Lectures on Modular Forms and Galois Representations
Wen-Ch’ing Winnie Li
at NCTS, Autumn, 2006.
From 9/13 to 11/29.
Introduction
0.1
Modular forms
The group SL2 (Z) acts on H = {z ∈ C | Im z > 0} by linear transformation, i.e.
az + b
a b
.
γ=
∈ SL2 (Z), γz =
c d
cz + d
I Let Γ be a subgroup of SL2 (Z) of finite index. Given a positive integer k, a modular form
f for Γ of weight k is a function satisfies:
(1) f is holomorphic on H,
a b
k
(2) f (γz) = (cz + d) f (z), ∀γ =
∈ SL2 (Z),
c d
(3) f is holomorphic at the cusps of Γ.
I Mk (Γ) = the space of modular forms of weight k for Γ. This is finite dimensional over C.
f ∈ Mk (Γ) is called a cusp form if
(4) f vanishes at all cusps of Γ.
Sk (Γ) = the space of cusp forms.
I We compactify the quotient Γ\H by adding cusps to get a modular curve XΓ . It is a
Riemmann surface of genus gΓ . The space Ω1 (XΓ ) of holomorphic differential forms on XΓ
is a vector space of dimension gΓ . The map f (z) 7→ f (z)dz is an isomorphism from S2 (Γ) to
Ω1 (XΓ ).
I Γ is called a congruence subgroup if it contains
(
a b
1 0
a b
≡
Γ(N ) =
∈ SL2 (Z)
c d
0 1
c d
)
(mod N )
for some N.
If Γ is a congrunce subgroup, then Mk (Γ) = Ek (Γ) ⊕ Sk (Γ) where Ek (Γ) denotes the space of
Eisenstein series.
1
I We are interested in two special kinds of congruence subgroups:
a b
Γ0 (N ) = {
∈ SL2 (Z) | c ≡ 0 (mod N )}
c d
∪
Γ1 (N ) = {
a b
c d
∈ SL2 (Z) |
a b
c d
≡
1 ∗
0 1
(mod N )}
.
×
Γ1 (N ) C Γ0 (N ) with Γ0 (N ) Γ1 (N ) ' Z/N Z .
I So Γ0 (N ) acts on Sk (Γ1 (N )). Then Sk (Γ1 (N )) decomposes into
M
Sk (Γ1 (N )) =
Sk (N, χ) where
χ∈((Z/N Z)× )∨
Sk (N, χ) =
∗ ∗
k
f ∈ Sk (Γ1 (N ))f (γz) = χ(d)(cz + d) f (z), ∀γ =
∈ SL2 (Z)
c d
= the space of cusp forms of weight k level N and character χ
I Theory of newforms =⇒ it suffices to understand all Sk (N, χ).
Let f ∈ Sk (N, χ). Then f (z + 1) = f (z) such that f has Fourier expansion
∞
X
f (z) =
an e2πinz .
n=1
We want to understand the arithmetic of the Fourier coefficients. Further, we associate to f
an L-function:
∞
X
L(s, f ) =
an n−s converges absolutely for Re s > k.
n=1
I For p - N, there is the Hecke operator Tp defined by
p−1
1X
Tp f (z) =
f
p
u=0
=
=
∞
X
n=1
∞
X
z+u
p
2πiz
anp e
+ χ(p)pk−1 f (pz)
+ χ(p)p
k−1
∞
X
an e2πinpz
n=1
anp + χ(p)pk−1 an/p e2πinz .
n=1
Here, ax = 0 if x 6∈ Z.
I For p | N, we have the operator Up defined by
Up f (z) =
∞
X
n=1
2
anp e2πinz .
Theorem Let f (z) =
∞
P
an e2πinz ∈ Sk (N, χ). Then L(s, f ) is Eulerian at p - N if and only
n=1
if f is an eigenfunction of Tp . Moreover, if Tp f = λp f, then


X
1
L(s, f ) = 
.
am m−s  ·
1 − λp p−s + χ(p)pk−1−2s
m≥1,p-m
I For p - N, r ≥ 1,
Tpr+1 = Tp Tpr − hpipk−1 Tpr −1 .
Here hpi represents the operator on Sk (Γ1 (N )) such that on Sk (N, χ) it is given by χ(p).
I For p | N define
Tpr = (Up )r .
For integer n = pe11 · · · pess ≥ 1 define Tn = Tpe1 · · · Tpess where T1 = id .
1
I For a congruence subgroup Γ with Γ1 (N ) ⊆ Γ ⊆ Γ0 (N ), we have
M
Sk (Γ) =
Sk (Γ, χ).
χtrivial on Γ/Γ1 (N )
Q Q
I Define the Hecke algebra
= Sk (Γ) to be the algebra over C generated by Tn , n ≥ 1 and
×
hdi for d ∈ Z/N Z on Sk (Γ).
Formally, we have
∞
X
n=1
Tn n−s =
Y
p-N
Y
1
1
.
1 − Tp p−s
1 − Tp p−s + hpipk−1−2s
p|N
I For p - N, Peterson inner product h·, ·i on Sk (Γ, χ), we have hTp f, gi = χ(p)hf, Tp gi.
In other words, the transpose of Tp is hpi−1 Tp as operators. Consequently, Tp on Sk (Γ, χ)
is diagonalizable. Since the Tp ’s commute, the operators Tp , p - N can be simutaneously
diagonalizable. Thus,
M
Sk (N, χ) =
(common eigenspace of Tp , p - N ).
If f is a non-zero function in a common eigenspace of dimension one, it is called a newform
of weight k, level N and character χ. A common eigensapce of dimension greater than one
N
is spanned by a newform g of weight k level M | N and character χ and g(mz) for m M
.
So it is sufficient to understand newforms of all levels. Each common eigenspace for p - N
containsPa unique subspace which is of one dimensional invariant under all operators in T. Let
g(z) = an (g)e2πinz be a non-zero common eigenform for T. Then, a1 (g) 6= 0. We normalize
it such that a1 (g) = 1. Then, Tm (g) = am (g)g for all m ≥ 1. The map Tm 7→ am (g) gives
rise to an algebra homomorphism T → C. Denote TQ(A) the algebra generated by Tm , hdi
over A (any ring ⊇ Q). This map Tm 7→ am (g) is an algebra homomorphism from TQ to
Kg = Q(am (g), m ≥ 1) the field of Fourier coefficients of g. In particular, g can be a newform.
We shall show that the field of coefficients of a normalized newform is a finite extension of Q,
and the eigenvalues of Tp on g for p - N are algebraic integers.
3
Theorem (Strong Multiplicity One Theorem)
For any two distinct common eigenspaces, the eigenvalues of Tp are different for inifintely
many p.
In particular, if for the two eigenspaces, the eigenvalues of Tp are equal for almost all p then
these two eigenspaces are the same.
I Fromally, we have
∞
X
Tn n−s =
n=0
Y
p-N
Y
1
1
.
−s
k−1−2s
1 − Tp p−s
1 − Tp p + hpip
p|N
∞
P
In summary, for a normalized newform f (z) =
an e2πinz of weight k level N and character
n=1
χ, we have
L(s, f ) =
Y
p-N
1 − ap
p−s
Y
1
1
.
k−1−2s
1 − ap p−s
+ χ(p)p
p|N
I Ramanujan Conjecture. What about the size of ap ?
For p | N, we have

(k−1)/2 if χ is not a character mod N ,

p
|ap | = p
ap = 0
if p2 | N and χ is a character mod Np ,

 2
ap = χ(p)pk−1 if p||N and χ is a character mod Np .
For p - N, there is the celebrated Ramanujan conjecture that |ap | ≤ 2p(k−1)/2 . Viewing 1 −
ap p−s + χ(p)pk−1 p−2s as a quadratic polynomial in p−s then this conjecture has the following
equivalent form. Write
1 − ap p−s + χ(p)pk−1 p−2s = (1 − αp p−s )(1 − βp p−s )
then
|ap | ≤ 2p(k−1)/2 ⇐⇒ |αp | = |βp | = p(k−1)/2 .
This conjecture is proved by Deligne-Serre for k = 1, Eichler-Shumura for k = 2 and Deligne
for k ≥ 3 using Galois representations.
P
I Sato-Tate Conjecture. Let f (z) =
an e2πinz be a normalized newform fo weight
n≥1
k level N and trivial character (i.e. f ∈ Sk (Γ0 (N ))). Write αp = p(k−1)/2 eiθp then βp =
p(k−1)/2 e−iθp . We may assume that θp ∈ [0, π]. How are the θp ’s distributed? Or, equivalently,
consider lp = αp + βp = p(k−1)/2 2 cos θp and put rp = lp /p(k−1)/2 so that rp ∈ [−2, 2]. We want
to know how rp ’s distributed on [−2, 2]?
Sato-Tate Conjecture:
Suppose that f does not have complex multiplication, i.e. L(s, f ) is not the L-function attached to any idéle class character of an imaginary quadratic extension of Q. Then, the rp ’s
are uniformly distributed with respect to the Sato-Tate measure:
r
1
x2
µst =
1−
dx on [−2, 2].
π
4
4
In other words, for any interval [a, b] ⊆ [−2, 2],
#{p ≤ x | rp ∈ [a, b]}
lim
=
x→∞
#{p ≤ x}
Z
b
µst .
a
In terms of θp ’s, this means that θp ’s are uniformly distributed with respect to
which arises from measure on conjugacy classes of SU 2 .
0.2
1
π
sin2 θ dθ,
Elliptic curves
Algebraically, an elliptic curve E defined over a field K is a smooth curve of genus one over K
containing at least one K-rational point. If K is a subfield of C, then E(C) can be identified
with a torus C/L for some lattice L in C. Up to equivalence, we may assume that L has a
Z-basis {τ, 1} with τ ∈ H. There is a group structure on E making it an abelian additive
group. For N ≥ 1, denote by E[N ] = {P ∈ E | N P = O} = the group of poitns of order
dividing N.
If char(K) - N then E[N ] ' Z/N Z × Z/N Z.
I (`-adic representation of GQ .)
Let E be an ellitpic curve defined over Q, and let ` be a prime. Clearly, E[`r+1 ] ⊃ E[`r ] for
all r ≥ 1. We let
T` (E) := lim E[`r ],
←−
called the Tate-module. Clearly, T` (E) ' Z` × Z` is a rank two Z` -module. The Galois
group GQ = Gal(Q/Q) acts on each E[`r ] and hence on T` (E). This yields a continuous
homomorphism
ρ`,E : GQ → Aut(T` (E)) = GL2 (Z` ) ,→ GL2 (Q` ).
This is a degree two `-adic representation of GQ .
Denote by N the conductor of E. Then, ρ`,E is unramified outside N `. More precisely, for each
prime p, we have the p-adic valuation on Q. Extending this to a valuation on Q (there are
many such extensions). The automorphisms in GQ preserving this valuation form a subgroup
Gp , called a decomposition group at p. All decomposition groups at p are conjugate in GQ .
The group Gp ' Gal(Qp /Qp ). Denote by Ip the inertia group of Gp such that
Gp /Ip ' Gal(Qur
p /Qp ) ' Gal(Fp /Fp ).
We have the following exact sequence of groups:
1 → Ip → Gp → Gal(Fp /Fp ) → 1.
Let Frobp be a preimage of the Frobenius automorphism in Gal(Fp /Fp ). Then, Frobp is welldefined mod Ip . When we change Gp , the Frobenius Frobp is changed by a conjugate (mod
inertia). A representation ρ is unramified at p if it is trivial on Ip . Then, ρ(Frobp ) is welldefined up to conjugacy. In particular, Tr ρ(Frobp ) and det ρ(Frobp ) are well-defined.
On the other hand, for p - N, the ellitpic curve E has good reduction mod p, i.e. there is a
defining equation of E such that after reducing modulo p, it defines an ellitpic curve Ep over
5
Fp . Let Ep (Fpr ) denote the set of Fpr -rational points on Ep . For ` 6= p, by a direct computation,
one has
ap := 1 + p − #E(Fp ) = Tr ρ`,E (Frobp ) which is independent of `
p = det ρ`,E (Frobp ) independent of `.
Moreover, the Leftschetz fixed point formula gives
#E(Fpr ) = 1 + pr − Tr ρ`,E (Frobrp ).
Recall the zeta function for Ep is defined as
∞
X
Z(Ep , p−s ) := exp
r=1
p−rs
#E(Fpr )
r
!
det(1 − ρ`,E (Frobp )p−s )
(1 − p−s )(1 − p1−s )
1 − ap p−s + p1−2s
=
(1 − p−s )(1 − p1−s )
=
Define the Hass-Weil L-function attached to E by
L(s, E) =
1
Y
p-N
1 − ap
p−s
Y
+
p1−2s
p|N
1
1 − ap pL−s
where for p | N,


