Galois representations associated to classical modular forms of weight at least 2: Deligne's theorem

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Galois representations associated to classical modular forms of weight at least 2: Deligne’s

theorem

Laia Amor´os i Caraf´ı

Universitat de Barcelona, University of Luxembourg

January 2015

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1. Introduction to Galois representations 2. Introduction to modular forms

3. Shimura’s construction for weight k = 2 4. Deligne’s construction for weight k ≥2

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1. INTRODUCTION TO GALOIS REPRESENTATIONS

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Introduction to `-adic Galois representations

LetGbe a profinite group and letk be a topological field. We will study

`-adic Galois representations, which are defined as follows:

• a representationof G(of dimensionn) is a continuous homomorphism of groups

ρ:G→GLn(k),

• ρis a Galoisrepresentation whenG=GK = absolute Galois group of a field K,

• is an `-adicrepresentation whenk⊆Q`,

• and`is just some prime number.

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Introduction to `-adic Galois representations

LetAdenote some abelian group andA[`ⁿ] its`ⁿ-torsion. One may construct an inverse systemA[`ⁿ⁺¹]A[`ⁿ]. Its inverse limit

T_`(A) := lim

←−

n

A[`ⁿ], is known as theTate module of A at `.

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Introduction to `-adic Galois representations

Example (The `-adic cyclotomic character)

LetK be a field of characteristicpandK its separable closure. Let`6=p be a prime. By choosing a compatible system of roots of unityµ_`n, we have an inverse systemµ_`n(K)µ_`n−1(K) given byx 7→x^`, and we can define the`-adic Tate module of K^×,

T`(K^×) = lim

←−

n

µ`ⁿ(K)∼= lim←−

n

(Z/`ⁿZ)∼=Z`.

GK acts compatibly onµ`ⁿ(K) for alln. We have a Galois representation: χ`: GK → Aut(T`(K^×))∼=Z`^×= GL1(Z`),→GL1(Q`).

σ 7→ x7→σ(x)

It is called the`-adic cyclotomic character(overK).

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Introduction to `-adic Galois representations

Example (The `-adic cyclotomic character)

LetK be a field of characteristicpandK its separable closure. Let`6=p be a prime. By choosing a compatible system of roots of unityµ_`n, we have an inverse systemµ_`n(K)µ_`n−1(K) given byx 7→x^`, and we can define the`-adic Tate module of K^×,

T`(K^×) = lim

←−

n

µ`ⁿ(K)∼= lim←−

n

(Z/`ⁿZ)∼=Z`.

GK acts compatibly onµ`ⁿ(K) for alln. We have a Galois representation:

χ`: GK → Aut(T`(K^×))∼=Z`^×= GL1(Z`),→GL1(Q`).

σ 7→ x7→σ(x)

It is called the`-adic cyclotomic character(overK).

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Introduction to `-adic Galois representations

Example (Galois representations and elliptic curves)

LetEbe an elliptic curve over K and`6=p. Consider the inverse system E[`ⁿ]E[`ⁿ⁻¹] given byP7→`·P and the corresponding `-Tate module of E, T`(E) = lim

←−E[`ⁿ]∼=Z²`. For eachn, the fieldQ(E[`ⁿ]) is a Galois number field, giving a restriction map and an injection

GK Gal(Q(E[`ⁿ])/Q) ,→ Aut(E[`ⁿ]).

σ 7→ σ|_Q(E[`ⁿ])

These maps are compatible for eachn.

Choosing basis ofE[`ⁿ] for eachnin a compatible way one can determine isomorphisms Aut(E[`ⁿ])∼= GL2(Z/`ⁿZ), and these combine to give Aut(T`(E))−→^∼ GL2(Z`). SinceGK acts on T`(E), we obtain a Galois representation

ρE,`:GK −→GL2(Z`)⊂GL2(Q`),

the2-dimensional `-adic Galois representation associated to E.

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Introduction to `-adic Galois representations

Example (Galois representations and elliptic curves)

LetEbe an elliptic curve over K and`6=p. Consider the inverse system E[`ⁿ]E[`ⁿ⁻¹] given byP7→`·P and the corresponding `-Tate module of E, T`(E) = lim

←−E[`ⁿ]∼=Z²`. For eachn, the fieldQ(E[`ⁿ]) is a Galois number field, giving a restriction map and an injection

GK Gal(Q(E[`ⁿ])/Q) ,→ Aut(E[`ⁿ]).

