstarting at t0and ending at
ht(Yt0(C),Z)H00. If we carry γalong P in a locallyn-1(Yht0(C),Z)H
-(n+1) 0)-1
for some h∈ H. Let γ∈ Hconstant fashion we end up with an element Pγ∈
Hwhere v( Pγ) = S(t)exp(±2pimA(0)/(n+1))S(t0v(γ), and so
th -(n+1) 0)exp(±2pimA(0)/(n+1))S(t)-1
(C),Z)H0
-(n+1) 0v( Pγ) = S(t
n-1(Yt0
n
(P1
0
V on P1
nomial (T- 1)n
1
→
th
n+1
P
1
VЧv(γ). In particular we see that the monodromy around infinity on H
is generated by exp(2piA(0)) with respect to a suitable basis. This matrix is Lemma 1.6 The mon
unipotent with minimal polynomial (T- 1). Let ζdenote a primitive (n+
The monodromy aro
1)n+1root of 1. The map t→ tgives a finite Galois etale cover
exactly n- 1. The mo
(n+1)roots of 1 (eac
- { 0,8} )Ч SpecC -→ (P1- { 0,8} )Ч SpecC with Galois group H/H. Thus the last but one paragra
sheaf V descends to a locally constant sheaf(C)- { 0,1,8} . Note that there is
V⊗ C around 8 can
a natural perfect alternating pairing:
some basis. The act
Because Pover Z[1/
lemma 1.4 that the m
n- 1. Because it pres
have determinant 1.
identity as else the m
at 0 or its inverse. ✷
10
-1h
V -→ Z.
p1(SpecC((1/T)) ~=
Gal(C((1/T 1/N
Zp.
lim←N
n. The monodromy around any element of µ n+1Corollary 1.7 The monodromy
))/C((1/T))) ~=
of V around 8 unipotent with minimal polynomial (T- 1)(C) is unipotent with 1
eigenspace of dimension exactly n- 1.
Corollary 1.8 Identify C((1/T)) = O∧ P1ЧC,8(T). Also identify
p
(SpecC((1/T)) on Vl|SpecC((1/T))
n
x
(resp. V[l]|SpecC((1/T))
l
Then the action of p1 for a unipotent matrix u. In the case of V) is via x→ unthen uhas minimal
polynomial (X- 1). There exists a constant D(n) depending only on nsuch
that for l>D(n), this is also true in the case of V[l].
1(Z) with minimal polynomial (X- 1)n
Proof: A unipotent matix u∈ GL
n
n(Fl
i
P1(C)- { 0,1,8} then let Sp( VVzz
1
1(P1
Vz
z
r
r
r
r
0(C) let Sp(Vt
t ⊗ C (resp. V[N]t)
which preserve the alternating form.
11
reduces modulo lfor all but finitely many primes lto an unipotent matrix in GL) with minimal
polynomial (X- 1). (If not for some 0 <i<nwe would have (u- 1)= 0 mod lfor infinitely many l.) ✷ It
seems likely that N.I.S.-B.’s resolution of Y would allow one to make
explicit the finite set of lfor which the last assertion fails. We would like to thank Nick Katz for telling
us that the following lemma is
true and providing a reference to [K2]. Because of the difficulty of comparing the notation of [K2]
with ours we have chosen to give a direct proof. If z∈⊗ C) denote the group of automorphisms
of ⊗ C which preserve the alternating form. Lemma 1.9 Ifz∈ P(C)- { 0,1,8} thentheimageofp(C)- {
0,1,8} ,z) in Sp(⊗ C) is Zariski dense. Proof: This follows from the previous lemma and the
results of [BH]. More
r
precisely let H denote the image of p (P1(C) - { 0,1,8} ,z) in Sp( V
r
z
t
⊗ C) and let Hdenote the normal subgroup generated by monodromy at 1. It follows from
proposition 3.3 of [BH] that H is irreducible and from theorem 5.8 of [BH] that H is also primitive.
Theorem 5.3 of [BH] tells us that His irreducible and then theorem 5.14 of [BH] tells us that His
primitive. (In the case n= 2 use the fact that His irreducible and contains a non-trivial unipotent
element.) His infinite. Then it follows from propositions 6.3 and 6.4 of [BH] that His Zariski dense
in Sp( V⊗ C). ✷ If t∈ T⊗ C) (resp. Sp(V[N])) denote the group of automorphisms of V
Corollary 1.10 If t∈ T0(C) then the image of p1(T0(C),t) in Sp(Vt⊗ C) is Zariski
dense.
Combining this with theorem 7.5 and lemma 8.4 of [MVW] (or theorem 5.1
of [N]) we obtain the following corollary. (In the spirit of full disclosure we
remark that theorem 7.5 of [MVW] relies on the classification of finite simple
groups and [N] does not pretend to give a complete proof of its theorem 5.1.)
Corollary 1.11 There is a constantC(n) such that if Nis an integerdivisible only
by primes p>C(n) and if t∈ T(C) then the map p1(T00(C),t) -→ Sp(V[N]t) is
surjective.
Let Fbe a number field and let Wbe a free Z/NZ-module of rank nwith a
continuous action of Gal( F/F) and a perfect alternating pairing
, W: WЧ W-→ Z/NZ(1- n). We may think of W as a lisse etale sheaf over
SpecF. Consider the functor
from T0Ч SpecF-schemes to sets which sends Xto the set of isomorphisms
between the pull back of Wand the pull back of V[N] which sends , to the
pairing we have defined on V[N]. This functor is represented by a finite etale
cover TW/T0WЧ SpecF. The previous corollary implies the next one. Corollary
1.12 If N is an integer divisible only by primes p>C(n) and if
W, , (C) is connected for any embedding F → C, i.e. TWWis as above, then
TWis geometrically connected. Lemma 1.13 Suppose that K/Qlis a finite
extension and that t∈ T(K). Then Vl,t0Kn+1and 1/(tis a de Rham representation
of Gal( K/K) with Hodge-Tate numbers { 0,1,.,,,,n- 1} . If t∈ O- 1) ∈ OKthen Vis
crystalline. Proof: Vl,t= Hn-1(YtЧ SpecK,Ql)H0l,t. The first assertion follows from
the comparison theorem and the fact that Hn-1 DR(Yt/K)Hhas one dimensional
graded pieces in each of the degrees 0,1,...,n- 1. The second assertion follows
as Yt/OKis smooth and projective. ✷0
Lemma 1.14 Suppose that l= 1 mod n+1. Then V[l]0~ =
1⊕ ω-1 l1-n l⊕ ...⊕ ω
as a module for IQl.
12
Proof: It suffices to prove that
Vl, ~ = 1⊕ ⊕ ...⊕ o1-n
0
ol
l.
(As l >nthe characters o0 ,...,o1-nthall have distinct reductions modulo l).
HoweverbecauselsplitsintheextensionofQobtainedbyadjoiningaprim
(n+1)root of 1, lemma 1.3 tells us that Vis the direct sum of ncharac
Gal(Ql/Qll,0)-module. These characters are crystalline and the Hodg
numbers are 0,1,...,n- 1. The results follows. ✷
Lemma 1.15 Suppose q = lare primes not dividing n+1, and suppos
K/Qqis a finite extension. Normalise the valuation vKon Kto have ima
Suppose that t∈ Khas v(t) <0. 1. The semisimplification of Vl,tand V[
unramified and Frobn-1Khas
eigenvaluesoftheforma,a#k(K),...,a(#k(K))forsomea∈ { ±1} , where k
denotes the residue field of K.
as exp(NtK
n
as exp(vK(t)NtK
l,t
t
t
~
p
Zp
n.
q
p1
), where N2 is a nilpotent endomorphism of Vwith minimal poly
. The inertia group acts on Vl,t The inertia group acts on V[l]), where N is a nilpotent endomo
V[l], and if l>D(n) then Nhas minimal polynomial T
Proof: First we prove the second and third parts. Let Wdenote t
vectors of Fand let F denote its field of fractions. We have a com
diagram:
-→
Zp
(Spec F((1/T))) ~-→ p=q
1(SpecW((1/T)))
~
ZpZp K
p
1
a
n
d
p=q p1
p1(Sp
1(Spec F((1/T)))
(S
ec
p=q
p
F((1/
ec
T))) =
W
lim←N
~
((1/T)))
1/N
↓
↓
p
↑ ↑ v(a) p(SpecFK) →→ .
