Lectures on Modular Forms and Galois...

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.

γ =(a bc d

)∈ SL2(Z), γz =

az + b

cz + d.

I Let Γ be a subgroup of SL2(Z) of finite index. Given a positive integer k, a modular formf for Γ of weight k is a function satisfies:(1) f is holomorphic on H,

(2) f(γz) = (cz + d)kf(z), ∀γ =(a bc d

)∈ SL2(Z),

(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 aRiemmann 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) toΩ1(XΓ).I Γ is called a congruence subgroup if it contains

Γ(N) =

(a bc d

)∈ SL2(Z)

∣∣∣∣∣(a bc d

)≡(

1 00 1

)(mod N)

for some N.

If Γ is a congrunce subgroup, then Mk(Γ) = Ek(Γ)⊕ Sk(Γ) where Ek(Γ) denotes the space ofEisenstein series.

1

I We are interested in two special kinds of congruence subgroups:

Γ0(N) = (a bc d

)∈ SL2(Z) | c ≡ 0 (mod N)

∪

Γ1(N) = (a bc d

)∈ SL2(Z) |

(a bc d

)≡(

1 ∗0 1

)(mod N)

Γ1(N) C Γ0(N) with Γ0(N)/Γ1(N) '

(Z/NZ)×.

I So Γ0(N) acts on Sk(Γ1(N)). Then Sk(Γ1(N)) decomposes into

Sk(Γ1(N)) =⊕

χ∈((Z/NZ)×)∨

Sk(N,χ) where

Sk(N,χ) =f ∈ Sk(Γ1(N))

∣∣∣∣f(γz) = χ(d)(cz + d)kf(z), ∀γ =(∗ ∗c d

)∈ SL2(Z)

= 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

f(z) =∞∑n=1

ane2πinz.

We want to understand the arithmetic of the Fourier coefficients. Further, we associate to fan L-function:

L(s, f) =∞∑n=1

ann−s converges absolutely for Re s > k.

I For p - N, there is the Hecke operator Tp defined by

Tpf(z) =1p

p−1∑u=0

f

(z + u

p

)+ χ(p)pk−1f(pz)

=∞∑n=1

anpe2πiz + χ(p)pk−1

∞∑n=1

ane2πinpz

=∞∑n=1

(anp + χ(p)pk−1an/p

)e2πinz.

Here, ax = 0 if x 6∈ Z.I For p | N, we have the operator Up defined by

Upf(z) =∞∑n=1

anp e2πinz.

2

Theorem Let f(z) =∞∑n=1

ane2πinz ∈ Sk(N,χ). Then L(s, f) is Eulerian at p - N if and only

if f is an eigenfunction of Tp. Moreover, if Tpf = λpf, then

L(s, f) =

∑m≥1,p-m

amm−s

· 11− λpp−s + χ(p)pk−1−2s

.

I For p - N, r ≥ 1,Tpr+1 = TpTpr − 〈p〉pk−1Tpr−1.

Here 〈p〉 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

1· · ·Tpes

swhere T1 = id .

I For a congruence subgroup Γ with Γ1(N) ⊆ Γ ⊆ Γ0(N), we have

Sk(Γ) =⊕

χtrivial on Γ/Γ1(N)

Sk(Γ, χ).

I Define the Hecke algebra∏

=∏Sk(Γ) to be the algebra over C generated by Tn, n ≥ 1 and

〈d〉 for d ∈(Z/NZ

)×on Sk(Γ).

Formally, we have

∞∑n=1

Tnn−s =

∏p-N

11− Tpp−s + 〈p〉pk−1−2s

∏p|N

11− Tpp−s

.

I For p - N, Peterson inner product 〈·, ·〉 on Sk(Γ, χ), we have 〈Tpf, g〉 = χ(p)〈f, Tpg〉.In other words, the transpose of Tp is 〈p〉−1Tp as operators. Consequently, Tp on Sk(Γ, χ)is diagonalizable. Since the Tp’s commute, the operators Tp, p - N can be simutaneouslydiagonalizable. Thus,

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 newformof weight k, level N and character χ. A common eigensapce of dimension greater than oneis spanned by a newform g of weight k level M | N and character χ and g(mz) for m

∣∣NM .

So it is sufficient to understand newforms of all levels. Each common eigenspace for p - Ncontains a unique subspace which is of one dimensional invariant under all operators in T. Letg(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) givesrise to an algebra homomorphism T → C. Denote TQ(A) the algebra generated by Tm, 〈d〉over A (any ring ⊇ Q). This map Tm 7→ am(g) is an algebra homomorphism from TQ toKg = 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 inifintelymany p.

In particular, if for the two eigenspaces, the eigenvalues of Tp are equal for almost all p thenthese two eigenspaces are the same.I Fromally, we have

∞∑n=0

Tnn−s =

∏p-N

11− Tpp−s + 〈p〉pk−1−2s

∏p|N

11− Tpp−s

.

In summary, for a normalized newform f(z) =∞∑n=1

ane2πinz of weight k level N and character

χ, we have

L(s, f) =∏p-N

11− app−s + χ(p)pk−1−2s

∏p|N

11− app−s

.

I Ramanujan Conjecture. What about the size of ap?For p | N, we have

|ap| = p(k−1)/2 if χ is not a character mod Np ,

ap = 0 if p2 | N and χ is a character mod Np ,

a2p = χ(p)pk−1 if p||N and χ is a character mod N

p .

For p - N, there is the celebrated Ramanujan conjecture that |ap| ≤ 2p(k−1)/2. Viewing 1 −app

−s +χ(p)pk−1p−2s as a quadratic polynomial in p−s then this conjecture has the followingequivalent form. Write

1− app−s + χ(p)pk−1p−2s = (1− αpp−s)(1− βpp−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 Delignefor k ≥ 3 using Galois representations.

I Sato-Tate Conjecture. Let f(z) =∑n≥1

ane2πinz be a normalized newform fo weight

k level N and trivial character (i.e. f ∈ Sk(Γ0(N))). Write αp = p(k−1)/2eiθp then βp =p(k−1)/2e−iθp . We may assume that θp ∈ [0, π]. How are the θp’s distributed? Or, equivalently,consider lp = αp +βp = p(k−1)/22 cos θp and put rp = lp/p

(k−1)/2 so that rp ∈ [−2, 2]. We wantto 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 at-tached to any idele class character of an imaginary quadratic extension of Q. Then, the rp’sare uniformly distributed with respect to the Sato-Tate measure:

µst =1π

√1− x2

4dx on [−2, 2].

4

In other words, for any interval [a, b] ⊆ [−2, 2],

limx→∞

#p ≤ x | rp ∈ [a, b]#p ≤ x

=∫ b

aµst.

In terms of θp’s, this means that θp’s are uniformly distributed with respect to 1π sin2 θ dθ,

which arises from measure on conjugacy classes of SU2.

0.2 Elliptic curves

Algebraically, an elliptic curve E defined over a field K is a smooth curve of genus one over Kcontaining at least one K-rational point. If K is a subfield of C, then E(C) can be identifiedwith a torus C/L for some lattice L in C. Up to equivalence, we may assume that L has aZ-basis τ, 1 with τ ∈ H. There is a group structure on E making it an abelian additivegroup. For N ≥ 1, denote by E[N ] = P ∈ E | NP = O = the group of poitns of orderdividing N.

If char(K) - N then E[N ] ' Z/NZ× Z/NZ.

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] forall 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 Galoisgroup GQ = Gal(Q/Q) acts on each E[`r] and hence on T`(E). This yields a continuoushomomorphism

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 eachprime p, we have the p-adic valuation on Q. Extending this to a valuation on Q (there aremany such extensions). The automorphisms in GQ preserving this valuation form a subgroupGp, 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(Qurp /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 well-defined mod Ip. When we change Gp, the Frobenius Frobp is changed by a conjugate (modinertia). A representation ρ is unramified at p if it is trivial on Ip. Then, ρ(Frobp) is well-defined 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 adefining 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

Z(Ep, p−s) := exp

( ∞∑r=1

#E(Fpr)p−rs

r

)

=det(1− ρ`,E(Frobp)p−s)

(1− p−s)(1− p1−s)

=1− app−s + p1−2s

(1− p−s)(1− p1−s)

Define the Hass-Weil L-function attached to E by

L(s,E) =∏p-N

11− app−s + p1−2s

∏p|N

11− appL−s

where for p | N,

ap =

1 if E has split multipicative reduction mod p−1 if E has non-split multiplicative reduction mod p0 if E has additive reduction mod p.

0.3 Galois representations and modular forms

We begin with the following result.

Theorem (Eichler-Shumura)

Let f(z) =∞∑n=1

ane2πinz be a normalized newform of weight 2 level N and trivial character

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

f(z) =∞∑n=1

ane2πinz of weight k level N and character χ, for a prime `, let K be a finite

6

extension of Q`. A representation ρ`,f : GQ → GL2(K) is called an `-adic representationattached 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 Breuil-Conrad-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 attachedto f.

Theorem (Deligne-Serre for k = 1)Given a normalized newform f of weight one, there exists complex degree two irreduciblerepresentation 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 odddegree two representation from

GQ → GL2(F`).

Serre’s conjecture:

(1) Any odd irreducible degree two representation ρ : GQ → GL2(F`) arises from reductionof an `-adic modular representation.

(2) An explicit recipe for the a newform f of lowest weight and level such that (1) holdsis given.