σ 7→ σ|_Q(E[`ⁿ])

These maps are compatible for eachn.

Choosing basis ofE[`ⁿ] for eachnin a compatible way one can determine isomorphisms Aut(E[`ⁿ])∼= GL2(Z/`ⁿZ), and these combine to give Aut(T`(E))−→^∼ GL2(Z`). SinceGK acts on T`(E), we obtain a Galois representation

ρE,`:GK −→GL2(Z`)⊂GL2(Q`),

the2-dimensional `-adic Galois representation associated to E.

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Introduction to `-adic Galois representations

Example (Galois representations and abelian varieties)

LetAbe an abelian variety of dimensiong overK and`6=p. Consider the inverse systemA[`ⁿ]A[`ⁿ⁻¹] given byP7→`·P and define the

`-adic Tate module of A, T_`(A) = lim

←−A[`ⁿ]. One can compatibly identifyA[`ⁿ] with (Z/`ⁿZ)^2g, yielding an isomorphism T_`(A)∼= (Z`)^2g.

ConsiderV`(A) := T`(A)⊗_Z_`Q`∼=Q^2g` . The Galois groupGK acts on T_`(A) and onV_`(A). This yields to the `-adic Galois representation associated to A,

ρ_A,`:G_K→Aut_Q_`(V_`(K))∼= GL2g(Q`).

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Introduction to `-adic Galois representations

Example (Galois representations and abelian varieties)

LetAbe an abelian variety of dimensiong overK and`6=p. Consider the inverse systemA[`ⁿ]A[`ⁿ⁻¹] given byP7→`·P and define the

`-adic Tate module of A, T_`(A) = lim

←−A[`ⁿ]. One can compatibly identifyA[`ⁿ] with (Z/`ⁿZ)^2g, yielding an isomorphism T_`(A)∼= (Z`)^2g. ConsiderV`(A) := T`(A)⊗_Z_`Q`∼=Q^2g` . The Galois groupGK acts on T`(A) and onV`(A). This yields to the `-adic Galois representation associated to A,

ρ_A,`:G_K →Aut_Q_`(V_`(K))∼= GL2g(Q`).

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Introduction to `-adic Galois representations

Example (Cohomology representations)

LetXbe an algebraic variety overK and`6=p. Attach toXthe ´etale cohomology groups Hⁱ(X_et,Z/`ⁿZ), i ∈Z. Then, one defines

Hⁱ(X,Z`) := lim

←−Hⁱ(X_et,Z/`ⁿZ), and H_`ⁱ(X,Q`) := Hⁱ(X,Z`)⊗_Z_`Q`.

The group Hⁱ_`(X,Q`) is aQ`-vector space on whichGK acts. It is finite dimensional, at least if char(K) = 0 or ifXis proper.

One thus gets an`-adic representation ofGK associated to Hⁱ_`(X,Q^`), thei-th `-adic Galois representation associated to X, for

0≤i≤2 dim(X).

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Introduction to `-adic Galois representations

Example (Cohomology representations)

LetXbe an algebraic variety overK and`6=p. Attach toXthe ´etale cohomology groups Hⁱ(X_et,Z/`ⁿZ), i ∈Z. Then, one defines

Hⁱ(X,Z`) := lim

←−Hⁱ(X_et,Z/`ⁿZ), and H_`ⁱ(X,Q`) := Hⁱ(X,Z`)⊗_Z_`Q`. The group Hⁱ_`(X,Q`) is aQ`-vector space on whichGK acts. It is finite dimensional, at least if char(K) = 0 or ifXis proper.

One thus gets an`-adic representation ofGK associated to Hⁱ_`(X,Q^`), thei-th `-adic Galois representation associated to X, for

0≤i≤2 dim(X).

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Introduction to `-adic Galois representations

LetKbe a number field. Then one may find not one, but a family of representations{ρ_`}_` attached toK, one for each`,

ρ_`: Gal(K/Q)→GL(Q`).

Since they come from the same object, they might be expected to be compatiblein some sense.