Herethelefthanduparrowisinduced
byT→ t.
Therighthanddownarrowisthe
natural projection and the right
hand up arrow is multiplication by
v(a). The isomorphisms p→
Zpresult from corollary XIII.5.3 of
[SGA1]. More precisely→
))/F((1/T)))
Gal(F((1/T
13
and p1(SpecW((1/T))) = lim←(N,q)=1
Gal(W((1/T 1/N ))/W((1/T))).
(Note that, as the fraction field of W[[1/T]]/(1/T) has characteristic
zero, the tame assumption in corollary XIII.5.3 is vacuous.) The final
surjection p1(SpecFK) →→ p=qZpcomes from
p1(SpecFK) →→
Gal(FK(̟1/N K )/FK),
lim←(N,q)=1
∧ P(T), the sheaves Vl|SpecW((1/T))and V[l]|1,8SpecW((1/T))corr
where ̟Kis a uniformiser in K.
representations of p1(SpecW((1/T))). Corollary 1.8 tells
Considering
of these representations to p1(Spec F((1/T))) ~= sends
matrix. Moreover in the case VlpZpnor in the case V[l] w
that this unipotent matrix has minimal polynomial (X- 1
Now we prove the first part. It is enough to consider V.
W((1/T)) = O
we see that FrobKl,thas eigenvalues a,a#k(K),...,a(#k(K
Qn-1. The alternating pairing shows that { a,a#k(K),...,a
a-1,a-1#k(K),...,a-1(#k(K))n-1} . Thus a= ±1. ✷
2 Some algebraic number theory
We briefly recall a theorem of Moret-Bailly [MB] (see a
Dieulefait tells us that he has also explained this slight
the result of [MB] in a conference in Strasbourg in July
Proposition 2.1 Let F be a number field and let S = S1
S2
1
v
⊂ T(Fv) is a non-empty open (for the v-topology)
subset and that for v ∈ S)-invariant subset .
Suppose finally that L/F is a finite extension.
be a finite set of places of F such that Scontains no infinite place.
Suppose that T/F is a smooth, geometrically connected variety.
Suppose also that for v∈ S, Ω
2, Ωv nr v⊂ T(F ) is a non-empty open Gal(Fnr v/Fv
'
'/F and a point P∈ T(F
Then there is a finite Galois extension F
14
) such that
•F
'
i splits
'w
completely in F'
is unramified in F'
' w).
1
v
1and
if wis a prime of F '
'
1
2 i
v
+
→Q ,
Ч
Ч M,S
'
•χ
)Ч
: (A8 M+
•χ
0
•ψ
:M
0
+
(M+
)Ч
)
n
O
Ч
Ч
=
M
,
S
χ
Ч→
Q
+|(M+)
= χ+|(AM +)8)ЧnOЧ M,S
|(A
/F is linearly disjoint from L/F; • every place vof Sabove vthen P∈ Ω⊂ T(F); •
every place vof Sand if wis a prime of Fabove vthen P∈ Ωn T(F
Proof: We may suppose that L/F is Galois. Let L ,...,Lr
v
i
1
⊃ F with L
:
'
O
2
2
Ч
S
+
denote the
intermediate fields L⊃ L/F Galois with simple Galois group. Combining
Hensel’s lemma with the Weil bounds we see that T has an F
rational point for all but finitely many primes vof F. Thus enlarging Sto include
one sufficiently large prime that is not split in each field L(the prime may
depend on i), we may suppress the first condition on F. Replacing Fby a finite
Galois extension in which all the places of Ssplit completely and in which the
primes of Sare unramified with sufficiently large inertial degree, we may
suppose that S= Ш . (We may have to replace the field Fwe obtain with its
normal closure over the original field F.) Now the theorem follows from
theorem 1.3 of [MB]. ✷
Lemma 2.2 Let Mbe an imaginary CM field with maximal totally real subfield
M, Sa finite set of finite places of Mand T⊃ San infinite set of finite places of
Mwith cT= T. Suppose that there are continuous characters:
→
,
Q
, such that
• if χis ramified at vthen T contains some place of Mabove v, • ψ|SM +8)ЧЧ , and •
χ. Then there is a continuous character
Ч-→ Q
ψ: (A8 M)Ч
such that
15
ψ• ψis unramified outside T, •
ψ|OЧ M,S0= χ, • ψ|S,
• and ψ|(A8 M+)Ч = χ+.
= v∈S
U0, ⊂ v∈S
Proof: Choose U0
v
MЧ =
OЧ
M,v
Ч
0n
(AЧ
+
+
+
M,v
v∈S
v
∈
S
0,v
8
v
0
Ч M,v
Ч M,S
v
Ч
v∈S
Uv
= (M
Ч
ЧM
Ч
be an open subgroup such that
U8 M)⊂ kerχand U= OЧ M,vfor v∈
T. Let V = Obe an open compact
subgroup such that V n µV⊂ (M)
= { 1} and V= OЧ M,vfor v∈ T. Let
Udenote the subset of
Uconsisting of elements u with
c(u)/u∈ V. Then U=with Uv= Ofor
v∈ T. Moreover Mn OU(A8
M+)Ч+)M/M. (For if alies in the
intersection then c(a)/a∈ ker(N+ :
O-→ OЧ M+)n OЧ M,SV = µ8(M)n
OЧ M,S)V = { 1} , so that a∈ (M.)
Define a continuous character
ψ: OЧ M,SU(A8 M+)Ч Ч-→ Q
to be χS
Ч on
. This is easily seen to be well
defined. Extend ψto MЧ M,SU(Aby
setting it equal to ψ
Ч
n(F ∨ O. This is well defined because M
8 M)Ч
8 M)Ч
OЧ M,S, to be 1 on Uand to be χ+ on (A8 M+)Ч
8 M+)Ч
Чn
F
OЧ M,SU(A8 M+ЧOЧ M,SU(A8 M+)Ч
→ CЧ
l~
v
nn(AF)
of weight 0 and type {
ρv}(AF
Ч
n MЧ
o
.
Now ext
end ψtoset of finite places of F and for v∈ Sthe ρbe an irreducible square-integrable+)Ч = (M )Ч
(Ain any
representation of GLv). Recall (see section 4.5 of [CHT]) that by a RAESDC
way. representation pof GLwe mean a cuspidal automorphic representation pof
(This isGL~ = χpfor some character χ: F) such that • pv∈S\ A8; • phas the same
possible
infinitessimal character as the trivial representation of GLn(F8);
as Mhas
16
finite
index in
(A.) This
ψsatisfie
s the
require
ments of
the
theorem
.✷
3
Pote
ntial
mod
ulari
ty
Let F
denote
a totally
real field
and na
positive
integer.
Let lbe a
rational
prime
and let i:
Q→ C.
Let Sbe
a
non-em
pty finite
• and for v∈ Sthe representation pv
is an unramified twist of
ρv
v}v∈S
n(AF
-1(rec(pv)⊗
BDR)Gal(F
w
K
l
rl,i
dimQl g i(rl,i(p)⊗t,Fv
r
(
1
n
)
/
2
|Art
1. For
every
prime v
|lof Fwe
have
WD(rl,i(p
)|Gal(
Fv/Fv))F-ss
=i
v/Fv)
= 1.
.
We say that phas level prime to lif for all places w|lthe representation
pis unramified.
Recall (see [TY] and section 4.5 of [CHT]) that if pis a RAESDC
representation of GL) of weight 0 and type { ρ(with S = Ш ), then
there is a continuous irreducible representation
(Q
).
(p) : Gal( F/F) -→ GL -1 K|
n
) with the following properties.
2. r l,i(χ) see [HT] or [TY].) 3. If v|lis a prime of Fthen r(p)|is potentially
= rl,i(p)on-1rl,i(χ). (Fo
semistable, and if pvGal( Fv/Flis unramified then it is crystalline. 4. If v|lis a
prime of Fand if t: F→ Qv)lies above vthen dimQlgri(rl,i(p)⊗t,FvBDR)Gal(
Fv/Fv)= 0 unless i∈ { 0,1,...,n- 1} in which case
(p) is conjugate to one into GLn(OQl). Reducing this modulo
themaximalidealandtaking thesemisimplificationgivesasemisimple
The representation rl,i continuous representation
rl,i(p) : Gal(F/F) -→ GLn(Fl) which is independent of the
choice of conjugate.