1
ap = −1


0
0.3
if E has split multipicative reduction mod p
if E has non-split multiplicative reduction mod p
if E has additive reduction mod p.
Galois representations and modular forms
We begin with the following result.
Theorem (Eichler-Shumura)
∞
P
an e2πinz be a normalized newform of weight 2 level N and trivial character
Let f (z) =
n=1
such that an ∈ Z for all n ≥ 1. Then there exists an elliptic curve E defined over Q such that
L(s, f ) = L(s, E)(for p - N ).
I We view ρ`,E as `-adic representation attached to f. In general, given a normalized newform
∞
P
f (z) =
an e2πinz of weight k level N and character χ, for a prime `, let K be a finite
n=1
6
extension of Q` . A representation ρ`,f : GQ → GL2 (K) is called an `-adic representation
attached to f if ρ`,f is unramified outside N ` and for each p - N `,
Tr ρ`,f (Frobp ) = ap ,
det ρ`,f (Frobp ) = χ(p)pk−1 .
An `-adic representation ρ is said to be modular if in addition to the above property,i.e.
ρ = ρ`,f , it is also odd (meaning det ρ`,f (complex conjugate) = −1 and it is irreducible.
I (Taniyama-Shimura Modularity Conjecture)
Given an elliptic curve E defined over Q there is a newform of weight 2 such that
L(s, f ) = L(s, E).
Wiles and Taylor-Wiles proved this conjecuture for E is semi-stable, completed by BreuilConrad-Diamond-Taylor.
Theorem (Eichler-Shimura for k = 2, Deligne for k ≥ 3)
Given a normalized newform f of weight k ≥ 2, there exists `-adic representation ρ`,f attached
to f.
Theorem (Deligne-Serre for k = 1)
Given a normalized newform f of weight one, there exists complex degree two irreducible
representation of GQ attached to f.
Note that for ρf : GQ → GL2 (C), Im ρf is a finite subgroup of GL2 (C).
I Call an `-adic representation ρ of GQ geometric if
(1) it is unramified outside a finite set of places of Q,
(2) the restriction of ρ to all decomposition groups are potentially semi-stable (p.s.s.)
(By Grothendieck, ρ is automatically p.s.s. at all p 6= `.)
I Fontain-Mazur Conjecture:
Let ρ be an odd irreducible degree two `-adic representation of GQ . Then, ρ is modular ⇐⇒
it is geometric.
I Mod ` representation of GQ .
Given a modular `-adic represenation of GQ , we can psss it to the redidue field to get an odd
degree two representation from
GQ → GL2 (F` ).
Serre’s conjecture:
(1) Any odd irreducible degree two representation ρ : GQ → GL2 (F` ) arises from reduction
of an `-adic modular representation.
(2) An explicit recipe for the a newform f of lowest weight and level such that (1) holds
is given.
7
1
1.1
Eichler-Shimura Relation and Galois Representation
Homology group and rationality
Let Γ be a finite index subgroup of SL2 (Z). The modular curve XΓ has universal cusp
H∗ = H ∪ {cusps} = H ∪ Q ∪ {i∞}
I dimC S2 (Γ) = gΓ and S2 (Γ) ' Ω1 (XΓ ).
I H1 (XΓ , Z) is a free Z-module of rank 2gΓ .
I Define the map from
H1 (XΓ , Z) × S2 (Γ) −→ C
Z
(c, f (z)) 7→
f (z)dz.
c
R
If c 6= 0, then ∃ f such that c f (z)dz
R 6= 0. Hence
R γτ0we may embed H1 (XΓ , Z)∗ into Hom∗C (S2 (Γ), C)
as a lattice Λ of rank 2gΓ . Here c f (z)dz = τ0 f (z)dz by lifting c to H . Since H is simply
connected, the integral is independent of the path from τ0 to γτ0 , and in fact, τ0 can be any
point of H∗ . Conversely, given γ ∈ Γ, we define a cycle cγ in H1 (XΓ , Z) as the image of τ0 to
γτ0 for any choice τ0 ∈ H∗ so that for any f ∈ S2 (Γ)
Z
Z γτ0
f (z)dz =
f (z)dz
t0
cγ
I Call c : Γ → H1 (XΓ , Z) the map γ →
7 cγ . For γ, γ 0 ∈ Γ, f ∈ S2 (Γ)
Z
Z γγ 0 τ0
f (z)dz =
f (z)dz
τ0
cγγ 0
Z
γ 0 τ0
=
Z
γγ 0 τ0
+
!
f (z)dz
γ 0 τ0
τ0
Z !
Z
+
=
cγ 0
f (z)dz
cγ
i.e. c(γγ 0 ) = c(γ) + c(γ 0 ) so that c is a homomorphism. If γ is elliptic (resp. parabolic), we
may choose τ0 to be the fixed point of γ in H (resp. cups), then c(γ) = 0.
I Denoted by Γep the subgroup generated by elliptic and parabolic elements in Γ. It is a
normal subgroup contained in ker c.
ab
to
Since H1 (XΓ , Z) is abelian, c factors through a surjective homomorphism from Γ/Γ
ep
H1 (XΓ , Z). In fact, this is an isomorphism.
I Shall represent elements in H1 (XΓ , Z) by [γ] for γ ∈ Γ. Now let Γ = Γ1 (N ) or Γ0 (N ).
We know that the Hecke algebra T acts on S2 (Γ).
For m
≥ 1, one way to define the Hecke
S
1 0
operator Tm is to consider the double coset Γ
Γ = Γαi . The action of Tm on
0 m
i
f ∈ S2 (Γ) defined before can be rephrased as
X
Tm (f (z)dz) =
f (αi z)d(αi z).
i
8
×
For d ∈ Z/N Z , choose a matrix Rd =
a b
c d
∈ Γ0 (N ). Then
< d > (f (z)dz) = f (Rd z)d(Rd z).
I We want to define the action of T on H1 (XΓ , Z) such that
(T c, f ) = (c, T f ) ∀ T ∈ T
Write c = [γ],
γτ0
Z
Tm f (z)dz =
([γ], Tm f ) =
τ0
=
f (αi z)d(αi z)
τ0
i
XZ
γτ0
XZ
αi γτ0
f (z)dz
αi τ0
Z
X γi αj(i) τ0
i
=
i
=
X Z
XZ
i
where
P R αj(i) τ0
i
αi τ0
γi αj(i) τ0
[γi ]
αj(i) τ0
Z
!
+
αj(i) τ0
i
=
f (z)dz, by writing αi γ = γi αj(i) for some γi ∈ Γ
αi τ0
f (z)dz
αi τ0
f (z)dz +
αj(i) τ0
XZ
i
f (z)dz
αi τ0
f (z)dz = 0, since αj(i) goes through all coset representatives as αi does. So,
we should defined
X
Tm [γ] =
[γi ],
where αi γ = γi αj(i) . Similarly
< d > [γ] = [Rd γRd−1 ].
This defines the action of TZ on H1 (XΓ , Z).
H∗ .
a b
c d
I The map τ 7→ −τ̄ dinfes an involution on
If γ2 = γτ1 for some γ =
∈ Γ,
a −b
then τ2∗ = γ ∗ τ1∗ with γ ∗ =
∈ Γ. So ∗ induces an involution on XΓ , and hence on
−c
d
H1 (XΓ , Z) := H1 (Z). Let H1 (R) = H1 (Z) ⊗Z R and H1 (C) = H1 (Z) ⊗Z C.
Extend the involution to H1 (R) and H1 (C) by linearity. We have H1 (R) = H1 (R)+ ⊕ H1 (R)−
and H1 (C) = H1 (C)+ ⊕ H1 (C)− as a direct sum of eigenspaces with eigenvalues ±1.
We have the following properties:
R
(1) H + , H − each have dimension gΓ and (c, f ) = c f (z)dz exhibits duality between H1 (C)+
and Ω1 (XΓ ) (or S2 (Γ)).
(2) T [γ ∗ ] = (T [γ])∗ , ∀ T ∈ T, so T preserves the eigenspaces. T |H1 (C)+ is the transpose of
T on S2 (Γ).
9
(3) H1 (Z) ∩ H1 (R)+ is lattice of rank gΓ and a Z-basis of this intersection is a basis of
H1 (C)+ .
Consequnce
(3)=⇒ every T ∈ T on H1 (C)+ can be represented by a gΓ × gΓ matrix with coefficients in Z.
(2)=⇒ Same holds for T ∈ T on S2 (Γ)
=⇒ the characteristic polynonmial of T is monic with coefficients in Z.
=⇒ all eigenvalues of T are algebraic integers.
=⇒ Any normalized newform of weight 2 has algebraic integral Fourier coefficients.
=⇒ TQ is finitely generated.
L
I Recall that S2 (Γ1 (N )) = (common eigenspace for Tp , p - N , all < d >). Each common
eigenspace contains a unique normalized newform of level dividing N . Let g1 , · · · , gt be the
normalized of weight 2 level N1 , · · · , Nt . Denote the common eigenspace containing gi by
N
S(gi ). Then S(gi ) has a basis gi (mz) for all positive divisors m of N
. On S(gi ), the Tp for
i
×
p - N acts as scalar multiple by ap (gi ) for d ∈ Z/N Z , < d > acts as multiple by χi (d) where
χi is the character of gi .
N
m
Observe that if a prime q m N
,
then
T
g
(mz)
=
g
q
i
i
q z . This shows that S(gi ) is a TC i
N
module generated by gi N
z . With respect to decomposition
i
S2 (Γ1 (N )) =
M
S(gi ).
i
The Hecke algebra TC =
L
TC |S(gi ) decomposes. Put together, let
g(z) = g1
N
z
N1
+ · · · + gt
N
z
Nt
∈ S2 (Γ1 (N )).
P
Then S2 (Γ1 (N )) as a TC -module generated by g. Moreover, if h =
an e2πinz is a nonzero
element in S2 (Γ1 (N )) with am 6= 0, then Tm h = am + higher terms 6= 0.
Proposition S2 (Γ1 (N )) is a free TC -module of rank 1. The same is true for S2 (Γ0 (N )),
that is, S2 (Γ0 (N )) is a also a free TC -module of rank 1.
I Now suppose gi is a newform of level N . Then S(gi ) is 1-dimensional and the action of Tm is
multiplication by am (gi ), the mth Fourier coefficient of gi . The algebra TQ |S(gi ) is nothing but
Q(am , m ≥ 1) = Kgi , the field of coefficients of gi . (Note that for p - N, ap2 = (ap )2 − χ(p)p,
×
so χ(d) ∈ Kgi , ∀d ∈ Z/N Z ). Since TQ is finitely generated ⇒ [Kgi : Q] < ∞.
I Let σ ∈ Gal(Q/Q). The map Tm 7→ σ(am ) and < d >7→ σ(χi (d)) is also a homomorphism
from P
TQ to Q. So there exists gj ∈ S2 (Γ) which realizes this homomorphism. In other words,
gj = σ(an )e2πinz = giσ is a newform of weight 2 level N and character χσi .
10
Theorem Let f (z) =
∞
P
an e2πinz be a newform of weight 2 level N and character χ. Then
n=1
(1) all Fourier coefficients of f are algebraic integers
(2) Kf = Q(an , n ≥ 1) is a finite extension of Q
P
(3) for any σ ∈ Gal(Q/Q), f σ =
σ(an )e2πinz is a normalized newform of weight 2 level
N and character χσ .
By the theorem above, there is a finite Galois extension K such that all forms in S2 (Γ1 (N ))
with Fourier coefficients in K span S2 (Γ1 (N )) and the set of such forms form a vector space
over K invariant under Gal(K/Q).
Using Hilbert Theorem 90, S2 (Γ1 (N )) has basis with coefficients in Q. The integrality property
of the newforms (and their push-ups) implies one can actually find a basis in Z.
Corollary S2 (Γ) has a basis with coefficients in Z.
Denote by S2 (Γ, Z) the set of forms with Fourier coefficients in Z. For any ring A, let
S2 (Γ, A) = S2 (Γ, Z) ⊗Z A.
Corollary S2 (Γ, Z) is invariant under TZ .
Proposition Let A be a ring. Then S2 (Γ, A)∨ := HomA (S2 (Γ, A), A) is a free TA -module of
rank 1.
Proof. It suffices to prove the case A = Z. Consider the pairing TZ × S2 (Γ, Z) → Z given by
(T, f ) 7→ a1 (T f ) the 1st Fourier coefficient of T f .
(i) It is TZ -equiinvariant. Given T, T 0 ∈ TZ , f ∈ S2 (Γ, Z), we have
(T 0 T, f ) = a1 (T 0 T f ) = a1 (T T 0 f ) = (T, T 0 f ).
(ii) The pairing is nondegenerate
If f is nonzero, say the mth Fourier coefficient am 6= 0, then
(Tm , f ) = a1 (Tm f ) = am 6= 0.
Suppose T ∈ TZ is such that (T, f ) = a1 (T f ) = 0, ∀ f , want to show T F = 0, ∀ f (Then
T = 0). If T f 6= 0 for some f , then T f has a nonzero Fourier coefficient, say am . We
have
(T, Tm f ) = (Tm , T f ) = a, 6= 0
a contradiction.
Corollary HomC (TC , C) ' HomC (S2 (Γ), C) ' TC as TC -module, i.e., TC is Gorenstein.
Recall that Λ = the image of H1 (XΓ , Z) in HomC (S2 (Γ), C) a Z-module of rank 2gΓ .
11
Corollary Λ ⊗ Q is a rank 2 TQ -module.
Proof. Consider Λ = Λ+ + Λ− , and
Λ ⊗ C = H1 (C)+ + H1 (C)−
||
+
Λ ⊗C
||
−
Λ ⊗C
Have H1 (C)+ ' S2 (Γ) ' H1 (C)− as TC -module.
This implies Λ± ⊗ Q is a free rank 1 TQ -module.
1.2
Jacobian of XΓ
I Let Γ be a finite index subgroup of SL2 (Z). Suppose XΓ has genus g ≥ 1. We know that
H1 (XΓ , Z) is embedded as a rank 2gΓ lattice Λ in HomC (S2 (Γ), C) ' CgΓ .
.
CgΓ Λ ' Hom(S2 (Γ), C) Λ
is the Jacobian of XΓ .
I Choose a basis f1 , · · · , fgΓ of S2 (Γ) and c1 , · · · , c2gΓ a Z-basis of H1 (XΓ , Z). The vectors
Z
Z
f1 (z)dz, · · · ,
fgΓ (z)dz ∈ CgΓ , for i = 1, · · · , 2gΓ
ci
ci
are linearly independent over R and they generate a rank 2gΓ lattice Λ̃ in CgΓ .
I Define a map
g
Φ : XΓ −→ C Γ Λ̃
Z τ
Z τ
fgΓ (z)dz
f1 (z)dz, · · · ,
τ 7→
τ0
τ0
for a fixed τ0 ∈ H∗ .
P
P
Extend this to Φ# : Div(XΓ ) → CgΓ /Λ̃ by Φ# ( nτ τ ) =
nτ Φ(τ ). Restrict this map to
0
Div (XΓ ).
0
/ CgΓ
Φ# |Div0 (XΓ ) : Div
eJJ (XΓ )
Λ̃
JJ
{=
JJ τ − τ
{
Φ(τ ){{
JJ
cH0
JJ
{
HH
x;
JJ
HH
xx {{{
JJ
x
H
x {
HH
JJ
xx {{
H
JJ
JJ H 8xx {{{
JJ τ {{
JJ
{
XΓ
since Φ(τ0 ) = 0.
12
Theorem (Jacobi
Inversion Theorem)
g
Given any v ∈ C Γ Λ̃, there exists gΓ points τ1 , · · · , τgΓ such that
v = Φ(τ1 ) + · · · + Φ(τgΓ ) = Φ# (τ1 + · · · + τgΓ − gΓ τ0 ).
So Φ# |Div0 (XΓ ) is surjective.
Theorem (Abel’s Thoerem)
0
The kernel of Φ# on Div.
(XΓ ) consists of the principal divisors. Hence Φ# induces an iso0
g
morphism from Div (XΓ ) Prin(XΓ ) to C Γ Λ̃.
I From now on, Γ = Γ1 (N ) or Γ0 (N ). The modular curve XΓ has a model defined over Q,
and it has a good reduction outside N .