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Introduction to `-adic Galois representations

Consider an arbitrary Galois extensionL/K/QandP/p/pprime ideals in these fields. Recall thedecomposition groupofP

D_P:={σ∈G_K |P^σ=P} ∼= Gal(L_P/K_p) and theinertia groupofP,

IP:={σ∈DP|x^σ≡x (modP), ∀x ∈ OL}.

There is an isomorphism

DP/IP∼= Gal(F(P)/F(p)) =hFrobpi,

and any representative inDP mapping to the Frobenius automorphism Frobp is called aFrobenius elementof Gal(L/K), denoted by Frobp.

Definition

An`-adic Galois representation ρ`:GK →Aut(V) is calledunramified at a primepofK if IP⊂kerρfor any place PofK extendingp.

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Introduction to `-adic Galois representations

Definition

An`-adic Galois representation ρ`:GK →Aut(V) is calledunramified at a primepofK if IP⊂kerρfor any place PofK extendingp.

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Introduction to `-adic Galois representations

Definition

An`-adic Galois representation ρ`:GK →Aut(V) is calledunramified at a primepofK ifIP⊂kerρfor any place PofK extendingp.

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Introduction to `-adic Galois representations

Letpbe an unramified prime with respect to some representationρ`. One defines thecharacteristic polynomial of Frobenius ofρ` at p as

P_p,ρ_`(T) := det(1−Tρ_`(Frob_p))∈Q`[T].

The representationρ` is calledrational(resp. integral) if

• it is unramified at all primes except for a finite setS, and

• Pp,ρ_`(T)∈Q[T] (resp. Z[T]).

We will consider from now onrational `-adic representations, so we will be able to compare different rational representations (even over different completions) just by comparing those polynomials.

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Introduction to `-adic Galois representations

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Introduction to `-adic Galois representations

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Introduction to `-adic Galois representations

Examples

The`-adic Galois representationsχ_`, ρ_E,` andρ_A,`ofG_K are integral representations.

In the first case one can take asS the setS` of places inK that lie over

`. In the second and third cases, one can take asS the union ofS` and the set of primes of bad reduction of E(resp. A).

The fact that the corresponding Frobenius has an integral characteristic polynomial (which is independent of`) is a consequence of Weil’s results on endomorphisms of abelian varieties.

Cohomology representations ofGK are known to be integral ifK =Fq

(Weil conjectures), and it is a well known open question in general.

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Introduction to `-adic Galois representations

Examples

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Introduction to `-adic Galois representations

Examples

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Introduction to `-adic Galois representations

Examples

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Introduction to `-adic Galois representations

Let`⁰ be a prime, and consider an`⁰-adic Galois representationρ⁰ ofGK. Assume thatρandρ⁰ are rational. Then,ρandρ⁰ arecompatibleif

• there exists a finite subset of primesS such that ρandρ⁰ are unramified outsideS and

• Pp,ρ(T) =Pp,ρ⁰(T), for all primes outside S.

Given a rational`-adic Galois representationρ:G_K →Aut(V) it is always possible to find a compatible representation withρwhich is semisimple. It is unique (up to isomorphism) and is called the semisimplificationofρ.

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Introduction to `-adic Galois representations

Let`⁰ be a prime, and consider an`⁰-adic Galois representationρ⁰ ofGK. Assume thatρandρ⁰ are rational. Then,ρandρ⁰ arecompatibleif

• there exists a finite subset of primesS such that ρandρ⁰ are unramified outsideS and

• Pp,ρ(T) =Pp,ρ⁰(T), for all primes outside S.

Given a rational`-adic Galois representationρ:GK →Aut(V) it is always possible to find a compatible representation withρwhich is semisimple. It is unique (up to isomorphism) and is called the semisimplificationofρ.

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Introduction to `-adic Galois representations

For each prime`, letρ_` be a rational`-adic representation ofG_K. The system{ρ`}` is called acompatible systemif any twoρ` andρ`⁰ are compatible for any primes`, `⁰.

LetS_`={primes over`} ⊂ O_K. The system{ρ_`}_`is said to bestrictly compatibleif there exists a finite subsetS ⊂ O_K of primes such that

• for everyp∈/S∪S`, ρ` is unramified atp andPp,ρ_`(T)∈Q(T).

• P_p,ρ_`(T) =P_p,ρ⁰

`(T), forp∈/ S∪S_`∪S_`⁰.