We will call a representation r: Gal( F/F) -→ GLn(Ql)
(resp.
r: Gal(F/F) -→ GLn(Fl)) which arises in this way for some p(resp. some po
level prime to l) and i
automorphic of weight 0 and type { ρv}v∈S. In the case of r, if phas level
prime to lthen we will say that ris automorphic of level prime to l.
We will call a subgroup ∆ ⊂ GL(V/ Fl) big if the following hold.
17
• Hi(∆,ad
h,a 0
i
: V → Vh,a (resp. ih,a
l
l
i
h,a
: Vh,a
h,a
◦ W◦ ih,a
w
n(Zl
◦ ih,a
l
i),ni
i
i
1
q))j
h,a
1. ri
i
i
i
4. Fkerad ri
i
i
.
V) = (0) for i= 0 and 1. • For all irreducible
Fwiththefollowingproperties. TheageneralisedeigenspaceV[∆]-submodules
W of adV we can find h∈ ∆ and a∈ Fof hon V is one dimensional. Let p→
V) denote the h-equivariant projection of V to V(resp. h-equivariant injection
of Vinto V). (So that ph,ah,a= 1.) Then p = (0).
Note that this only depends on the image of ∆ in PGL(V/ F). Theorem 3.1
Suppose that F/F0,...,nis a Galois extension of totally real fields and that nrare
even positive integers. Suppose that l>max{ C(n} is a prime which is
l
unramified in F and satisfies l= 1 mod n+ 1 for i= 1,...,r. Let vqbe a prime of F
above a rational prime q = l such that (#k(v= 1 mod lfor j = 1,...,max{ n} and q
| (n+ 1) for i= 1,...,r. Let L be a finite set of primes of F not containing primes
above lq.Suppose also and that for i= 1,...,r ri: Gal(F/F) -→ GSp) is a
continuous representation with the following properties.
has multiplier o1-nli. 2. rramifies at only finitely many primes. 3. Let rdenote
the semisimplification of the reduction of r. Then the image rGal(F/F(ζl)) is
big.
d
o
e
s
n
o
t
c
o
n
t
a
i
n
F
(
ζ
l
)
.
5
.
r
i
s
)
t,Fw
v q /Fv q
are unramifiedandri ss Gal( vq
ss Gal(
F|
q))ni-1.
F|
q
18
7. ri ) has eigenvalues 1,(#k(v)),...,(#k(v
)
and ri|Gal(F
v q /Fv q
)
/Fv q )(Frobvq
Then there is a totally real field F'/Fwhich is Galois over F0
ker ri
'
' (1)}{wq}.
q
vi
i ')
i|
i
i
q
i
ni/2-1 j=0
i
i
j
/Q). Choose a prime p
0
i|MЧ i
•
ψi|(A8
M+ i)
an
d linearly independent from the compositum of the Fover F.
Moreover all primes of L and all primes of Fabove lare unramified
in F. Finally there is a prime wof Fover vsuch that each rnGal(F/Fis
automorphic of weight 0 and type { Sp
Proof: Before proving this theorem, we remark that following
the proof the reader can find some brief comments which may
help in navigating the technical complexities of the proof.
Let E/Q be an imaginary quadratic field. For i= 1,....,rlet M/Q be
a cyclic Galois imaginary CM field of degree nover Q such that •
land the primes below L are unramified in M; • and the
compositum of E and the normal closure of F/Q is linearlydisjoint
from the compositum of the M’s. Choose a generator tof
Gal(Mwhich is inert but unramified in Mand split completely in
EF. For i= 1,...,rchoose a continuous homomorphismi
ψi : (A8 )Ч -→ M Чi
M
i
with the following properties.
/2 i(ani-1-j).
)tj+n
• ψ(a) = Ч = | |tj i1-nvi(a.j
• ψi is unramified at land the primes below L.
OЧ Mi ,pi = ψtij|iOЧ Mi ,pi
i for j= 1,...,n- 1.
• ψi| i
i
i
•
ψi
'
only ramifies above rational primes which split in E. The
existence of such a character ψ
denote a finite extension of
follows easily from lemma 2.2. Let
M
M
1
r n1(n1+1)
nr(nr+1)
. Choose a
prime l
which is Galois over Q and contains the image of ψ
) such that
>8((ni +2)/4)ni/2+1 for all i;
'
19
•l
which splits in EF
... (ζ
,...,ζ
M
M
'
i >C(ni
is unramified above l'
ψi,l' : MЧ i\ (A8 Mi ) for all i; • ldoes not divide the class number of E;= wl',i|Mi
ni/2-1• each riis unramified above l'; • each ψ; • l |pnii- 1 l ' ,i
for all i; • l = l, l' = qand l',idenote a prime of
Miabove l'does not lie below L. Let wand let wl',i
• l'
Ч i,ew
'.
Define a continuous character
'
l'
b
y
ψi,l' (a) = ψi(a)
a-j t- j iwl ' ,iaj+1-nti- j+n i
j=0
)Ч -→
M
.
l ' ,i
/2iw
Composing this with the Artin reciprocity map and
reducing modulo wwe obtain a character
i
G
a
l
(
M
i
/
Q
)
→
F
iθ c i=
o1-n lwith the following
for j= 0,...,n/2- 1.
properties. • θi. • θ|IM i,tj i'w l' ,i= o'-jisl unramified above land the primes below
L.
• θ i|
θl :
'Ч
• θi
= θtj |IM i,pi for j= 1,...,n- 1.
IM i,pi
i
• θi only ramifies above primes above rational primes which split in
E.
Gal( Mi/Q) Gal( b
Define an alternating pairing on
Mi/Mi)θi
Ind
y
ϕ,ϕ' =
o(s)ni-1ϕ(s)ϕ(cs)
s∈Gal(Mi/Mi
)\Gal(Mi/Q)
20
where cis any complex conjugation. (It is alternating because n iis even.) This gives rise to a
homomorphism
I( θi) : Gal(Q/Q) -→ GSpni(F' ). Consider Kl(resp. Klkerrs i' ) the fixed field of Gal( F/EF(ζll))
i,s∈Gal(F/F0)
|
i,s∈Gal(F/F0)
(resp.
)|Gal( F/EF(ζl ' kerI(θi
))).
n Kl' /Fis unramified at land l'. Let Hl
⊂ Gal(Kl/F) (resp. H'l
l
vq ' l j
' |(ni
lKl' /F). Choose sl
∈ Hl l(resp. sl' ∈ H'
'
)/Q)). Let vlq
l
1...Mr
'
lK l
l
=
q,
that
q'
j
i
l
i
')n
Then Kl ' ⊂ Gal(K' /F)) denote the inertia group at a prime above l(resp. l). Then HЧ
H' ⊂ Gal(K' ) which maps to a generator of Gal(Q(ζ)/Q) (resp. Gal(Q(ζ' be a
place of F which is split over a rational prime qand which is unramified in
K' with Frob= sЧ sl' . We may further assume that qdoes not lie below a
prime of L, and that q'splits in EM+1) for i= 1,...,r. Then we see that • q'); •
(q= 1 mod lfor j= 1,...,max{ n')} ; • (q= 1 mod l'for j= 1,...,max{ ni(Frobvq ')
has eigenvalues 1,q',...,(q} ; • ri-1;
q
,...,(q')ni-1
(Q
'
'
q'
(
t
ni[l
q'
]
)
<
0
a
n
d
v
= 0 mod l. Then Frobvq' ∈ T'.