I The Jacobian Jac(XΓ ) also has a model over Q as an abelian variety. The map τ 7→ τ − τ0
gives an embedding of XΓ to Jac(XΓ ) (for gΓ ≥ 1), which is Q-rational. Jac(XΓ ) has the
following universal property:
For every F : XΓ → T a holomorphic and Q-rational map from XΓ to a complex torus T
/
w; T
w
w
ww
ww f
w
w
F : XΓL
LL
τ DD LLLLL
DD
LL%
"
τ − τ0 ∈ Jac(XΓ )
∃ f : Jac(XΓ ) → T , Q-rational such that f (τ − τ0 ) = F (τ ) − F (τ0 ).
I We know that the Hecke algebra acts on S2 (Γ), hence on HomC (S2 (Γ), C) ⊃ Λ, and Λ is
invariant under the action of T.PSo T acts on the quotient, which is Jac(XΓ ). More concretely,
given Tm , then Tm (f (z)dz) = f (αi z)d(αi z). This gives map
i
P
P
/
/
τ
i αi τ
i Φ(αi τ )
/ Div(XΓ )
/ Jac(XΓ )
r9
LLL
rrr
LLL
r
r
r
LLL
%
rrr Tm
#
Tm
: XΓL
Jac(XΓ )
By the universal property of Jac(XΓ ), ∃ Tm : Jac(XΓ ) → Jac(XΓ ). In fact, Tm maps
Z τ
Z τ
g
f1 (z)dz, · · · ,
fgγ (z)dz ∈ C Γ Λ̃
Φ(τ ) =
τ0
to
Z
τ0
τ
Z
τ
Tm (f1 (z)dz), · · · ,
τ0
τ0
In particular, Tm and < d > are Q-rational maps.
13
Tm (fgγ (z)dz)
I Γ = Γ1 (N ) or Γ0 (N )
Let Λ be the image of H1 (XΓ , Z) ,→ HomC (S2 (Γ), C).
.
0
JΓ = Jac(XΓ ) = Div (XΓ ) {principal divisors}
.
' HomC (S2 (Γ), C) Λ
g
' C Γ Λ̃
R
where Λ̃ generated by the periods ( cj fi (z)dz) if we choose a basis f1 (z), · · · , fgΓ (z) of S2 (Γ).
Choose fi (z)’s with Fourier coefficients in Z. Let T ∈ T be a Hecke operator, i.e. T = Tn or
< d >. Note that
/ Φ(τ )
/τ τ
/ Jac(XΓ )
/ Div(XΓ )
8
HH
rrr
r
HH
r
HH
rr
H$
rrr
XΓ H
H
τ DD
DD
"
τ − τ0 ∈ Jac(XΓ )
and we have
τ
#
Tm
/
P
i αi τ
/
P
i Φ(αi τ )
/ Div(XΓ )
: XΓL
/ Jac(XΓ )
9
rrr
r
r
rr
rrr T
LLL
LLL
LLL
%
Jac(XΓ )
Choose τ0 to be i∞. We have T sends
Z τ
Z
Φ(τ ) =
f1 (z)dz, · · · ,
τ0
to
Z
τ
fgΓ (z)dz
τ0
τ
Z
τ
T (f1 (z)dz), · · · ,
T Φ(τ ) =
τ0
T (fgΓ (z)dz) .
τ0
Each T can be represented by a gΓ × gΓ matrix with entries Z, hence it is fixed over Q, i.e.,
T ∈ EndQ (JΓ ).
I For Γ = Γ1 (N ) or Γ0 (N ), the quotient YΓ = Γ\H has an interpretation as a moduli space.
I For Γ = Γ1 (N ), YΓ1 (N ) parametrizes equivalent classes of (E, P ), where E is an elliptic
curve over C and P is a point of order N . Each class is representated by
1
C
,
(E, P ) =
/L<τ,1> ,
N
where L<τ,1> = Zτ ⊕ Z with τ ∈ H. (E, P ) ∼ (E 0 , P 0 ) if there is an isomorphic mapping E to
E 0 and P to P 0 .
14
I For Γ = Γ0 (N ), YΓ0 (N ) parametrizes the equivalent classes of (E, C), where E is an elliptic
curve over C and C a cyclic group of order N . The class of (E, C) may be represented by
1
(E, C) = C/L<τ,1> , h i .
N
Call this Γ-structures.
×
I For d ∈ Z/N Z , < d > maps (E, P ) to (E, dP ) and (E, C) to (E, dC) = (E, C).
I For p - N, L<τ,1> has p + 1 superlattices containing L<τ,1> as sublattice of index p, given
by Lh τ +i ,1i for i = 0, 1, · · · , p − 1, and Lhτ, 1 i .
p
p
Starting from (E, P ) = C/L<τ,1> , N1 , we get p + 1 isogenies of (E, P ) of degree p, given by
1
C
for i = 0, 1, · · · , p − 1.
(Ei , Pi ) =
/L τ +i ,
h p ,1i N
and
(E∞ , P∞ ) =
C/L 1 , 1
hτ, i N
'
p
1
C/L
<pτ,1> , N
If (E, P ) is represented by τ is XΓ , then
1 i
τ
(Ei , Pi ) =
0 p
p 0
τ
(E∞ , P∞ ) = < p >
0 1
I If p | N , the last one (E∞ , P∞ ) is no longer a point in XΓ , so Tp for p | N has only p points.
So on the divisor level, Tp is given by
X
E/C , P mod C
Tp ((E, P )) =
C
where C goes through all subgroups of E of order p.
1.3
Eichler-Shimura Relation
I Let ϕ be a function from a curve X to itself. The graph of ϕ is {(x, ϕ(x)) | x ∈ X} ⊂ X ×X.
If ϕ is multi-valued, {(x, ϕ(x)) | x ∈ X} is called a correspondence, if it intersects each
horizontal and vertical line in X × X finitely many times.
I For example, we can view the Hecke operator Tp as a correspondence
{((E, P ), (Ei , Pi )) | i = 0, 1, · · · , p − 1, ∞, (E, P ) ∈ YΓ1 (N ) } ⊂ YΓ1 (N ) × YΓ1 (N ) .
Its closure in XΓ1 (N ) × XΓ1 (N ) is called a Hecke correspondence of Tp .
I If T = {(x, y)} is a correspondence in X × X, its transpose T 0 = {(y, x) | (x, y) ∈ T }.
Observe that for p - N, < p > Tp0 = Tp
((E, P ), (E 0 , P 0 )) ∈ Tp ⇐⇒ ((E 0 , P 0 ), (E, P )) ∈ Tp0 .
15
α
β
By definition, there exists a p-isogeny (E, P ) −→ (E 0 , P 0 ) and the dual isogeny (E 0 , P 0 ) −→
(E, pP ). The moduli space interpretation of XΓ allows it to have a model over Spec(Z). Thus
one can consider reductions of XΓ .
I It has a good reduction at all p - N . The reduction XΓ/Fp now parametrizes (E, P ) for
E defined over Fp , P of order N . The Hecke operator Tp at a ordinary point (E, P ) is given
by the same formula. The Jacobian Jac(XΓ ) also has good reduction at p - N . Moreover,
Jac(XΓ )/Fp may be identified with Jac(XΓ/Fp ). So Tp also acts on Jac(XΓ/Fp ) = Jac(XΓ )/Fp .
I For a variety X over Fp , the Frobenius map ,denoted by φx , sends a point to another point
whose coordinate are pth power. Let (E, P ) be an ordinary point on XΓ/Fp , φE maps (E, P )
to an isogney of degree p denoted by (E∞ , P∞ ) (since j(τ )p ≡ j(pτ ) (mod p)).
The Frobenius map φXΓ/Fp is a function on XΓ/Fp of degree p. Let F denote its graph in
XΓ/Fp × XΓ/Fp , and by F 0 its transpose. These two correspondence induce the Frobenius map
F on the Jac(XΓ/Fp ) and its dual F 0 in the sense of abelian varieties.
I In terms of divisors
F 0 ((E, P )) = (E0 , P0 ) + · · · + (Ep−1 , Pp−1 ).
Let φEi be the isogeny sending (Ei , Pi ) to (E, P ). The dual isogeny sends (E, P ) to (Ei , pPi )
for i = 0, 1 · · · , p − 1. Since E is ordinary, (Ei , pPi ), i = 0, 1, · · · , p − 1 and (E∞ , P∞ ) are the
p + 1 degree p isogenies of (E, P ). So
Tp ((E, P )) = (E0 , pP0 ) + · · · + (Ep−1 , pPp−1 ) + (E∞ , P∞ )
= (F + < p > F 0 )((E, P ))
This shows that
Tp = F + < p > F 0 on ordinary (E, P )
Since these points dense on XΓ/Fp , we have
Tp = F + < p > F 0 on XΓ/Fp
Thus on Jac(XΓ/Fp ). We have shown
Theorem (Eichler-Shimura congruence relation)
For p - N , on Jac(XΓ1 (N ) )/Fp , we have
Tp = F + < p > F 0 .
1.4
Galois representation
I Remark on rationality:
The curve XΓ is defined over Q. So S2 (Γ) = Ω1 (XΓ ) has a natural Q-rational structure
using Q-rational differentials. The fundamental q-expansion principle says that the Q-rational
differentials arise from the cusp forms with Fourier coefficients in Q. The Hecke operators are
Q-rational. So they commute with the action of the Galois group GQ = Gal(Q/Q).
16
.
I Regard JΓ = HomC (S2 (Γ), Q) Λ. For a prime `, the group of `n -torsion points are
JΓ [`n ] =
.
2gΓ
Λ ' Z/`n Z
1
`n Λ
on which TZ and GQ act. Taking projective limit, we obtain the Tate module
T` (JΓ ) = lim JΓ [`n ] ' Z`2gΓ
←−
n
on which TZ and GQ act. Call V` = V` (JΓ ) = T` (JΓ ) ⊗Z` Q` on which TQ` and GQ act. Notice
that V` ' Λ ⊗Z Q` . We saw before that Λ = Λ+ ⊕ Λ− , Λ± ⊗Z Q is a free TQ -module of rank 1.
Proposition V` is a free TQ` -module of rank 2.
I So the action of GQ on V` can be viewed as a representation
GQ −→ AutTQ` (Vl ) ' GL2 (TQ` ).
For each ` and n ≥ 1, there is a perfect Weil pairing
JΓ [`n ] × JΓ [`n ] −→ Z/`n Z(' µ`n ).
Passing the projective limit, we get a perfect pairing
h , i : T` (JΓ ) × T` (JΓ ) −→ Z` .
The Hecke operators are not self-adjoint w.r.t h , i.
0 −1
. wN preserves the pairing
N 0
i.e. hx, yi = hwN x, wN yi. It turns out that the adjoint of a Hecke operator T is wN T wN i.e.
I On XΓ , there is the Atkin-Lehner involution wN =
hT x, yi = hx, wN T wN yi.
For x ∈ T` (JΓ ), define φx ∈ HomZ` (T` (JΓ ), Z` ) by φx (y) = hx, wN yi. Then
φT x (y) = hT x, wN yi = hx, wN T wN wN yi = hx, wN T yi = φx (T y) = T φx (y).
Proposition The map x 7→ φx from T` (JΓ ) to HomZ` (T` (JΓ ), Z` ) is an isomorphism as
TZ -algebra. Consequently, we have
V` ' V`∨ = HomQ` (V` , Q` )
as TQ` -module.
Proposition Fp and < p > Fp0 are adjoint w.r.t. twisted pairing h , wN i i.e.
hFp x, wN yi = hx, wN < p > Fp0 yi.
17
Proof. We use the geometric interpretation of wN . On XΓ , it sends (E, P ) to E/hP i, Q
where hP, QiN = ζN . Here h , iN is the Weil pairing on elliptic curve E. We first show that
wN ◦ Fp = Fp ◦ < p >−1 ◦ wN . By definition,
(p)
wN ◦ Fp ((E, P )) = wN (E (p) , P (p) ) = E /hP (p) i, Q , where hP (p) , QiN = ζN .
Fp ◦ < p >−1 ◦ wN ((E, P )) = Fp ◦ < p >−1 E/hP i, Q0 , where hP, Q0 iN = ζN
= Fp E/hP i, Q00 where pQ00 = Q0
E (p)/hP (p) i, Q00(p)
=
Want to show Q00
Compute
(p)
= Q.
(p)
hP (p) , Q00 iN
= hP, Q00 ipN = hP, pQ00 iN
= hP, Q0 iN = ζN
= hP (p) , QiN .
By the property of Weil pairing, we have Q00
(p)
= Q. Now
hFp x, wN yi = hwN ◦ Fp x, yi
= hFp ◦ < p >−1 ◦ wN x, yi
= hx, wN ◦ < p > ◦Fp0 yi as desired.
Since φFp x (y) = hFp x, wN yi = hx, wn < p > Fp0 yi =< p > Fp0 φx (y), we have
Corollary det(I − Fp t on V` ) = det(I− < p > Fp0 t on V` ). Also Tr Fp = Tr < p > Fp0 on V` .
Corollary The eigenvalues of Fp and < p > Fp0 on V` are algebraic integers with absolute
1
value p 2 .
Proof. Consider the zeta function of XΓ/Fp
Z(XΓ/Fp , t) =
det(I − Fp t : T` (JΓ ))
.
(1 − t)(1 − pt)
By Weil conjecture for curves, det(I − Fp t : T` (JΓ )) is a polynomial in Z[t] of degree 2gΓ and
2g
QΓ
1
it is equal to
(1 − αi t) with |αi | = p 2 (Riemann hypothesis). The αi ’s are the eigenvalues
i=1
of Fp on T` (JΓ )
We are ready to show
18
Theorem Viewed as an element in GL2 (TQ` ), the Frobenius automorphism Fp has characteristic polynomial X 2 − Tp X+ < p > p.
Proof. Since Fp Fp0 = p and Tp = Fp + < p > Fp0 , we have
X 2 − Tp X+ < p > p = (X − Fp )(X− < p > Fp0 ).
In particular, Fp satisfies this degree two polynomial. If we can show Tr Fp = Tp , then
X 2 − Tp X+ < p > p is the characteristic polynomial of Fp . Since Tr Fp = Tr < p > Fp0 , we
have 2 Tr Fp = Tr Fp + Tr < p > Fp0 = Tr Tp = 2Tp .
∞
P
I Let f (z) =
an e2πinz be a newform of weight 2 level N and character χ. Let Kf denote
n=1
its field of coefficients. Let λ be a place of Kf , dividing `. Consider V` ⊗Q` Kf,λ on which
TKf,λ acts. Then V` ⊗ Kf,λ has 2-dimensional space U over Kf,λ invariant under TKf,λ arising
from f . Since GQ commutes with TKf,λ , U is GQ -invariant. The characteristic polynomial of
Fp on U is X 2 − ap X + χ(p)p, ∀ p - N `. This proves
P
Theorem Given any newform f =
an e2πinz of weight 2 level N and character χ, there
exists a compatible family `-adic representations ρ` of GQ attached to f i.e.
Tr ρ` (Frobp ) = ap
det ρ` (Frobp ) = χ(p)p, ∀ p - N `
1
Corollary |ap | ≤ 2p 2 .
1.5
Shimura’s construction
Let f (z) =
∞
P
an e2πinz be a normalized newform of weight 2 level N and character χ. ¿From
n=1
f we obtain a surjective homomorphism λf : TQ → Kf = Q(an , n ≥ 1) given by
λf (Tn ) = an ,
λf (< d >) = χ(d).
P
I For each embedding σ of Kf into C, there is a newform f σ =
σ(an )e2πinz of weight 2
level N and character χσ . Let [f ] denote the collection of the Galois orbit of f . Then
#[f ] = [Kf : Q].
To this f , we shall construct an abelian variety Af /Q (actually depending only on [f ]) of
dimension equal to [Kf : Q].
I Note that if f has coefficients in Q, then Af is an elliptic curve.
I Let If = (ker λf ) ∩ TZ , which is an ideal of TZ . The image If (JΓ ) (Γ = Γ1 (N )) is an
abelian subvariety defined over Q. The desired variety Af is defined as JΓ/If (JΓ ).
I Note that the Hecke
L operators act on Af . Let’s study Af .
Recall that S2 (Γ) =
S(gi ), where gi are the normalized weight 2 newform of level Ni | N ,
i
19
N
. For i 6= i0 , S(gi )⊥S(gi0 ) with respect to the Petersson
each S(gi ) has dimension σ0 N
i
inner product. To decompose S2 (Γ) with respect to the Q-structure, we group together the
gi ’s which are conjugate under the action of Gal(Q/Q) to get
M
S[gi ] = S[f ] ⊕ orthogonal complement.
S2 (Γ) =
[gi ]
We saw before that on S[f ] with respect to the basis f σ ,