There is a smallest finite setS having these properties. We call it the exceptional setof the compatible system.

Examples

The systems of`-adic representations {χ`}`, {ρE,`} and{ρA,`}are strictly compatible. The exceptional set of the first one is empty. The exceptional sets of{ρE,`}and{ρA,`}are the set of places where the elliptic curve (resp. abelian variety) has bad reduction.

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Introduction to `-adic Galois representations

• for everyp∈/S∪S`,ρ` is unramified atp andPp,ρ_`(T)∈Q(T).

• P_p,ρ_`(T) =P_p,ρ⁰

`(T), forp∈/ S∪S_`∪S_`⁰.

Examples

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Introduction to `-adic Galois representations

• for everyp∈/S∪S`,ρ` is unramified atp andPp,ρ_`(T)∈Q(T).

• P_p,ρ_`(T) =P_p,ρ⁰

`(T), forp∈/ S∪S_`∪S_`⁰.

Examples

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Introduction to `-adic Galois representations

Theorem

Let`be a prime and letEbe an elliptic curve overQwith conductor N.

The Galois representation

ρE,`:G_Q→GL2(Q`)

is unramified at every prime p-`N. For any such p, let p⊆Zbe any maximal ideal over p. Then the characteristic equation ofρ_E,`(Frob_p)is

x²−ap(E)x+p= 0, where ap=p+ 1−#eE(Fp).

The Galois representation is irreducible.

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Introduction to `-adic Galois representations

Unramified: Let p-`Nand let plie overp. There is a commutative diagram

Dp //

Aut(E[`ⁿ])

∼ G_F_p //Aut(eE[`ⁿ])

The inertia groupIp is contained in the kernel of↓_→. Sincep-`N,Ehas good reduction atpand the reduction preserves the`ⁿ-torsion structure. ThusIp is contained in the kernel of→↓ and

Ip⊂ker(Dp→Aut(E[`ⁿ])) = ker(Dp→GL2(Z/`ⁿZ))

⇒I_p ⊂ker(D_p→GL₂(Z`))⊂kerρ_E,`.

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Introduction to `-adic Galois representations

Unramified: Let p-`Nand let plie overp. There is a commutative diagram

Dp //

Aut(E[`ⁿ])

∼

G_F_p //Aut(eE[`ⁿ])

The inertia groupIp is contained in the kernel of↓_→. Sincep-`N,Ehas good reduction atpand the reduction preserves the`ⁿ-torsion structure.

ThusIp is contained in the kernel of→_↓and

Ip⊂ker(Dp→Aut(E[`ⁿ])) = ker(Dp→GL2(Z/`ⁿZ))

⇒Ip ⊂ker(Dp→GL2(Z`))⊂kerρE,`.

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Introduction to `-adic Galois representations

Characteristic polynomial: Letp-`N.

• detρE,`(Frobp): Letρn:G_Q→GL2(Z/`ⁿZ) be the nth entry ofρE,`

forn∈Z⁺. Using that, for allσ∈G_Q, detρ_n(σ) =χ_`,n(σ), we obtain that detρ_E,`(σ) =χ_`(σ) inZ^∗`. In particular,

detρ_E,`(Frob_p) =p.

• trρE,`(Frobp): the characteristic equation ofρE,`(Frobp) =:Ais A²−trA+p= 0, so trA=A+pA⁻¹. Using that

ap(E) = Frobp+p(Frob⁻¹_p )∈End(Pic⁰(eE)),

and that Frob_p acts onE[`e ⁿ] asFrob_p acts onE[`ⁿ], we have that a_p(E) = Frob_p+pFrob⁻¹_p ≡Frob_p+pFrob⁻¹_p (mod`ⁿ), ∀n.

⇔ap(E)≡A+pA⁻¹(mod`ⁿ), ∀n.

⇔trA=a_p(E)⇔trρ_E,`(Frob_p) =a_p(E).

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Introduction to `-adic Galois representations

Characteristic polynomial: Letp-`N.

• detρE,`(Frobp): Letρn:G_Q→GL2(Z/`ⁿZ) be the nth entry ofρE,`

forn∈Z⁺. Using that, for allσ∈G_Q, detρ_n(σ) =χ_`,n(σ), we obtain that detρ_E,`(σ) =χ_`(σ) inZ^∗`. In particular,

detρ_E,`(Frob_p) =p.

