Arithmetic of modular forms

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Dipartimento di Matematica

Corso di Laurea Magistrale in Matematica

Tesi di Laurea

Arithmetic of

modular forms

22 settembre 2017

Candidata Relatori

Nirvana Coppola Prof. Massimo Bertolini Prof. Roberto Dvornicich

Controrelatore

Dr. Davide Lombardo

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Introduction

The main purpose of this work is to present Deligne-Serre’s Theorem and some of its consequences. Suppose given a modular form f of given level, and of type (1, ε), where ε is a Dirichlet character. Suppose also that f is an eigenform for certain operators called Hecke operators. Then there exists a Galois representation, namely a representation of the Galois group of an algebraic closure of Q over Q, that takes values in GL2(C), which is

associated to the modular form f , in a sense to be explained in Chapter 2 of this work.

The thesis is structured as follows. In Chapter 1, a series of tools are presented, in order to understand the statement of Deligne-Serre’s Theorem. In particular, we define modular forms of given level and weight, cusp forms, Dirichlet characters, Hecke operators of first and second type, Eisenstein series and particular spaces of cusp forms, namely the space of oldforms and the space of newforms. The latter is particularly important, since it has a basis of Eigenfunction for all but finitely many Hecke operators.

In Chapter 2, we introduce Galois representations, which are linear rep-resentations of the absolute Galois group of Q. More precisely, they are continuous homomorphisms: Gal( ¯Q/Q) → GLd(K), where K is either the

complex field C, a finite field or a `-adic field, each with a suitable topol-ogy. Then we state and prove Deligne-Serre’s Theorem, following the paper in [4]. In the proof we will use a result by Rankin, giving a bound to the sumP_|a

p|2p−s, a theorem on `-adic representations that allows us to find a

family of representation over finite fields attached to a given modular form, and finally a bound on the order of the subgroups of GL2(F`) which satisfy

a given condition.

In Chapter 3, we explain Serre’s Conjecture, which states a problem that is a converse to Deligne-Serre’s Theorem: given a representation ρ of the absolute Galois group of Q, is it possible to find a modular form to which

ρ is attached in the sense of Deligne-Serre’s Theorem? Serre conjectures

that, if ρ : G → GL₂(F ) is an odd irreducible representation modulo p, then the answer is positive. In particular, he states that it is possible to find a form that is also a newform, and he finds a recipe for the level, weight and character of it. This conjecture has been fully proved by Khare and Wintenberger. In this thesis, first we will explain Serre’s recipe and

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and Ribet’s Theorem, and a more recent result, proved by Diamond. Then we will briefly show how Serre’s Conjecture can be proven in two particular cases. Finally, as an interesting application, we will explain Fermat’s Last Theorem and how it is related to Serre’s Conjecture.

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Tools

In this chapter we provide most of the tools that are necessary for the proof of Deligne-Serre’s Theorem and to explain Serre’s Conjectures.

1.1 Modular Forms

Let us consider the group SL2(Z), also called modular group, consisting of

all 2-by-2 matrices with integer entries and determinant 1:

SL₂_{(Z) =} ( a b c d ! : a, b, c, d ∈ Z, ad − bc = 1 ) .

We can see each element of this group as an automorphism of the Rie-mann Sphere ˆC = C ∪ {∞} in the following way:

a b c d ! (z) = az + b cz + d where ∞ maps to a c and − d

c maps to ∞, if c 6= 0; otherwise ∞ maps to ∞.

Let H be the upper half plane in C, H = {z ∈ C : =(z) > 0}. Then it is easy to show that for each γ ∈ SL₂_{(Z), γ(H) ⊆ H: if γ =} a b

c d ! and z = u + iv ∈ H: =(γz) = = _{az + b} cz + d = = _{au + b + iva} cu + d + ivc = v det(γ) (cu + d)2_{+ (vc)}2 = v |cz + d|2.

Thus if v = =(z) > 0 then also =(γz) > 0. Moreover, the group SL₂_(Z) acts on H, since Iz = z and (γγ0)z = γ(γ0z). The first equality is obvious,

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the second holds because: aa 0_{z + b}0 c0z + d0 + b ca 0_{z + b}0 c0_{z + d}0 + d = (aa 0_{+ bc}0_{)z + ab}0_{+ bd}0 (ca0+ dc0)z + (cb0+ dd0) and a b c d ! a0 b0 c0 d0 ! = aa 0_{+ bc}0 _ab0_{+ bd}0 ca0+ dc0 cb0+ dd0 ! .

Now we are ready to define modular forms.

Definition 1.1. Let k be an integer; a meromorphic function f : H → C is a weakly modular form of weight k if

f (γ(z)) = (cz + d)kf (z)

for γ = a b c d

!

∈ SL₂_{(Z) and z ∈ H.}

Such a function is Z-periodic - this is easily seen setting γ = 1₀ 1₁

!

in the definition above - and so there exists a function g : D0_{→ C such that}

f (z) = g(e2πiz), where

D0 _{= {q ∈ C : |q| < 1} \ {0}.}

If in addition f is holomorphic on H, then g is holomorphic on D0 by composition, so it has a Laurent expression

g(q) = X

n∈Z

anqn

for q ∈ D0. Moreover, q → 0 as =(z) → ∞, so it is natural to say that f is

holomorphic at ∞ if g extends holomorphically to q = 0. In this case, f has

a Fourier expansion: f (z) = g(e2πiz) = ∞ X n=0 an(f )e2πiz.

Definition 1.2. Let k be an integer; a function f : H → C is a modular

form of weight k if it is holomorphic on H, weakly modular of weight k and

holomorphic at ∞.

A special class of modular forms is given by the cusp forms, which are the modular forms whose Fourier expansion has leading coefficient a0(f ) = 0.

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The set of modular forms of weight k is denoted Mk(SL2(Z)), the subset

of cusp form is S_k(SL2(Z)). In particular, a modular form is a cusp form if

lim

=(z)→∞f (z) = 0.

We also call the limit point ∞ the cusp of SL2(Z) and say that a cusp

form vanishes at the cusp.

In general we can replace the modular group SL₂_{(Z) with a subgroup} Γ in the above definitions, thus obtaining the notion of (weakly) modular form with respect to Γ. Our main concern will be the case where Γ is a

congruence subgroup.

Definition 1.3. Let N be a positive integer; the principal congruence

sub-group of level N is Γ(N ) = ( a b c d ! ∈ SL2(Z) : a ≡ d ≡ 1 (mod N ), b ≡ c ≡ 0 (mod N ) ) .

A subgroup Γ of SL₂_{(Z) is a congruence subgroup if there exists N ∈ Z}+ such that Γ(N ) ⊆ Γ; if N is the smallest integer with this property, Γ is said to be of level N .

The most important congruence subgroups of level N are the following.

Γ0(N ) = ( a b c d ! ∈ SL2(Z) : c ≡ 0 (mod N ) ) , Γ1(N ) = ( a b c d ! ∈ SL2(Z) : a ≡ d ≡ 1 (mod N ), c ≡ 0 (mod N ) ) .

We can also re-define weakly modular forms introducing the factor of automorphy and the weight-k operator of a matrix γ ∈ SL₂_(Z).

Definition 1.4. Given γ = a b c d

!

∈ SL2(Z), the factor of automorphy

j(γ, z) for z ∈ H is

j(γ, z) = cz + d;

for an integer k, the weight-k operator [γ]k on functions f : H → C is given

by

(f [γ]k)(z) = j(γ, z)−kf (γz)

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Hence we can say that a meromorphic function f : H → C is weakly modular of weight k with respect to Γ if it satisfies the relation

f [γ]k= f

for every γ ∈ Γ. Some basic properties of these operators are stated in the following lemma.