7

1 Eichler-Shimura Relation and Galois Representation

1.1 Homology group and rationality

Let Γ be a finite index subgroup of SL2(Z). The modular curve XΓ has universal cuspH∗ = 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

(c, f(z)) 7→∫

cf(z)dz.

If c 6= 0, then ∃ f such that∫c f(z)dz 6= 0. Hence we may embedH1(XΓ,Z) into HomC(S2(Γ),C)

as a lattice Λ of rank 2gΓ. Here∫c f(z)dz =

∫ γτ0τ0

f(z)dz by lifting c to H∗. Since H∗ is simplyconnected, the integral is independent of the path from τ0 to γτ0, and in fact, τ0 can be anypoint of H∗. Conversely, given γ ∈ Γ, we define a cycle cγ in H1(XΓ,Z) as the image of τ0 toγτ0 for any choice τ0 ∈ H∗ so that for any f ∈ S2(Γ)∫

cγ

f(z)dz =∫ γτ0

t0

f(z)dz

I Call c : Γ→ H1(XΓ,Z) the map γ 7→ cγ . For γ, γ′ ∈ Γ, f ∈ S2(Γ)∫cγγ′

f(z)dz =∫ γγ′τ0

τ0

f(z)dz

=

(∫ γ′τ0

τ0

+∫ γγ′τ0

γ′τ0

)f(z)dz

=

(∫cγ′

+∫

cγ

)f(z)dz

i.e. c(γγ′) = c(γ) + c(γ′) so that c is a homomorphism. If γ is elliptic (resp. parabolic), wemay 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 anormal subgroup contained in ker c.

Since H1(XΓ,Z) is abelian, c factors through a surjective homomorphism from(Γ/Γep

)abto

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

operator Tm is to consider the double coset Γ(

1 00 m

)Γ =

⋃i

Γαi. The action of Tm on

f ∈ S2(Γ) defined before can be rephrased as

Tm(f(z)dz) =∑i

f(αiz)d(αiz).

8

For d ∈(Z/NZ

)×, choose a matrix Rd =

(a bc d

)∈ Γ0(N). Then

< d > (f(z)dz) = f(Rdz)d(Rdz).

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],

([γ], Tmf) =∫ γτ0

τ0

Tmf(z)dz =∑i

∫ γτ0

τ0

f(αiz)d(αiz)

=∑i

∫ αiγτ0

αiτ0

f(z)dz

=∑i

∫ γiαj(i)τ0

αiτ0

f(z)dz, by writing αiγ = γiαj(i) for some γi ∈ Γ

=∑i

(∫ γiαj(i)τ0

αj(i)τ0

+∫ αj(i)τ0

αiτ0

)f(z)dz

=∑i

∫[γi]

f(z)dz +∑i

∫ αj(i)τ0

αiτ0

f(z)dz

where∑i

∫ αj(i)τ0αiτ0

f(z)dz = 0, since αj(i) goes through all coset representatives as αi does. So,

we should definedTm[γ] =

∑[γi],

where αiγ = γiαj(i). Similarly< d > [γ] = [RdγR−1

d ].

This defines the action of TZ on H1(XΓ,Z).

I The map τ 7→ −τ dinfes an involution on H∗. If γ2 = γτ1 for some γ =(a bc d

)∈ Γ,

then τ∗2 = γ∗τ∗1 with γ∗ =(

a −b−c d

)∈ Γ. So ∗ induces an involution on XΓ, and hence on

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:

(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 ofT on S2(Γ).

9

(3) H1(Z) ∩ H1(R)+ is lattice of rank gΓ and a Z-basis of this intersection is a basis ofH1(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.

I Recall that S2(Γ1(N)) =⊕

(common eigenspace for Tp, p - N , all < d >). Each commoneigenspace contains a unique normalized newform of level dividing N . Let g1, · · · , gt be thenormalized of weight 2 level N1, · · · , Nt. Denote the common eigenspace containing gi byS(gi). Then S(gi) has a basis gi(mz) for all positive divisors m of N

Ni. On S(gi), the Tp for

p - N acts as scalar multiple by ap(gi) for d ∈(Z/NZ

)×, < d > acts as multiple by χi(d) where

χi is the character of gi.Observe that if a prime q

∣∣m∣∣ NNi, then Tqgi(mz) = gi

(mq z). This shows that S(gi) is a TC-

module generated by gi(NNiz). With respect to decomposition

S2(Γ1(N)) =⊕i

S(gi).

The Hecke algebra TC =⊕

TC |S(gi) decomposes. Put together, let

g(z) = g1

(N

N1z

)+ · · ·+ gt

(N

Ntz

)∈ S2(Γ1(N)).

Then S2(Γ1(N)) as a TC-module generated by g. Moreover, if h =∑ane

2πinz is a nonzeroelement in S2(Γ1(N)) with am 6= 0, then Tmh = 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 ismultiplication by am(gi), the mth Fourier coefficient of gi. The algebra TQ |S(gi) is nothing butQ(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/NZ)×

). 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 homomorphismfrom TQ to Q. So there exists gj ∈ S2(Γ) which realizes this homomorphism. In other words,gj =

∑σ(an)e2πinz = gσi is a newform of weight 2 level N and character χσi .

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Theorem Let f(z) =∞∑n=1

ane2πinz be a newform of weight 2 level N and character χ. Then

(1) all Fourier coefficients of f are algebraic integers

(2) Kf = Q(an, n ≥ 1) is a finite extension of Q

(3) for any σ ∈ Gal(Q/Q), fσ =∑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 spaceover K invariant under Gal(K/Q).Using Hilbert Theorem 90, S2(Γ1(N)) has basis with coefficients in Q. The integrality propertyof 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, letS2(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 ofrank 1.

Proof. It suffices to prove the case A = Z. Consider the pairing TZ × S2(Γ,Z) → Z given by(T, f) 7→ a1(Tf) the 1st Fourier coefficient of Tf .

(i) It is TZ-equiinvariant. Given T, T ′ ∈ TZ, f ∈ S2(Γ,Z), we have

(T ′T, f) = a1(T ′Tf) = a1(TT ′f) = (T, T ′f).

(ii) The pairing is nondegenerateIf f is nonzero, say the mth Fourier coefficient am 6= 0, then

(Tm, f) = a1(Tmf) = am 6= 0.

Suppose T ∈ TZ is such that (T, f) = a1(Tf) = 0, ∀ f , want to show TF = 0, ∀ f (ThenT = 0). If Tf 6= 0 for some f , then Tf has a nonzero Fourier coefficient, say am. Wehave

(T, Tmf) = (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.

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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 thatH1(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(∫

ci

f1(z)dz, · · · ,∫

ci

fgΓ(z)dz)∈ CgΓ , for i = 1, · · · , 2gΓ

are linearly independent over R and they generate a rank 2gΓ lattice Λ in CgΓ .I Define a map

Φ : XΓ −→ CgΓ/Λ

τ 7→(∫ τ

τ0

f1(z)dz, · · · ,∫ τ

τ0

fgΓ(z)dz)

for a fixed τ0 ∈ H∗.Extend this to Φ# : Div(XΓ) → CgΓ/Λ by Φ#(

∑nττ) =

∑nτΦ(τ). Restrict this map to

Div0(XΓ).

Φ# |Div0(XΓ): Div0(XΓ) // CgΓ/Λ

XΓ

==

eeJJJJJJJJJJJJJJJJJJJJJJJJ

τ − τ0 Φ(τ)

τ

ccHHHHHHHHHH 8

;;xxxxxxxxx

since Φ(τ0) = 0.

12

Theorem (Jacobi Inversion Theorem)Given any v ∈ CgΓ/

Λ, there exists gΓ points τ1, · · · , τgΓ such that

v = Φ(τ1) + · · ·+ Φ(τgΓ) = Φ#(τ1 + · · ·+ τgΓ − gΓτ0).

So Φ# |Div0(XΓ) is surjective.

Theorem (Abel’s Thoerem)The kernel of Φ# on Div0(XΓ) consists of the principal divisors. Hence Φ# induces an iso-morphism from Div0(XΓ)

/Prin(XΓ) to CgΓ/

Λ.

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 − τ0gives an embedding of XΓ to Jac(XΓ) (for gΓ ≥ 1), which is Q-rational. Jac(XΓ) has thefollowing universal property:For every F : XΓ → T a holomorphic and Q-rational map from XΓ to a complex torus T

F : XΓ//

%%LLLLLLLLLL T

Jac(XΓ)f

;;wwwwwwwwwτ

""DDDD

τ − τ0 ∈

∃ 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 isinvariant under the action of T. So T acts on the quotient, which is Jac(XΓ). More concretely,given Tm, then Tm(f(z)dz) =

∑if(αiz)d(αiz). This gives map

τ //∑

i αiτ //

∑i Φ(αiτ)

T#m : XΓ

//

%%LLLLLLLLLLDiv(XΓ) // Jac(XΓ)

Jac(XΓ)Tm

99rrrrrrrrrr

By the universal property of Jac(XΓ), ∃Tm : Jac(XΓ)→ Jac(XΓ). In fact, Tm maps

Φ(τ) =(∫ τ

τ0

f1(z)dz, · · · ,∫ τ

τ0

fgγ (z)dz)∈ CgΓ/

Λ

to (∫ τ

τ0

Tm(f1(z)dz), · · · ,∫ τ

τ0

Tm(fgγ (z)dz))

In particular, Tm and < d > are Q-rational maps.