• trρE,`(Frobp): the characteristic equation ofρE,`(Frobp) =:Ais A²−trA+p= 0, so trA=A+pA⁻¹. Using that

ap(E) = Frobp+p(Frob⁻¹_p )∈End(Pic⁰(eE)),

and that Frob_p acts onE[`e ⁿ] asFrob_p acts onE[`ⁿ], we have that a_p(E) = Frob_p+pFrob⁻¹_p ≡Frob_p+pFrob⁻¹_p (mod`ⁿ), ∀n.

⇔ap(E)≡A+pA⁻¹(mod`ⁿ), ∀n.

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2. INTRODUCTION TO MODULAR FORMS

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Introduction to modular forms

Recall themodular group SL2(Z) and its congruence subgroups for N∈Z>0:

Γ0(N) = _{a b}

c d

∈SL2(Z) : ^{a b}_{c d}

≡(^{∗ ∗}₀_∗) (modN) ,

∪

Γ1(N) = _{a b}

c d

∈SL2(Z) : ^{a b}_{c d}

≡(¹_{0 1}^∗) (modN) ,

∪

Γ(N) = _{a b}

c d

∈SL2(Z) : ^{a b}_{c d}

≡(^{1 0}_{0 1}) (modN) . They act on theupper half planeh={τ ∈C: Im(τ)≥0}.

Notation:

For any matrixγ= _{c d}^{a b}

∈SL₂(Z) define thefactor of automorphy j(γ, τ)∈Cforτ∈hto bej(γ, τ) =cτ+d, and for any integerk, define theweight-k operator [γ]_k on functionsf :h→Cby

(f[γ] )(τ) =j(γ, τ)^−kf(γ(τ)), τ∈h.

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Introduction to modular forms

Definition

Let Γ be a congruence subgroup of SL2(Z) and letk be an integer. A functionf :h→Cis a modular of weightk with respect to Γ if

• f is holomorphic,

• f is weight-k invariant under Γ, i.e.,f[γ]k =f, ∀γ∈Γ,

• f[α]_k is holomorphic at∞for allα∈SL₂(Z).

If in addition,

• a0= 0 in the Fourier expansion off[α]k for allα∈SL2(Z), thenf is acusp formof weightk with respect to Γ.

The set of modular forms of weightk with respect to Γ is denoted by Mk(Γ), the cusp formsSk(Γ).

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Introduction to modular forms

Let’s recall the modular interpretation for the congruence subgroups Γ₀(N) and Γ₁(N).

Theorem (Modular interpretation)

Let N be a positive integer. Then there are isomorphisms Γ0(N)/h −→^∼ Y0(N)(C)

Γ0(N)τ 7−→ [C/Λτ,h[_N¹]Λ_τi]

and

Γ₁(N)/h −→^∼ Y₁(N)(C) Γ1(N)τ 7−→ [C/Λτ,[_N¹]Λτ].

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Introduction to modular forms

A nonzero modular formf ∈ Mk(Γ₁(N)) that is an eigenform for the Hecke operatorsT_n andhnifor alln∈Z⁺is a Hecke eigenformor simplyeigenform. The eigenformf(τ) =P

n=0a_n(f)qⁿ isnormalised whena₁(f) = 1. Anewformis a normalised eigenform inS_k(Γ₁(N))^new.

The set of newforms inSk(Γ₁(N))^newis an orthogonal basis of this space. Each such newform lies in an eigenspaceS_k(N, χ) and satisfies

T_nf =a_n(f)f, for alln∈Z⁺, i.e., its Fourier coefficients are itsTn-eigenvalues.

Letf ∈ S₂(Γ₁(N)) be a normalised eigenform for the Hecke operators T_p. Then the eigenvaluesa_n(f) are algebraic integers.

TheHecke algebra overZis the algebra of endomorphisms of S2(Γ₁(N))

TZ:=Z[{T_n,hni:n∈Z⁺}].

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Introduction to modular forms

n=0a_n(f)qⁿ isnormalised whena₁(f) = 1. Anewformis a normalised eigenform inS_k(Γ₁(N))^new. The set of newforms inSk(Γ₁(N))^new is an orthogonal basis of this space.