Lemma 1.5. For all γ, γ0 ∈ SL₂_{(Z) and z ∈ H,}

(a) j(γγ0, z) = j(γ, γ0z)j(γ0, z); (b) (γγ0)z = γ(γ0z); (c) [γγ0]_k= [γ]_k[γ0]_k; (d) =(γz) = =(z) |j(γ, z)|2; (e) d(γz) dz = 1 j(γ, z)2.

Proof. Properties (a) − (d) easily follow from what proven above. For (e)

we can write γz0− γz = det(γ)(z 0_{− z)} j(γ, z)j(γ, z0) thus γz 0_{− γz} z0_{− z} = 1 j(γ, z)j(γ, z0₎ and so d(γz) dz = limz0_→z γz0− γz z0_{− z} = 1 j(γ, z)2,

and this gives (e).

In order to define modular forms with respect to a congruence subgroup Γ, consider h minimal such that Γ contains the matrix 1 h

0 1

!

; such h always exists and divides N if Γ is of level N . Thus every function f : H → C which is weakly modular of weight k with respect to Γ is h-periodic and there exists g : D0 _{→ C such that}

f (z) = g(e2πiz/h);

if such a function f is holomorphic, then also g is holomorphic and has a Laurent expansion

g(qh) =

X

n

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where qh = e2πiz/h; if g extends holomorphically at qh = 0, f is said to be

holomorphic at ∞, so it has a Fourier expansion

f (z) = g(e2πiz/h) =

∞

X

n=0

ane2πiz/h.

The limit points, or cusps, of a congruence subgroup Γ are not only ∞, but also all the rational points in Q ⊆ C, identified under Γ-equivalence, i.e.

r, s ∈ Q are equivalent if and only if there is γ ∈ Γ such that γr = s. For

example, the only cusp of SL2(Z) is ∞. We now need to clarify what is the

meaning of holomorphic at a cusp; writing any s ∈ Q as s = α(∞), where

α ∈ SL2(Z), we have that f [α]k is weakly modular of weight k with respect

to α−1Γα, holomorhpic on H, and we can check whether it is holomorphic at ∞; in this case, we say that f is holomorphic at the cusp s. Now we are ready to give the final definition of a modular form.

Definition 1.6. Let Γ be a congruence subgroup of SL₂_{(Z) and let k be an} integer. A function f : H → C is a modular form of weight k with respect

to Γ if:

• f is holomorphic;

• f is weakly modular of weight k with respect to Γ; • f [α]_k is holomorphic at ∞ for all α ∈ SL2(Z).

It is called a cusp form of weight k with respect to Γ if in addition a0= 0

in the Fourier expansion of f [α]_k for all α ∈ SL₂(|Z).

The set of modular forms of weight k with respect to Γ is denoted M_k(Γ), the subset of the cusp forms is S_k(Γ).

The modular forms involved in Deligne-Serre’s Theorem are the ones of type (k, ε) with respect to Γ0(N ), which are a particular class of modular

forms of weight k with respect to Γ₁(N ). Here, ε is a Dirichlet character mod N , as defined below.

Definition 1.7. A Dirichlet character mod N is a homomorphism ε : Z N Z ∗ → C∗. It is called even if ε(−1) = 1, odd if ε(−1) = −1.

Let k be an integer and ε a Dirichlet character of the same parity, that is to say ε(−1) = (−1)k. We know that a modular form of weight k with respect to Γ1(N ) satisfies the relation

f

_{az + b} cz + d

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for a b

c d !

= γ ∈ Γ1(N ). Now, if f is such a function and γ ∈ Γ0(N ),

the form f [γ]k only depends on d as an element of

Z

N Z ∗

. In particular, if this dependence is given by a Dirichlet character ε, we get modular forms of type (k, ε).

Definition 1.8. A modular form of weight k with respect to Γ1(N ) is called

of type (k, ε) with respect to Γ0(N ) if for each a b

c d ! ∈ Γ1(N ): f _{az + b} cz + d = ε(d)(cz + d)kf (z).

1.2 Hecke Operators

We now define the double coset operator; the Hecke operators will be partic-ular cases of it. Let Γ1 and Γ2 be congruence subgroups of SL2(Z). We can

also view them as subgroups of GL+₂_{(Q), that is the subgroup of GL}₂_(Q) consisting of the matrices whose determinant is positive.

Definition 1.9. For each α ∈ GL+₂_{(Q), the set}

Γ1αΓ2 = {γ1αγ2: γi∈ Γi}

is a double coset in GL+₂_(Q).

The group Γ1acts on Γ1αΓ2 by left multiplication; an orbit of this action

takes the form Γ₁β, where β = γ1αγ2 is a representative of the orbit; the

orbit space Γ1αΓ2 Γ1

is the disjoint union of the orbits,S

iΓ1βi, for a suitable

choice of representatives βi. As a generalization of weight-k operators, given

β ∈ GL+₂(Q), we can define the weight-k β operator [β]k such that

(f [β]_k)(z) = (det β)k−1j(β, z)−kf (βz),

with the same notations as in the previous section.

Definition 1.10. For congruence subgroups Γ1 and Γ2 of SL2(Z) and α ∈

GL+₂_{(Q), the weight-k Γ}1αΓ2 operator takes functions f ∈ Mk(Γ1) to

f [Γ1αΓ2]k=

X

i

f [βi]k,

where Γ₁αΓ2 =SiΓ1βi is a disjoint union.

First of all, we have to show that the sum is finite. In order to do so, we will prove the following lemmas.

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Lemma 1.11. Let Γ be a congruence subgroup of SL2(Z) and let α ∈

GL+₂_{(Q). Then α}−1Γα ∩ SL2(Z) is another congruence subgroup.

Proof. Let Ñ be an integer such that Γ( Ñ ) ⊆ Γ and the matrices Ñ α and

˜

N α−1 have integer entries. Set N = ˜N3; then αΓ(N )α−1⊆ Γ( ˜N ), indeed: αΓ(N )α−1⊆ α(I + N M2(Z))α−1 =

= I+ Ñ ( Ñ α)M2(Z)( ˜N α−1) ⊆ I + Ñ M2(Z),

thus αΓ(N )α−1⊆ SL₂_{(Z) ∩ I + ˜}N M2(Z) = Γ( ˜N ).

This proves that Γ(N ) ⊆ α−1Γα ∩ SL2(Z), so the latter is a congruence

subgroup of SL₂_(Z).

Lemma 1.12. Let Γ1 and Γ2 be two congruence subgroups of SL2(Z) and

let α in GL+₂_{(Q). Set Γ}3 = α−1Γ1α ∩ Γ2; then left multiplication by α:

Γ2 → Γ1αΓ2

induces a natural bijection from Γ2

Γ3

to Γ1αΓ2

Γ1

.

Proof. The left multiplication by α:

Γ₂ → Γ₁αΓ2

γ2 7→ αγ2

induces a surjective map

Γ2 →

Γ1αΓ2

Γ₁

γ2 → Γ1αγ2.

To prove injectivity of the induced map from Γ2 Γ₃ to

Γ1αΓ2

Γ₁ , we can see that Γ1αγ2 = Γ1αγ20 if and only if γ20γ

−1

2 ∈ α−1Γ1α; moreover γ20γ −1 2 ∈ Γ2,

and so the condition holds if and only if γ₂0γ₂−1 ∈ Γ₃. The lemma follows.

Lemma 1.13. Any two congruence subgroups Γ₁and Γ₂are commensurable, meaning that

[Γ1 : Γ1∩ Γ2] and [Γ2: Γ1∩ Γ2]

are finite.

Proof. Let N₁, N₂ be natural numbers such that Γ(N_i) ⊆ Γ_i and let N₃ = lcm(N1, N2).

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We can notice that for each N ∈ Z+, the homomorphism SL2(Z) → SL2 Z N Z a b c d ! 7→ ¯a ¯b ¯ c d¯ !

is surjective with kernel Γ(N ); hence [SL₂_{(Z) : Γ(N )] =} SL₂ Z N Z is

finite. In particular, for each congruence subgroup of SL2(Z) of level N , the

index [Γ : Γ(N )] is finite.