Then Frobq' acts on Vni[ll'
qq(tq)
q'
(
t
q
⊕ a i,q' q' ⊕ ...⊕ ai,q' (q')ni-1 for some ai,q' ∈ (Z/ll'Z)Ч with a2
i,q'
Ч
di
i,q
i,q
∈ { ±1} . Let t
q
0
q
]s
st
i,q
. Let t∈ T0(F•vq) with vq(tq) <0 and vacts on Vss tqas a⊕ ai,qq⊕ ...⊕ ai,qqni-1for '
I( θi)(Frobvq ') has eigenvalues 1,q some a' ) with v' ) = 0 mod llas a= 1. Choose a character
: Gal( Q/Q) -→ { ±1} ⊂ (Z/lZ)
i,q
,q,q' and which takes Frob to ai,q and
'Z)Ч
Frobq
21
'i
which is unramified at l,l ' to a' . Also choose a character d: Gal( Q/Q) -→ { ±1} Ч { ±1} ⊂ (Z/ll
q'
and which takes Frobq' to ai,q
i
corresponding to ri ⊗
i
di
which is unramified at l,l',q' ' . Let Wbe the free Z/lZ-module of rank n. It comes with a perfect
alternating pairing
Ч W -→ (Z/lZ)(1- n
).
The scheme is geometrically connected. Let S1q,vqdenote the infinite primes union { v' } ,
TWi
and let S2denote L union the set of places of F above ll'. Let Ωi,vq(resp. Ωi,vq
') denote the preimage in TW i(Fvq) (resp. TW i(F0(Fvq) : vq)) of { t∈ T(t) <0}
(resp. { t∈ T0(Fvq ') : vqvq '' (t) <0} ). These sets are open and non-empty
(because they contain a point above tq(resp. t' )). If wis an infinite place of
Flet Ωi,w= TWi(Fwq). This is non-empty as all elements of GSpni(Z/lZ) of
order two and multiplier -1 are conjugate. If w∈ Sdenote the set of
elements of TWi(Fnr w) above { t∈ T0(Fnr wn) : w(1- ti2let Ω+1i,w) = 0} . Then
Ωi,wis open, Gal(Fnr w/F0(Fnr ww)-invariant and non-empty (as it contains a
point above 0 ∈ T)). Let Kdenote the compositum of the fixed fields of the
ker ri, the kerI(θi), the diand the d' i. By proposition 2.1 we can find
recursively totally real fields Fi/Fand point ti∈ TWi(Fi) such that • Fi/Fis
Galois, • F/Fis unramified above L and above ll'qand vq' split in F, • vii, • Fis
linearly disjoint from KF1...Fover F, • and tilies in Ωi,wfor all w∈ S1i-1∪ S. Let
F= F1...Fr2iaGaloisextensionofFwhichistotallyreal,inwhichallprimes of
S1split completely and in which all primes of S2
linearly disjoint from Kover F. Let ti
∈ T0are unramified. Then Fis( F) denote the image of t
di)|Gal( F/ e F). Moreover Yni,tiihas good reduction above
crystalline above l. If wis a prime of Fabove vq(resp
semisimplification of Vni,l,ti|Gal( e Fw/ e Fw)is unramified an
eigenvalues a,a(#k(vq)),...,a(#k(vq))ni-1(resp. a,aq',...,a
modulo l we see that a= ai,q(resp. a' ). Let W ' ibe the f
rank nicorresponding to ( riЧ I(θ))⊗ d' i. The module W
perfect alternating pairingi
' iЧ W -→ (Z/ll'Z)(1- ni).
W' i
22
i
i
W
i
in Gal(F '/F). Then Vni,l,ti
''
ni
i
''
i
ni
ni(1)}{wq}
' {wq}
F''
ni(1)}{w|vq}
i
ni(1)}{wq}
ti
t
i
'
'
'
2Lemma
3.2 Let n∈
Z(Fl=1
n-1
Proof: We have
ad r~ = 1⊕
(Symm2
⊗
det
-1)⊕
(Symm 4 ⊗
det
-2)⊕
...⊕
(Symm
2(Fl
2n-2
⊗
det
1-n).
is also automorphic over Fof weight
d
0 and type { Sp(1)}and level prim
ecomposition group of wq l(see lemma 4.5.2 of [CHT]). Thus V[l]is automorphic over F. Again a
theorem 5.2 of [T3] we see that rof weight 0 and type { Spis automor
overof weight 0 and type { Spand level prime to l. This implies that rii
automorphic over Fof weight 0 and type { Spand level prime to l. ✷ W
that the following informal remarks may help guide the reader
through the apparent complexity of the proofs of Theorems 3.1 and 3
complexity is imposed upon
local supercuspidal condition at piis sacrificed for reasons connected
us by three circumstances.above. A similar reason, though unrelated to (b), motivates the introd
of the auxiliary prime q(unrelated to the previous q) in the proof of the
(a) The modularity
3.3, below. Before proving the next theorem we need the following lit
theorems proved in [CHT] lemma.
and [T3] only apply to l-adic
representations which, at
and let l>2n+1 be a prime. Suppose that ∆
some finite place v,
subgroup of GL) which contains SL). Let rdenote the representation S
correspond under the local∆. Then ∆ is big with respect to r and the maximal abelian quotient of
Langlands correspondencehas order dividing 2.
to discrete series
24
representations. It is
possible that further
developments of the stable
trace formula will make this
hypothesis unnecessary.
(b) In the second place, our
knowledge of the bad
reduction of the Calabi-Yau
hypersurfaces Y'considered
in section 1 is only sufficient
to provide inertial
representations of
Steinberg type (with
maximally unipotent
mondromy), as in Lemma
1.15; this explains our local
hypotheses at the primes
denoted qand q. (c) The
monomial representations I(
θ) considered in the proof
of Theorem 3.1 can never
be locally of Steinberg type,
but they can be locally of
supercuspidal type, and are
chosen to be so at the
primes denoted p. The local
hypothesis at pis used in
the proof of theorem 5.4 of
[T3]. The auxiliaryprime qis
introducedintheaboveprooft
opreservethelocaldiscrete
series condition when the
As 2n- 2 = l- 1 each factor in this decomposition is irreducible. Thus PGL2(Fl)
⊃ ad r∆ ⊃ PSL2(F0) and the second assertion follows. Moreover H(∆,ad0lr) =
(0). Let Bdenote the subgroup of upper triangular matrices in ∆, let Udenote
the Sylow l-subgroup of Band let T denote the subgroup of diagonal matrices
in ∆. Then
H1(∆,ad0r) → H1(U,ad0r)B. Let χ1(resp. χ2) denote the character of Btaking a
matrix to its top left (resp. lower right) entry. Let 0 = i= n- 1. As 2i= l- 1 we
see that
H1(U,Symm 2i ⊗ det-i) = Fl (χ2i+1 2χ-1 1
1
0r)B
(Fl
1(∆,ad0r)
= (0). Let
D= (ad r)T
Symm
(U,ad). (As Uis cyclic.) As l- 1 >2i+1 we see that H= (0) and hence H
. Let tbe a generator of Tn SL2). As n<lwe can decompose
n-1 = V0
⊕ ⊕ ...⊕
V1 Vn-1
where the Vi
D= n-1 are the eigenspaces of tand each is one dimensional. Let id
injection Vj→ Symmn-1and pt,jt,jn-1denote the t-equivariant pro
Symm→→ Vj. Thus pt,jit,j= 1. As 2n<l+1 we see that
Hom(Vj,Vj)
j=0
-i
2ihas
dimension nand that, for i= 0,...,n- 1
dimDn (Symm⊗ det
2i
-i)
onto Hom(Vj,Vj) is non-trivial. Then
pt,j(Dn (Symm2i⊗ det-i
t,j
1
i),2ni
q
i
q))j
) = 1. For each i= 0,...,n- 1 choose jsuch that the projection of (Dn (Symm⊗
det
,...,nt
))i = (0). ✷
Theorem 3.3 Suppose that Fis a totally real field and that n
= 1 mod lfor j= 1,...,max{ n} . Suppose also and that
(Zl
r: Gal( F/F) -→ GL
25
are even positive integers. Suppose also that l>max{ C(n+1} is a prime
which is unramified in F and that vis a prime of F above a rational prime q = l
such that (#k(v
2
) is a continuous representation with the following
properties.