Each such newform lies in an eigenspaceSk(N, χ) and satisfies T_nf =a_n(f)f, for alln∈Z⁺,

i.e., its Fourier coefficients are itsTn-eigenvalues.

Letf ∈ S₂(Γ₁(N)) be a normalised eigenform for the Hecke operators T_p. Then the eigenvaluesa_n(f) are algebraic integers.

TZ:=Z[{T_n,hni:n∈Z⁺}].

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Introduction to modular forms

n=0a_n(f)qⁿ isnormalised whena₁(f) = 1. Anewformis a normalised eigenform inS_k(Γ₁(N))^new. The set of newforms inSk(Γ₁(N))^new is an orthogonal basis of this space.

Each such newform lies in an eigenspaceSk(N, χ) and satisfies T_nf =a_n(f)f, for alln∈Z⁺,

i.e., its Fourier coefficients are itsTn-eigenvalues.

Letf ∈ S2(Γ₁(N)) be a normalised eigenform for the Hecke operators T_p. Then the eigenvaluesa_n(f) are algebraic integers.

TZ:=Z[{Tn,hni:n∈Z⁺}].

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3. SHIMURA’S CONSTRUCTION FOR WEIGHT k = 2

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Shimura’s construction

We will see that one may associate a 2-dimensional Galois representation ofG_Q to each normalised cuspidal eigenform. The following theorem is due to Shimura fork= 2 and due to Deligne fork ≥2.

Theorem

Letf∈ S_k(N, χ)be a normalised eigenform with number field K_f. Let` be prime. For each maximal idealλof O_K_f lying over` there is an irreducible 2-dimensional Galois representation

ρf,λ:G_Q→GL2(Kf,λ).

This representation is unramified at all primes p-`N. For anyp⊂Z lying over such p, the characteristic equation ofρf,λ(Frobp)is

x²−ap(f)x+χ(p)p^k−1= 0.

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Shimura’s construction

We will see that one may associate a 2-dimensional Galois representation ofG_Q to each normalised cuspidal eigenform. The following theorem is due to Shimura fork= 2 and due to Deligne fork ≥2.

Theorem

Letf∈ S_k(N, χ)be a normalised eigenform with number field K_f. Let` be prime. For each maximal idealλofO_K_f lying over` there is an irreducible 2-dimensional Galois representation

This representation is unramified at all primes p-`N. For anyp⊂Z lying over such p, the characteristic equation ofρf,λ(Frobp)is

x²−ap(f)x+χ(p)p^k−1= 0.

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Shimura’s construction

We will sketch the construction ofρ_f_,λfor a modular formf of weight 2.

LetN be a positive integer and`be a prime. Consider the modular curve X₁(N) and letg denote its genus. The curveX₁(N)_Ccan also be viewed as a compact Riemann surface, and its Jacobian is ag-dimensional complex torus

J₁(N) = Jac(X₁(N)_C)^def=S2(Γ₁(N))^∧/H₁(X₁(N)_C,Z)∼=C^g/Λ_g. ThePicard groupofX1(N) is the abelian group of divisor classes on the points ofX1(N),

Pic⁰(X₁(N)) = Div⁰(X₁(N))/DivP(X₁(N)).

We can think of Pic⁰(X1(N)) as a subgroup of Pic⁰(X1(N)_C), and using Abel’s theoremwe have a natural isomorphism

Pic⁰(X₁(N)_C)∼= Jac(X₁(N)_C).

Thus, there is an inclusion of`ⁿ-torsion,

ιn: Pic⁰(X1(N))[`ⁿ],→Pic⁰(X1(N)_C)[`ⁿ]∼= (Z/`ⁿZ)^2g.

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Shimura’s construction

Denote byXe1(N) the reduction ofX1(N) atp. ByIgusa’s theoremwe know thatX1(N) has good reduction at primesp-N, so there is a natural surjective map Pic⁰(X1(N))Pic⁰(eX1(N)) restricting to

π_n: Pic⁰(X₁(N))[`ⁿ]Pic⁰(eX₁(N))[`ⁿ].

It turns out that bothι_n andπ_nare isomorphisms. Consider the`-adic Tate module of X₁(N), T_`(Pic⁰(X₁(N))) := lim

←−ⁿ{Pic⁰(X₁(N))[`ⁿ]}.