Now Γ(N3) ⊆ Γ1∩ Γ2 and, since [Γ1: Γ(N3)] and [Γ2: Γ(N3)] are finite,

also [Γ1 : Γ1∩ Γ2] and [Γ2: Γ1∩ Γ2] are.

In particular, since α−1Γα ∩ SL₂_{(Z) is a congruence subgroup of SL}₂_(Z) by Lemma 1.11, the space Γ2

Γ₃ is finite and so is

Γ₁αΓ2

Γ₁ , by Lemma 1.12. Hence in Definition 1.10 the sum is taken over a finite set of representatives {βi}i.

Now we can prove that the weight-k Γ₁αΓ2 operator is well defined.

Lemma 1.14. If β and β0 are two representatives for the same orbit of

Γ₁αΓ2, then

f [β]k= f [β0]k

for every f ∈ M_k(Γ1).

Proof. If Γ₁β = Γ1β0, then writing β = γ1αγ2 and β0= γ10αγ20 we have that

αγ2∈ Γ1αγ20, hence

δ = (αγ2)(αγ20)−1∈ Γ1.

If f ∈ Mk(Γ1), then f [γφ]k = f [φ]k for all γ ∈ Γ1 and φ ∈ GL+2(Q).

Thus if f ∈ M_k(Γ1), then f [δ(αγ20)]k = f [αγ20]k, which means f [αγ2]k =

f [αγ₂0]_k. Moreover

f [β]k= f [γ1αγ2]k= f [αγ2]k= f [αγ02]k= f [γ01αγ 0

2]k = f [β0]k,

and the lemma follows.

Next we show that the weight-k Γ₁αΓ2 operator takes modular forms

with respect to Γ₁ to modular forms with respect to Γ₂ and cusp forms to cusp forms.

Lemma 1.15. For each f ∈ M_k(Γ₁) the transformed f [Γ₁αΓ2]k is Γ2

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Proof. First we notice that each γ2 ∈ Γ2 permutes the orbits of

Γ1αΓ2

Γ1

acting by right multiplication, thus if {βi}i is a set of representatives, so is

{β_iγ2}i. Hence

(f [Γ₁αΓ2]k)[γ2]k=

X

i

f [βiγ2]k= f [Γ1αΓ2]k,

so f [Γ₁αΓ2]k is weight-k invariant under Γ2.

Lemma 1.16. For each f ∈ M_k(Γ₁) the transformed f [Γ₁αΓ2]k is

holo-morphic at the cusps.

Proof. We can note that if f ∈ Mk(Γ1), then for each γ ∈ GL+2(Q), the

function g = f [γ]_k is holomorphic at ∞. Moreover the sum of a finite number of functions that are holomorphic at ∞ is holomorphic at ∞. Now, if δ ∈ SL2(Z), then (f [Γ1αΓ2]k)[δ]k is a sum of functions of the form gi =

f [βiδ]k, which are holomorphic at ∞ by what just remarked.

Finally, if f ∈ Sk(Γ1), the function f [γ]k vanishes at ∞ for every γ ∈

GL+₂_{(Q). Proceeding as in the proof of Lemma 1.16 we can show that}

f [Γ1αΓ2]k∈ Sk(Γ2).

We now introduce two operators on M_k(Γ₁(N )), where N is a positive integer. The map Γ0(N ) →

Z N Z ∗ taking a b c d ! to d is a surjective homomorphism, with kernel Γ1(N ), thus it induces a isomorphism

Γ₀(N ) Γ₁(N ) → Z N Z ∗ ,

and Γ1(N ) is a normal subgroup of Γ0(N ). If we take Γ1 = Γ2 = Γ1(N )

and α ∈ Γ₀(N ), the double coset operator [Γ₁αΓ2]k acts as [α]k, defining

an isomorphism of Mk(Γ1(N )). Indeed Γ1αΓ1= Γ1αα−1Γ1α = Γ1α, so the

only representative for the orbits of Γ1α is α itself. Thus the group Γ0(N )

acts on M_k(Γ₁(N )) and the restriction of the action to Γ₁(N ) is trivial, thus defining an action of the quotient

Z

N Z ∗

on M_k(Γ₁(N )).

Definition 1.17. For each d ∈ Z such that (d, N ) = 1, we define a

di-amond operator (or Hecke operator of the first kind) hdi as follows. Let

α = a b c δ

!

∈ Γ0(N ) be a matrix such that δ ≡ d (mod N ); then hdi is

given by hdi : M_k(Γ1(N )) → Mk(Γ1(N )) f 7→ f [α]k for α = a b c δ ! ∈ Γ₀(N ), δ ≡ d (mod N ).

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Notice that by construction hdi depends only on d modulo N , in partic-ular it is independent on the choice of α.

Another particular case of weight-k double coset operator is defined when Γ1 = Γ2 = Γ1(N ) as above, and α =

1 0

0 p

!

and p is prime. Hence we have the following definition.

Definition 1.18. Let p be a prime number. The operator Tp, or second

type of Hecke operator is given by:

Tp: Mk(Γ1(N )) → Mk(Γ1(N )) f 7→ f " Γ₁(N ) 1 0 0 p ! Γ₁(N ) # k .

It is easy to see that the double coset here is given by

( γ ∈ M2(Z) : γ ≡ 1 ∗ 0 p ! (mod N ), det γ = p ) .

We will need some properties of these two kinds of Hecke operators.

Lemma 1.19. The operators hdi and T_p _{for d ∈ Z and p prime commute.}

Proof. Let α = 1 0

0 p

!

. First we can see that for γ ∈ Γ0(N ), it holds:

γαγ−1≡ a b 0 d ! 1 0 0 p ! d −b 0 a ! ≡ 1 ∗ 0 p ! (mod N ),

and so the double cosets Γ1(N )αΓ1(N ) and Γ1(N )γαγ−1Γ1(N ) are equal.

So, considering also that Γ1(N ) is normal in Γ0(N ):

Γ1(N )αΓ1(N ) = Γ1(N )γαγ−1Γ1(N ) = γΓ1(N )αΓ1(N )γ−1 = = γ[ i Γ1(N )βiγ−1 = [ i Γ1(N )γβiγ−1, where Γ₁(N )αΓ₁(N ) = S iΓ1(N )βi. Hence SiΓ1(N )βiγ = SiΓ1(N )γβi.

Thus for each f ∈ M_k(Γ₁(N )) and γ ∈ Γ₀(N ) such that the lower right entry is δ ≡ d (mod N ), X i f [βiγ]k= X i f [γβi]k, so hdiTp(f ) = Tphdi(f ).

To find an explicit representation of T_pwe now prove the following propo-sition.

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Proposition 1.20. Let N ∈ Z+, let Γ1 = Γ2 = Γ1(N ) and let α =

1 0

0 p

!

where p is prime. The operator Tp = [Γ1αΓ2]k on Mk(Γ1(N )) is

given by Tp(f ) =              Pp−1 i=0f " 1 i 0 p !# k p | N, Pp−1 i=0f " 1 i 0 p !# k + f " m n N p ! p 0 0 1 !# k p - N,

and in the latter formula it holds mp − nN = 1.

Proof. To prove the proposition it suffices to show that a set of

representa-tives for Γ1αΓ2 Γ₁ is:              ( 1 i 0 p ! : 0 ≤ i < p ) p | N, ( 1 i 0 p ! : 0 ≤ i < p ) ∪ ( m n N p ! p 0 0 1 !) p - N.

We know from Lemma 1.12 that a set of representatives for Γ1αΓ2 Γ2

corresponds to one for Γ2 Γ3

via the left multiplication by α. In our case, Γ₂ = Γ₁(N ) and Γ₃= Γ₁(N ) ∩ α−1Γ₁(N )α.

Let Γ0(p) and Γ0₁(N, p) be as follows.