am


..


.
Kf −→ Kfσ

,
Tm = 
.

aσm
am 7→ aσm


..
.
TQ |S2 (Γ) =
Y
TQ |S[gi ]
[gi ]
So λf on TQ |S[f ] is an isomorphism, and λf on TQ |S[gi ] with [gi ] 6= [f ] is zero map. The
kernel of λf is
TQ |s[f ]⊥ .
The orthogonal projection from S2 (Γ) → S[f ] is given by an element in TQ .
I Let πf denote the induced projection from HomC (S2 (Γ), C) → HomC (S[f ], C), and let
V[f ] = HomC (S[f ], C). Let πf (Λ) be the image of Λ under πf . It is a lattice of rank 2[Kf : Q].
Let π^
f (Λ) be the lattice generated by the periods of a Z-basis in S[f ].
Proposition Af over C is isomorphic to Vf πf (Λ).
I Similarly, each [gi ] yields an abelian variety A[gi ] .
Remark. In general, JΓ is isogenious to
Q
[gi ]
σ (N/Ni )
A[g0i ]
, but it is not isomorphic to it.
Example. When Γ = Γ0 (26), S2 (Γ0 (26)) is 2-dimensional with two newforms f1 and f2 of
level 26 and Fourier coefficients in Z. Moreover, f1 ≡ f2 mod 2.
The natural projection form JΓ to Af1 × Af2 is an isogeny with kernel Z/2Z × Z/2Z. This is
2
because a Z-basis of S2 (Γ0 (26)) is given by f1 +f
2 , f1 (or f2 ). So f1 , f2 generate a sublattice of
index 2. In this case, we can’t have JΓ ' Af1 × Af2 because
(1) Mazur has a general result saying that a Jacobian does not decompose into a direct
product of two principally polarized abelian varieties.
(2) Suppose JΓ ' Af1 ×Af2 . Since the only homomorphism from Af1 to Af2 is the zero map,
this isomorphism would imply that JΓ is a direct product of two principally polarized
abelian varieties.
20
I Let Tf denote the action of Hecke algebra on S[f ]. Then Tf acts on V[f ] and on Af . For a
prime `, we have Tate-module T` (Af ), which is a Z` -module of rank 2[Kf : Q]. Tf,Q ⊗Q Q` =
Kf ⊗Q Q` acts on T` (Af ) ⊗Z` Q` .
Proposition T` (Af ) ⊗Z` Q` is a free Kf ⊗Q Q` = Tf,Q ⊗Q Q` module of rank 2.
I Note that The action of GQ on T` (Af ) ⊗Z` Q` can then be viewed as a representation
GQ → GL2 (Kf ⊗Q Q` ). Note that
Y
Kf,λ , λ : places.
Kf ⊗Q Q` =
λ|`
Each piece gives a degree 2 `-adic representation of GQ .
I Since XΓ1 (N ) has good reduction at p - N, JΓ1 (N ) has good reduction at p - N and hence
Af has good reduction at p - N .
I For p - N `, the characteristic polynomial of the Frobenius Fp is X 2 − Tf,p X+ < p > p.
By Weil, the number of Fp -rational points on Af /Fp is given by
Nf,p = det(I − Fp on T` (Af /Fp ) = T` (Af ))
Y
=
(1 − aσp + χ(p)σ p)
σ
= NKf/ (1 − ap + χ(p)p).
Q
We have shown
Proposition The number Nf,p of the Fp -rational points on Af mod Fp is
Nf,p = NKf/ (1 − ap + χ(p)p).
Q
In particular, if Kf = Q, then Af is an elliptic curve, and we have shown
#Af /Fp (Fp ) = 1 − ap (f ) + p.
On the other hand, the ap for the elliptic curve Af is defined as 1 + p − #Af /Fp (Fp ) = ap (f ).
This proves
Theorem (Eichler-Shimura) Let f (z) =
∞
P
an e2πinz be a weight 2 newform of level N and
n=1
trivial character with Fourier coefficients in Q (hence in Z). Then there is an elliptic curve
E defined over Q such that
L(s, f ) = L(s, E)
locally at all primes p - N .
Remark. Let E = Af . We have a Q-morphism
ϕ : X0 (N ) −→JΓ0 (N ) −→E = Af
i∞
7→
O
7→
O
which sends i∞ to O. If ω is a nonzero Q-rational holomorphic differential on E, then
ϕ∗ (ω) = cf (z)dz for some c ∈ Q∗ .
21
I Call such E a Weil curve.
Proposition (Ribet) EndQ (Af ) ⊗Z Q is isomorphic to Kf . Hence Af is a simple abelian
variety over Q. Also, EndQ (Af ) ⊃ TZ/If , which is an order of Kf .
I In general, an abelian variety A over Q is said to be of GL2 -type if EndQ (A) contains an
order of a number field whose degree over Q is the dimension of A.
I Generlaized Modularity Conjecture
Any abelian variety over Q of GL2 -type is modular, i.e. isogeny to Af .
22
2
2.1
Galois Representations and Modularity
Galois representations
Let F be a field of characteristic zero or a finite field. Then Galois group GF = Gal(F /F ) is
a profinite group, obtained by
lim Gal(L/F ).
←−
[L:F ]<∞
L/ Galois
F
I It is endowed with the Krull topology such that an open neighborhood system of the
identity is given by Gal(F /L) as L runs through finite Galois extensions of F . Thus GF is
compact and totally disconnected.
I Let ` be a prime not dividing char F . Then F contains `n th roots of unity for all n ≥ 1.
Let ζ`n be an element of order `n and σ ∈ GF . Then σ(ζ`n ) = ζ`σnn for some σn ∈ Z/`n Z.
Since σn ≡ σn−1 (mod `n−1 ), we can take the inverse limit lim σn , which is an element in Z×
` ,
←−
n
denoted by ε` (σ). Clearly we have ε` (στ ) = ε` (σ)ε` (τ ). Hence ε` defines a character from GF
ab
to Z×
` , called the `th cyclotomic character. It factors through GF .
b which contains the cyclic subgroup generExample. 1 F = Fp . The Galois group GFp ' Z
ated by the Frobp as a dense subgroup. For ` 6= p, ε` (Frobp ) = p.
Example. 2 F = Qp .Qp contains the maximal unramified extension Qunr
p .
Gal(Qunr
p /Qp ) ' Gal(Fp /Fp ),
the Frobp in GFp lifts to Frobp in Gal(Qunr
p /Qp ). The restriction map
res : GQp −→ Gal(Qunr
p /Qp ) ' GFp
is surjective with kernel Ip the inertia subgroup of GQp . The local class field theory says that
the totally ramified abelian extensions of Qp are parametrized by Z×
p.
Theorem (Kronecker-Weber)
Qunr
=
p
[
Qp (ζm ).
p-m
ram ab
Qtotal
=
p
[
Qp (ζpn )
n≥1
This can be rephrased as
ab
×
res ×εp : Gab
Qp = Gal(Qp /Qp ) −→ GFp × Zp .
is an isomorphism.
23
For ` 6= p, ε` (Ip ) = 1, ε` (Frobp ) = p.
A finite extension L of Qp is tamely ramified if the ramification index is prime to p (i.e., the
order of p in L is prime to p).
ram containing
All finite tamely ramified extensions of Qp generate a Galois extension Qtame
p
tame ram ) = P is the maximal pro-p-subgroup of I and
Qunr
p
p
p . The group Gal(Qp /Qp
Ip/P '
p
Y
Z` (1),
`6=p
where Z` (1) is the Tate twist. Pp is called wild inertia group.
Pp



Qp


















ram
Qtame
p
Ip










unr
Qp
Qp
In fact, there is a finer filtration of Ip into a decreasing chain of closed subgroups Ipu indexed
by u ∈ [−1, ∞] such that
(i) Ipu = Ip for −1 ≤ u ≤ 0.
(ii) Ipu ⊃ Ipv if u ≤ v; Ip∞ = {id}.
T v
(iii) Ipu =
Ip .
v<u
(iv) Pp =
S
u>0
Ipu .
We get εp (Ipu ) = 1 + pdue Zp ⊂ Z×
p.
Example. 3 F = Q. For each prime p, there is a decomposition subgroup Gp which is
isomorphic to GQp and the different choice of Gp are conjugate under GQ . So there is Frobp ∈
Gp/I .
p
An extension L of Q is said to be unramified at p if Gal(Q/L) contains Ip and its conjugates.
So if L is unramified at p and Galois over Q, then Frobp is a well-defined element in Gal(L/Q)
up to conjugacy. Let [Frobp ] denote its conjugacy class.
Theorem If L is a finite extension of Q, then L is ramified at only finitely many places of
Q. If L 6= Q, the L is ramified somewhere.
Theorem (Čebotarov Density Theorem)
24
(1) Let L be a finite Galois extension of Q. Let C be a conjugacy class of Gal(L/Q). Then
|C|
{p[Frobp ] = C} has density equal to | Gal(L/Q)|
.
(2) Let
S L be a Galois extension of Q unramified outside a finite set S of places of Q. Then
[Frobp ] is dense in Gal(L/Q).
p6∈S
×
Remark. If L = Q(ζm ), then Gal(L/Q) ' Z/mZ . A conjugacy class C is represented by
an element a mod m, (a, m) = 1.
{p[Frobp ] = C} = {p | p ≡ a mod m}
1
. This is Dirichlet’s Theorem of primes in arithmetic
The density of {p | p ≡ a mod m} is φ(m)
progressions.
S
Kronecker-Weber Theorem. Qab =
Q(ζm ). This is equivalent to
m≥1
Y
εp : Gab
Q −→
Y
Z×
p.
p
p
is an isomorphism.
Pp