1. detr= o-1 l. 2. rramifies at only finitely many primes. 3. Let rdenote the
reduction rmod l. Then r: Gal(F/F) →→ GL2(F). 4. If w|lis a prime of Fthen
r|we have dimQlGal( Fgrjw(ri/Fw⊗)is crystalline and for t: Ft,FwBDR) = 1 for j= 0,1
and = 0 otherwise.wl→ Ql
'' )ss Gal( Fof F split above qfor which v q /Fv q )
Gal( F/F
r|(Frobvq) has eigenvalues 1,#k(vq
q
'')
ni
s
s
G
a
l
(
F
5. There is a
prime vq
v q/Fv q).is unramified and r|
Then there is a Galois totally real field F/F over F and in which lis
unramified, and a prime wof F''over vqsuch that each Symm-1
is automorphic of weight 0 and type { Spn(1)}{wq}.
r|
Proof: Choose a rational prime and a prime vq' of Fabove q'
q'
n +1,...,ζ t+1
i
i
26
in F, • ris
unramified above q'vq ', •
r(Frob) has eigenvalues
1,q'', • q | (n'i+1) for i=
1,...,t, • q = qand q' = l.
Also choose a prime l'
such that • q
'splits
|
(
q
'
)
j
which splits in Q(ζ 1
• li),n'i), • l'is unramified in F,
• l |(#k(vq))j- 1 for j= 1,...,max{ n'} , • l
f
''>max(C(n,
o
r
j
=
1
,
.
.
.
,
max{ n'} +, • and ris unramified at l.
Choose an elliptic curve E1/Fsuch that
) and such that • l = l, q, or q'
1
n
• Ehas multiplicative reduction at v1q; • Ehas
goodordinary reductionabovel', but H1; • Gal( F/F)
→→ Aut(H1(E1Ч F,Z/l'Z)). The existence of such an
E1
1q' ;
Ч F,Z/l'
'
vq
1
1
q' (j)
' and
< 0 and val
1
over Ql
' .) If necessary we can twist E 1
v
q
• E11has good reduction above l; • Ehas
multiplicative reduction at vq' , but H1(E1
Ч F,Z/l
Z) is unramified at v
(E
'
over F
q
q
l
'
Ч F,Z/l'
1(E1
Z)(-1) be a perfect alternating
pairing. Thus Wgives a lisse etale sheaf over SpecF.
/SpecFdenote the moduli space for the functor which takes a locally
noetherian F-scheme Sto the set of isomorphism classes of pairs (E,i),
where p: E→ Sis an elliptic curve and where
Z) takes , to the duality
coming from the cup product. Then Xis a fine moduli space (as ll>2). It is a
smooth, geometricallyconnected, affine curve. Let Sdenote the set of palces
of Fabove ll
) corresponding to elliptic curves with multiplicative
reduction. Over Fwe can find an elliptic curve Ewith split
multiplicative reduction such that HGal( Fv q
Z) is
tamely ramified at l
results from the form of Hilbert irreducibility with weak approximation
(see [E]). (The existence of such an Eresults from taking a j-invariant
with val' (j) = 0 mod l. The existence of such an E' results from taking
the canonical lift of an ordinary elliptic curve over Fby a quadratic
character to ensure that it has split multiplicative reduction at v' . Let
W denote the free rank two Z/llZ module with Gal( F/F)-action
corresponding to rЧ H'Z) and let , : WЧ W-→ (Z/ll
~-→ R1p*(Z/ll'
W
Let XW
i: W
v
'
q
1
1(Eq
Ч Qq,Z/ll'Z) ~= W|
v
q
/F
denote the set of infinite places of Funion { vq,vq' } . Let S'. Let
Ωvqdenote the open subset of XW(Fvq2
q
)
0
vq
2
Z). We take j(Eq) ∈ Q.) This gives an
Fvqrational point of XW
is
non-empty.
'
1
,
e
2
27
v
(with alternating pairings). (Suppose that (Z/ll'Z)(1) has basis eand
that W has basis ewhere Gal( Fvq/Fvq) acts trivially on e/Fvq(ζq)) satisfy
(sll 'v q)/ll 'v q = e. Suppose that se = 1. Let s ∈ Gal( F= e+ae1for some
a∈ (Z/ll'qwith val, which lies in Ω)) = -amod ll. Hence Ωq
0
-Z
LetΩvq '
W denotetheopensubsetofXW
1
W
K
(Fvq '
vq '
v
v
W(F
)
(
χ
)
nr v
nr v/F
b
y
µ
l
IF v
over IFv
K
8
1
(
F
∨
)correspondingtoellipticcurves with multiplicative reduction. Again we see
that Ωis open and non-empty. If vis an infinite place of F then any elliptic
curve over F
rational point of X . Take Ωv = XW (F ). It is non-empty as GL2(Z/ll'
nr v
'
let Ωv
v
~ = ω-1
2
v
nr v/Fv
gives a
Fv
l
)
/
0
-l 2⊕ ωor there isZan exact sequence (0) -→
o
l
Z/lZ -→ W[l] -→ (Z/lZ)(-1)
-→ (0)
v
)
e
(
r
χ
H1
)
k
b
y
µ
l
IF v
Z) has
a unique conjugacy class of elements of order 2 and determinant -1.If vis a
place of F above l⊂ X) consist of pairs (E,i) such that Ehas good reduction.
This set is open and Gal(F)-invariant. It is also non-empty: For instance take
E= E. If vis a place of F above llet Ω⊂ X) consist of pairs (E,i) such that Ehas
good reduction. This set is open and Gal(F)-invariant. It is also non-empty:
From the theory of Fontaine-Lafaille we see that either W[l]|
. In the first case any lift to the ring of integers of a finite extension of
Fofasupersingularellipticcurveover k(v)willgiveapointofΩ. Soconsider the
second case. Let k/k(v) be a finite extension and E/kan ordinary elliptic curve
such that Frobacts trivially on E[l](k). Let Kdenote the unramified extension
of Fvwith residue field k. Enlarging kif necessary we can assume that
Frobalso acts trivially on W. Let χgive the action of Gal( k/k) on E[l]( k). By
Serre-Tate theory, liftings of E to O-1are parametrised by extensions of (Q8
(χ∨. If the l-torsion in such an extensionisisomorphic(overK)toW,
thecorrespondingliftingEwillsatisfy H(EЧ K,Z/lZ) ~= W. Extensions of (Ql/Zl-18
(χ1are parametrised by H(Gal( K/K),Zl(ol-2χ2)) (as χ = 1). The representation
WcorrespondstoaclassinH1(Gal( K/K),(Z/lZ)(ol))whichis‘peu-ramifiґe’. We
must show that this class is in the image of
(Gal( K/K),Zl(ol -2χ) -→ H1(Gal(K/K),(Z/lZ)(ol
2
l )(χ )) -→ H1
H
/Zl
l
-→ (Q
28
O
K
v v
)o
2
'
)) coming from the fact that χ= 1 mod l. By local duality, this image is the
annihilator of the image of the map
2
(Gal( K/K),(Q /Zl
(Gal(K/K),Z/lZ) coming from the exact sequence
/Zl 2)(χ) ll
)(χ
(0) -→ Z/lZ -→ (Q
2
'
) -→ (0). Because χis
unramified, this image consists of unramified homomorphisms, which
annihilate any ‘peu-ramifiґe’ class.
Byproposition2.1wecanfindafiniteGaloisextensionF/Fandanelliptic curve
E/Fwith the following properties.
ker(Gal( F/F)→Aut(W))
• F 'over F. • Fqand vqis totally real. • v' split in F''. • All
is linearly disjoin
primes above llare unramified in F''. • Ehas good
reduction at all places above l. • Ehas good reduction at
all places above l. • Ehas split multiplicative reduction
above vqand v1' . • H(EЧ F,Z/lZ) ~= r|Gal( F/F').q
• H1(EЧ F,Z/l'Z) is unramified above vq' and tamely ramified
above l'. By theorem 3.1 (and lemma 3.2) we see that there is
a totally real field
i
'' above vq
and a prime wq' of F ''
'F ker(Gal( F/F)→Aut(W))
nn(1)}=
'
Symmni{w-1qr|}(1)}{wq}G
''
''
'
''
from F
''
'
'
'
ni-1H1
n(1)}{wq ' }
q
beaprimeofF''
q
l
'
l
'
''
'''
n⊃
F'. Each
Symmni-1H1(1)}{wniq,wq ' }-1H1
'' /F' ' such that: • F/Fis Galois. • land lare unramified in Fis linearly disjoint over F.