Γ0(p) = ( a b c d ! ∈ SL2(Z) : a b c d ! ≡ ∗ 0 ∗ ∗ ! (mod p) ) , Γ0₁(N, p) = Γ1(N ) ∩ Γ0(p).

Then it is easy to see that Γ₃ = Γ0₁(N, p). • If γ ∈ Γ3, then it takes the form

1 0 0 p−1 ! a b c d ! 1 0 0 p ! ∈ Γ1(N ), where a b c d !

∈ Γ₁(N ). So the top right entry of γ satisfies bp ≡ 0 (mod p), hence γ ∈ Γ0₁(N, p).

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• Vice versa, if a b

c d !

∈ Γ0

1(N, p), then it clearly belongs to Γ1(N ).

We only need to show that α a b

c d ! α−1 ∈ Γ₁(N ). We have 1 0 0 p ! a b c d ! 1 0 0 p−1 ! = a bp −1 pc d ! , and if a b c d ! ≡ 1 ∗ 0 1 !

(mod N ), then so does a bp

−1 pc d ! . Thus Γ3 = Γ1(N ) ∩ ( a b c d ! ∈ SL2(Z) : b ≡ 0 (mod p) ) . Let us con-sider the set

( γi = 1 i 0 1 ! : 0 ≤ i < p ) .

Clearly the orbits Γ3γi are all disjoint: if γγi ∈ Γ3γi, γ0γj ∈ Γ3γj and

γγi = γ0γj, then γjγ−1_i should belong to Γ3, but it does not if j 6= i, because

j − i 6≡ 0 (mod p).

Now let γ = a b

c d !

∈ Γ₁(N ). Then γ ∈ Γ₃γiif and only if γγi−1∈ Γ3,

that happens if and only if the top right entry of γγ_i−1 is a multiple of p. We can see that

γγ_i−1 = a b − ia

c d − ic !

,

so the condition we need is b−ia ≡ 0 (mod p). If p - a, then i ≡ ba−1 (mod p) satisfies it. Otherwise, there is no such i: if there were, then p | b as well and so p | det γ = 1, which is impossible. Using the fact that a ≡ 1 (mod N ), we can show that p | a if and only if p - N . In this case, let

γ∞= mp n

N 1

! ,

where mp − N n = 1. Then if p | a, γγ_∞−1 ∈ Γ₃, since its top right entry is

bmp − an, a multiple of p.

To sum up, a set of representatives for Γ1(N ) Γ3

is

(

γ0, . . . , γp−1 p | N,

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Using the correspondence between Γ1(N )αΓ1(N ) Γ1(N )

and Γ1(N ) Γ3

recalled above, we find the corresponding representatives for Γ1(N )αΓ1(N )

Γ1(N ) , which are βi = 1 i 0 p ! for 0 ≤ i < p, β∞= m n N p ! p 0 0 1 ! as we wanted.

Using Proposition 1.20, we can describe the Fourier coefficients of T_p(f ), where f ∈ M_k(Γ₁(N )). First we can notice that 1 1

0 1

!

∈ Γ₁(N ), so f has period 1 and a Fourier expansion

f (z) =

∞

X

n=0

an(f )qn, q = e2πiz.

Proposition 1.21. Let f ∈ M_k(Γ₁(N )) with Fourier expansion as above.

Let 1_N be the trivial Dirichlet character modulo N , such that 1_N(p) = 0 if

p | N , 1N(p) = 1 if p - N ; then Tp(f ) has Fourier coefficients:

an(Tpf ) = anp(f ) + 1N(p)pk−1an/p(hpif ),

where a_n/p_{= 0 if p - n.}

Proof. Let us first suppose that p | N . Then Tp(f ) =Pp−1j=0f

" 1 j 0 p !# k . We know that f " 1 j 0 p !# k (z) = pk−1p−kf _{z + j} p = 1 p ∞ X n=0 an(f )qnpµnjp ,

where q_p = e2πiz/p and µ_p= e2πi/p. Since Pp−1

j=0µnjp = ( 0 _{p - n} p p | n, then p−1 X j=0 f " 1 j 0 p !# k (z) = 1 p ∞ X n=0 an(f )qpn p−1 X j=0 µnj_p = 1 p X n≡0 (p) an(f )qpnp = = ∞ X n=0 anp(f )qn.

Hence if p | N then the proposition follows; if p - N , there is another term:

f " m n N p ! p 0 0 1 !# k (z) = (hpif ) " p 0 0 1 !# k (z) = = pk−1(hpif )(pz) = pk−1 ∞ X n=0 an(hpif )qpn,

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We can see that if ε is another Dirichlet character modulo N and if M_k(N, ε) is the ε-eigenspace of the diamond operator:

Mk(N, ε) = f ∈ Mk(Γ1(N )) : hdif = ε(d)f ∀d ∈ Z N Z ∗ ,

then if f ∈ M_k(N, ε) also T_p(f ) ∈ M_k(N, ε) and

an(Tpf ) = anp(f ) + ε(p)pk−1an/p(f ).

Notice that it holds M_k(Γ₁(N )) = L

εMk(N, ε) (see [6], §4 and 5 for

the proof).

Since the operators Tp and hdi commute, Tp(f ) is a ε-eigenfunction and

applying the result of Proposition 1.21 to hdif the formula follows.

Finally, we will show some other commutation properties of the Hecke operators.

Proposition 1.22. Let d and e be elements of

Z

N Z ∗

and p and q two prime numbers. Then

• hdiTp = Tphdi;

• hdihei = heihdi = hdei; • TpTq= TqTp.

Proof. We have already proven the first property; since both the Hecke

operators map M_k(N, ε) to itself and since M_k(Γ₁(N )) =L

εMk(N, ε) we

only need to check the other two properties on arbitrary f ∈ Mk(N, ε).

Since for each d ∈ Z

N Z, hdif = ε(d)f , then

hdihei(f ) = ε(d)ε(e)f = ε(de)f, which is symmetric in d and e.

As for the last equality, by the previous formula we have:

an(Tp(Tqf )) = anp(Tqf ) + ε(p)pk−1an/p(Tqf ) =

=anpq(f ) + ε(q)qk−1anp/q(f ) + ε(p)pk−1(anq/p(f ) + ε(q)qk−1an/pq(f )) =

=a_npq(f ) + ε(q)qk−1a_np/q(f ) + ε(p)pk−1a_nq/p(f ) + ε(p)ε(q)(pq)k−1a_n/pq(f ),

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1.2.1 A generalization

It is possible to extend the definition of Hecke operators to hni and T_n for all n ∈ Z+. We know how to define the diamond operator hdi for d ∈ Z

N Z.

Let n ∈ Z+with (n, N ) = 1; then hni = h¯ni where ¯n is the class of n modulo N and the latter is the usual diamond operator. If (n, N ) 6= 1, we just set

hni = 0. Hence the mapping n 7→ hni satisfies: hnmi = hnihmi = hmihni.

For the Hecke operator of second type, we need to proceed inductively. Set T1 = 1, that is the identity operator; then, we already know how to

define Tp for a prime number p. For n = pr, we set

Tpr = T_pT_pr−1− pk−1hpiT_pr−2.

As in Proposition 1.22, we can show that Tpr and T_qs commute for all

prime numbers p and q and positive integers r and s. Finally, we extend the definition to n setting Tn= Y T_pei i where n = Y pei i .

Again Tn and Tm commute if (n, m) = 1. With the same techniques as

in Proposition 1.21, we can compute the Fourier expansion of Tn(f ), where

f ∈ Mk(Γ1(N )). The coefficients are given in the following Proposition.

Proposition 1.23. Let f ∈ M_k(Γ₁(N )) have Fourier expansion

f (z) =

∞

X

m=0

am(f )qm

where q = e2πiz_{. Then for all n ∈ Z}+, T_n(f ) has Fourier expansion

(Tnf )(z) = ∞ X m=0 am(Tnf )qm, where a_m(T_nf ) =P d|(n,m)dk−1amn/d2(hdif ). In particular if f ∈ M_k(N, ε), then am(Tnf ) = X d|(n,m) ε(d)dk−1a_mn/d2(f )

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1.3 The Petersson inner product

In order to define newforms, we need to put an inner product on cusp forms, as follows.