Qp


















ram
Qtame
p
Ip










unr
Qp
Qp
I Pp is maximal pro-p subgroup of Ip . Also, Ip/Pp '
Z` = lim Z/mZ = lim F×
. There is a
←−
←− pn
`6=p
Q
p-m
n
natural surjection for every n ≥ 1
ψn : Ip −→ Ip/Pp −→ F×
pn .
called fundamental character of Ip . When n = 1, ψ1 is εp |Ip mod p.
I Recall that a representation ρ of a topological group G is a continuous homomorphism
from G to Aut(V ), where V is a finite-dimensional vector space. When G = GQp (or GQ ),
we say that ρ is unramified at p if ρ is trivial on Ip (and its conjugates). For a subgroup H
ofG, denoted by ρH = V H the subspace of elements in V fixed by H, and by ρH the space
V hv − h(v) | h ∈ H, v ∈ V i. If H is a normal subgroup of G, then we obtain two representations of G/H , on ρH and ρH , also denoted by ρH and ρH . In particular, when H = Ip , the
inertia subgroup, then we can talk about ρIp (Frobp ) and ρIp (Frobp ), unique up to conjugacy.
I We are interested in representations of following types.
25
I. Artin representation ρ : GQ → GLn (C). Since GQ is compact and totally disconnected, so is ρ(GQ ), hence is finite. Let L be the fixed field of ker ρ. So L is a finite
extension of Q, hence is ramified at finitely many places. So ρ(Frobp ) makes sense for
almost all p. Such ρ is semi-simple.
II. Mod ` representation ρ : GQ → GLn (κ), where κ is a finite field of characteristic `.
Again, ρ is unramified at almost all places, but ρ may not be semi-simple.
III. `-adic representation ρ : GQ → GLn (K), where K is a finite extension of Q` . As
not all such representations are unramified outside a finite set of places of Q, we require
that ρ is unramified outside finitely many places so that we can talk about ρ(Frobp ) for
almost all p. Also, ρ may not be semi-simple.
Proposition
(I) For an Artin representation ρ : GQ → GLn (C), it is determined by Tr ρ(Frobp ) for
almost all p.
(II) For a semi-simple mod ` representation ρ : GQ → GLn (κ), it is determined
by the trace
Vi n
n
κ for i = 1, · · · , n
of ρ(Frobp ) on the space κ as well as on all exterior products
and for almost all p. If char κ = ` > n, then it is determined by Tr ρ(Frobp ) for almost
all p. (Remark. If n = 2, for ` ≥ 3. Tr ρ(Frobp ) is enough for almost p. If n = 2 and
` = 2, Tr ρ(Frobp ) and det ρ(Frobp ) determine the representation for almost all p.)
(III) For a semi-simple `-adic representation ρ : GQ → GLn (K). ρ is determined by Tr ρ(Frobp )
for almost all p.
I Let ρ be a representation of GQ of type I,II and III as above. We want to define the
conductor of ρ at p (p 6= ` for type II and III):
Z ∞
Z ∞
v
Iv
Ip
codim ρI dv.
mp (ρ) =
codim ρ dv = codim ρ +
−1
0
v
v
If ρ is unramified at p, then ρIp is the whole space, and so is ρI for v ≥ −1. So codim ρI = 0
and mp (ρ) = 0. Assume that ρ ramifies at p. First look at the case where ρ(GQ ) is finite, i.e.,
type I and type II, if v1 < v2 < · · · ,
Ip ⊃ I v1 ⊃ I v2 ⊃ · · · = I ∞ = {id}
ρ(Ip ) ⊃ ρ(I v1 ) ⊃ ρ(I v2 ) ⊃ · · · = {id}.
Since ρ is continuous, we have ρ(I v ) = {id} for some v < ∞. So the
proper integral, mp is well-defined.
R∞
−1 codim ρ
I v dv
is a
I Suppose ρ : GQ → GLn (K) is an `-adic representation. Since GQ is compact, ρ(GQ ) is a
compact subgroup of GLn (K). By choosing a suitable basis of K n (i.e., conjugating ρ(GQ )
by a matrix in GLn (K)), we may assume that ρ(GQ ) ⊂ GLn (O), where O is the ring of
integral elements in K. Thus we can take reduction mod the maximal ideal of O to get a
representation ρ̄ : GQ → GLn (κ) over the residue field κ of K. The kernel of the reduction
GLn (O) → GLn (κ) is a pro-` group.
26
I Recall that Pp is a pro-p group, so is ρ(Pp ). For p 6= `, ρ(Pp ) intersects the kernel of the
reduction trivially, thus ρ(Pp ) ' ρ̄(Pp ) is finite. By the same argument, mp (ρ) is well-defined.
It is known that mp (ρ) is an integer. Define the conductor of a representation ρ of GQ to be
Y
pmp (ρ) ,
N (ρ) =
p
where p is over all primes if ρ is an Artin representation, an all p 6= ` otherwise.
Remark. The representation ρ̄ may depend on the choice of basis, but its semi-simple filtration (i.e., the Jordon-Hölder factor) is independent of the conjugation.
The analysis above proves
Proposition Let ρ : GQ → GLn (K) be an `-adic representation of GQ and ρ̄ a reduction of
ρ. For p 6= `, we have
Z ∞
Z ∞
v
Iv
codim ρ dv =
codim ρ̄I dv.
0
0
Moreover,
N (ρ) = N (ρ̄)
Y
pdim ρ̄
Ip
−dim ρIp
.
p6=`
In particular, N (ρ̄)N (ρ).
2.2
Galois representations associated to elliptic curves
Let E be an elliptic curve defined over Q. For each prime `, we have the Tate module
T` (E) = lim E[`n ] ' Z` × Z` .
←−
n
GQ acts on T` (E). We get an `-adic representation
ρE,` : GQ −→ Aut(T` (E)) = GL2 (Z` ) ,→ GL2 (Q` )
Its reduction mod ` is
ρ̄E,` : GQ −→ Aut(E[`]) ' GL2 (F` ).
Proposition
(a) det ρE,` = ε` .
(b) (Serre) ρE,` is absolutely irreducible. For each E, ρ̄E,` is absolutely irreducible for `
large.
(c) (Serre) If E does not have complex multiplication, then ρE,` and hence ρ̄E,` is surjective
for almost all `. Concerning the irreducibility of ρ̄E,` , Mazur showed that ρ̄E,` is irreducible for ` > 163 for all E; if E is semi-stable, then ρ̄E,` is irreducible for ` > 7, and
ρ̄E,` is irreducible for ` > 3 if ρE,2 is trivial.
27
I Local behavior of ρE,` and ρ̄E,` .
Proposition Suppose E has a good reduction at p.
(a) If ` 6= p, then ρE,` is unramified at p, and
Tr ρE,` (Frobp ) = 1 + p − #E/Fp (Fp ) = ap (E) = ap .
and ap ∈ Z is independent of `.
(b) For all n ≥ 1, there is a finite flat group scheme Fn/Zp such that
E[pn ](Qp ) ' Fn (Qp )
as G` -modules.
(c1) If E has a good ordinary reduction at p (i.e., p - ap ), then E/Fp [p] ' Z/pZ and
εp χ ∗
ρE,p |Gp ∼
0 χ−1
for an unramified character χ of GQ . In particular
εp ∗
.
ρE,p |Ip ∼
1
(c2) If E has a good supersingular reduction (i.e., p | ap ), then E/Fp [p] is trivial and
ρ̄E,p |Ip : Ip −→ Aut(E[p](Qp )) ' GL2 (Fp )
is isomrophic to
Ip
/5 F×2 
/ / Ip/P
p
p
/ GL2 (Fp )
ψ2
arising from ψ2 . Moreover, ρ̄E,p is absolutely irreducible and ρE,p is irreducible.
I Sketch of proof.
(b) Let E/Fp be the Néron model of E/Qp arising from the minimal Weierstrass equation.
The finite flat group scheme is E[pn ].
(c1) It suffices to see that Tp (E) has a rank 1 Zp -module invariant under the action of GQ
such that on the quotient the Galois action is unramified. By assumption, the Néron
model E/Zp has a closed fiber E/Fp which is an ordinary elliptic curve. It means that
the p-divisible group of E/Zp has a closed fibre with non-trivial connected and étale
factors. Thus the p-divisible group of E/Zp has non-trivial connected and étale factors
since the formation of the connected-étale sequence of a finite flat group scheme over Zp
is compatible with the base change Zp → Fp .
Passing to the generic fiber over Qp and their Qp -points, the non-trivial connected-étale
sequence over Zp gives rise to the desired decomposition of ρE,p . In particular, the
unramified quotient of Tp (E) can be interpreted as Tp (E/Fp ) under the reduction map.
28
(c2) The first statement is a hard theorem, using finite flat group schemes and Raynaud’s
results. Assume this, we show that ρ̄E,p is absolutely irreducible. Assume not, then
there is a 1-dimensional subspace invariant under the action of Gp , given by a character.
Its restriction to Ip is either ψ2 or ψ2p . Recall that for any σ ∈ Ip and any preimage Frobp
p
of the Frobp in Gp , we have Frobp ◦σ ◦ Frob−1
p = σ . This shows that Frobp permutes
p
ψ2 and ψ2 , hence there is no way to extend the character on Ip to a character of Gp .
If ρE,p is reducible, then there is a 1-dimensional subspace invariant under the action
of Gp . We may choose a suitable vector in this space so that after adjoining another
vector, ρE,p (Gp ) has image over O. Then ρ̄E,p could not be absolutely irreducible.
I Next consider the case where E has a multiplicative reduction at p. In this case the
j-invariant of E is NOT p-adically integral. Recall that the q = e2πiz expansion of j
j=
1
+ 744 + 196884q + · · ·
q
has coefficients in Z. We can express q in powers of j:
q = j −1 + 744j −2 + 750420j −3 + · · · , with coefficients in Z
This series converges p-adically to q = qE ∈ pZp , called the Tate p-adic period of E.
I Recall that
.
C× q Z , where q = e2πiz .
exp
E(C) = C L<τ,1> −→
z
e2πiz
7→
Tate showed that there is a p-adic analytic isomorphism
×
∼
Φ : Qp q Z −→ E(Qp )
E
×
such that σΦ(x) = Φ σ(x)δ(σ) , ∀σ ∈ GQp and x ∈ Qp /q Z . Here δ is a character from GQp to
{±1} defined as follows: δ is the trivial character if E has split multiplicative reduction (i.e.
with rational tangents), and δ is the unique unramified character of GQp otherwise.


Qp 



H




Qp (ζp2 −1 )
Qp
Proposition Suppose E has multiplicative reduction at p. Let δ be the character defined
above. Then
δε` ∗
(1) ρE,` |Gp ∼
for all ` and if ` 6= p, mp (ρE,` ) = 1.
0 δ
29
.
δ ε̄` Φ
×
`
(2) ρ̄E,` |Gp ∼
, where Φ ∈ H 1 (Gp , Z/`Z(1)) ' Qp (Q×
corresponding to the
p)
0 δ
.
×
`
image of q = qE in Qp (Q×
.
p)
(3) if ` 6= p, then ρ̄E,` is unramified at p if and only if p | ordp (q) = − ordp (j) = ordp (∆E ).
(4) There is a finite flat group scheme F/Zp such that E[p](Qp ) ' F (Qp ) as Gp -modules if
and only if p | ordp (q) = ordp (∆E ) = − ordp (j).
Look at the case ρE,p |Gp . Tp (E) = lim E[pn ]. The usual Galois action on ζpn is given by
←−
n
σ(ζpn ) =
ε (σ)
ζpnp ,
×
∀ n ≥ 1. Imbed ζpn into Qp /q Z , on which the Galois action is
ε (σ) δ(σ)
σ ◦ ζpn = σ(ζpn )δ(σ) = (ζpnp
)
.
×
Hence {ζpn } determines a rank 1 Zp -module in Qp /q Z on which Gp acts by the character εp δ.
Then the action of Gp on the quotient is given by δ since det = εp and δ 2 = id.
Proposition Suppose E has additive reduction at p. Then for all ` 6= p, the conductor of
ρE,` |Gp is at least 2 and equal to 2 if p > 3.
Remark. The conductor of ρE,` at p 6= ` is the p-factor of the conductor of E.
As we consider before E/Q , ρE,` : GQ → Aut(T` (E)) ' GL2 (Z` ) ,→ GL2 (Q). Study the
behavior of ρE,` |Gp and ρ̄E,` |Gp .
Proposition Suppose E has a good reduction at p
(a) If ` 6= p, then ρE,` is unramified at p, and Tr ρE,` (Frobp ) = ap (E) = 1 + p − #E/Fp (Fp ).
(b) For all n ≥ 1, there is a finite flat group scheme Fn /Zp such that
E[pn ](Qp ) ' Fn (Qp ) as Gp -modules.
(1) If E has ordinary reduction at p, i.e., p - ap , then
χεp
∗
ρE,p |Gp ∼
.
χ−1
for some unramified character χ of GQ .
(2) If E has supersingular reduction at p, then
ρ̄E,p |Ip : Ip −→ Aut(E[p](Qp )) ' GL2 (Fp )
is isomorphic to
Ip
/5 F×2 
/ / Ip/P
p
p
ψ2
30
/ GL2 (Fp )
Remark.
(1) A precise definition of ψ2 : Let ω be any uniformizer of Zp . Let
of ω. For σ ∈ Ip
2 √
σ( p −1 ω)
= ψ2 (σ).
√
p2 −1
ω
√
p2 −1
ω be a p2 − 1 root
(2) The characteristic polynomial of ρE,` (Frobp ) for ` 6= p is
X 2 − ap X + p.
Factor this over Qp : X 2 − ap X + p = (X − α)(X − β). If p - ap , exactly one of α, β is a
p-adic unit.
E = the Néron model obtained from minimal Weiestrass equation. E[p] is a finite
flat group scheme over Zp of order p2 . There is a short exact sequence of finite flat
commutative group scheme:
0 −→ µp −→ E[p] −→ Z/pZ −→ 0
Gp is unramified on Z/pZ, the action of Frobp there is given by multiplication by the
p-adic unit.
(3) In case of (2), ρE,p ⊗Fp Fp ' ψ2 ⊕ ψ2p .
×
Q
Proposition Suppose E has a multiplicative reduction at p (E(Qp ) ' p q Z (δ)). Let δ
be the character from GQ → {±1} which is trivial if split, and δ is the unique unramified
quadratic character if nonsplit, σ · x = σ(x)δ(σ) .
δε` ∗
and mp (ρE,` ) = 1 if p 6= `.
(a) ρE,` |Gp ∼
δ
(b) If ` 6= p, then ρ̄E,` |Gp is unramified (at p) ⇔ ` | ordp (∆E ) = ordp (q), i.e., mp (ρ̄E,` ) = 0.
(c) There is a finite flat group scheme F/Zp such that
E[p](Qp ) ' F (Qp ) as Gp -modules ⇐⇒ p | ordp (∆E ) = ordp (q).
Remark. The conductor of ρE,` is the non-`-part of the conductor of E.
I Let ` be an odd prime, O be a complete discrete valuation ring. Let G(= G` ) be a
topological group. By a finite O[G]-module M we mean an O-module with finite cardinality
and discrete topology on which G acts continuously. A profinite O[G]-module M is an inverse
limit of finite O[G]-modules.
A profinite O[G]-module M is said to be
good: if for every finite discrete quotient M 0 of M , there is a finite flat group scheme F/Z`
31
such that M 0 ' F (Q` ) as Z` [G` ]-modules
ordinary: if there is an exact sequence
0 −→ M (−1) −→ M −→ M (0) −→ 0
of O[G` ]-modules such that I` acts trivially on M (0) , and by the cyclotomic character ε on
M (−1) .
semi-stable: if it is either good or ordinary.
I Suppose that R is a complete Noetherian local O-algebra whose residue field is the same as
the residue field of O. We call a continuous homomorphism ρ : G` → GL2 (R) good, ordinary
or semi-stable if
(1) det ρ |I` = ε.
(2) The underlying module, denoted by Mρ (' R2 ), is good, ordinary, or semi-stable.
Denote by ρ̄ the reduction of ρ modulo the maximal ideal of R. As a consequence of the
Nakayama lemma, we have
(−1)
Lemma If Mρ and ρ̄ are ordinary, then Mρ
and ρ is ordinary.
(0)
and Mρ
are each free R-modules of rank 1,
Proposition Let E be an elliptic curve over Q and O = Z` . Then
(i) if E has good (resp. semi-stable) reduction at `, then ρE,` |G` and ρ̄E,` |G` are good (resp.
semi-stable).
(ii) ρE,` |G` is ordinary ⇔ ρ̄E,` |G` is ordinary ⇔ E has good ordinary or multiplicative
reduction at `.
(iii) ρE,` |G` is good (resp. semi-stable) ⇒ E has good (resp. semi-stable) reduction at `.
Lemma
(a) If a mod ` representation ρ̄ : G` → GL2 (κ) is good, then either ρ̄ is ordinary or ρ̄ |I`
⊗F` ' ψ2 ⊕ ψ2` .
(b) If ρ : G` → GL2 (R) is such that Mρ is good and ρ̄ is ordinary, then ρ is good and
ordinary.
I An `-adic representation ρ : G` → GL2 (K) is good ordinary or semi-stable if after conjugating ρ to a representation from G` to GL2 (OK ), where OK is the ring of integers of the
`-adic field K, the representation is good, ordinary or semi-stable.
32
2.3
Galois representations attached to modular forms
∞
P
an e2πinz be a normalized newform of weight 2 level N and character χ. Denoted
×
→ S 1 . By class field
by Kf = Q(an , n ≥ 1). Here χ is a Dirichlet character χ : Z/N Z
theory, we may regard χ as a character of GQ , unramified outside N such that χ(Frobp ) = χ(p)
for p - N , and χ(complex conjugation) = χ(−1) = (−1)wt (= 1).
Let f (z) =
n=1
I Shimura constructed an abelian variety Af over Q as a quotient Jac(XΓ1 (N ) ). For each
prime `, both GQ and the Hecke algebra Tf ⊗ Z` act on the Tate module T` (Af ), which gives
a representation
ρ : GQ −→ Aut(T` (Af )) ' GL2 (Tf ⊗ Z` )
L
Recall that Tf × Q` ' Kf ⊗ Q` =
Kf,λ . So ρ can be viewed as a representation
λ|`
GQ −→ GL2 (Kf ⊗ Q` ).
We get an `-adic representation ρf,` : GQ → GL2 (Kf,λ ). We summarize the properties of ρf,`
as follows:
Proposition Let f (z) =
∞
P
an e2πinz be a normalized newform of weight 2 level N and
n=1
character χ. For each prime `, let K be a finite extension of Q` containing Kf . Then we have
an `-adic representation
ρf,` : GQ −→ GL2 (K)
(i) ρf,` is unramified at each p - N `. Moreover, ρf,` (Frobp ) has characteristic polynomial
X 2 − ap X + pχ(p).
(ii) det ρf,` = χε` , and ρf,` (complex conjugation) '
1
0
0 −1
. In other words, ρf,` is odd.
(iii) ρf,` is absolutely irreducible.
(iv) The conductor of ρf,` is the non-`-part of N .
(v) Suppose p 6= ` and pkN . Let η be the unramified character of Gp such that η(Frobp ) = ap .
If p - condχ, then
ηε` ∗
ρf,` |Gp ∼
.
η
If p | condχ, then
ρf,` |Gp ' η −1 ε` χ |Gp ⊕η.
(vi) If ` - 2N , then ρf,` |G` is good. Moreover, ρf,` |G` is ordinary if and only if a` is a unit
(0)
in K, in which case ρI` (Frob` ) (i.e., ρ on Mρf,` ) is the unit root of X 2 − a` X + `χ(`).
(vii) If ` is odd and `kN , but ` - cond χ, then ρf,` |G` is ordinary and ρI` (Frob` ) = a` .
33
N
Remark. Associated to the newform f is a cuspital automorphic representation π = 0 πv
v
N
of GL2 (AQ ) such that L(s, f ) = L(s, π) = π∞ ⊗0
πp . Conductor of π = level N of f . χ is a
p
×
× Q Q ei
Q
Z/pei Z , N = p . So we may write χ =
character of Z/N Z '
χp , where χp is
i
i
p|N
×
a character of Z/pe Z . At a place pkN , the theory of newform tell us