F
• F(and hence Fis linearly disjoint over Ffrom Fker r). • Each Symm(EЧ F,Zl' )
is automorphic over Fand level prime to ll'of weight 0, type { Sp.
LetwabovevqandletF'''denotethefixedfieldofthedecomposition group of win
Gal(F''/F). Thus F(EЧ F,Z' ) is infact automorphic over F''of weight 0, type {
Spand level prime to ll. By lemma4.5.2 of[CHT] wesee that eachSymm(EЧ
F,Z' ) is also automorphic over F'''of weight 0, type { Spand level prime to ll.
Thus each Symmni-1H1(EЧ F,Z/lZ) ~is automorphic over Fnof weight 0 and
type { Sp-1. By theorem 5.2 of [T3] (and lemma 3.2) we see that each
Symmris automorphic over Fof weight 0 and type { Sp(1)}{wq}. Thus each
Symmni-1ris automorphic over Fof weight 0 and type { Spn(1)}{wq}. ✷ We will
need another little lemma, which is proved exactly as in corollary
4.5.4 of [CHT].
29
n
Lemma 3.4 Suppose that nis an even positiveinteger and that l>max{ n,3} is a
rational prime. Suppose also that ∆ is a subgroup of GSpn(F) containing
Spn(Fll). Let rdenote the standard (n-dimensional)representation of ∆. Then ad
r(∆) has no abelian quotient of order more than 2 and ∆ is big.
Theorem 3.5 Suppose that F is a totally real field and that nis an even positive
integer. Suppose also that l>max{ C(n),D(n),n} is a rational prime which is
unramified in F. Let vbe a prime of Fabove a rational prime q = l such that
(#k(vq))jq= 1 mod lfor j= 1,...,nand q |n+1. Suppose also that
r: Gal( F/F) -→ GSpn(Zl)
is a continuous representation with the following properties. 1. rhas multiplier
o1-n l. 2. rramifies at only finitely many primes. 3. Let rdenote the
semisimplification of the reduction of r. Then the image
rGal(F/F(ζl)) is big.
Q
w/
)
t is crystalline and for t: F
does not contain F(ζl
l
F
4. Fkerad r
DR
Fnr w
Gal( F
w
⊗
w
l
t
,
F
B
w
i
with
w(tn+1 w
/Fv q)
q
Gal( F/F qis
unramified and r|ss Gal(v q /Fv q )(Frobvq
q
''
F))n-1
Fv q
q
''
.
q}
n(1)}
). 5. If w|lis a prime of Fthen r|we ) = 1 for j= 0,...,n- 1 and = 0 otherwise. Moreover there is a
have dimj(ri
O- 1) = 0 such that
w
r|IF w ~ =
.
Vn[l]
ss Gal( 6. r| ) has eigenvalues 1,#k(v),...,(#k(v. Moreover r(I) is either trivial or
Fv q
contains a regular unipotent.
Then there is a totally real Galois extension F
/F and a place w
above v such that each r|
'')
{w
of Fis automorphic of weight 0 and type { Sp
which splits in Q(ζn+1) such that
Proof: Choose a rational prime l'
30
is unramified in Fand ris unramified above l'
' |(#k(vq))j
n+1 1
~ = 1⊕ o'-1⊕ ...⊕ o'1-n l
'
l
', and vq(t1
']t1|IF w
q(t1
'
q
'
1
n
[l']t1
• l' , • l>max{ n,C(n)} , • l- 1 for j= 1,...,n. Choose tthen w(t∈ Fwith the following
properties. • If w|llthen Vn[l- 1) = 0. • If w|lq(t1) <0, and v. • v) = 0 mod l) = 0
mod lif and only if Vn[l]t1is unramified at v. • Gal( F/F) → GSp(V) is
surjective. The existence of such an tresults from the form of Hilbert
irreducibility with weak approximation (see [E]). (One may achieve the
second condition by taking t1l'-adically close to zero.) On Vv qv
ss
n,l,t1 1
),...,a(#k(v ))n-1 |
Gal( qF
has eigenvalues
a,a#k(v
v q/F
q
1
'
(Fw
1
0(F
. As before we see that a= ±1. Let dbe a quadratic
character of Gal( Q/Q) which is unramified at llqand
which takes Frobqto a. Let W be the free rank two
Z/ll'n[l']t1Z-module with Gal( F/F)-action corresponding
to ( r⊗ d)Ч V. It comes with a prefect alternating
pairing , : WЧ W-→ (Z/llZ)(1- n). The scheme TWis
geometrically connected. Let Sqdenote the union of
the infinite places of F and { v} . Let S2'denote the set
of places of F above ll. For wan infinite place of F let
Ωw= TW) which is non-empty as all elements of order
two in GSpn(Z/ll'Z) with multiplier -1 are conjugate. Let
Ωvq⊂ TW(Fvq) denote the open subset of points lying
above { t∈ T) : vq(t) < 0} . It is non-empty as it contains
a point above tlet Ωw⊂ TW(Fnr w) denote the preimage
of { t∈ Tnr wn+1- 1) = 0} . It is open, Gal(Fnr w/Fw) :
w(t)-invariant and non-empty. Thus we may find a
finite Galois totally real extension F'0(F'/F and a point
t∈ T'are unramified in F') with the following properties. •
land lqsplits in F'.. • v
'
ker(Gal( F/F)→Aut(W))
over F.
• F is linearly disjoint from F 31
• Vn[l]t ~ = (r⊗ d)|Gal( F/F
'
n,l',t
n
'
')
'
'
'
'
'
w
/F' w )
h
a
s
for some
a'
e
i
g
e
n
v
a
l
u
e
s
a
. • Vis unramified above land crystalline above
labove lthen V[l']t. • If wis a place of F|IF 'w
above vqthen Vn,l',tss Gal( F. • If wis a place of
Frob#k(vq),...,a'(#k(vq))n-1= ±1. Moreover Vn
According to theorem 3.1 (and lemma 3.4)
and a place wq|vq of F ''
n(1)}{wq}
'
''
''
''
'
n,l',t
⊃ F'
of weight 0, type { Sp''' Ll
λ n(1
)}{
'''
wq}
'''
'''
'
'
in
Ga
l(F
qn ',t
,
'
l n(1)}'''{wq
{wq}
{wq}
n[l]t
n
detH1(AЧ F,Ql
n(1)}
t )⊗
ension F''/F'
L with the following properties. • F/Fis Galois. • land la
automorphic over F. • Vof weight 0, type { Spand lev
Fdenote the fixed field of the decomposition group f
According to lemma 4.5.2 of [CHT] the representatio
over Fand level prime to ll. Hence Vand rare automo
and type { Sp. Finally theorem 5.2 of [T3] tells us tha
Fof weight 0
and type {
Sp(1)}. Hence
ris
automorphic
over F''. ✷of
weight 0 and
type { Sp
4
A
p
pl
ic
at
i se that Fand L⊂ R are totally real fields and that A/F
equipped with an embedding i: L→ End(A/F). Recal
o1.10, proposition 1.4 and the discussion just before
nthat Aadmits a polarisation over F whose Rosati inv
Thus if λis a prime of Labove a rational prime lthe
siL.
= Lλ(o-1 l).
0S
up
po
Suppose also that mis a positive integer. For each finite place vof
Fthere is a two dimensional Weil-Deligne representation WDv(A,i) over
Lsuch that
32
for each prime λof Lwith residue characteristic ldifferent from the residue characteristic of vwe
have
WD(H1(AЧ F,Ql)|Gal( Fv/Fv)
⊗L Lλ) ~= WDv(A,i).
l
L(Symm mWDv(A,i),s).
We define an L-series
L(Symmm(A,i)/F,s) =
v |8
mIt converges absolutely, uniformly on compact sets, to a non-zero holomorphic function in
Res>1+m/2. We say that Symmv(A,i) is automorphic of type { ρ}v∈S, if there is a RAESDC
representation of GLm+1(Av) of weight 0 and type { ρ}v∈Ssuch thatF
-m/2
rec(pv)|Art -1
= Symm mWDv(A,i)
K K|
for all finite places vof F. Note that the following are equivalent.