13

I Γ = Γ1(N) or Γ0(N)Let Λ be the image of H1(XΓ,Z) → HomC(S2(Γ),C).

JΓ = Jac(XΓ) = Div0(XΓ)/principal divisors

' HomC(S2(Γ),C)/Λ

' CgΓ/Λ

where Λ generated by the periods (∫cjfi(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

τ // τ // Φ(τ)

XΓ//

$$HHHHHHHHH Div(XΓ) // Jac(XΓ)

Jac(XΓ)

88rrrrrrrrrrτ

""DDDD

τ − τ0 ∈

and we have

τ //∑

i αiτ //

∑i Φ(αiτ)

T#m : XΓ

//

%%LLLLLLLLLLDiv(XΓ) // Jac(XΓ)

Jac(XΓ)T

99rrrrrrrrrr

Choose τ0 to be i∞. We have T sends

Φ(τ) =(∫ τ

τ0

f1(z)dz, · · · ,∫ τ

τ0

fgΓ(z)dz)

to

TΦ(τ) =(∫ τ

τ0

T (f1(z)dz), · · · ,∫ τ

τ0

T (fgΓ(z)dz)).

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 ellipticcurve over C and P is a point of order N . Each class is representated by

(E,P ) =(

C/L<τ,1>,1N

),

where L<τ,1> = Zτ ⊕ Z with τ ∈ H. (E,P ) ∼ (E′, P ′) if there is an isomorphic mapping E toE′ and P to P ′.

14

I For Γ = Γ0(N), YΓ0(N) parametrizes the equivalent classes of (E,C), where E is an ellipticcurve over C and C a cyclic group of order N . The class of (E,C) may be represented by

(E,C) =(

C/L<τ,1>, 〈1N〉).

Call this Γ-structures.I For d ∈

(Z/NZ)×, < 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, givenby L〈 τ+i

p,1〉 for i = 0, 1, · · · , p− 1, and L〈τ, 1

p〉.

Starting from (E,P ) =(C/L<τ,1>,

1N

), we get p+ 1 isogenies of (E,P ) of degree p, given by

(Ei, Pi) =(

C/L〈 τ+ip,1〉,

1N

)for i = 0, 1, · · · , p− 1.

and

(E∞, P∞) =(

C/L〈τ, 1p〉,

1N

)'(

C/L<pτ,1>,1N

)If (E,P ) is represented by τ is XΓ, then

(Ei, Pi) =(

1 i0 p

)τ

(E∞, P∞) = 

(p 00 1

)τ

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

Tp((E,P )) =∑C

(E/C, P modC

)where C goes through all subgroups of E of order p.

1.3 Eichler-Shimura Relation

I Let ϕ be a function from a curveX 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 eachhorizontal 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 ′ = (y, x) | (x, y) ∈ T.Observe that for p - N, T ′p = Tp

((E,P ), (E′, P ′)) ∈ Tp ⇐⇒ ((E′, P ′), (E,P )) ∈ T ′p.

15

By definition, there exists a p-isogeny (E,P ) α−→ (E′, P ′) and the dual isogeny (E′, P ′)β−→

(E, pP ). The moduli space interpretation of XΓ allows it to have a model over Spec(Z). Thusone can consider reductions of XΓ.I It has a good reduction at all p - N . The reduction XΓ/Fp

now parametrizes (E,P ) forE defined over Fp, P of order N . The Hecke operator Tp at a ordinary point (E,P ) is givenby 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 pointwhose 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/Fpof degree p. Let F denote its graph in

XΓ/Fp×XΓ/Fp

, and by F ′ its transpose. These two correspondence induce the Frobenius mapF on the Jac(XΓ/Fp

) and its dual F ′ in the sense of abelian varieties.I In terms of divisors

F ′((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 thep+ 1 degree p isogenies of (E,P ). So

Tp((E,P )) = (E0, pP0) + · · ·+ (Ep−1, pPp−1) + (E∞, P∞)= (F+ F ′)((E,P ))

This shows thatTp = F+ F ′ on ordinary (E,P )

Since these points dense on XΓ/Fp, we have

Tp = F+ F ′ 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+ F ′.

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 structureusing Q-rational differentials. The fundamental q-expansion principle says that the Q-rationaldifferentials arise from the cusp forms with Fourier coefficients in Q. The Hecke operators areQ-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] = 1`n Λ/Λ '

(Z/ nZ)2gΓ

on which TZ and GQ act. Taking projective limit, we obtain the Tate module

T`(JΓ) = lim←−n

JΓ[`n] ' Z2gΓ`

on which TZ and GQ act. Call V` = V`(JΓ) = T`(JΓ)⊗Z`Q` on which TQ`

and GQ act. Noticethat 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/ nZ(' µ`n).

Passing the projective limit, we get a perfect pairing

〈 , 〉 : T`(JΓ)× T`(JΓ) −→ Z`.

The Hecke operators are not self-adjoint w.r.t 〈 , 〉.

I On XΓ, there is the Atkin-Lehner involution wN =(

0 −1N 0

). wN preserves the pairing

i.e. 〈x, y〉 = 〈wNx,wNy〉. It turns out that the adjoint of a Hecke operator T is wNTwN i.e.

〈Tx, y〉 = 〈x,wNTwNy〉.

For x ∈ T`(JΓ), define φx ∈ HomZ`(T`(JΓ),Z`) by φx(y) = 〈x,wNy〉. Then

φTx(y) = 〈Tx,wNy〉 = 〈x,wNTwNwNy〉 = 〈x,wNTy〉 = φx(Ty) = Tφx(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 F ′p are adjoint w.r.t. twisted pairing 〈 , wN 〉 i.e.

〈Fpx,wNy〉 = 〈x,wN F ′py〉.

17

Proof. We use the geometric interpretation of wN . On XΓ, it sends (E,P ) to(E/〈P 〉, Q

)where 〈P,Q〉N = ζN . Here 〈 , 〉N is the Weil pairing on elliptic curve E. We first show thatwN Fp = Fp −1 wN . By definition,

wN Fp((E,P )) = wN (E(p), P (p)) =(E(p)

/〈P (p)〉, Q), where 〈P (p), Q〉N = ζN .

Fp −1 wN ((E,P )) = Fp −1(E/〈P 〉, Q′

), where 〈P,Q′〉N = ζN

= Fp

(E/〈P 〉, Q′′

)where pQ′′ = Q′

=(E(p)

/〈P (p)〉, Q′′(p))

Want to show Q′′(p)= Q.

Compute

〈P (p), Q′′(p)〉N = 〈P,Q′′〉pN = 〈P, pQ′′〉N= 〈P,Q′〉N = ζN

= 〈P (p), Q〉N .

By the property of Weil pairing, we have Q′′(p)= Q. Now

〈Fpx,wNy〉 = 〈wN Fpx, y〉= 〈Fp −1 wNx, y〉= 〈x,wN F ′

py〉 as desired.

Since φFpx(y) = 〈Fpx,wNy〉 = 〈x,wn F ′

py〉 = F ′pφx(y), we have

Corollary det(I −Fpt on V`) = det(I− F ′pt on V`). Also TrFp = Tr F ′

p on V`.

Corollary The eigenvalues of Fp and F ′p on V` are algebraic integers with absolute

value p12 .

Proof. Consider the zeta function of XΓ/Fp

Z(XΓ/Fp, t) =

det(I − Fpt : T`(JΓ))(1− t)(1− pt)

.

By Weil conjecture for curves, det(I − Fpt : T`(JΓ)) is a polynomial in Z[t] of degree 2gΓ and

it is equal to2gΓ∏i=1

(1− αit) with |αi| = p12 (Riemann hypothesis). The αi’s are the eigenvalues

of Fp on T`(JΓ)

We are ready to show

18

Theorem Viewed as an element in GL2(TQ`), the Frobenius automorphism Fp has charac-

teristic polynomial X2 − TpX+ p.

Proof. Since FpF ′p = p and Tp = Fp+ F ′

p, we have

X2 − TpX+ p = (X − Fp)(X− F ′p).

In particular, Fp satisfies this degree two polynomial. If we can show TrFp = Tp, thenX2 − TpX+ p is the characteristic polynomial of Fp. Since TrFp = Tr F ′

p, wehave 2TrFp = TrFp + Tr F ′

p = TrTp = 2Tp.

I Let f(z) =∞∑n=1

ane2πinz be a newform of weight 2 level N and character χ. Let Kf denote

its field of coefficients. Let λ be a place of Kf , dividing `. Consider V` ⊗Q`Kf,λ on which

TKf,λacts. Then V`⊗Kf,λ has 2-dimensional space U over Kf,λ invariant under TKf,λ

arisingfrom f . Since GQ commutes with TKf,λ

, U is GQ-invariant. The characteristic polynomial ofFp on U is X2 − apX + χ(p)p, ∀ p - N`. This proves

Theorem Given any newform f =∑ane

2πinz of weight 2 level N and character χ, thereexists 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`

Corollary |ap| ≤ 2p12 .

1.5 Shimura’s construction

Let f(z) =∞∑n=1

ane2πinz be a normalized newform of weight 2 level N and character χ. ¿From

f we obtain a surjective homomorphism λf : TQ → Kf = Q(an, n ≥ 1) given by

λf (Tn) = an, λf (< d >) = χ(d).