Let H be the upper half plane of C; we can define the hyperbolic measure on it, given by

dµ(z) = dx ∧ dy y2 ,

where z = x + iy. This measure is invariant under the action of the au-tomorphism group GL+₂_{(R) of H, and in particular it is SL}₂_{(Z)-invariant;} indeed dµ(z) = d z+¯z 2 ∧ d z−¯z 2i z−¯z 2i 2 = 2dz ∧ d¯z i(z − ¯z)2.

We can compute that d(γz) = det γ

j(γ, z)2dz, d(γ ¯z) =

det γ

j(γ, ¯z)2d¯z and γz − γ ¯z =

det γ

j(γ, z)j(γ, ¯z)(z − ¯z), thus dµ(γz) = dµ(z).

Since the set Q ∪ {∞} is countable, its measure is 0, so we can use dµ to integrate over H∗ _{= H ∪ Q ∪ {∞}. A fundamental domain of H}∗ under the action of SL₂_{(Z) is}

D∗ = D ∪ {∞} = {z ∈ H : |<(z)| ≤ 1/2, |z| ≥ 1} ∪ {∞}. To show this, let us take z ∈ H and repeatedly apply one of 1 1

0 1

!

and 1 −1

0 1

!

to translate z into the vertical strip {|<(z)| ≤ 1/2}. Now, if z /∈ D, then |z| < 1 and so = −1 z = = − z¯ |z|2 = = _z |z|2 > =(z). So we can replace z by −1 z = 0 −1 1 0 !

(z) and repeat the process until the resulting z belongs to D. Let us see why this process ends after a finite number of steps. We can notice that there are only finitely many matrices γ ∈ SL2(Z) such that =(γz) > =(z): it holds

=(γz) = =(z)

j(γ, z)2

and so =(γz) > =(z) if and only if j(γ, z) < 1. Now, there are finitely many pairs (c, d) such that |cz + d| < 1, hence there are finitely many matrices

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a b c d

!

∈ SL2(Z) which increase the imaginary part. In particular, there

is a matrix γ such that j(γ, z) is minimal, thus =(γz) is maximal; we can apply one the matrices 1 ±1

0 1

!

a finite number of times to move γz into the stripe {|<(z)| < 1/2}, obtaining a new element z0 that is SL₂ (Z)-equivalent to z. Now, if |z0| < 1, then

−1 z0 > 1, but −1 z is SL2(Z)-equivalent

to z and its imaginary part is greater than =(γz), a contradiction.

As for the cusps, we already know that for each s ∈ Q there exists

γ ∈ SL2(Z) such that γs = ∞.

Moreover for any continuous, bounded function φ : H → C and any

α ∈ SL2(Z), the integral

Z

D∗φ(αz)dµ(z)

converges. Now let Γ be a congruence subgroup of SL₂_{(Z), and let {α}_j}_j be a subset of SL2(Z) such that the following holds:

SL2(Z) =

[

j

{±I}Γαj.

and the union is disjoint. Let φ be Γ-invariant. Then P

j

R

D∗φ(αjz)dµ(z)

does not depend on the choice of the representatives αj and is equal to

Z

S

αj(D∗)

φ(z)dµ(z).

We also write R

X(Γ)φ(z)dµ(z) to indicate the integral above1.

In particular, setting φ = 1, we obtain what is called the volume of X(Γ):

VΓ=

Z

X(Γ)

dµ(z).

Note that the volumes VΓ and VSL2(Z) are related by the following

VΓ= [SL2(Z) : {±I}Γ]VSL2(Z). (1.1)

To construct the Petersson inner product of two cusp forms f, g ∈ Sk(Γ),

let

φ(z) = f (z)g(z)=(z)k for z ∈ H.

1

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This function is continuous, being the composition of continuous func-tions, and Γ-invariant: for any γ ∈ Γ,

φ(γz) = f (γz)g(γz)=(γz)k=

j(γ, z)k(f [γ]k)(z)j(γ, z)kg(γz)=(z)k|j(γ, z)|−2k=

(f [γ]_k)(z)(g[γ]_k)z=(z)k= f (z)g(z)=(z)k since f and g are Γ-invariant.

Now we need to show that such φ is bounded on C, which reduces to show that it is bounded onS

jαj(D). Since this union is finite, it suffices to

prove that for each α ∈ SL₂_{(Z), the function φ ◦ α is bounded on D. Being} continuous, it is bounded in every compact subset of D; near ∞, we can use its Fourier expansion. Remembering that

(f [α]k)(z) = ∞ X n=1 an(f [α]k)qhn, (g[α]k)(z) = ∞ X n=1 an(g[α]k)qhn,

where qh = e2πiz/h for some h ∈ Z+, and since |qh| = e−2π=(z)/h, we have

that

|(f [α]k)(z)| · |(g[α]k)(z)| · |=(z)|k

tends to 0 as =(z) → ∞. Hence, the following inner product is well defined.

Definition 1.24. Let Γ ⊂ SL2(Z) be a congruence subgroup; the Petersson

inner product is given by:

h, i_Γ: S_k(Γ) × S_k_{(Γ) → C} (f, g) 7→ 1 VΓ Z X(Γ) f (z)g(z)=(z)kdµ(z).

The Petersson inner product is linear in f , conjugate linear in g, Hermi-tian symmetric, and positive definite. Moreover the factor 1

VΓ

ensures that if Γ0 ⊆ Γ, then h, i_Γ0 = h, i_Γ on S_k(Γ).

1.3.1 The adjoints of Hecke operators

Using this inner product we can determine the adjoints of Hecke operators; if T is one such operator, its adjoint is T∗ such that

hT f, gi = hf, T∗gi, ∀f, g ∈ Sk(Γ1(N )).

First, we need some technical results. Let Γ be a congruence subgroup of SL₂_{(Z) and let {α}_i}_i be a set of representatives of SL2(Z)

{±I}Γ, that means such that

SL2(Z) =

[

i

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let α ∈ GL+₂_{(Q); then} H ∗ α−1_Γα = S iα−1αi(D∗). Defining Z H∗ α−1Γα φ(z)dµ(z) =X i Z D∗φ(α −1_α iz)dµ(z),

where φ : H → C is continuous, bounded and invariant with respect to

α−1Γα, then clearly Z X(Γ) φ(z)dµ(z) = Z H∗ α−1Γα φ(αz)dµ(z).

Moreover, if also α−1Γα is a subgroup of SL2(Z), then the volumes

V_α−1_Γα and V_Γ are equal (just set φ = 1 in the equality above), and so

[SL2(Z) : α−1Γα] = [SL2(Z) : Γ],

which follows from Formula (1.1) and the fact that −I ∈ α−1Γα if and only if it belongs to Γ. Hence

[SL2(Z) : α−1Γα ∩ Γ] = [SL2(Z) : Γ ∩ αΓα−1],

and similarly [Γ : α−1Γα ∩ Γ] = [Γ : Γ ∩ αΓα−1]. Let n be this number. So there are γ1, . . . , γn and ˜γ1, . . . , ˜γn such that

Γ =[

i

(α−1Γα ∩ Γ)γ_i =[

i

(αΓα−1∩ Γ)˜γ_i−1.

and both the unions are disjoint. We know from Lemma 1.12, applied to Γ₁ = Γ₂ = Γ and taking respectively α and α−1 as the α in the lemma, that

ΓαΓ =[ i Γαγi Γα−1Γ = [ i Γα−1˜γ_i−1,

and the latter is equivalent to ΓαΓ =S

i˜γiαΓ.