1. Symmm(A,i) is automorphic over Fof type { ρv}v∈S. 2. For all finite places λof L, if lis the residue
characteristic of λ, then
Lλ
Lλ
'
v}v∈S
vSy
mm
Lλ
m(H1
(AЧ
F,Ql
)⊗Ll
}v∈S
m(H1(AЧ F,Ql)⊗Ll
) is automorphic over F of weight 0 and type { ρ. 3. For some rational
prime land some place λ|lof Lthe representation) is automorphic over F of
weight 0 and type { ρ. (The first statement implies the third. The second
statement implies the
first (by the strong multiplicity one theorem). We will check that the third ~ : Ql' '
be the prime o
implies the second. Suppose that Symm) arises from an RAESDC
representation p and an isomorphism i: Lλ ~→ C. Let lbe a rational prime
and let i
'
rl',i
= Symm l '→ C. Let λ'
λ
' )⊗L
Symmm(H1(AЧ F,Ql)⊗Ll
m
v
}
v∈S(AЧ
F,Q.)
'corresponding
l
' (p) ~
to (i)-1 . Then from the Cebotarev density theorem we see that
Lλ
m(H1(AЧ F,Ql' )⊗Ll '
(H1
L
33
' ). Thus Symm' ) is also automorphic over F of weight 0 and type { ρ
◦ i|L
0Theorem
4.1 LetFandLbetotallyrealfields. LetA/Fbeanabelianvariety of
dimension [L: Q] and suppose that i: L→ End(A/F). Let N be a finite set of even
positive integers. Fix an embedding L→ R. Suppose that Ahas multiplicative
reduction at some prime v'qof F. There is a Galois totally real field F'/F such
that for any n∈ N and any intermediate field F⊃ F''⊃ F with F'/F''soluble,
Symmn-1''Ais automorphic over F.
Proof: TwistingbyaquadraticcharacterifnecessarywemayassumethatA
hassplit multiplicativereductionatvqi.e. Frobvqhas eigenvalues1and#k(v1) on
H(AЧ F,Qlss Gal(F)|v q/Fv q)qfor all ldifferent from the residue characteristic of vq.
Choose lsufficiently large that
• lis unramified in F, • l>max{ n,C(n)}q))jn∈N, • l |
(#k(v1- 1 for j= 1,...,maxN , • Ahas good reduction at
all primes above l, • Gal( F/F) →→ Aut(H(AЧ
F,Z/lZ)/OL/lOL), • and lsplits completely in L.
1(If this were not possible then for all but finitely many primes lwhich split
completely in Lthere would be a prime λ|l of Lsuch that Gal( F/F) → Aut(H(AЧ
F,Z/lZ)⊗OLOL/λOLЧ) is not surjective. Note that for almost all such lthe
determinant of the image is (Z/lZ)(look at inertia at l) and the image contains a
non-trivial unipotent element (look at inertia at v1). Thus for all but finitely many
primes lwhich split completely in Lthere is a prime λ|l of Lsuch that the image
of Gal( F/F) → Aut(H(AЧ F,Z/lZ)⊗qOLOL/λO) is contained in a Borel subgroup
of GL2L1(Z/lZ) and its semisimplification has abelian image. It follows from
theorem 1 of section 3.6 of [Se2] that the image of Gal( F/F) → Aut(H(AЧ F,Ql)
⊗LLλ) is abelian for all land λ. This contradicts the multiplicative reduction at
vq'.) Choose a prime λ|lof L. Theorem 3.3 tells us that there is a Galois totally
real field F/Fin which l is unramified and a prime wqof F'n-1such that for any n ∈
N , Symm(H1(AЧ F,Ql) ⊗LlLλabove vq) is automorphic over F'nof weight 0, type
{ Sp(1)}{wn-1q}and level prime to l. By lemma 4.5.2 of [CHT] we see that
Symm(H1(AЧ F,Ql) ⊗LlLλ) is also automorphic over any F''as in the
34
''.
✷
n(1)}{wq}
j andlevelprimetol.
n-1
HenceSymm
'
/
F
0
q
=1
m
=1
'
'
aj
Gal(F''/F)
Gal(F/Fj)χj
j
j
) → CЧof GL2(AFj
L(Symmm
j
j
j
A is automorphic over F
t
heoremofweight0, type{ Sp
Theorem 4.2 LetFandLbetotallyrealfields. LetA/Fbeanabelianvariety of
dimension [L: Q] and suppose that i: L→ End(A/F). Fix an embedding L→ R.
Suppose that Ahas multiplicative reduction at some prime vthe function
L(Symmof F. Then for all m∈ Z(A,i),s) has meromorphic continuation to the
whole complex plane, satisfies the expected functional equation and is
holomorphic and non-zero in Res= 1+m/2.
Proof: We argue by induction on m. The assertion is vacuous if m<1.
Suppose that m∈ Zis odd and that the theorem is proved for 1 = m/Fbe as
in the conclusion of that theorem. Write<m. We will prove the theorem for
mand m+ 1. Apply theorem 4.1 with N = { 2,m+1} . Let F1 = Ind
j
j soluble, and χ
'/F
⊃
∈ Z, F'
j
j
⊃ F with F
F
j
2m(A,i)Ч
j
F(AFjj). Then we
see that
j
and
L(Symm
where is a homomorphism Gal(F. Then (A,i)Ч Fis automorphic arising from an
aj
RAESDC representation s), and Symmis automorphic arising from an
RAESDC representation pof GL
(A,i),s) =
L(pj ⊗
◦ Art F ),s)aj
(χ
aj
m-1
Fj
j
◦ Art ))Ч sj,s)
),s)aj
Fj
j
(A,i),s) =
and L(Symm
2pj
(A,i),s)L(Sym
2
m
m+1
L((Symm
)⊗ (χ L((p⊗ (χj◦ Art(A,i),s- 1) =
(See [T2] for similar calculations.) Our theorem for mand m+1 follows (for instance) from
[CP] and theorem 5.1 of [Sh] (and in the case m+1 = 2 also from [GJ]). ✷
Theorem 4.3 Let Fbe a totally real field. Let E/Fbe an elliptic curve with multiplicative
reduction at some prime vqof F. For all but finitely many places vof F we may write
ifev )(1- (#k(v))1/2
-ifev )
#E(k(v)) = (1- (#k(v))1/2
35
for a unique fv∈ [0,p]. Then the fpare uniformly distributed with respect
to the measure 2p sin2fdf.
Proof: This follows from theorem 4.2 and the corollary to theorem 2 of
[Se1], as explained on page I-26 of [Se1]. ✷
(2p)(2p)2-ns/2n(n-2)/8n/2Now fix an even positive integer n. Finally let us consider
the L-functions of the motives Vtfor t∈ Q. More precisely for each pair of
rational primes land pthere is a Weil-Deligne representation WD(Vl,t|Gal( Qp/Qp))
of Wassociated to the Gal( Qp/Qp)-module Vl,tQp(see for instance [TY]).
Moreover for all but finitely many pthere is a Weil-Deligne representation
WDp(V) of W Qptover Q such that for each prime l = pand each embedding
Q → Qthe Weil-Deligne representation WDp(Vtl) is equivalent to the
Frobenius semisimplificationWD(Vl,t|Gal( Qp/Qp))F-ss.
LetS(V)denotethefinitesetofprimesp forwhichnosuchrepresentationWDp(Vtt) =
Ш . If indeed S(Vt) = Ш , then we set L(Vt)exists. ItisexpectedthatS(V,s) equal
toG(s)G(s- 1)...G(s+1- n/2) pL(WDp(Vtt),s)
o(WDp(Vt),ψp,µp,s),
p
and o(Vt,s) = i-n/2
where
ψ is the additive Haar measure on Qp defined by µp(Zp
p
µp
) = 1, and ψp:
Qp→ C is the continuous homomorphism defined by
-2pix(x+y) = e
. Thefunctiono(Vt
t
t
t
L(V,s) = o(V,s)L(V,n- s).
36
,s)isentire. Theproductdefining
L(V,s) converges absolutely uniformly in
f
compact subsets of Res>1+m/2 and hence gives a holomorphic function
orx∈ Z[1/p]andy∈ Z
Res>1+m/2.