Choosing bases of Pic⁰(X₁(N)) compatibly for allnshows that T_`(Pic⁰(X₁(N)))∼=Z^2g` .

Any automorphismσ∈G_Q defines an automorphism of Div⁰(X1(N)), (P

nP(P))^σ=P

nP(P^σ). Since div(f)^σ= div(f^σ) for any f ∈Q(X1(N)), the automorphism descends to Pic⁰(X1(N)).

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Shimura’s construction

For eachn, there is a commutative diagram G_Q

ww ''

Aut(Pic⁰(X₁(N))[`ⁿ])oo Aut(Pic⁰(X₁(N))[`ⁿ⁺¹]) This leads to a continuous homomorphism

ρ_X₁_(N),`:G_Q→GL2g(Z`)⊂GL2g(Q`).

This is the 2g-dimensional representation associated to X₁(N).

The representationρ_X₁_(N),`is unramified at every primep-`N. For any suchp, letp⊂Zbe any maximal ideal overp. Thenρ_X₁_(N),`(Frobp) satisfies the polynomial equation

x²−T_px+hpip= 0.

(To be proved later)

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Shimura’s construction

To continue, we consider a normalised eigenformf∈ S2(N, χ) and denote byAf the abelian variety associated tof. There is an isomorphism

TZ/If

−→ O∼ f=Z[{an(f) :n∈Z⁺}].

Under this isomorphism, each Fourier coefficientap(f) acts onAfas Tp+If. The ringOf generates thenumber field of f, denoted byKf. The extension degreed= [Kf:Q] is also the dimension ofAf as a complex torus. Consider the`-adic Tate module ofA_f

T_`(A_f) = lim

←−

n

{Af[`ⁿ]} ∼=Z^2d` .

The action ofOfonAf is defined on`-power torsion and thus extends to an action on T`(Af). Using that the map

Pic⁰(X1(N))[`ⁿ]Af[`ⁿ]

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Shimura’s construction

The action ofG_Q commutes with the action ofOfsince theG_Q-action and theTZ-action commute on T`(Pic⁰(X1(N))). Choosing coordinates appropriately it gives a Galois representation

ρA_f,`:G_Q→GL2d(Q`).

The representationρ_A_f_,` has the following properties:

• It is continuous becauseρX₁(N),` is continuous and ρ⁻¹_X

1(N),`(U(n,g))⊂ρ⁻¹_A

f,`(U(n,d)),

where U(n,g) = ker(GL2g(Z`)→GL2g(Z/`ⁿZ)).

• It is unramified at all primes p-`N since kerρX₁(N),`⊆kerρA_`,`.

• For any unramified primep, letp⊂Zbe any maximal ideal overp.

At the level of Abelian varieties, since T_p acts asa_p(f) andhpiacts asχ(p),ρ_A_f_,`(Frob_p) satisfies the polynomial equation

x²−a_p(f)x+χ(Frob_p)p= 0.

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Shimura’s construction

The Tate module T`(Af) has rank 2d overZ^`. Since it is anOf-module, the tensor productV`(Af) = T`(Af)⊗Qis a module over

Of⊗Q`=Kf⊗_QQ`. It turns out that G_QactsKf⊗_QQ`-linearly on V`(Af), andV`(Af)∼= (Kf⊗_QQ`)². Choose a basis ofV`(Af) to get a homomorphismG_Q→GL2(Kf⊗_QQ`). We have that

Kf⊗_QQ`∼=Y

λ|`

Kf,λ,

so for eachλwe can compose the homomorphism with a projection to get a continuous Galois representation

ρ_f,λ:G_Q→GL₂(K_f,λ).

We have proved the following.

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Shimura’s construction

Theorem

Letf∈ S₂(N, χ)be a normalised eigenform with number field K_f. Let` be a prime. For each maximal idealλofO_K_f lying over `there is a 2-dimensional Galois representation

This representation is unramified at every prime p-`N. For any such p letp⊂Zbe any maximal ideal lying over p. Thenρf,λ(Frobp)satisfies the polynomial equation

x²−ap(f)x+χ(p)p= 0.

In particular, iff∈ S2(Γ0(N)), the relation is x²−ap(f)x+p= 0.