It is easy to see that Γαγi∩ ˜γiαΓ is never empty: if it were, there would

exist j such that

Γαγj ⊆ [ i6=j ˜ γiαΓ, thus multiplying by Γ, ΓαΓ ⊆ [ i6=j ˜ γiαΓ,

but this is impossible. Now, if we take βi∈ Γαγi∩˜γiαΓ for every i = 1, . . . , n,

we have ΓαΓ =[ i Γβi = [ i βiΓ.

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Using these facts it is easy to show that given α ∈ GL+₂_{(Q) and setting}

α0 = det(α)α−1, then if α−1Γα ⊆ SL2(Z) it holds

hf [α]k, giα−1_Γα= hf, g[α0]_ki_Γ (1.2)

for every f ∈ S_k(Γ) and g ∈ S_k(α−1Γα). Moreover for all f, g ∈ S_k(Γ),

hf [ΓαΓ]k, gi = hf, g[Γα0Γ]ki. (1.3)

This two equalities combine to show the following theorem.

Theorem 1.25. In Sk(Γ1(N )), the adjoints of the Hecke operators hpi and

Tp for p a prime number that does not divide N are

hpi∗= hpi−1, T_p∗ = hpi−1Tp.

Proof. Since Γ1(N ) is normal in Γ0(N ), if α ∈ Γ0(N ) has lower right entry

equal to p modulo N then by (1.2) hpi∗= [α]∗_k= [α−1]_k = hpi−1. As for the operator Tp, by (1.3) we have T_p∗= " Γ 1 0 0 p ! Γ #∗ k = " Γ p 0 0 1 ! Γ # k .

To compute this, we can note that

p 0 0 1 ! = 1 n N mp !−1 1 0 0 p ! p n N m ! ,

where mp − nN = 1. Hence, noticing that the first matrix in the triple product lies in Γ₁(N ) while the third is in Γ₀(N ),

Γ1(N ) p 0 0 1 ! Γ1(N ) = Γ1(N ) 1 0 0 p ! p n N m ! Γ1(N ) = = Γ₁(N ) 1 0 0 p ! Γ₁(N ) p n N m ! .

So we have that this operator is given by Tp[β]k, where β =

p n N m

!

has lower right entry equal to p−1 modulo N , thus T_p∗ = hp−1iT_p2_.

2

Notice that even if p−1is not integer, it is an element of

_Z

N Z

∗

, hence hp−1i is well defined.

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Knowing this, we can compute the adjoints of Hecke operators of type hni and Tn for (n, N ) = 1. If this condition does not hold, we can compute

adjoints anyway.

As for the diamond operator, if (n, N ) 6= 1 then hni = 0 and so hni∗ = 0. We will now prove that for every n, the adjoint of operator Tnis

T_n∗= w_NTnw−1N ,

where wN is the operator

"

0 −1

N 0

!#

k

. It suffices to show this property for n = p a prime number. By (1.3) we know that the adjoint of T_p is given by the double coset operator

" Γ1(N ) p 0 0 1 ! Γ1(N ) # k . Let γ = 0 −1 N 0 ! , with γ−1= 0 N −1 −1 0 !

. Then the equality

γ−1 a b N c d ! γ = d −c −N b a !

shows that γ−1Γ1(N )γ = Γ1(N ); moreover, using this equality we can also

prove that p 0 0 1 ! = γ−1 1 0 0 p ! γ. So " Γ₁(N ) p 0 0 1 ! Γ₁(N ) # k = = " Γ1(N )γ−1 1 0 0 p ! γΓ1(N ) # k = =[γ−1]k " Γ1(N ) 1 0 0 p ! Γ1(N ) # k [γ]k= =w_NTpwN−1,

and that is the adjoint of Tp.

General facts of linear algebra guarantee that the space of cusp forms has an orthogonal basis of simultaneous eigenforms for the Hecke operators hni, Tn, for (n, N ) = 1, since they all commute with their adjoints.

1.4 Oldforms and newforms

We now want to study the space of cusp forms S_k(Γ1(N )), splitting it into

two orthogonal subspaces, namely oldforms and newforms. In order to con-struct them, we need a class of maps that allow us to move between cusp

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forms at different levels. Let

αd=

d 0

0 1

!

be the multiplication by d. If M | N and d | _MN, then the map f 7→ f [α_d]_k takes Sk(Γ1(M )) to Sk(Γ1(N )).

For each divisor d of N , let us consider the map i_dgiven by

id: Sk(Γ1(N d−1)) × Sk(Γ1(N d−1)) → Sk(Γ1(N ))

(f, g) 7→ f + g[αd]k.

Definition 1.26. The subspace of oldforms at level N is Sk(Γ1(N ))old=

X

p|N, prime

Im(ip).

The subspace of newforms at level N is the orthogonal complement of oldforms with respect to the Petersson inner product,

S_k(Γ1(N ))new = Sk(Γ1(N ))old ⊥

The Hecke operators respect the decomposition into oldforms and new-forms, as shown in the following proposition.

Proposition 1.27. The subspaces S_k(Γ₁(N ))old and S_k(Γ₁(N ))new are sta-ble under the Hecke operators Tn and hni for every n ∈ Z+.

Proof. Let us first consider oldforms; by definition, an oldform is an element

of

X

p|N, prime

ip(Sk(Γ1(N p−1)) × Sk(Γ1(N p−1))).

Let p be a prime dividing N and let T be a Hecke operator. We will now prove that if f, g ∈ Sk(Γ1(N p−1)), then T ◦ ip(f, g) is an element in Im(ip).

If T is the diamond operator hdi we have two cases: if (d, N ) 6= 1, then hdi = 0 and clearly 0 is an oldform; if otherwise (d, N ) = 1, then let

α = a b c δ

!

be a matrix representing hdi, where δ ≡ d (mod N ); then

hdi ◦ ip(f, g) = hdi(f + g[αp]k) = hdif + g[αpα]k.

It is an easy calculation to check that αpα = α0αp, where

α0 = a pb

cp−1 δ !

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represents the operator hdi on Γ1(N p−1). Hence

hdif + g[α_pα]k= hdif + g[α0αp]k= ip hdif + g[α0]k.

Now, let p0 6= p be another prime number. Then

Tp0g =              Pp0−1 i=0 g " 1 i 0 p0 !# k p0 | N Pp0−1 i=0 g " 1 i 0 p0 !# k + g " m n N p0 ! p0 0 0 1 !# k p0 - N,

where in the second equality mp0− nN = 1. By direct calculation,

p 0 0 1 ! 1 i 0 p0 ! = 1 ip 0 p0 ! p 0 0 1 ! ,

and since the numbers {ip : i = 0, . . . , p0− 1} represent all the elements of Z p0 Z , it holds   p0₋₁ X i=0 g " 1 i 0 p0 !# k  [αp]k= p0₋₁ X i=0 g[αp]k " 1 i 0 p0 !# k .

If p0 - N , we also need to consider the additional term in the second equality; we can note that

p 0 0 1 ! m n N p0 ! p0 0 0 1 ! = m np N p−1 p0 ! p0 0 0 1 ! p 0 0 1 ! ,

and the matrix m np

N p−1 p0 !

p0 0

0 1

!

represents the additional term in

Tp0g at level N p−1. Hence T_p0 and i_p commute, precisely

Tp0◦ i_p(f, g) = i_p(T_p0f, T_p0g)

and so T_p0 takes oldforms to oldforms. Let us consider T_p. We claim that

Tp◦ ip(f, g) = ip(Tpf + pk−1g, −hpif ).

Let us first suppose that p2- N . Then,

ip(Tpf + pk−1g, −hpif ) = = p−1 X i=0 f " 1 i 0 p !# k + f " m np N p−1 p !# k + pk−1g − f [ααp]k,

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where α represents the diamond operator hpi. Choosing α ≡ mp

−1 _np

N p−2 p !