Theorem 4.4 Suppose that t∈ Q - Z. Then S(V) = Ш and the function
L(V,s) hasmeromorphic continuationto the whole complex plane
andsatisfies the functional equation
Proof: Chooseaprimeqdividing thedenominatoroft. Bylemma1.15and, for
instance, proposition 3 of [Sc] (see also [TY]), we see that Gal( Q/Q) acts
irreducibly on Vl,t. Let Gdenote the Zariski closure of the image of Gal(Q/Q)
in GSp(Vl,t) and let G0 lldenote the connected component of the identity in G.
Then G0 lln. Moreover as the action of Gal( Q/Q) on Vis reductive and (by
lemma 1.15) contains a unipotent element with minimal polynomial (T1)1-nhas multiplier o, we see that the multiplier map from G0 lto Gml,tis
dominating. By theorem 9.10 of [K1] (see also [Sc] for a more conceptual
argument due to Grojnowski) we see that G0 lis either GSpnor (GmЧ
GL2)/Gn-1y. (Here Gmembedded via (x,y) → xSymm→ GmЧ GL2mvia z→
(z1-n,z).) In either case we also see that Gl= G0 l. (In the second case use the
fact that any automorphism of SL2is inner.) Let Gdenote the image of
Gal(Q/Q) in PGSp(V[l]tl). The main theorem of [L] tells us there is a set Sof
rational primes of Dirichlet density zero, such that if l∈ Sthen either
PSp(V[l]t) ⊂ Gl ⊂ PGSp(V[l]t
Symm PSL2
al) ⊂ Gl ⊂ Symm qn-1PGL2(Fl
t
t
∈Q
t
{v|q}
n(1)}
'
'
n(A
F
F' )
l,i(p
~= V
|Gal( F'/F
j)
) or
n-1
l (F
in Sp(V[l]
q
l
Q
Gal( F/F)
kerV[l]
{v|q}
l~
). Choose a prime l∈ S such that l |val(t), l>max{ 2n+ 1,D(n)} and l = q. By
lemma 1.15 we see that the image of In) contains a unipotent element with
minimal polynomial (T- 1). Combining the above discussion with lemmas 3.2
and 3.4, we see that the image of Gal( Q/Q(ζ)) in GSp(V[l]) is big and that ζ.
Thus theorem 3.5 tells us that we can find a Galois totally real field F/Q such
that Vl,t|is automorphic of weight 0 and type { Sp. If Fis any subfield of F with
Gal(F/F) soluble, we see that there is a RAESDC representation p' of GLF' ) of
weight 0 and type { Spsuch that for any rational prime land any isomorphism
i: Qn(1)}→ C we have rl,t'). As a virtual representation of Gal(F/Q) write
1=
jInd Gal(F/Q) χj,
Gal(F/F
j
with Gal(F/Fj
jj
j
◦ Art ) soluble
Fj)).
∈ Z, where F ⊃ F
j)
where
a
rl,i(pFj
37
Gal(F/Fj) → CЧis a homomorphism. Then, for all rational primes land for
all isomorphisms i: Ql ~→ C, we have (as virtual representations of Gal(
Q/Q)) Vl,t= ajIndGal(F/Q) Gal(F/F⊗ (χ
j:
We deduce that, in the notation of [TY], WD(Vl,t|Gal( Qp/Qp))ssis independent of l
= p. Moreover by theorem 3.2 (and lemma 1.3(2)) of [TY], we see
thatWD(Vl,t|Gal( Qp/Qp))F-ssis pure. Hence by lemma 1.3(4) of [TY] we deduce
that WD(Vl,t|Gal( Qp/Qp))F-ssis independent of l = p, i.e. S(Vt) = Ш . Moreover
L(Vt,s) =
L(pFj ⊗
◦ Art Fj ),s)aj ,
j
(χj
from which the rest of the theorem follows. ✷
References
[BH] F.Beukers and G.Heckman, Monodromy for the hypergeometric
functionnFn-1, Invent. Math. 95 (1989), 325-354. [CHT] L. Clozel, M. Harris, and
R. Taylor, Automorphy for some l-adic lifts
of automorphic mod lGalois representations, preprint. [CP] J. Cogdell
and I. Piatetski-Shapiro, Remarks on Rankin-Selberg convolutions, in “Contributions to automorphic forms, geometry, and
number theory”, Johns Hopkins Univ. Press, Baltimore, MD, 2004.
[CPS] E. Cline, B. Parshall, and L. Scott, Cohomology of finite groups of Lie
type I, Inst. Hautes tudes Sci. Publ. Math. 45 (1975), 169–191.
[DMOS] P. Deligne, J. S. Milne, A. Ogus, K.-Y. Shih, Hodge cycles, motives
and Shimura varieties, LNM 900, Springer 1982.
[DT] F. Diamond and R. Taylor, Nonoptimal levels of mod lmodular
representations, Invent. Math. 115 (1994), 435-462.
[E] T.Ekedahl, An effective version of Hilbert’s irreducibility theorem, in
“Sґeminaire de Thґeorie des Nombres, Paris 1988–1989”, Progress in Math.
91, BirkhЁauser 1990.
[G] P. A. Griffiths, On the periods of certain rational integrals I, Ann. of Math.
90, (1969) 460–495.
[GJ] S. Gelbart and H. Jacquet, A relation between automorphic
representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. 11 (1978),
471–542.
38
[GPR] B. Green, F. Pop, and P. Roquette, On Rumely’s local-global principle,
Jahresber. Deutsch. Math.-Verein. 97 (1995), 43–74.
[HT] M. Harris and R. Taylor, The geometry and cohomology of some simple
Shimura varieties, Annals of Math. Studies 151. PUP 2001.
[I] Y.Ihara, On modular curves over finite fields, in “Discrete subgroups of Lie
groups and applications to moduli” OUP, Bombay, 1975.
[K1] N.M.Katz,Gausssums, Kloostermansums, andmonodromygroups, Annals
of Math. Studies 116, PUP 1988.
[K2] N. M. Katz, Exponential Sums and Differential Equations, Annals of Math.
Studies 124, PUP 1990.
[L] M. Larsen, Maximality of Galois actions for compatible systems, Duke
Math. J. 80 (1995), 601-630.
[LSW] W. Lerche, D-J. Smit, and N. P. Warner, Differential equations for
periods and flat coordinates in two-dimensional topological matter theories,
Nuclear Phys. B 372 (1992), 87–112.
[M] D. R. Morrison, Picard-Fuchs equations and mirror maps for
hypersurfaces, in “Essays on mirror manifold” International Press 1992.
[MB] L. Moret-Bailly, Groupes de Picard et probl`emes de Skolem II, Ann.
Scient. Ec. Norm. Sup. 22 (1989), 181–194.
[MVW] C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler, Congruence
properties of Zariski dense subgroups I, Proc. Lon. Math. Soc. 48 (1984),
514–532.
[N] M.Nori, On subgroupsofGLn(Fp), Invent.Math.88(1987), 257–275. [R] M.
Rapoport, Compactifications de l’espace de modules de HilbertBlumenthal, Compositio Math. 36 (1978), 255–335. [Sc] A. J. Scholl,
On some l-adic representations of Gal( Q/Q) attached
to noncongruence subgroups, to appear Bull. London Math. Soc.
[Se1] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves, W.
A. Benjamin (1968). [Se2] J.-P. Serre, Propriґetґes galoisiennes des
points d’ordre fini des courbes
elliptiques, Invent. Math. 15 (1972), 259-331.
39
[SGA1] A. Grothendieck, Revetements ґetales et groupe fondamental, LNM
224, Springer 1971.
[SGA7] P. Deligne and N. Katz, Groupes de monodromie en gґeomґetrie
algґebrique II, LNM 340, Springer 1973.
[Sh] F. Shahidi, On certain L-functions, Am. J. Math. 103 (1981), 297– 355.
[T1] R. Taylor, Remarks on a conjecture of Fontaine and Mazur, J. Inst
Math.Jussieu 1 (2002), 1–19.
[T2] R. Taylor, On the meromorphic continuation of degree two Lfunctions, to
appear Documenta Math.
[T3] R. Taylor, Automorphy of some l-adic lifts of automorphic mod l
representations II, preprint.
[TW] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke
algebras, Ann. of Math. 141 (1995), 553–572.
[TY] R. Taylor and T.Yoshida, Compatibility of local and global Langlands
correspondences, to appear JAMS.
[W] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math.
141 (1995), 443–551.
40