1
 |ap | = p 2 , if p | cond χ ←→ πp is a principal induced form (η −1 χp | · |−1 , η)

a2p = χ(p), if p - cond χ ←→ πp is Steinberg representation induced form (η| · |−1 , η).
where η is the unramified character of Q×
p s.t. η(p) = ap .
I Let ρ̄f,` be the semi-simplification of ρf,` :
ρ̄f,` : GQ −→ GL2 (κ),
where κ is the residue field of K. ρ̄f,` has the same properties as ρf,` except
(iii)0 ρ̄f,` may not be absolutely irreducible. If ` is odd, then
ρf,` is irreducible ⇐⇒ it is absolutely irreducible.
(iv)0 The conductor of ρ̄f,` divides the non-`-part of N .
I For newforms of weight≥ 2, there are associated `-adic representations. For newforms of
weight 1, we have
Theorem (Deligne-Serre)
∞
P
Let g =
an e2πinz be a normalized newform of weight 1 level N and character χ. Then
n=1
there is an irreducible representation
ρg : GQ −→ GL2 (C)
with conductor equal to N such that at p - N , the characteristic polynomial of ρg (Frobp ) is
X 2 − ap X + χ(p)
which implies det ρg = χ.
I The key idea of the proof:
Up to conjugation, we may assume that ρg (GQ ) ⊂ GL2 (Kg ),where Kg = Q(an , n ≥ 1). Let K
be a finite extension of Q` containing Kg . We may regard
ρg : GQ −→ GL2 (Kg ) ,→ GL2 (K)
as an `-adic representation, so we have the mod ` representation ρ̄g . Deligne and Serre showed
that there is a normalized newform f of weight 2 level Nf | N ` and character χf such that
ρ̄g ' ρ̄f,` over κf .
That is, ap (g) ≡ ap (f ) and χ(p) ≡ pχf (p) modulo maximal ideal. So ρ̄f,` gives the (would-be)
ρ̄g . Then lift this to a suitable `-adic field with finite image and hence put it into C.
34
2.4
The modularity of Galois representations
We have seen representations of GQ over C, `-adic fields and finite fields. We want to describe
these representations. In other words, given a representation of GQ over C, `-adic fields or
finite fields, want to know when they are isomorphic to those arising from modular forms, i.e.,
modular in short.
The first is
I Artin’s Conjecture: Every irreducible representation ρ : GQ → GLd (C) with d ≥ 2 has
a holomorphic L-function L(s, ρ).
L(s, ρ) = Γ(s, ρ)
Y
Lp (s, ρ) = Γ(s, ρ)
p
Y
p
1
det(1 − ρ(Frobp )p−s |V Ip )
If ρ is a degree d irreducible representation with d ≥ 2, and χ is any character of GQ , then ρ⊗χ
is also an irreducible degree d representation. In case d = 2, Artin’s conjecture ⇒ L(s, ρ) and
L(s, ρ ⊗ χ) are holomorphic ∀ χ. Apply the converse theorem for GL2 (AQ ), this means that
there is a cuspidal automorphic representation π for GL2 (AQ ) such that L(s, π) = L(s, ρ).
In other words, there is an automorphic form f such that L(s, f ) = L(s, ρ). If ρ is odd,
Γ(s, ρ) = ΓC (s) = (2π)−s Γ(s), in this case, f is a holomorphic cusp newform of weight 1. If
s
ρ is even, Γ(s, ρ) = ΓR (s)2 or ΓR (s + 1)2 , where ΓR (s) = π − 2 Γ 2s . Then f is Maass wave
form.
I We may restate Artin’s conjecture for odd degree 2 irreducible representations as
Artin’s Conjecture0 . Every odd degree 2 irreducible rerpresentation ρ of GQ is isomorphic
to a ρg for a newform g of weight 1.
I What’s known?
The image of degree 2 irreducible Artin’s representations, when passed to PGL2 (C), falls in
one of following types:
cyclic Cn ,
dihedral D2n , A4 , S4 , A5 .
The type Cn , type D2n cases are easy. In 70’s, Langlands proved the conjecture for type A4
representations using “base change”. In 1981, Tunnell extended it to type S4 .
Theorem If an odd irreducible representation ρ : GQ → GL2 (C) has image ρ(GQ ) solvable,
then ρ is isomorphic to ρg .
Recently, the case of A5 was proved by Buzzud, Dickinson, Stephand-Barron and Taylor under
some technical conditions.
I Recall Artin’s conjecture0 . If ρ : GQ → GL2 (C) is odd, irreducible, then ρ ' ρg for some
newform g of weight 1.
This is true if ρ(GQ ) is solvable. In this case, ρ̄g,` ' ρ̄f,` for a newform of weight 2.
I mod ` representations:
Original Serre’s conjecture:
35
(1) Let ρ̄ : GQ → GL2 (κ) be an odd irreducible representation, where κ is a finite field of
characteristic `. Then ρ̄ ' ρ̄f,` for some newform f .
(2) A recipe for the weight, level and character of such f. Through the efforts of Mazur,
Ribet, Gross, this conjecture is shown to be equivalent to the following for ` odd.
(If ` = 3, need the assumption that ρ̄ |GQ(√−3) is not induced from a character of
√
Q( −3).)
I Serre’s Conjecture Let ρ̄ : GQ → GL2 (κ) be a mod `, odd, irreducible representation.
Then ρ̄ ' ρ̄f,` for a newform f of weight 2. If this is the case, we say ρ̄ is modular.
Proposition Serre’s conjecture is known to hold in the following case: if
(i) κ = F3 ;
(ii) the projective image of ρ̄ is dihedral.
Proof.
√
√
(i) Consider the imaginary quadratic
extension
Q(
−2).
Its
ring
of
integers
is
Z[
−2], in
√
√
which 3 splits
√ (since (3) = (1 + −2)(1 − −2)). Regard F3 as the
√ residue field of the
prime (1 √
+ −2). Therefore we have the surjective map GL2 (Z[ −2]) → GL2 (F3 ) by
mod 1 + −2. This homomorphism has a lifting given by sending
−1 1
−1 1
1 −1
1
−1
√
√
7→
and
7→
.
−1 0
−1 0
1
1
− −2 −1 + −2
So we have
√
ρ̄ : GQ −→ GL2 (F3 ) ,→ GL2 (Z[ −2]) ,→ GL2 (C).
The group GL2 (F3 ) acts on P1 (F3 ) by fractional linear transformations
az + b
a b
(z) =
.
c d
cz + d
The elements in GL2 (F3 ) fixing all elements in P1 (F3 ) are the scalars. So we have an
imbedding of PGL2 (F3 ) ,→ S4 . This is surjective by counting cardinalities. So ρ̄, as an
Artin representation, is isomorphic to ρg for a newform g of wieght 1. By remark before,
as mod 3 representation, ρ̄ ' ρ̄g,3 ' ρ̄f,3 for some newform f of weight 2.
(ii) By assumption, ρ̄ is induced. In other words, there is a Galois extension F of Q and a
G ¯
¯ so the image
Let n be the order of ξ,
character ξ¯ : GF → κ× such that ρ̄ = IndGQF ξ.
×
¯
ξ(GF ) is a cyclic group of order n contained in κ . Let K be an `-adic field with residue
field κ. Lift the nth roots of unity in κ to the group hζn i generated by nth roots of unity
G
in K. We obtain a character ξ : GF → hζn i ⊂ K. Let ρ = IndGQF ξ. Replacing ξ be a
twist by a quadratic character of GF if necessary, we obtain an odd irreducible degree
2 representation ρ. We know that ρ ' ρg for a newform g of weight 1. By the same
argument as before, ρ̄ is modular.
36
Theorem (Diamond, Gross, Edixhoven)
Suppose that ` is odd and the mod ` representation ρ̄ : GQ → GL2 (κ) is modular. If ` = 3,
assume also that ρ̄ |GQ(√−3) is absolutely irreducible. Then there is a newform f of weight 2
such that
(i) ρ̄ ' ρf,` over κf ;
(ii) the level of f is N (ρ̄)`δ(ρ̄) with δ(ρ̄) = 0, 1, 2 defined as follows;
(iii) ` does not divide the order of the character of f .
I δ is of the following form:
(i) δ(ρ̄) = 0 if ρ̄ |G` is good.
(ii) δ(ρ̄) = 1 if ρ̄ |G` is not good and ρ̄ |I` ⊗κ κ̄ is of the form
a
a
ψ2
0
ε` ∗
ε` ∗
.
,
,
εa`
1
0 ψ2`a
for some positive integer a < `. Here ψ2 is the fundamental character.
(iii) δ(ρ̄) = 2 otherwise.
In particular, δ(ρ̄) ≤ 1 if ρ̄ |G` is semi-stable.
I `-adic representations:
I Fontaine-Mazur Conjecture: let ρ : GQ → GL2 (K) be an irreducible `-adic representation (ramified at only finitely many places by requirement). Assume ρ is not a Tate twist
of an even representation which factors through a finite quotient of GQ . Then
ρ ' ρf,` for some newform f
⇔ ρ is potentially semi-stable everywhere,
⇔ ρ is potentially semi-stable at `. (Grothendieck)
I Such f is unique. Since
ρ ' ρf,` ⇒ Tr ρ(Frobp ) = Tr ρf,` (Frobp ) = ap (f ), ∀ p
f is unique by strong multiplication theorem.
Here ρ is potentially semi-stable if there is a finite extension F of Q such that ρ |GF is semistable ⇔ for every place v of F dividing `, ρ |Gv is semi-stable.
I Call an `-adic representation ρ modular if ρ ' ρf,` for some newform f of weight 2.
I Weak Fontaine-Mazur Conjecture: If an irreducible `-adic representation ρ : GQ →
GL2 (K) is such that ρ |G` is semi-stable, then ρ is modular. (Recall that if ρ |G` is semi-stable,
then det ρ |I` = ε` by definition.)
I Wiles: Under some conditions, ρ̄ modular ⇒ ρ modular.
37
Remark. Under the Weak Fontaine-Mazur conjecture, if ρ ' ρf,` , it is expected to find a
newform f for Γ1 (N (ρ)) ∩ Γ0 (`) (resp. Γ1 (N (ρ))) if ρ |G` is semi-stable (resp. good).
I Taniyama-Shimura Conjecture: Every elliptic curve E defined over Q is modular,
i.e., E is isogenous to Af for some newform f of weight 2 trivial character and Z-Fourier
coefficients. That is,
L(s, E) = L(s, f ), ρE,` ' ρf,` for all `.
Proposition The following are equivalent.
(a) Elliptic curve E/Q is modular.
(b) ρE,` is modular for all `.
(c) ρE,` is modular for some `.
Proof. If E is modular, then L(s, E) = L(s, f ) for some newform f of weight 2. Then
ρE,` ' ρf,` , ∀ `.
(c)⇒(b): Suppose ρE,` ' ρf,` for some cusp form f of weight 2 and for some `. Then at almost
all p where both representations are unramified, we have
(∗)
Tr ρE,` (Frobp ) = ap (E) = 1 + p − #E/Fp (Fp ) = Tr ρf,` (Frobp ) = ap (f ) ∈ Z.
and ap (f ) is independent of f . For any prime `0 , we have
Tr ρE,`0 (Frobp ) = Tr ρf,`0 (Frobp )
as long as both representations are unramified at p. Since the `0 -adic representations are
determined by trace at almost all p, this implies ρE,`0 ' ρf,`0 for all `0 .
(b)⇒(a): We have ρE,` ' ρf,` (∀ `) for some weight 2 cusp form f . We know that the
conductor of ρE,` is the non-`-part of the conductor N of E, and the conductor of ρf,` is the
non-`-part of the level Nf of f . So N = Nf . By (∗), we know ap (f ) ∈ Z for all p - Nf . ¿From
det ρf,` = det ρE,` = ε` , we know that f has trivial character. From the theory of newforms,
we know that ap = 0 or ±1 for p | N . So f has Fourier coefficients in Z. This shows that Af
is an elliptic curves. L(s, E) = L(s, Af ) implies E and Af are isogenous by Falting’s theorem.
Corollary The Weak Fontaine-Mazur conjecture ⇒ Taniyama-Shimura conjecture.
Proof. Choose ` so that E has good reduction at `. Then ρE,` |G` is semi-stable.
Serre proved that
Theorem If mod ` Serre’s conjecture holds for infinitely many `, then Taniyama-Shimura
holds.
38
3
The Modularity of Semi-Stable Elliptic Curves
3.1
Reduction of the problem
Let E be an elliptic curve defined over Q, semi-stable. Want to show ρE,` is modular for some
`. If ρE,` is modular, so is ρ̄E,` . Look for ` such that ρ̄E,` is modular. If ρ̄E,3 is irreducible,
then we know it is modular. What if ρ̄E,3 is reducible?
I Assume ρ̄E,3 reducible. Recall that ρ̄E,3 : GQ → Aut(E[3](Q)) ' GL3 (F3 ), ρ̄E,3 is reducible
⇒ E has a subgroup of order 3 invariant under GQ . If ρ̄E,5 is also reducible, then E has a
subgroup of order 5 invariant under GQ . Put together, E has a cyclic subgroup C of order 15
defined over Q. This means that (E, C) is a Q-rational point on the modular curve X0 (15). On
the other hand, X0 (15) is known to have 4 non-cusp rational points, none of which corresponds
to a semi-stable elliptic curves.
Conclusion. If ρ̄E,3 is reducible, then ρ̄E,5 is irreducible.
Lemma There is an auxiliary semi-stable elliptic curve A defined over Q satisfying
(i) A[5] ' E[5] as GQ -module (i.e., ρ̄A,5 ' ρ̄E,5 ).
(ii) A[3] is an irreducible GQ -module (i.e. ρ̄A,3 is irreducible).
Proof. Recall that Y (5) = Γ(5)\H parametrizes the equivalence classes of (E, P, Q) where E
is an elliptic curve over C, P, Q ∈ E[5] with Weil pairing hP, Qi = ζ5 . For a number field K,
the K-rational points on Y (5) are parametrized by (E, P, Q) all defined over K. We define a
twist of Y (5). Let Y 0 (5) be the curve over Q parametrizes equivalent classes of elliptic curve
A over C together with an isomorphism A[5] ' E[5] which preserves the Weil pairings. If
we fix P, Q ∈ E[5] with hP, Qi = ζ5 . An isomorphism A[5] ' E[5] as described amounts to
a choice of generators P 0 , Q0 ∈ A[5] such that hP 0 , Q0 i = ζ5 . Y 0 (5) has a Q-rational point
x0 = (E, P, Q) and its Q-rational points are from the A’s over Q such that A[5] ' E[5] as
GQ -modules and preserving the Weil pairings. If P and Q are defined over K, then Y 0 (5) and
Y (5) are isomorphic over K. It was shown by Kline that X(5) over C is P1 (C). The compatified Y 0 (5), denoted by X 0 (5) over Q, is isomorphic to P1 (Q) since it has a Q-rational point
x0 . Therefore, X 0 (5) is infinite. Define another curve Y 0 (5, 3) which parametrizes the ellitpic
curve A with isomorphism A[5] ' E[5] preserving Weil pairings and a cyclic subgroup of order
3. Since Y 0 (5, 3) has genus > 1, by a theorem of Falting’s Y 0 (5, 3)(Q) is finite. Hence there
are infinite many points in Y 0 (5)(Q) which do not lie in the image of Y 0 (5, 3)(Q) → Y 0 (5)(Q).
In other words, there are infinitely many elliptic curves A/Q satisfying (i) and (ii). Choose