(mod N p−1) - note that all this numbers exist in Z

N p−1

Z

, under the assump-tion p2 _{- N - the two terms f}

"

m np N p−1 p

!#

k

and f [αα_p]_k are equal and

so drop out, giving i_p(T_pf + pk−1g, −hpif ) =Pp−1

i=0f " 1 i 0 p !# k + pk−1g.

On the other hand,

Tp◦ ip(f, g) = p−1 X i=0 f " 1 i 0 p !# k + p−1 X i=0 g " αp 1 i 0 p !# k ,

so we need to compute this last term. First we note that α_p 1 i

0 p ! = pI ◦ 1 i 0 1 ! ; we have g " pI ◦ 1 i 0 1 !# k = g[pI]k " 1 i 0 1 !# k = pk−2g " 1 i 0 1 !# k = pk−2g; so Tp◦ ip(f, g) =Pp−1i=0 f " 1 i 0 p !# k + pk−1g, as wanted.

Finally, if p2 | N , the operator hpi is 0 and i_p(T_pf + pk−1g, −hpif ) = Pp−1 i=0f " 1 i 0 p !# k

+ pk−1g = Tp ◦ ip(f, g) as before. So far, we have

considered all the possible Hecke operators and hence proved that oldforms are stable under them. As for newforms, we proceed as follows. If f is a newform, g is an oldform and T is a Hecke operator, then

hf, T∗gi = hT f, gi;

if T∗g is still an oldform, then the inner product above is equal to 0, hence T f is a newform as well. So we just need to show that oldforms are stable

under the adjoints of Hecke operators. The adjoint of a diamond operator is another diamond operator, so there is nothing to show. Moreover, we know from previous section that T_n∗ = wNTnw−1_N , where wN =

" 0 1 −N 0 !# k . Then it suffices to show that this last operator preserves oldforms; to do so,

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we claim that wN ◦ ip(f, g) = ip(pk−2wN p−1g, w_{N p}−1f ). It holds: ip(pk−2wN p−1g, cw_{N p}−1f ) = ip pk−2g " 0 1 −N p−1 ₀ !# k , f " 0 1 −N p−1 ₀ !# k ! = pk−2g " 0 1 −N p−1 0 !# k + f " 0 1 −N p−1 0 ! p 0 0 1 !# k ; on the other hand

wN◦ ip(f, g) = f " 0 1 −N 0 !# k + g " p 0 0 1 ! 0 1 −N 0 !# k .

Direct calculation shows that 0 1

−N p−1 ₀ ! p 0 0 1 ! = 0 1 −N 0 ! , and so the terms involving f are the same in the two expressions, and

pk−2g " 0 1 −N p−1 0 !# k (z) = (−1)−kN−1pk−1z−kg − p N z ; g " p 0 0 1 ! 0 1 −N 0 !# k = (−1)−kN−1pk−1z−kg − p N z .

This concludes the proof.

In particular the spaces S_k(Γ1(N ))oldand Sk(Γ1(N ))newhave orthogonal

bases of eigenforms for the Hecke operators hni and T_n with (n, N ) = 1. 1.4.1 Eigenforms

This section uses a theorem due to Atkin and Lehner; we omit the proof, which can be found in [6] (see Theorem 5.7.1), and use it to prove some basic properties of newforms. Note that in this section the meaning of the term newform will be slightly different from that in the previous section, as will be specified in Definition 1.29.

Theorem 1.28 (Atkin-Lehner). If f ∈ S_k(Γ₁(N )) has Fourier expansion

f (z) =X

n

an(f )qn, q = e2πiz/N

with an(f ) = 0 whenever (n, N ) = 1, then f takes the form

f =X

p|N

ιpfp

with each fp ∈ Sk(Γ1(N p−1)); the map ιp is defined as follows:

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We already know that the spaces Sk(Γ1(N ))old and Sk(Γ1(N ))new

have orthogonal bases of eigenforms for the Hecke operators hni and Tn

with (n, N ) = 1. Our aim is now to show that if f is an element of such a basis for Sk(Γ1(N ))

new

, then it is an eigenform for all Hecke operators. In the case of the diamond operator, there is nothing to show: if (n, N ) 6= 1, then hni is simply the zero operator and clearly f is an eingenform for it. We shall now consider Hecke operators of the second type.

Definition 1.29. A nonzero modular form f ∈ Mk(Γ1(N )) that is an

eigenform for all the Hecke operators is a (Hecke) eigenform. The Hecke eigenform f (z) = ∞ X n=0 an(f )qn, q = e2πiz/N

is normalized when a₁(f ) = 1. A newform is a normalized Hecke eigenform

in Sk(Γ1(N ))new.

The following theorem will prove our statement and give a relation be-tween the eigenvalues of Hecke operators and Fourier coefficients.

Theorem 1.30. Let f ∈ S_k(Γ₁(N ))new

be a nonzero eigenform for the Hecke operators Tn and hni for all n with (n, N ) = 1. Then

• f is a Hecke eigenform; a suitable scalar of f is a newform;

• if ˜f satisfies the same conditions as f and has the same Tn-eigenvalues,

then ˜f = cf for some constant c.

The set of newforms in the space S_k(Γ₁(N ))new

is an orthogonal basis of it. Each such newform lies in a space Sk(N, ε) and satisfies Tnf = an(f )f

for all n ∈ Z+: the Fourier coefficients of f are its Tn-eigenvalues.

Proof. Let f ∈ S_k(Γ₁(N )) be an eigenform for T_n and hni where (n, N ) = 1; thus there exist some eigenvalues c_n and d_n_{∈ C such that}

Tnf = cnf hnif = dnf

for every such n. The map n 7→ d_n defines a Dirichlet character ε, being a homomorphism, and so f ∈ Sk(N, ε). Now, using the formulas in

Proposi-tion 1.23, we can compute that a₁(T_nf ) = an(f ) for n ∈ Z+. On the other

hand, for (n, N ) = 1, it holds a₁(T_nf ) = cna1(f ), hence

an(f ) = cna1(f ) for (n, N ) = 1.

In particular, if a1(f ) = 0, then an(f ) = 0 for all n such that (n, N ) = 1,

and so f ∈ Sk(Γ1(N )) old

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Sk(Γ1(N ))new then a1(f ) 6= 0 and we may assume it is equal to 1, so that

f is normalized. For m ∈ Z+, let

gm= Tmf − am(f )f.

This element belongs to S_k(Γ₁(N ))new

and by direct calculation it is an eigenform for Tn and hni with (n, N ) = 1, indeed

Tn(gm) = TnTmf − Tnam(f )f = TmTnf − am(f )Tnf = cngm,

hni(gm) = hniTmf − hniam(f )f = Tmdnf − am(f )dnf = dngm.

Moreover a₁(g_m) = a₁(T_mf ) − a1(am(f )f ) = am(f ) − am(f )a1(f ) =

am(f ) − am(f ) = 0. Hence gm ∈ Sk(Γ1(N )) new

∩ S_k(Γ₁(N ))old

= {0}. This proves that Tmf = am(f )f for every m ∈ Z+, hence f is a Hecke

eigenform (a newform if we normalize it), with eigenvalues for the operators

Tngiven by its Fourier coefficients; if ˜f satisfies the same conditions as f and

has the same eigenvalues, then the Fourier coefficients of f and ˜f only differ

by a constant. To prove that the set of newforms is an orthogonal basis of S_k(Γ₁(N ))new

we just need to show that it is linearly independent. Let by contradiction

n

X

i=1

cifi= 0

for some ci ∈ C, all different from 0, with as few terms as possible (in

particular n ≥ 2). Then for any prime p, applying Tp− ap(f1) to the relation

gives

n

X

i=2

ci(ap(f1) − ap(fi))fi = 0;

this linear relation involves n − 1 elements, so it must be 0, thus a_p(f_i) =

ap(f1) for every i. Since p is arbitrary, this gives fi = f1 for all i, and this is

a contradiction since the original relation involved at least two terms.