I For each embedding σ of Kf into C, there is a newform fσ =∑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 ]) ofdimension equal to [Kf : Q].I Note that if f has coefficients in Q, then Af is an elliptic curve.I Let If = (kerλf ) ∩ TZ, which is an ideal of TZ. The image If (JΓ) (Γ = Γ1(N)) is anabelian subvariety defined over Q. The desired variety Af is defined as JΓ/If (JΓ).I Note that the Hecke operators act on Af . Let’s study Af .Recall that S2(Γ) =

⊕iS(gi), where gi are the normalized weight 2 newform of level Ni | N ,

19

each S(gi) has dimension σ0

(NNi

). For i 6= i′, S(gi)⊥S(gi′) with respect to the Petersson

inner product. To decompose S2(Γ) with respect to the Q-structure, we group together thegi’s which are conjugate under the action of Gal(Q/Q) to get

S2(Γ) =⊕[gi]

S[gi] = S[f ]⊕ orthogonal complement.

We saw before that on S[f ] with respect to the basis fσ,

Tm =

am

. . .aσm

. . .

,Kf −→ Kσ

f

am 7→ aσm.

TQ |S2(Γ)=∏[gi]

TQ |S[gi]

So λf on TQ |S[f ] is an isomorphism, and λf on TQ |S[gi] with [gi] 6= [f ] is zero map. Thekernel 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 letV[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 πf (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∏[gi]

Aσ0(N/Ni)[gi]

, but it is not isomorphic to it.

Example. When Γ = Γ0(26), S2(Γ0(26)) is 2-dimensional with two newforms f1 and f2 oflevel 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 isbecause a Z-basis of S2(Γ0(26)) is given by f1+f2

2 , f1 (or f2). So f1, f2 generate a sublattice ofindex 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 directproduct 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 polarizedabelian varieties.

20

I Let Tf denote the action of Hecke algebra on S[f ]. Then Tf acts on V[f ] and on Af . For aprime `, 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

Kf ⊗Q Q` =∏λ|`

Kf,λ, λ : places.

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 henceAf has good reduction at p - N .I For p - N`, the characteristic polynomial of the Frobenius Fp is X2 − Tf,pX+ 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 ))

=∏σ

(1− aσp + χ(p)σp)

= NKf/Q(1− ap + χ(p)p).

We have shown

Proposition The number Nf,p of the Fp-rational points on Af mod Fp is

Nf,p = NKf/Q(1− ap + χ(p)p).

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) =∞∑n=1

ane2πinz be a weight 2 newform of level N and

trivial character with Fourier coefficients in Q (hence in Z). Then there is an elliptic curveE 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, thenϕ∗(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 abelianvariety 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 anorder of a number field whose degree over Q is the dimension of A.

I Generlaized Modularity ConjectureAny abelian variety over Q of GL2-type is modular, i.e. isogeny to Af .

22

2 Galois Representations and Modularity

2.1 Galois representations

Let F be a field of characteristic zero or a finite field. Then Galois group GF = Gal(F/F ) isa profinite group, obtained by

lim←−[L:F ]<∞

L/F Galois

Gal(L/F ).

I It is endowed with the Krull topology such that an open neighborhood system of theidentity is given by Gal(F/L) as L runs through finite Galois extensions of F . Thus GF iscompact and totally disconnected.I Let ` be a prime not dividing charF . Then F contains `nth roots of unity for all n ≥ 1.Let ζ`n be an element of order `n and σ ∈ GF . Then σ(ζ`n) = ζσn

`n for some σn ∈ Z/ nZ.Since σn ≡ σn−1 (mod `n−1), we can take the inverse limit lim←−

n

σn, which is an element in Z×` ,

denoted by ε`(σ). Clearly we have ε`(στ) = ε`(σ)ε`(τ). Hence ε` defines a character from GFto Z×` , called the `th cyclotomic character. It factors through Gab

F .

Example. 1 F = Fp. The Galois group GFp ' Z which contains the cyclic subgroup gener-ated by the Frobp as a dense subgroup. For ` 6= p, ε`(Frobp) = p.

Example. 2 F = Qp.Qp contains the maximal unramified extension Qunrp .

Gal(Qunrp /Qp) ' Gal(Fp/Fp),

the Frobp in GFp lifts to Frobp in Gal(Qunrp /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)Qunrp =

⋃p-m

Qp(ζm).

Qtotal ram abp =

⋃n≥1

Qp(ζpn)

This can be rephrased as

res×εp : GabQp

= Gal(Qabp /Qp) −→ GFp × Z×p .

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., theorder of p in L is prime to p).All finite tamely ramified extensions of Qp generate a Galois extension Qtame ram

p containingQunrp . The group Gal(Qp/Qtame ram

p ) = Pp is the maximal pro-p-subgroup of Ip and

Ip/Pp '∏` 6=p

Z`(1),

where Z`(1) is the Tate twist. Pp is called wild inertia group.

Qp

Qtame ramp

Qunrp

Qp

Pp

Ip

In fact, there is a finer filtration of Ip into a decreasing chain of closed subgroups Iup indexedby u ∈ [−1,∞] such that

(i) Iup = Ip for −1 ≤ u ≤ 0.

(ii) Iup ⊃ Ivp if u ≤ v; I∞p = id.

(iii) Iup =⋂v0

Iup .

We get εp(Iup ) = 1 + pdueZp ⊂ Z×p .

Example. 3 F = Q. For each prime p, there is a decomposition subgroup Gp which isisomorphic to GQp and the different choice of Gp are conjugate under GQ. So there is Frobp ∈Gp/Ip.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 ofQ. If L 6= Q, the L is ramified somewhere.

Theorem (Cebotarov Density Theorem)

24

(1) Let L be a finite Galois extension of Q. Let C be a conjugacy class of Gal(L/Q). Thenp∣∣[Frobp] = C has density equal to |C|

|Gal(L/Q)| .

(2) Let L be a Galois extension of Q unramified outside a finite set S of places of Q. Then⋃p6∈S

[Frobp] is dense in Gal(L/Q).

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

The density of p | p ≡ a mod m is 1φ(m) . This is Dirichlet’s Theorem of primes in arithmetic

progressions.Kronecker-Weber Theorem. Qab =

⋃m≥1

Q(ζm). This is equivalent to

∏p

εp : GabQ −→

∏p

Z×p .

is an isomorphism.

Qp

Qtame ramp

Qunrp

Qp

Pp

Ip

I Pp is maximal pro-p subgroup of Ip. Also, Ip/Pp '∏` 6=p

Z` = lim←−p-m

Z/mZ = lim←−n

F×pn . There is a

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 homomorphismfrom 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 Hof G, denoted by ρH = V H the subspace of elements in V fixed by H, and by ρH the spaceV/〈v − h(v) | h ∈ H, v ∈ V 〉. If H is a normal subgroup of G, then we obtain two representa-

tions of G/H, on ρH and ρH , also denoted by ρH and ρH . In particular, when H = Ip, theinertia 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 discon-nected, so is ρ(GQ), hence is finite. Let L be the fixed field of ker ρ. So L is a finiteextension of Q, hence is ramified at finitely many places. So ρ(Frobp) makes sense foralmost 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`. Asnot all such representations are unramified outside a finite set of places of Q, we requirethat ρ is unramified outside finitely many places so that we can talk about ρ(Frobp) foralmost 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) foralmost all p.

(II) For a semi-simple mod ` representation ρ : GQ → GLn(κ), it is determined by the traceof ρ(Frobp) on the space κn as well as on all exterior products

∧i κn for i = 1, · · · , nand for almost all p. If charκ = ` > n, then it is determined by Tr ρ(Frobp) for almostall 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 theconductor of ρ at p (p 6= ` for type II and III):

mp(ρ) =∫ ∞

−1codim ρI

vdv = codim ρIp +

∫ ∞

0codim ρI

vdv.

If ρ is unramified at p, then ρIp is the whole space, and so is ρIv

for v ≥ −1. So codim ρIv

= 0and 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 ⊃ Iv1 ⊃ Iv2 ⊃ · · · = I∞ = idρ(Ip) ⊃ ρ(Iv1) ⊃ ρ(Iv2) ⊃ · · · = id.

Since ρ is continuous, we have ρ(Iv) = id for some v < ∞. So the∫∞−1 codim ρI

vdv is a

proper integral, mp is well-defined.I Suppose ρ : GQ → GLn(K) is an `-adic representation. Since GQ is compact, ρ(GQ) is acompact subgroup of GLn(K). By choosing a suitable basis of Kn (i.e., conjugating ρ(GQ)by a matrix in GLn(K)), we may assume that ρ(GQ) ⊂ GLn(O), where O is the ring ofintegral elements in K. Thus we can take reduction mod the maximal ideal of O to get arepresentation ρ : GQ → GLn(κ) over the residue field κ of K. The kernel of the reductionGLn(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 thereduction trivially, thus ρ(Pp) ' ρ(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

N(ρ) =∏p

pmp(ρ),

where p is over all primes if ρ is an Artin representation, an all p 6= ` otherwise.

Remark. The representation ρ may depend on the choice of basis, but its semi-simple fil-tration (i.e., the Jordon-Holder factor) is independent of the conjugation.

The analysis above proves

Proposition Let ρ : GQ → GLn(K) be an `-adic representation of GQ and ρ a reduction ofρ. For p 6= `, we have ∫ ∞

0codim ρI

vdv =

∫ ∞

0codim ρI

vdv.

Moreover,N(ρ) = N(ρ)

∏p6=`

pdim ρIp−dim ρIp.

In particular, N(ρ)∣∣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←−n

E[`n] ' Z` × Z`.