such a point x in Y 0 (5)Q very close to x0 under the 5-adic topology. Then the elliptic curve
A correspond to x is semi-stable (why??)
The main result of Wiles is
Theorem Let ` is an odd prime. Let ρ : GQ → GL2 (K) be an `-adic representation with
reduction ρ̄. If
(a) ρ̄ is irreducible and modular.
39
(b) for p 6= `, ρ |Ip ∼
1 ∗
0 1
.
(c) ρ |G` is semi-stable.
(d) det ρ = ε` .
Then ρ is modular.
Lemma There is an auxiliary semi-stable elliptic curve A defined over Q satisfying
(1) A[5] ∼
= E[5] as GQ -modules. (i.e.,ρA,5 ∼
= ρE,5 )
(2) A[3] is an irreducible GQ -modules. (i.e., ρA,3 is irreducible)
Corollary Taniyama-Shimura conjecture holds for semi-stable elliptic curves E/Q .
Proof. If ρ̄E,3 is irreducible, then it is modular, then ρE,` is modular.
If ρ̄E,3 is reducible, then ρ̄E,5 is irreducible. Let A be an elliptic curve as in Lemma. Since
ρ̄A,3 is irreducible, we know ρA,3 is modular. So ρA,5 is modular and we have ρ̄A,5 ' ρ̄E,5 is
modular. Hence ρE,5 is modular.
3.2
The strategy
Let K be a finite extension of Q` with ring of integers O, maximal ideal λ, and residue field
κ = O/λ. Given representation ρ : GQ −→ GL2 (κ) with det ρ = ε` mod `. If f is a normalized
newform of weight 2 level N and trivial character such that the attached `-adic representation
ρf,` has ρf,` w ρ, then ρf,` satisfies :
• It is irreducible
• det ρf,` = ε`
• It is unramified outside a finite set of places
Assume
• this representation |G` is semi-stable
• ` : odd
I Denote by < the set of isomorphic classes of `-adic degree 2 continuous representations
ρ of GQ satisfying the above conditions and ρ lifts ρ. In other words, think of < as a set of
”plausibly modular” liftings of ρ. Denote by = the set of isomorphic classes of ρf,` , where
f is a newform of weight 2 level N and trivial character, ρf,` satisfying the conditions and
ρf,` w ρ. Hence = is the set of modular liftings of ρ parameterized by the newforms f as
described. Clearly < ⊇ =.
I If we assume ρ modular and ρ |G` semi-stable, then the set = is nonempty. Ribet showed
that in this case = is an infinite set, and so is <. We are led to a filtration of <. SFor a finite
set Σ of primes, defines representations of type Σ. Let =Σ = = ∩ <Σ . Have < = <Σ . Thus
40
it suffices to show <Σ = =Σ for all Σ. In this case, =Σ is a finite set.
I The preliminary definition of a representation ρ to have type Σ is in terms of the conductor
N (ρ)
of ρ and conductor of ρ, i.e. Σ should contain all primes dividing N
(ρ) . In other words, ρ
outside Σ is very nice.
I If we assume N (ρ) square-free, (which is ok if ρ = ρE,` for a semi-stable elliptic curve
E/Q ), then this is a suitable definition of type Σ. But then we should extend O to complete
Noetherian local O-algebras. For this purpose it is convenient to work in the context of Mazur’s
deformation theory, thereby introducing more structure into the problem, and enabling us to
use tools from commutative algebra.
I The desired quality <Σ = =Σ is stated by Wiles as the “main conjecture/theorem” as
follows. Denote by <Σ the maximal deformation ring which parameterizes representation of
type Σ with a fixed residual representation ρ, and by TΣ the Hecke algebra which parameterizes
the newforms of weight 2 and trivial character, of type Σ which reduce to ρ. Then the ring
homomorphism <Σ → TΣ from deformation theory is an isomorphism.
Wiles’ idea is to prove the isomorphism for Σ = ∅ first, then deduce the case of general Σ
from this.
3.3
Deformation of representations
I Denote by CO the category where objects are complete Noetherian local O-algebras with
residue field κ, the morphisms are O-algebra homomorphisms which send maximal ideal to
maximal ideal (called local). Let ρ be an irreducible representation from G → GLd (κ), fixed.
Let D0 denote the category of profinite O[G]-modules with continuous morphisms. Denote
by D a full subcategory of D0 which is closed under taking sub-objects, quotients and direct
products and whichTcontains Mρ . Note that if M is an object in D0 and Mi is a collection
of sub-objects with Mi = {0} such that M/Mi are objects in D, then M is also in D since
Q
M ,→ M/Mi .
I Let χ: G −→ O× be a character of G such that det ρ = χ mod λ. By a lifting of ρ of
type D = ( O, χ , D ) we shall mean an object R of CO and a continuous representation ρ :
G −→ GLd (R) such that
(a) ρ mod mR is ρ,
(b) det ρ = χ,
(c) Mρ is an object in D.
Theorem (Mazur for certain D’s, Ramakrishna for all D)
There is a lifting ρuniv
: G → GLd (R0 ) of type D such that if ρ : G → GLd (R) is a lifting of type
D
D then there is a unique morphism φ : R0 → R such that ρ = φ ◦ ρuniv
D . The representation
ρuniv
is
called
the
universal
deformation
of
type
D.
D
Proof. (Sketch of Falting’s proof)
Let g1 , · · · , gr be a fixed set of topological generators of G. Let A1 , · · · , Ar be elements in
GLd (O) which lift ρ(g1 ), · · · , ρ(gr ). Define a mapping c : Md (O) → Md (O)r by x 7→ (xA1 −
A1 x, · · · , xAr − Ar x). Since c has torison free cokernel, we can decompose Md (O)r = Im c ⊕ V
41
for some O-submodule V in Md (O)r . If ρ : G → GLd (R) is a lifting of ρ of type D, set
υρ = (ρ(g1 ) − A1 , · · · , ρ(gr ) − Ar ). Then ρ is uniquely determined by υρ . Note that υρ ≡ 0
mod mR . We say that ρ is well-placed if υρ ∈ V ⊗O R
The crucial point is
Lemma If ρ : G → GLd (R) is a lifting of ρ of type D, then there is a unique conjugate ρ0 of
ρ which is well-placed.
This is proved first for the case mnR = 0 by induction on n. Then deduce the general case
from this case.
s
P
I Let e1 , · · · , es be an O-basis of V , we may write υρ =
υρ,i ei , where υρ,i ∈ mR . Consider
i=1
a homomorphism θρ : O[[T1 , · · · , Ts ]] → R defined by θρ (Ti ) = υρ,i (So that υρ =
s
P
Note that υρ is completely determined by θρ , and ρ(gj ) = Aj +
s
P
θρ (Ti )ei ).
i=1
θρ (Ti )eij for j = 1, · · · , r,
i=1
where ei = (ei1 , · · · , eir ), eij ∈ Md (O). Let I denote the intersection
of all .ideals
J of
O[[T
,
·
·
·
,
T
]]
1
s J of type
O[[T1 , · · · , Ts ]] such that there is a representation ρJ : G → GLd
.
s
P
D with ρJ (gj ) = Aj +
θρJ (Ti )eij for j = 1, · · · , r. Let RD = O[[T1 , · · · , Ts ]] I and define
i=1
the representation ρuniv
: G → GLd (RD ) by ρuniv
D
D (gj ) = Aj +
s
P
Tij eij for j = 1, · · · , r.
i=1
I Properties of RD
ρ : G → GLd (κ), κ = O/λ. If K 0 is a finite extension of K with ring of integers O0 and residue
field κ0 ⊃ κ, ρ : G → GLd (κ) ,→ GLd (κ0 ), D = (O, χ, D), D0 = (O0 , χ, D0 ). Here D0 is the full
subcategory of the category of profinite O0 [G]-modules such that the underlying object of D0
is actually an object of D.
Lemma We have universal deformation rings RD , RD0 and universal representations ρuniv
D
univ ⊗ 1.
0
univ
and ρuniv
D0 . Then RD0 = RD ⊗O O and ρD0 = ρD
I Let ρ : G → GLd (O/λn ) be a continuous representation. Let R0 = (O/λn )[ε]/ε2 be the
tangent space of O/λn . Let ρ̃ : G → GLd (R0 ) be a lifting of ρ. Then for each g ∈ G, write
ρ̃(g) = (1 + εξ(g))ρ(g) , where ξ(g) ∈ Md (O/λn ).
ρ̃(g1 g2 ) = ρ̃(g1 )ρ̃(g2 ), ∀g1 , g2 ∈ G ⇐⇒ ξ(g1 g2 ) = ξ(g1 ) + ρ(g1 )ξ(g2 )ρ(g1 )−1 , ∀g1 , g2 ∈ G.(∗)
I Given ρ, we have a representation induced from ρ acting on Md (O/λn ) by conjugation.
This representation is called the adjoint representation of ρ. The relation (∗) says that ξ :
G → Ad(ρ) is a 1-cocycle. If ρ̃0 is another lifting such that ρ̃0 is obtained from ρ̃ by conjugation
by a matrix of the form 1 + M , where M ∈ Md (O/λn ) (say that ρ̃ and ρ̃0 are equivalent),
then it is easy to check that ξ and ξ 0 differ by a 1-coboundary. Therefore H 1 (G, Ad(ρ))
parameterizes the equivalent classes of liftings of ρ to R0 .
42
I Md (O/λn ) contains a subspace of trace zero, which is invariant under Ad(ρ). Denote by
Ad0 (ρ), the restriction of Ad(ρ) to the trace zero subspace.
Note that
det ρ̃(g) = det(1 + εξ(g)) det ρ(g) = (1 + εT rξ(g)) det ρ(g).
So H 1 (G, Ad0 (ρ)) parameterizes the equivalent classes of liftings of ρ preserving det.
Remark. If ` - d, then H 1 (G, Ad(ρ)) = H 1 (G, Ad0 (ρ))⊕Hom(G, O/λn ). For ξ ∈ Hom(G, O/λn ),
det(1 + εξ(g)) = 1 + εdξ(g). H 1 (G, Ad(ρ)) has another interpretation. Denote by Mρ the underlying space of ρ and Mρ̃ the space for a lifting ρ̃. Note that Mρ̃/εMρ̃ ' Mρ , and multiply by
ε gives an isomorphism from Mρ to εMρ̃ . Consider the short exact sequence of O/λn -modules
0 −→ εMρ̃ −→ Mρ̃ −→ Mρ̃/εMρ̃ −→ 0
So ρ̃ gives rise to an extension of Mρ to Mρ . Check that if ρ̃0 is equivalent to ρ̃, then the
resulting extensions are isomorphic. So we obtain H 1 (G, Ad(ρ)) → Ext0 (Mρ , Mρ ), which is an
isomorphism.
1 (G, Ad0 (ρ̄)) the subset of
Now consider ρ : G → GLd (O/λ), i.e. n = 1. Denote by HD
H 1 (G, Ad(ρ̄)) corresponding to the extension in Ext0 (Mρ̄ , Mρ̄ ) which are objects in D.
1 (G, Ad0 (ρ̄)) → Hom (mR /
Lemma There is an isomorphism HD
D (λ, m 2 ), κ) where
κ
RD
m
Homκ ( RD/(λ, m 2 ), κ) is the tangent space of RD as κ-modules.
RD
1 (G, Ad0 (ρ̄)) classifies liftings of ρ̄ to GL (κ[ε]/ 2 ) of type D. To each such ρ, there
Proof. HD
d
ε
is a unique O-algebra homomorphism φ: RD → κ[ε]/ε2 such that ρ = φ ◦ ρuniv
D . The desired
0
1
m
R
map from HD (G, Ad (ρ̄)) to Homκ ( D/(λ, mR2 ), κ) is given by φ 7→ φ |mRD .
D
I Now suppose θ: RD −→ O is a surjective O-algebra homomorphism with kernel ℘. Let
ρ = θ ◦ ρuniv
D . Set
−n
1
1
HD
(G, Ad0 (ρ) ⊗ K/O) = lim HD
(G, Ad0 (ρ) ⊗ λ /O)
−→
−n
−n
1 (G, Ad0 (ρ) ⊗ K/ )).
(Note that λ /O ' O/λn and H1D (G, Ad0 (ρ) ⊗ λ /O) ⊂ HD
O
Lemma There is a canonical O-linear isomorphism
1
HomO ((℘/℘2 ), K/O) ' HD
(G, Ad0 (ρ) ⊗ K/O).
I Recall the notations:
K ⊃ O with residue field κ.
CO objects: complete noetherian local O-algebra with residue field κ.
G: pro-finite topopogical finitely generated group.
ρ̄ : G → GLd (κ) continuous representation.
D0 objects: O[G]-modules.
D: full subcategory of D0 , containing Mρ̄ .
Fix χ : G → O× character such that det ρ̄ = χ mod mO .
D = (O, χ, D).
I A representation ρ : G → GLd (R) is of type D if
43
(1) det ρ = χ.
(2) ρ mod mR is ρ̄.
(3) Mρ lies in D.
Theorem (Mazur)
There is a lifting ρuniv
: G → GLd (RD ) of type D such that if ρ : G → GLd (R) is a lifting of
D
type D, then there is a unique O-morphism φ : RD → R such that φ ◦ ρuniv
is conjugate to ρ.
D
3.4
Deformation of Galois representations
I Let ` be an odd prime and ρ̄ : GQ → GL2 (κ) is a continuous mod ` absolutely irreducible
representation of GQ . Suppose ρ̄ satisfies
• det ρ̄ = ε` mod `.
• ρ̄ |G` is semi-stable.
• For p 6= `, #ρ̄(Ip ) | `.
Remark. These conditions are satisfied by ρ̄E,` for a semi-stable elliptic curve E over Q if
it is irreducible.
I Let Σ be a finite set of primes. For an object R in CO , a continuous lifting ρ : GQ → GL2 (R)
of ρ̄ is said to be of type Σ if
• det ρ = ε` .
• ρ |G` is semi-stable.
• If ` 6∈ Σ and ρ̄ |G` is good, then so is ρ |G` .
• If p 6∈ Σ ∪ {`} and ρ̄ is unramified at p, so is ρ
1 ∗
• If p 6∈ Σ ∪ {`}, ρ̄ |Ip ∼
(i.e. ordinary).
0 1
That is, for p 6∈ Σ, we want ρ as unramified as ρ̄. Clearly, if Σ ⊂ Σ0 and ρ is a lifting of type
Σ, then it is also of type Σ0 .
I If E is a semi-stable elliptic curve over Q, then ρE,` is of type Σ if Σ contains all primes
dividing conductor of E.
Theorem Given Σ, there is a universal lifting ρuniv
: GQ → GL2 (RΣ ) of type Σ such that if
Σ
ρ : GQ → GL2 (R) is a lifting of type Σ, then there is a unique O-morphism φ from RΣ to R
such that ρ is conjugate to φ ◦ ρuniv
Σ . Moreover, the following hold:
0 is the universal deformation ring, then R0 =
a) If K 0 is a finite extension of K, and RΣ
Σ
0
univ ⊗ 1.
RΣ ⊗O O0 and ρuniv
=
ρ
Σ
Σ
44
b) RΣ can be topologically generated as as an O-module by dimκ HΣ1 (GQ , Ad0 ρ̄) elements.
c) If φ : RΣ O is a O-algebra homomorphism with kernel ℘ and ρ is the lifting φ ◦ ρuniv
Σ ,
then
Hom(℘/℘2 , K/O) ' HΣ1 (GQ , Ad0 ρ ⊗ K/O) as O-linear isomorphism.
I Sketch of proof.
Let L0 be the fixed field of ker ρ̄. For n ≥ 1, let Ln be the maximal elementary abelian `∞
S
Ln ,
extension of Ln−1 unramified outside Σ∪{`}∪{places where ρ̄ is ramified}. Let L∞ =
n=0
which is Galois over Q. Note that any lifting of ρ̄ of type Σ factors through G = Gal(L∞ /Q).
Further, Gal(L∞ /L0 ) is a pro-`-group and its maximal abelian quotient is Gal(L1 /L0 ), which
is finite by Hermite-Minkowski’s thoerem since L1/L0 is ramified at finitely many places with
elementary abelian `-group as a Galois group.
GL (R)
; 2
ww
w
w
ww
ww
w
R
w
GL2 ( /m2 )
w
R
ww
w
w
ρ w
w
w
ww
ww
w
R
w
GL
(
2 /m1 )
ww
ww
w
w
ww
ww
/ GL2 (κ) = GL2 (R/ )
ρ̄ : GQ
5
mR
k
KKK
kk
kkk
KKK
k
k
kkk
KK
%
kkk ρ̄
L∞