1.5 Eisenstein series

In this final section, we will introduce Eisenstein series, which can be viewed as a complement to cusp forms in the space of modular forms. Extending the Petersson inner product to each pair of modular forms (f, g) such that the integral

Z

X(Γ)

f (z)g(z)=(z)kdµ(z)

converges, we will be able to prove that Eisenstein series and cusp forms are orthogonal.

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Definition 1.31. An Eisenstein series with parameter associated to the congruence group Γ(N ) is a function

E_kv¯(z, s) = _N X γ∈ Γ(N )δ P+∩Γ(N) =(z)s[γ]_k, <(k + 2s) > 2, where: • N = 1/2 if N = 1, 2 and 1 if N > 2; • ¯v = (cv, dv) ∈ Z N Z 2 , of order N ;

• δ is a matrix in SL₂_{(Z) whose bottom row is a lift of ¯}_{v to Z}2_;

• P₊ is the subgroup of SL₂_{(Z) consisting of the matrices} 1 n

0 1

! , n ∈ Z.

Here, the weight-k operators act on the function =(z)s, which depends on two parameters, and is defined, in general, as

f [γ]k(z, s) = j(γ, z)−kf (γz, s).

We can analytically continue the function E_kv¯(z, s) at s = 0, though not necessarily obtaining a holomorphic function: for example at weight k = 2 the continued series is nonholomorphic, but linear combinations cancel away the nonholomorphic terms. Hence we can give the following definition.

Definition 1.32. The Eisenstein space with respect to Γ(N ) is the subspace

E_k(Γ(N )) of M_k(Γ(N )) given by the holomorphic functions in

Span ( E_k¯v(z, 0) : ¯v ∈ Z N Z 2 , ord(¯v) = N ) .

It can be shown that this space is isomorphic to the quotient Mk(Γ(N )) Sk(Γ(N ))

(see [6], §5.11) hence we can write

M_k(Γ(N )) ∼= Sk(Γ(N )) ⊕ Ek(Γ(N )).

Eisenstein series as defined here are the general case of classical Eisen-stein series, which are

Gk(z) =

X

(c,d)∈Z2_\{(0,0)}

(cz + d)−k;

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Moreover, as mentioned at the beginning of this section, cusp forms and Eisenstein series are orthogonal with respect to the Petersson inner product. To show this we will first need some notation. Let P₊(N ) = P₊∩ Γ(N ), D∗

N

be a fundamental domain for the orbit space H

∗

P+(N )

, where D_N∗ = {z ∈ H∗_{∩ C : 0 ≤ <(z) ≤ N} ∪ {∞}.} Now let αi, βj satisfy

Γ(N ) P+(N ) =S iP+(N )αi and SL2(Z) Γ(N ) = S jΓ(N )βj, so that SL₂_(Z) P+(N ) =[ i,j P+(N )αiβj, D∗N = [ i,j αiβj(D∗).

If f is a cusp form in Sk(Γ(N )), then the condition a0(f ) = 0 implies

that

Z N

0

f (x + iy)dx = 0

for y > 0 and so for any s ∈ C such that <(k + 2s) ≥ 0 it holds

Z ∞

y=0

Z N

x=0

f (x + iy)yk+s−2dxdy = 0,

that can be re-written as

Z D∗ N f (z)=(z)k+sdµ(z); it follows 0 =X i,j Z D∗f (αiβjz)=(αiβjz) k+s_{dµ(z) =} =X i,j Z D∗f (βjz)j(αi, βjz) k =(βjz)k+s |j(α_i, βjz)|2(k+s) dµ(z) = =X j Z D∗f (βjz)E (0,1) k (βjz, s)/N=(βjz)kdµ(z),

where the last equality holds when <(k + 2s) > 2 (in which case the sum over i converges absolutely and passes through the integral). In particular, this shows that the Petersson inner product off (z) and E_k(0,1)(z, s) is 0 for all s such that <(k + 2s) > 2. This relation analytically continues to s = 0, giving

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Now, for ¯v ∈

Z

N Z 2

of order N , there is γ ∈ SL2(Z) such that ¯v =

(0, 1)γ. So, for any cusp form f ∈ S_k(Γ(N )):

hf, E_k(0,1)γi_{Γ(N )}= hf, E_k(0,1)[γ]kiΓ(N ) = hf [γ−1]k, E_k(0,1)iΓ(N ),

where the last equality follows from Formula (1.2). Since also f [γ−1]_k is a cusp form, hf, E_k(0,1)γi_{Γ(N )} = 0. This proves that Eisenstein series as here defined and cusp forms are orthogonal. For any congruence subgroup Γ of SL2(Z) at level N we define

Ek(Γ) = Ek(Γ(N )) ∩ Mk(Γ)

and for a Dirichlet character ε modulo N :

Ek(N, ε) = Ek(Γ1(N )) ∩ Mk(N, ε).

In particular, this proves that any modular form is the sum of an Eisen-stein series and a cusp form. Our main concern about EisenEisen-stein series is the following theorem, which will be used in the proof of Deligne-Serre’s Theorem.

Theorem 1.33. Let N ∈ Z+ and let A_N,1 be the set of all ({ψ, ϕ}, t) such that ψ and ϕ are Dirichlet character modulo u, v respectively, with uv | N , ψϕ is odd, and t ∈ Z+ is such that tuv | N . Set

E₁{ψ,ϕ}(z) = δ(ϕ)L(0, ψ) + δ(ψ)L(0, ϕ) + 2 ∞ X n=1 σ{ψ,ϕ}₀ (n)e2πinz, where • δ(ϕ) = 1 if ϕ is trivial, 0 otherwise;

• L(s, ϕ) is the Dirichlet series associated to ϕ, namely

L(s, ϕ) = ∞ X n=1 ϕ(n) ns ; • σ{ψ,ϕ}₀ (n) =P m|nψ(n/m)ϕ(m).

If we also set E₁{ψ,ϕ},t(z) = E₁{ψ,ϕ}(tz), then the set of all E₁{ψ,ϕ},t forms a basis of the space E1(Γ1(N )); for any character ε modulo N , the set of the

E₁{ψ,ϕ},t with ψϕ = ε is a basis of E1(N, ε). Finally, if p - N is prime, then

TpE {ψ,ϕ},t 1 = (ψ(p) + ϕ(p))E {ψ,ϕ},t 1 ∀E {ψ,ϕ},t 1 .

This is a particular form of a more general theorem, which can be found in [6] (see Theorems 4.5.2, 4.6.2 and 4.8.1.).

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Deligne-Serre’s Theorem

In this chapter, we introduce Galois representations, and then we will state and prove Deligne-Serre’s Theorem.

Let ¯_{Q be an algebraic closure of Q and let G = Gal(}_{Q/Q) the Galois}¯ group of the (infinite) field extension ¯_{Q/Q, that is called absolute Galois}

group. Let K be one of the following fields:

• the field C, with the discrete topology; • a finite field, with the discrete topology;

• a `-adic field, where ` is a prime number, with its natural topology, which is to be explained later.

Definition 2.1. A Galois representation is a linear representation of G, i.e. a continuous homomorphism

ρ : G → GLd(K),

where K is as above.

2.1 A review of Number Theory

Before defining the `-adic topology, we will summarize some of the basic properties of `-adic integers and `-adic fields. What is stated in the following section can be found in [1], §1.

Let us consider the sequences (a1, a2, a3, . . . ) : an∈ Z

`n

Z, an+1

≡ an (mod `n) ∀n;

these form a ring, which is isomorphic to the inverse limit of Z

`n_Z, for n ∈ Z

+_.

Arithmetic of modular forms

Tesi di Laurea

Arithmetic of

modular forms

22 settembre 2017

Contents

Introduction

Tools

1.1

Modular Forms

1.2

Hecke Operators

1.3

The Petersson inner product

1.4

Oldforms and newforms

1.5

Eisenstein series

Deligne-Serre’s Theorem

2.1

A review of Number Theory