GQ acts on T`(E). We get an `-adic representation

ρE,` : GQ −→ Aut(T`(E)) = GL2(Z`) → GL2(Q`)

Its reduction mod ` isρE,` : GQ −→ Aut(E[`]) ' GL2(F`).

Proposition

(a) det ρE,` = ε`.

(b) (Serre) ρE,` is absolutely irreducible. For each E, ρE,` is absolutely irreducible for `large.

(c) (Serre) If E does not have complex multiplication, then ρE,` and hence ρE,` is surjectivefor almost all `. Concerning the irreducibility of ρE,`, Mazur showed that ρE,` is irre-ducible for ` > 163 for all E; if E is semi-stable, then ρE,` is irreducible for ` > 7, andρE,` is irreducible for ` > 3 if ρE,2 is trivial.

27

I Local behavior of ρE,` and ρE,`.

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/Zpsuch 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

ρE,p |Gp∼(εpχ ∗0 χ−1

)for an unramified character χ of GQ. In particular

ρE,p |Ip∼(εp ∗

1

).

(c2) If E has a good supersingular reduction (i.e., p | ap), then E/Fp[p] is trivial and

ρE,p |Ip : Ip −→ Aut(E[p](Qp)) ' GL2(Fp)

is isomrophic toIp // //

ψ2

55Ip/Pp// F×p2

// GL2(Fp)

arising from ψ2. Moreover, ρE,p is absolutely irreducible and ρE,p is irreducible.

I Sketch of proof.

(b) Let E/Fpbe the Neron 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 GQsuch that on the quotient the Galois action is unramified. By assumption, the Neronmodel E/Zp

has a closed fiber E/Fpwhich is an ordinary elliptic curve. It means that

the p-divisible group of E/Zphas a closed fibre with non-trivial connected and etale

factors. Thus the p-divisible group of E/Zphas non-trivial connected and etale factors

since the formation of the connected-etale sequence of a finite flat group scheme over Zpis compatible with the base change Zp → Fp.Passing to the generic fiber over Qp and their Qp-points, the non-trivial connected-etalesequence over Zp gives rise to the desired decomposition of ρE,p. In particular, theunramified 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’sresults. Assume this, we show that ρE,p is absolutely irreducible. Assume not, thenthere is a 1-dimensional subspace invariant under the action of Gp, given by a character.Its restriction to Ip is either ψ2 or ψp2 . Recall that for any σ ∈ Ip and any preimage Frobpof the Frobp in Gp, we have Frobp σ Frob−1

p = σp. This shows that Frobp permutesψ2 and ψp2 , 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 actionof Gp. We may choose a suitable vector in this space so that after adjoining anothervector, ρE,p(Gp) has image over O. Then ρE,p could not be absolutely irreducible.

I Next consider the case where E has a multiplicative reduction at p. In this case thej-invariant of E is NOT p-adically integral. Recall that the q = e2πiz expansion of j

j =1q

+ 744 + 196884q + · · ·

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

E(C) = C/L<τ,1> exp−→ C×/qZ, where q = e2πiz.

z 7→ e2πiz

Tate showed that there is a p-adic analytic isomorphism

Φ : Q×p

/qZE

∼−→ E(Qp)

such that σΦ(x) = Φ(σ(x)δ(σ)

), ∀σ ∈ GQp and x ∈ Q

×p/qZ. 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

Qp(ζp2−1)

Qp

H

Proposition Suppose E has multiplicative reduction at p. Let δ be the character definedabove. Then

(1) ρE,` |Gp∼(δε` ∗0 δ

)for all ` and if ` 6= p, mp(ρE,`) = 1.

29

(2) ρE,` |Gp∼(δε` Φ0 δ

), where Φ ∈ H1(Gp,Z/ Z(1)) ' Q×

p

/(Q×

p )` corresponding to the

image of q = qE in Q×p

/(Q×

p )`.

(3) if ` 6= p, then ρE,` is unramified at p if and only if p | ordp(q) = − ordp(j) = ordp(∆E).

(4) There is a finite flat group scheme F/Zpsuch 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←−n

E[pn]. The usual Galois action on ζpn is given by

σ(ζpn) = ζεp(σ)pn , ∀n ≥ 1. Imbed ζpn into Q×

p/qZ, on which the Galois action is

σ ζpn = σ(ζpn)δ(σ) = (ζεp(σ)pn )δ(σ).

Hence ζpn determines a rank 1 Zp-module in Q×p/qZ on which Gp acts by the character εpδ.

Then the action of Gp on the quotient is given by δ since det = εp and δ2 = id.

Proposition Suppose E has additive reduction at p. Then for all ` 6= p, the conductor ofρE,` |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 thebehavior of ρE,` |Gp and ρE,` |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/Zpsuch that

E[pn](Qp) ' Fn(Qp) as Gp-modules.

(1) If E has ordinary reduction at p, i.e., p - ap, then

ρE,p |Gp∼(χεp ∗

χ−1

).

for some unramified character χ of GQ.

(2) If E has supersingular reduction at p, then

ρE,p |Ip : Ip −→ Aut(E[p](Qp)) ' GL2(Fp)

is isomorphic toIp // //

ψ2

55Ip/Pp// F×p2

// GL2(Fp)

30

Remark.

(1) A precise definition of ψ2: Let ω be any uniformizer of Zp. Let p2−1√ω be a p2 − 1 root

of ω. For σ ∈ Ipσ( p2−1

√ω)

p2−1√ω

= ψ2(σ).

(2) The characteristic polynomial of ρE,`(Frobp) for ` 6= p is

X2 − apX + p.

Factor this over Qp: X2 − apX + p = (X − α)(X − β). If p - ap, exactly one of α, β is ap-adic unit.

E = the Neron model obtained from minimal Weiestrass equation. E[p] is a finiteflat group scheme over Zp of order p2. There is a short exact sequence of finite flatcommutative 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 thep-adic unit.

(3) In case of (2), ρE,p ⊗Fp Fp ' ψ2 ⊕ ψp2 .

Proposition Suppose E has a multiplicative reduction at p (E(Qp) ' Q×p

/qZ(δ)). Let δ

be the character from GQ → ±1 which is trivial if split, and δ is the unique unramifiedquadratic character if nonsplit, σ · x = σ(x)δ(σ).

(a) ρE,` |Gp∼(δε` ∗

δ

)and mp(ρE,`) = 1 if p 6= `.

(b) If ` 6= p, then ρE,` |Gp is unramified (at p) ⇔ ` | ordp(∆E) = ordp(q), i.e., mp(ρE,`) = 0.

(c) There is a finite flat group scheme F/Zpsuch 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 atopological group. By a finite O[G]-module M we mean an O-module with finite cardinalityand discrete topology on which G acts continuously. A profinite O[G]-module M is an inverselimit of finite O[G]-modules.A profinite O[G]-module M is said to begood: if for every finite discrete quotient M ′ of M , there is a finite flat group scheme F/Z`

31

such that M ′ ' F (Q`) as Z`[G`]-modulesordinary: 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 onM (−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 asthe residue field of O. We call a continuous homomorphism ρ : G` → GL2(R) good, ordinaryor semi-stable if

(1) det ρ |I`= ε.

(2) The underlying module, denoted by Mρ(' R2), is good, ordinary, or semi-stable.

Denote by ρ the reduction of ρ modulo the maximal ideal of R. As a consequence of theNakayama lemma, we have

Lemma If Mρ and ρ are ordinary, then M (−1)ρ and M (0)

ρ are each free R-modules of rank 1,and ρ is ordinary.

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,` |Gànd ρE,` |G`

are good (resp.semi-stable).

(ii) ρE,` |Gìs ordinary ⇔ ρE,` |G`

is ordinary ⇔ E has good ordinary or multiplicativereduction at `.

(iii) ρE,` |Gìs good (resp. semi-stable) ⇒ E has good (resp. semi-stable) reduction at `.

Lemma

(a) If a mod ` representation ρ : G` → GL2(κ) is good, then either ρ is ordinary or ρ |I`⊗F` ' ψ2 ⊕ ψ`2.

(b) If ρ : G` → GL2(R) is such that Mρ is good and ρ is ordinary, then ρ is good andordinary.

I An `-adic representation ρ : G` → GL2(K) is good ordinary or semi-stable if after con-jugating 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

Let f(z) =∞∑n=1

ane2πinz be a normalized newform of weight 2 levelN and character χ. Denoted

by Kf = Q(an, n ≥ 1). Here χ is a Dirichlet character χ :(Z/NZ

)× → S1. By class fieldtheory, 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).I Shimura constructed an abelian variety Af over Q as a quotient Jac(XΓ1(N)). For eachprime `, both GQ and the Hecke algebra Tf ⊗Z` act on the Tate module T`(Af ), which givesa representation

ρ : GQ −→ Aut(T`(Af )) ' GL2(Tf ⊗ Z`)

Recall that Tf ×Q` ' Kf ⊗Q` =⊕λ|`Kf,λ. So ρ can be viewed as a representation

GQ −→ GL2(Kf ⊗Q`).

We get an `-adic representation ρf,` : GQ → GL2(Kf,λ). We summarize the properties of ρf,às follows:

Proposition Let f(z) =∞∑n=1

ane2πinz be a normalized newform of weight 2 level N and

character χ. For each prime `, let K be a finite extension of Q` containing Kf . Then we havean `-adic representation

ρf,` : GQ −→ GL2(K)

(i) ρf,` is unramified at each p - N`. Moreover, ρf,`(Frobp) has characteristic polynomial

X2 − apX + pχ(p).