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Shimura’s construction

Lemma

The characteristic equation ofρX₁(N),`(Frobp)is x²−T_px+hpip= 0.

In order to do this, we need a description of the Hecke operatorTp at the level of Picard groups of reduced modular curves,

Te_p : Pic⁰(eX₁(N))→Pic⁰(eX₁(N)).

Suppose this reduction exists, i.e., that there is a commutative diagram Pic⁰(X₁(N)) ^T^p //

Pic⁰(X₁(N))

Pic⁰(eX (N)) ^T^e^p //Pic⁰(eX (N)).

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Shimura’s construction

The way to compute the reductionTep on Pic⁰(eX1(N)) is to compute it first in the moduli space environment.

Tp: Div(Y1(N)) → Div(Y1(N)) [E,Q] 7→ Tp[E,Q] =P

C[E/C, Q+C], where the sum is taken over allp subgroupsC⊂Esuch that C∩ hQi={0}.

IfEhas good reduction atp, then so do all the curvesE/C. We will describe the right-hand side overFp rather than overQ.

LetC0denote the kernel of the reduction map E[p]→eE[p], an orderp subgroup ofE. Then, for any orderpsubgroupC ofE,

[E/C,g Q^+C] =







[eE^Frob^p,Qe^Frob^p] ifC=C0

[eE^Frob

−1

p ,[p]Qe^Frob⁻¹^p ] ifC6=C0

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Shimura’s construction

Define the moduli space diamond operator in characteristicpto be hdif : Ye₁(N) → Ye₁(N)

[E,Q] 7→ [E,[d]Q], (d,N) = 1.

There arep+ 1 orderpsubgroups C ofE, one of which is C0.

Summing the previous formula over all orderpsubgroupsC ⊂Egives for a curveEwith ordinary reduction at p,

P

C[E/Cg,Q^+C] = [eE^Frob^p,Qe^Frob^p] +P

C[eE^Frob

−1

p ,[p]Qe^Frob⁻¹^p ]

= [eE^Frob^p,Qe^Frob^p] +phpif[eE^Frob

−1

p ,Qe^Frob⁻¹^p ]

= Frobp[eE,Q] +e phpiFrobf ⁻¹_p [eE,Qe]

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Shimura’s construction

This can be extended to curves with good reduction atp, and one obtains a commutative diagram (where the primes mean to avoid finitely many points):

Y1(N)⁰_gd ^T^p //

Div(Y1(N)⁰_gd)

Ye1(N)⁰_gd

Frob_p+phpiFrobf ⁻¹_p

//Div(eY1(N)⁰_gd).

This extends to degree-0 divisors, and to Picard groups:

Div⁰(Y1(N)⁰_gd) ^T^p //

Div⁰(Y1(N)⁰_gd)

Div(eY1(N)⁰_gd)

Frob_p+phpiFrobf ⁻¹_p

//Div⁰(eY1(N)⁰_gd)

Pic⁰(eX₁(N))^Frob^p,∗^+f^hpi^∗^Frob

∗

p//Pic⁰(eX₁(N)).

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Shimura’s construction

Theorem (Eichler-Shimura Relation)

Let p-N. The following diagrams commute

Pic⁰(X1(N)) ^T^p //

Pic⁰(X1(N))

Pic⁰(eX1(N))

Frobp,∗+fhpiFrob^∗_p

//Pic⁰(eX1(N)).

Pic⁰(X0(N)) ^T^p //

Pic⁰(X0(N))

0 Frobp,∗+Frob^∗_p

// ⁰

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Shimura’s construction

The Eichler-Shimura relation restricts to`ⁿ-torsion, and we obtain Tp= Frobp+hpipFrob⁻¹_p ⇔Frob²_p−TpFrobp+hpip= 0 on Pic⁰(X₁(N))[`ⁿ]. This holds for alln, so the equality extends to T_`(Pic⁰(eX₁(N))).

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3. DELIGNE’S CONSTRUCTION FOR WEIGHT k≥ 2

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Deligne’s construction

Fork≥2,

k = 2 k >2

J1(N) Kuga-Sato varietyW1(N)

Af Mf= Scholl motiv associated tof Tp(J1(N)) ´etale cohomology ofW1(N)

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