L1



unramified outside Σ ∪ {`}



L0
Gal(L0 /Q)
Q
mR = (π, x1 , · · · , xr ) ⊃ m1 = (π 2 , πxi , x1 , · · · , xr ) ⊃ m2 = (π 2 , πxi , x1 xi , x2 , · · · , xr ) ⊃ · · · ⊃ m2R = mr
0 −→ 1 + M2 (mR/m1 ) −→ GL2 (R/m1 ) −→ GL2 (R/mR ) −→ 0
where 1 + M2 (mR/m1 ) is an elementary `-group.
b its maximal elementary abelian quotient. If h1 , · · · , hr
Lemma Let Hbe a pro-`-group and H
b
in H generates H, then h1 , · · · , hr topologically generate H.
Since Gal(L1 /L0 ) is finite, by lemma, we know that Gal(L∞ /L0 ) is topologically finitely generated. Since Gal(L0 /Q) is finite, so Gal(L∞ /Q) = G is also topologically finitely generated.
I Next we define D. Let D denote the category of profinite O[G]-modules M satisfying
• M is semi-stable as O[G` ]-module
• if ` 6∈ Σ, and if ρ̄ |G` is good, then M is a good O[G` ]-module.
• if p 6∈ Σ ∪ {`} and if ρ̄ is ramified at p, then there exists an exact sequence of O[Ip ]modules
0 −→ M (−1) −→ M −→ M (0) −→ 0.
such that Ip acts trivially on M (−1) and M (0) .
45
Then a lifting ρ : GQ → GL2 (R) of ρ̄ is of type Σ if and only if
• det ρ = ε` .
• ρ factors through G.
• Mρ is an object of D.
By Mazur’s theorem, ρuniv
exists. The properties (a), (b), and (c) follow from the properties
Σ
(A), (B), and (C) before.
Corollary Suppose ρ̄ |GQ(√−3) is absolutely irreducible if ` = 3. Then RΣ can be topologically
generated as an O-module by
X
dimκ H 0 (Gp , ad0 ρ̄(1))
dimκ HΣ1 (GQ , ad0 ρ̄(1)) + d` +
p∈Σ\{`}
elements, where
δ` =
1 (G , ad0 ρ̄) − dim H 1 (G , ad0 ρ̄) , if ` ∈ Σ
dimκ Hss
κ f
`
`
0
, if ` ∈
6 Σ.
I Next we consider RΣ where Σ has a special property. Namely Σ is a finite set consisting
of special primes q satisfying
• q ≡ 1 (mod `).
• ρ̄ is unramified at q such that ρ̄(Frobq ) has two distinct eigenvalues in κ.
To distinguish this kind of set, call it Q. The goal is to compare RQ and Rφ .
I For each q, let ∆q denote the maximal quotient of (Z/qZ)× whose order is a power of `.
Then ∆q is naturally a quotient of GQ and Gq :
/ Gal(Q(ζq )/Q) ' (Z/ )×
qZ
χq : GQ
and
χq : Gq
/ Gal(Qq (ζq )/Qq )
/ / ∆q
/ / ∆q
Gal(Qq /Qq )
Q
Q
Let ∆Q =
∆q and χQ =
χq : GQ → ∆Q . Note that O[∆Q ] is isomorphic to
q∈Q
q∈Q
.
O[[sq , q ∈ Q]] ((1 + s )#∆q − 1, q ∈ Q) by Iwasawa theory argument. For q ∈ Q, let αq and
q
βq be the two eigenvalues of ρ̄(Frobq ).
Lemma For q ∈ Q,
ρuniv
|Gq
Q
is conjegate to
¯
that ξ(Frob
q ) = αq .
46
ξ
0
0 ε` ξ −1
for some character ξ of Gq such
 



Qq








wild; pro-q-group



 




Iq Pq









tame







Qunr


q
Proof. Let f be a lifting of Frobq in Gq . The characteristic polynomial of ρuniv
Q (f ) is a monic
polynomial of degree 2 with coefficients in RQ . Modulo mRQ , it is the characteristic polynomial
of ρ̄(Frobq ), which has two distinct roots αq and βq . Let α
eq and βeq be the Hensel lifting of
αq and βq to roots of the characteristic polynomial of ρuniv
eq 6= βeq . So the
Q (f ). Note that α
univ
univ
representation space ρQ has an RQ -basis which diagonalizes ρQ (f ). Next we show that
this basis also diagonalizes ρuniv
Q (σ) for all σ ∈ Iq . Since Iq hf i is dense in Gq , this will show
the lemma. With respect to this basis, for σ ∈ Iq , we may write
a b
univ
, where a, b, c, d ∈ mRQ .
ρQ (σ) = I2 +
c d
univ
univ
−1 = ρuniv (σ)q . That is,
Since ρuniv
|Gq is tamely ramified, we have ρuniv
Q
Q (f )ρQ (σ)ρQ (f )
Q
α
eq
q
−1 α
eq
a b
a b
I2 +
,
= I2 +
c d
c d
βeq
βeq
i.e.,

I2 + 
This implies
since
α
eq
βeq
α
eq
βeq
− 1 and
a
βe
c αeqq
α
e
βq
b eq
d

qa qb
qc qd
a b
≡ I2 +
c d
 ≡ I2 +
mod m2RQ
mod m2RQ .
e
β
− 1 b and αeqq − 1 c lie in mRQ (b, c). It then implies (b, c) = mRQ (b, c)
βeq
α
eq
− 1 are units. Hence by Nakayama lemma, b = c = 0.
×
I So for each q ∈ Q, we have a character ξ : Gq → RQ
, denote it by ξq,Q . Observe that ξq,Q |Iq
is a tame character, which factors through the maximal tame abelian extension of Qq , which is
×
Qq (ζq ). So ξq,Q |Iq factors through χq . Let ξ : GQ → RQ
be the character unramified outside Q
and ξ |Gq = ξq,Q . Thus the ramified part of ξ factors through χQ . We
. regard RQ as an O[∆Q ]
−2
algebra by ξ . Denote by IQ the ideal (sq : q ∈ Q) of O[[sq , q ∈ Q]] ((1 + s )#∆q − 1, q ∈ Q).
q
Then Rφ ' RQ/IQ RQ .
We have shown
∼
Corollary The canonical map from RQ to Rφ yields the isomorphism RQ/IQ RQ −→ Rφ .
47
Lemma
(a) If q ∈ Q, then
H 0 (GFq , ad0 ρ̄) = H 0 (GFq , ad0 ρ̄(1)) = κ.
and
H 1 (GFq , ad0 ρ̄) = H 1 (GFq , ad0 ρ̄(1)) = κ.
(b) RQ can be topologically generated as an O-module by
1
#Q + dimκ HQ
(GQ , ad0 ρ̄(1)) elements.
∼
(c) If Hφ1 (GQ , ad0 ρ̄(1)) −→
L
q∈Q
H 1 (GFq , ad0 ρ̄(1)), then #Q = dimκ Hφ1 (GQ , ad0 ρ̄(1)) and
RQ can be topologically generated as an O-module by #Q elements.
Proof.
αq 0
(a) Since ρ(Frobq ) =
, ad0 ρ(Frobq ) acts semi-simply with eigenvalues x, 1, x−1 ,
0 βq
α
where x = βqq ∈ κ \ {0, 1}. Same with Ad0 ρ̄(1). Thus
H 0 (GFq , Ad0 ρ̄) = κ = H 0 (GFq , Ad0 ρ̄(1)).
Denote by M the representation space of Ad0 ρ̄ or Ad0 ρ̄(1). Since GFq = hFrobq i,
κ ' H 1 (GFq , Ad0 ρ̄) = M (Ad0 (ρ̄)(Frobq ) − 1)M .
Same with H 1 (GFq , Ad0 ρ̄(1)).
I To continue, we will need
Theorem Let H be a finite subgroup of PGL2 (κ). Then one of the following holds
• H is conjugate to a subgroup of the upper triangular matrices.
• H is conjugate to PSL2 (F`r ) or PGL2 (F`r ).
• H is isomorphic to A4 , S4 , A5 or to the dihedral group D2r with r > 1 and ` - r.
Lemma Let F be a finite field with odd characteristic. If #F 6= 5, then
where M20 (F) =
(
a
b
c −a
H 1 (SL2 (F), M20 (F)) = 0,
)
a, b, c ∈ F .
48