(ii) det ρf,` = χε`, and ρf,`(complex conjugation) '(

1 00 −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 p‖N . Let η be the unramified character of Gp such that η(Frobp) = ap.If p - condχ, then

ρf,` |Gp∼(ηε` ∗

η

).

If p | condχ, thenρf,` |Gp' η−1ε`χ |Gp ⊕η.

(vi) If ` - 2N , then ρf,` |Gìs good. Moreover, ρf,` |G`

is ordinary if and only if a` is a unitin K, in which case ρI`(Frob`) (i.e., ρ on M

(0)ρf,`) is the unit root of X2 − a`X + `χ(`).

(vii) If ` is odd and `‖N , but ` - condχ, then ρf,` |Gìs ordinary and ρI`(Frob`) = a`.

33

Remark. Associated to the newform f is a cuspital automorphic representation π =⊗v

′πv

of GL2(AQ) such that L(s, f) = L(s, π) = π∞ ⊗′⊗pπp. Conductor of π = level N of f . χ is a

character of(Z/NZ

)× ' ∏(Z/peii Z)×

, N =∏peii . So we may write χ =

∏p|N

χp, where χp is

a character of(Z/peZ)×. At a place p‖N , the theory of newform tell us |ap| = p

12 , if p | condχ←→ πp is a principal induced form (η−1χp| · |−1, η)

a2p = χ(p), if p - condχ←→ 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 ρf,` be the semi-simplification of ρf,`:

ρf,` : GQ −→ GL2(κ),

where κ is the residue field of K. ρf,` has the same properties as ρf,` except

(iii)′ ρf,` may not be absolutely irreducible. If ` is odd, then

ρf,` is irreducible⇐⇒ it is absolutely irreducible.

(iv)′ The conductor of ρf,` divides the non-`-part of N .

I For newforms of weight≥ 2, there are associated `-adic representations. For newforms ofweight 1, we have

Theorem (Deligne-Serre)

Let g =∞∑n=1

ane2πinz be a normalized newform of weight 1 level N and character χ. Then

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

X2 − apX + 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 Kbe 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 ρg. Deligne and Serre showedthat there is a normalized newform f of weight 2 level Nf | N` and character χf such that

ρg ' ρf,` over κf .

That is, ap(g) ≡ ap(f) and χ(p) ≡ pχf (p) modulo maximal ideal. So ρf,` gives the (would-be)ρg. 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 describethese representations. In other words, given a representation of GQ over C, `-adic fields orfinite fields, want to know when they are isomorphic to those arising from modular forms, i.e.,modular in short.The first isI Artin’s Conjecture: Every irreducible representation ρ : GQ → GLd(C) with d ≥ 2 hasa holomorphic L-function L(s, ρ).

L(s, ρ) = Γ(s, ρ)∏p

Lp(s, ρ) = Γ(s, ρ)∏p

1det(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⊗χis also an irreducible degree d representation. In case d = 2, Artin’s conjecture ⇒ L(s, ρ) andL(s, ρ ⊗ χ) are holomorphic ∀χ. Apply the converse theorem for GL2(AQ), this means thatthere 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)−sΓ(s), in this case, f is a holomorphic cusp newform of weight 1. Ifρ is even, Γ(s, ρ) = ΓR(s)2 or ΓR(s + 1)2, where ΓR(s) = π−

s2 Γ(s2

). Then f is Maass wave

form.I We may restate Artin’s conjecture for odd degree 2 irreducible representations asArtin’s Conjecture′. Every odd degree 2 irreducible rerpresentation ρ of GQ is isomorphicto 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 inone 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 undersome technical conditions.

I Recall Artin’s conjecture′. If ρ : GQ → GL2(C) is odd, irreducible, then ρ ' ρg for somenewform g of weight 1.This is true if ρ(GQ) is solvable. In this case, ρg,` ' ρf,` for a newform of weight 2.I mod ` representations:Original Serre’s conjecture:

35

(1) Let ρ : GQ → GL2(κ) be an odd irreducible representation, where κ is a finite field ofcharacteristic `. Then ρ ' ρf,` 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 ρ |GQ(

√−3)

is not induced from a character ofQ(√−3).)

I Serre’s Conjecture Let ρ : GQ → GL2(κ) be a mod `, odd, irreducible representation.Then ρ ' ρf,` for a newform f of weight 2. If this is the case, we say ρ is modular.

Proposition Serre’s conjecture is known to hold in the following case: if

(i) κ = F3;

(ii) the projective image of ρ 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 0

)7→(−1 1−1 0

)and

(1 −11 1

)7→(

1 −1−√−2 −1 +

√−2

).

So we haveρ : GQ −→ GL2(F3) → GL2(Z[

√−2]) → GL2(C).

The group GL2(F3) acts on P1(F3) by fractional linear transformations(a bc d

)(z) =

az + b

cz + d.

The elements in GL2(F3) fixing all elements in P1(F3) are the scalars. So we have animbedding of PGL2(F3) → S4. This is surjective by counting cardinalities. So ρ, as anArtin representation, is isomorphic to ρg for a newform g of wieght 1. By remark before,as mod 3 representation, ρ ' ρg,3 ' ρf,3 for some newform f of weight 2.

(ii) By assumption, ρ is induced. In other words, there is a Galois extension F of Q and acharacter ξ : GF → κ× such that ρ = IndGQ

GFξ. Let n be the order of ξ, so the image

ξ(GF ) is a cyclic group of order n contained in κ×. Let K be an `-adic field with residuefield κ. Lift the nth roots of unity in κ to the group 〈ζn〉 generated by nth roots of unityin K. We obtain a character ξ : GF → 〈ζn〉 ⊂ K. Let ρ = IndGQ

GFξ. Replacing ξ be a

twist by a quadratic character of GF if necessary, we obtain an odd irreducible degree2 representation ρ. We know that ρ ' ρg for a newform g of weight 1. By the sameargument as before, ρ is modular.

36

Theorem (Diamond, Gross, Edixhoven)Suppose that ` is odd and the mod ` representation ρ : GQ → GL2(κ) is modular. If ` = 3,assume also that ρ |GQ(

√−3)

is absolutely irreducible. Then there is a newform f of weight 2such that

(i) ρ ' ρf,` over κf ;

(ii) the level of f is N(ρ)`δ(ρ) with δ(ρ) = 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(ρ) = 0 if ρ |Gìs good.

(ii) δ(ρ) = 1 if ρ |Gìs not good and ρ |I` ⊗κκ is of the form(

εa` ∗1

),

(ε` ∗

εa`

),

(ψa2 00 ψà2

).

for some positive integer a < `. Here ψ2 is the fundamental character.

(iii) δ(ρ) = 2 otherwise.

In particular, δ(ρ) ≤ 1 if ρ |Gìs semi-stable.

I `-adic representations:I Fontaine-Mazur Conjecture: let ρ : GQ → GL2(K) be an irreducible `-adic represen-tation (ramified at only finitely many places by requirement). Assume ρ is not a Tate twistof 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 semi-stable ⇔ 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 |Gìs semi-stable,

then det ρ |I`= ε` by definition.)I Wiles: Under some conditions, ρ modular ⇒ ρ modular.

37

Remark. Under the Weak Fontaine-Mazur conjecture, if ρ ' ρf,`, it is expected to find anewform 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-Fouriercoefficients. 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. ThenρE,` ' ρf,`, ∀ `.(c)⇒(b): Suppose ρE,` ' ρf,` for some cusp form f of weight 2 and for some `. Then at almostall 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 `′, we have

Tr ρE,`′(Frobp) = Tr ρf,`′(Frobp)

as long as both representations are unramified at p. Since the `′-adic representations aredetermined by trace at almost all p, this implies ρE,`′ ' ρf,`′ for all `′.(b)⇒(a): We have ρE,` ' ρf,` (∀ `) for some weight 2 cusp form f . We know that theconductor of ρE,` is the non-`-part of the conductor N of E, and the conductor of ρf,` is thenon-`-part of the level Nf of f . So N = Nf . By (∗), we know ap(f) ∈ Z for all p - Nf . ¿Fromdet 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 Afis 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-Shimuraholds.

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 ρE,`. Look for ` such that ρE,` is modular. If ρE,3 is irreducible,then we know it is modular. What if ρE,3 is reducible?I Assume ρE,3 reducible. Recall that ρE,3 : GQ → Aut(E[3](Q)) ' GL3(F3), ρE,3 is reducible⇒ E has a subgroup of order 3 invariant under GQ. If ρE,5 is also reducible, then E has asubgroup of order 5 invariant under GQ. Put together, E has a cyclic subgroup C of order 15defined over Q. This means that (E,C) is a Q-rational point on the modular curve X0(15). Onthe other hand, X0(15) is known to have 4 non-cusp rational points, none of which correspondsto a semi-stable elliptic curves.Conclusion. If ρE,3 is reducible, then ρE,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., ρA,5 ' ρE,5).

(ii) A[3] is an irreducible GQ-module (i.e. ρA,3 is irreducible).

Proof. Recall that Y (5) = Γ(5)\H parametrizes the equivalence classes of (E,P,Q) where Eis an elliptic curve over C, P,Q ∈ E[5] with Weil pairing 〈P,Q〉 = 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 atwist of Y (5). Let Y ′(5) be the curve over Q parametrizes equivalent classes of elliptic curveA over C together with an isomorphism A[5] ' E[5] which preserves the Weil pairings. Ifwe fix P,Q ∈ E[5] with 〈P,Q〉 = ζ5. An isomorphism A[5] ' E[5] as described amounts toa choice of generators P ′, Q′ ∈ A[5] such that 〈P ′, Q′〉 = ζ5. Y ′(5) has a Q-rational pointx0 = (E,P,Q) and its Q-rational points are from the A’s over Q such that A[5] ' E[5] asGQ-modules and preserving the Weil pairings. If P and Q are defined over K, then Y ′(5) andY (5) are isomorphic over K. It was shown by Kline that X(5) over C is P1(C). The compat-ified Y ′(5), denoted by X ′(5) over Q, is isomorphic to P1(Q) since it has a Q-rational pointx0. Therefore, X ′(5) is infinite. Define another curve Y ′(5, 3) which parametrizes the ellitpiccurve A with isomorphism A[5] ' E[5] preserving Weil pairings and a cyclic subgroup of order3. Since Y ′(5, 3) has genus > 1, by a theorem of Falting’s Y ′(5, 3)(Q) is finite. Hence thereare infinite many points in Y ′(5)(Q) which do not lie in the image of Y ′(5, 3)(Q)→ Y ′(5)(Q).In other words, there are infinitely many elliptic curves A/Q satisfying (i) and (ii). Choosesuch a point x in Y ′(5)Q very close to x0 under the 5-adic topology. Then the elliptic curveA 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 withreduction ρ. If

(a) ρ 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 ρE,3 is irreducible, then it is modular, then ρE,` is modular.If ρE,3 is reducible, then ρE,5 is irreducible. Let A be an elliptic curve as in Lemma. SinceρA,3 is irreducible, we know ρA,3 is modular. So ρA,5 is modular and we have ρA,5 ' ρE,5 ismodular. 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 fieldκ = O/λ. Given representation ρ : GQ −→ GL2(κ) with det ρ = ε` mod `. If f is a normalizednewform of weight 2 level N and trivial character such that the attached `-adic representationρf,` 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 representationsρ 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,`, wheref is a newform of weight 2 level N and trivial character, ρf,` satisfying the conditions andρf,` w ρ. Hence = is the set of modular liftings of ρ parameterized by the newforms f asdescribed. Clearly < ⊇ =.I If we assume ρ modular and ρ |G`

semi-stable, then the set = is nonempty. Ribet showedthat in this case = is an infinite set, and so is <. We are led to a filtration of <. For a finiteset 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 conductorof ρ and conductor of ρ, i.e. Σ should contain all primes dividing N(ρ)

N(ρ) . In other words, ρoutside Σ is very nice.I If we assume N(ρ) square-free, (which is ok if ρ = ρE,` for a semi-stable elliptic curveE/Q), then this is a suitable definition of type Σ. But then we should extend O to completeNoetherian localO-algebras. For this purpose it is convenient to work in the context of Mazur’sdeformation theory, thereby introducing more structure into the problem, and enabling us touse tools from commutative algebra.I The desired quality <Σ = =Σ is stated by Wiles as the “main conjecture/theorem” asfollows. Denote by <Σ the maximal deformation ring which parameterizes representation oftype Σ with a fixed residual representation ρ, and by TΣ the Hecke algebra which parameterizesthe newforms of weight 2 and trivial character, of type Σ which reduce to ρ. Then the ringhomomorphism <Σ → TΣ from deformation theory is an isomorphism.Wiles’ idea is to prove the isomorphism for Σ = ∅ first, then deduce the case of general Σfrom this.

3.3 Deformation of representations

I Denote by CO the category where objects are complete Noetherian local O-algebras withresidue field κ, the morphisms are O-algebra homomorphisms which send maximal ideal tomaximal 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. Denoteby D a full subcategory of D0 which is closed under taking sub-objects, quotients and directproducts and which contains Mρ. Note that if M is an object in D0 and Mi is a collectionof sub-objects with

⋂Mi = 0 such that M/Mi are objects in D, then M is also in D since

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 oftype 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

D : G→ GLd(R0) of type D such that if ρ : G→ GLd(R) is a lifting of typeD then there is a unique morphism φ : R0 → R such that ρ = φ ρuniv

D . The representationρunivD is called the universal deformation of type D.

Proof. (Sketch of Falting’s proof)Let g1, · · · , gr be a fixed set of topological generators of G. Let A1, · · · , Ar be elements inGLd(O) which lift ρ(g1), · · · , ρ(gr). Define a mapping c : Md(O) → Md(O)r by x 7→ (xA1 −A1x, · · · , xAr−Arx). 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, setυρ = (ρ(g1) − A1, · · · , ρ(gr) − Ar). Then ρ is uniquely determined by υρ. Note that υρ ≡ 0mod 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 ρ′ ofρ 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.

I Let e1, · · · , es be an O-basis of V , we may write υρ =s∑i=1

υρ,iei, where υρ,i ∈ mR. Consider

a homomorphism θρ : O[[T1, · · · , Ts]]→ R defined by θρ(Ti) = υρ,i (So that υρ =s∑i=1

θρ(Ti)ei).

Note that υρ is completely determined by θρ, and ρ(gj) = Aj +s∑i=1

θρ(Ti)eij for j = 1, · · · , r,

where ei = (ei1, · · · , eir), eij ∈ Md(O). Let I denote the intersection of all ideals J of

O[[T1, · · · , Ts]] such that there is a representation ρJ : G→ GLd(O[[T1, · · · , Ts]]

/J

)of type

D with ρJ(gj) = Aj +s∑i=1

θρJ (Ti)eij for j = 1, · · · , r. Let RD = O[[T1, · · · , Ts]]/I and define

the representation ρunivD : G→ GLd(RD) by ρuniv

D (gj) = Aj +s∑i=1

Tijeij for j = 1, · · · , r.

I Properties of RDρ : G→ GLd(κ), κ = O/λ. If K ′ is a finite extension of K with ring of integers O′ and residuefield κ′ ⊃ κ, ρ : G→ GLd(κ) → GLd(κ′), D = (O, χ,D), D′ = (O′, χ,D′). Here D′ is the fullsubcategory of the category of profinite O′[G]-modules such that the underlying object of D0

is actually an object of D.

Lemma We have universal deformation rings RD, RD′ and universal representations ρunivD

and ρunivD′ . Then RD′ = RD ⊗O O′ and ρuniv

D′ = ρunivD ⊗ 1.

I Let ρ : G → GLd(O/λn) be a continuous representation. Let R0 = (O/λn)[ε]/ε2 be thetangent space of O/λn. Let ρ : G → GLd(R0) be a lifting of ρ. Then for each g ∈ G, writeρ(g) = (1 + εξ(g))ρ(g) , where ξ(g) ∈Md(O/λn).

ρ(g1g2) = ρ(g1)ρ(g2),∀g1, g2 ∈ G⇐⇒ ξ(g1g2) = 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 ρ′ is another lifting such that ρ′ is obtained from ρ by conjugationby a matrix of the form 1 + εM , where M ∈ Md(O/λn) (say that ρ and ρ′ are equivalent),then it is easy to check that ξ and ξ′ differ by a 1-coboundary. Therefore H1(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 byAd0(ρ), the restriction of Ad(ρ) to the trace zero subspace.

Note that

det ρ(g) = det(1 + εξ(g)) det ρ(g) = (1 + εTrξ(g)) det ρ(g).

So H1(G,Ad0(ρ)) parameterizes the equivalent classes of liftings of ρ preserving det.

Remark. If ` - d, thenH1(G,Ad(ρ)) = H1(G,Ad0(ρ))⊕Hom(G,O/λn). For ξ ∈ Hom(G,O/λn),det(1 + εξ(g)) = 1 + εdξ(g). H1(G,Ad(ρ)) has another interpretation. Denote by Mρ the un-derlying space of ρ and Mρ the space for a lifting ρ. Note that Mρ/εMρ 'Mρ, and multiply byε gives an isomorphism from Mρ to εMρ. Consider the short exact sequence of O/λn-modules

0 −→ εMρ −→Mρ −→Mρ/εMρ −→ 0

So ρ gives rise to an extension of Mρ to Mρ. Check that if ρ′ is equivalent to ρ, then theresulting extensions are isomorphic. So we obtain H1(G,Ad(ρ))→ Ext′(Mρ,Mρ), which is anisomorphism.

Now consider ρ : G → GLd(O/λ), i.e. n = 1. Denote by H1D(G,Ad0(ρ)) the subset of

H1(G,Ad(ρ)) corresponding to the extension in Ext′(Mρ,Mρ) which are objects in D.

Lemma There is an isomorphism H1D(G,Ad0(ρ))→ Homκ(mRD/(λ,mR2

D), κ) where

Homκ(mRD/(λ,mR2D), κ) is the tangent space of RD as κ-modules.

Proof. H1D(G,Ad0(ρ)) classifies liftings of ρ to GLd(κ[ε]/ε2) of type D. To each such ρ, there

is a unique O-algebra homomorphism φ: RD → κ[ε]/ε2 such that ρ = φ ρunivD . The desired

map from H1D(G,Ad0(ρ)) to Homκ(mRD/(λ,mR2

D), κ) is given by φ 7→ φ |mRD

.

I Now suppose θ: RD −→ O is a surjective O-algebra homomorphism with kernel ℘. Letρ = θ ρuniv

D . Set

H1D(G,Ad0(ρ)⊗K/O) = lim−→H1

D(G,Ad0(ρ)⊗ λ−n/O)

(Note that λ−n/O ' O/λn and H1D(G,Ad0(ρ)⊗ λ−n/O) ⊂ H1

D(G,Ad0(ρ)⊗K/O)).

Lemma There is a canonical O-linear isomorphism

HomO((℘/℘2),K/O) ' H1D(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.ρ : G→ GLd(κ) continuous representation.D0 objects: O[G]-modules.D: full subcategory of D0, containing Mρ.Fix χ : G→ O× character such that det ρ = 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 ρ.

(3) Mρ lies in D.

Theorem (Mazur)There is a lifting ρuniv

D : G → GLd(RD) of type D such that if ρ : G → GLd(R) is a lifting oftype D, then there is a unique O-morphism φ : RD → R such that φ ρuniv

D is conjugate to ρ.

3.4 Deformation of Galois representations

I Let ` be an odd prime and ρ : GQ → GL2(κ) is a continuous mod ` absolutely irreduciblerepresentation of GQ. Suppose ρ satisfies

• det ρ = ε` mod `.

• ρ |Gìs semi-stable.

• For p 6= `, #ρ(Ip) | `.

Remark. These conditions are satisfied by ρE,` for a semi-stable elliptic curve E over Q ifit 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 ρ is said to be of type Σ if

• det ρ = ε`.

• ρ |Gìs semi-stable.

• If ` 6∈ Σ and ρ |Gìs good, then so is ρ |G`

.

• If p 6∈ Σ ∪ ` and ρ is unramified at p, so is ρ

• If p 6∈ Σ ∪ `, ρ |Ip∼(

1 ∗0 1

)(i.e. ordinary).

That is, for p 6∈ Σ, we want ρ as unramified as ρ. Clearly, if Σ ⊂ Σ′ and ρ is a lifting of typeΣ, then it is also of type Σ′.I If E is a semi-stable elliptic curve over Q, then ρE,` is of type Σ if Σ contains all primesdividing conductor of E.

Theorem Given Σ, there is a universal lifting ρunivΣ : GQ → GL2(RΣ) of type Σ such that if

ρ : GQ → GL2(R) is a lifting of type Σ, then there is a unique O-morphism φ from RΣ to Rsuch that ρ is conjugate to φ ρuniv

Σ . Moreover, the following hold:

a) If K ′ is a finite extension of K, and R′Σ is the universal deformation ring, then R′

Σ =RΣ ⊗O O′ and ρuniv

Σ′ = ρuniv

Σ ⊗ 1.

44

b) RΣ can be topologically generated as as an O-module by dimκH1Σ(GQ,Ad0 ρ) elements.

c) If φ : RΣ O is a O-algebra homomorphism with kernel ℘ and ρ is the lifting φ ρunivΣ ,

thenHom(℘/℘2,K/O) ' H1

Σ(GQ,Ad0 ρ⊗K/O) as O-linear isomorphism.

I Sketch of proof.Let L0 be the fixed field of ker ρ. For n ≥ 1, let Lnbe the maximal elementary abelian `-

extension of Ln−1 unramified outside Σ∪`∪places where ρ is ramified. Let L∞ =∞⋃n=0

Ln,

which is Galois over Q. Note that any lifting of ρ 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), whichis finite by Hermite-Minkowski’s thoerem since L1/L0 is ramified at finitely many places withelementary abelian `-group as a Galois group.

L∞

L1

L0

Q

unramified outside Σ ∪ `

GL2(R)

GL2(R/m2

R)

GL2(R/m1)

ρ : GQ //

%%KKKKKKKK

ρ

;;wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwGL2(κ) = GL2(R/mR

)

Gal(L0/Q)ρ

55kkkkkkkkkkkkkk

mR = (π, x1, · · · , xr) ⊃ m1 = (π2, πxi, x1, · · · , xr) ⊃ m2 = (π2, πxi, x1xi, 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.

Lemma Let Hbe a pro-`-group and H its maximal elementary abelian quotient. If h1, · · · , hrin 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 gen-erated. 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 ρ |G`is good, then M is a good O[G`]-module.

• if p 6∈ Σ ∪ ` and if ρ 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).

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Then a lifting ρ : GQ → GL2(R) of ρ 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, ρunivΣ exists. The properties (a), (b), and (c) follow from the properties

(A), (B), and (C) before.

Corollary Suppose ρ |GQ(√−3)

is absolutely irreducible if ` = 3. Then RΣ can be topologicallygenerated as an O-module by

dimκH1Σ(GQ, ad0ρ(1)) + d` +

∑p∈Σ\`

dimκH0(Gp, ad0ρ(1))

elements, where

δ` =

dimκH1ss(G`, ad

0ρ)− dimκH1f (G`, ad

0ρ) , if ` ∈ Σ0 , if ` 6∈ Σ.

I Next we consider RΣ where Σ has a special property. Namely Σ is a finite set consistingof special primes q satisfying

• q ≡ 1 (mod `).

• ρ is unramified at q such that ρ(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:

χq : GQ // Gal(Q(ζq)/Q) ' (Z/qZ)× // // ∆q

andχq : Gq // Gal(Qq(ζq)/Qq) // // ∆q

Gal(Qq/Qq)

Let ∆Q =∏q∈Q

∆q and χQ =∏q∈Q

χq : GQ → ∆Q. Note that O[∆Q] is isomorphic to

O[[sq, q ∈ Q]]/((1 + sq)#∆q − 1, q ∈ Q) by Iwasawa theory argument. For q ∈ Q, let αq and

βq be the two eigenvalues of ρ(Frobq).

Lemma For q ∈ Q, ρunivQ |Gq is conjegate to

(ξ 00 ε`ξ

−1

)for some character ξ of Gq such

that ξ(Frobq) = αq.

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Qq

Pq

Qunrq

wild; pro-q-group

Iq

tame

Proof. Let f be a lifting of Frobq in Gq. The characteristic polynomial of ρunivQ (f) is a monic

polynomial of degree 2 with coefficients in RQ. Modulo mRQ, it is the characteristic polynomial

of ρ(Frobq), which has two distinct roots αq and βq. Let αq and βq be the Hensel lifting ofαq and βq to roots of the characteristic polynomial of ρuniv

Q (f). Note that αq 6= βq. So therepresentation space ρuniv

Q has an RQ-basis which diagonalizes ρunivQ (f). Next we show that

this basis also diagonalizes ρunivQ (σ) for all σ ∈ Iq. Since Iq〈f〉 is dense in Gq, this will show

the lemma. With respect to this basis, for σ ∈ Iq, we may write

ρunivQ (σ) = I2 +

(a bc d

), where a, b, c, d ∈ mRQ

.

Since ρunivQ |Gq is tamely ramified, we have ρuniv

Q (f)ρunivQ (σ)ρuniv

Q (f)−1 = ρunivQ (σ)q. That is,(

αqβq

)(I2 +

(a bc d

))(αq

βq

)−1

=(I2 +

(a bc d

))q,

i.e.,

I2 +

a beαqeβq

ceβqeαq

d

≡ I2 +(qa qbqc qd

)mod m2

RQ

≡ I2 +(a bc d

)mod m2

RQ.

This implies( eαqeβq− 1)b and

( eβqeαq− 1)c lie in mRQ

(b, c). It then implies (b, c) = mRQ(b, c)

since eαqeβq− 1 and

eβqeαq− 1 are units. Hence by Nakayama lemma, b = c = 0.

I So for each q ∈ Q, we have a character ξ : Gq → R×Q, 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 isQq(ζq). So ξq,Q |Iq factors through χq. Let ξ : GQ → R×

Q be the character unramified outside Qand ξ |Gq= ξq,Q. Thus the ramified part of ξ factors through χQ. We regard RQ as an O[∆Q]

algebra by ξ−2. Denote by IQ the ideal (sq : q ∈ Q) of O[[sq, q ∈ Q]]/((1 + sq)#∆q − 1, q ∈ Q).

Then Rφ ' RQ/IQRQ.We have shown

Corollary The canonical map from RQ to Rφ yields the isomorphism RQ/IQRQ∼−→ Rφ.

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Lemma

(a) If q ∈ Q, thenH0(GFq , ad0ρ) = H0(GFq , ad0ρ(1)) = κ.

andH1(GFq , ad0ρ) = H1(GFq , ad0ρ(1)) = κ.

(b) RQ can be topologically generated as an O-module by

#Q+ dimκH1Q(GQ, ad0ρ(1)) elements.

(c) If H1φ(GQ, ad0ρ(1)) ∼−→

⊕q∈Q

H1(GFq , ad0ρ(1)), then #Q = dimκH

1φ(GQ, ad0ρ(1)) and

RQ can be topologically generated as an O-module by #Q elements.

Proof.

(a) Since ρ(Frobq) =(αq 00 βq

), ad0ρ(Frobq) acts semi-simply with eigenvalues x, 1, x−1,

where x = αq

βq∈ κ \ 0, 1. Same with Ad0 ρ(1). Thus

H0(GFq ,Ad0 ρ) = κ = H0(GFq ,Ad0 ρ(1)).

Denote by M the representation space of Ad0 ρ or Ad0 ρ(1). Since GFq = 〈Frobq〉,

κ ' H1(GFq ,Ad0 ρ) = M/(Ad0(ρ)(Frobq)− 1)M.

Same with H1(GFq ,Ad0 ρ(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

H1(SL2(F),M02 (F)) = 0,

where M02 (F) =

(a bc −a

) ∣∣∣∣∣ a, b, c ∈ F

.

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