A look at the Fontaine-Mazur conjecture and some related results

(1)

Stating of the Fontaine-Mazur Conjecture

1. Basic definitions

1.1. Representations. Let G be a profinite group, which is a projective limit of finite groups with the topology induced by the discrete one on the finite groups. By a representation of G, we mean a finite dimensional vector space V over a field F with a continous group homomorphism

ρ : G → GL(V )

More explicitly, we can call this an F -representation; it defines an action of g on V by g.v = ρ(g)v. By fixing an F -basis of V we can write ρ as an homomorphism

ρ : G → GLn(F )

We will be mainly interested in the case where G is the absolute Galois group of a number field (finite extension of Q) or a p-adic field (finite extension of Qp), where by the absolute Galois group of a field K we mean the group GK = Gal(Ks/K), where Ks is a separated closure of K. If K is a number field, we will sometimes take G to be tha Galois group GK,S = Gal(KS/K), where S is a finite set of finite places of K and KS is the maximal algebraic extension of K unramified outside the places of S. We will usually take V to be a Ql vector space for some prime l; in particular, if G = GK with K a p-adic field, we will call the representation an l-adic representation if l 6= p and a p-adic one if l = p.

We can extend the previous definition to include the situation in which V is a free module over a ring A: we will sometimes be interested in Zp-representations of G.

We call a representation V of G irreducible if V doesn’t have any non trivial subspace stable by the action of G. Given a representation V , we denote by VG the subspace of G-invariant elements of V .

Given a profinite group G and a field F , we can define the category Rep_F(G) of F -representations of G: a morphism between two representations V and W will be a G-equivariant map f : V → W , in the sense that it satisfies f (g.v) = g.f (v). We can check that RepF(G) is an abelian category. Moreover, given two representations V , W , we can define their tensor product by an action of G on the usual tensor product V ⊗ W as g(v ⊗ w) = g.v ⊗ g.w; we can also define a dual representation V∗ given by an action of G on the usual dual space V∗= {f : V → F, f F -linear} as g.f (v) = f (g−1_{.v). It can be showed that these two operations give Rep}

F(G) the structure of a Tannakian category.

Example 1.1. Given a Galois extension of fields K ⊂ L, we have a natural representation of its Galois group Gal(L/K) on the one dimensional L-vector space L, given by g.x = g(x). Less trivially, if K is a field and l a prime number, different

(4)

from the residue characteristic of K if K is local, we can identify the group µl∞

of l∞ roots of 1 in ¯_{K with a free Z}l-module of rang 1 with an action of GK given by the restriction to µl∞ of its natural action on ¯K. We call this representation

ρ : G → GL(Zl) = Z∗l the cyclotomic character and we denote it by Zl(1). We can view it as a Ql-representation by tensoring by Ql: we take the space Zl⊗Qlwith the action g.(v ⊗w) = ρ(g)v ⊗w; we denote this representation by Ql(1). We denote the dual representation of Zl(1) by Zl(−1), and the same for the Ql-representations.

Thanks to this last example, we can define an important way of altering a representation of the absolute Galois group of a field:

Definition 1.1. If K is a field, V a Ql-representation of GK and n a non-negative integer, we define the n-th Tate twist of V by V (n) = V ⊗ Ql(1)⊗n (we tensor by Zl(1) if we are considering Zl-representations). If n is a negative integer, we define V (n) = V ⊗ Ql(−1)⊗n.

In particular, Ql(n) is simply the tensor product of n copies of Ql(±1). If K is an algebraic extension of Q, we call a representation of GK odd if, for a chosen (hence for every) immersion K ,→ C, the image of complex conjugation is the opposite of the identity; we call it even if it is the identity.

1.2. Witt vectors. We recall here a construction which will prove necessary througout the following chapters; a reference can be found for example in [Bou83]. If A is a ring and p a prime number, we define the ring Wn(A) to be An as a set, with the only ring structure that makes the following map a ring homomorphism:

ρn: Wn(A) → An

(a0, a1, ...an−1) 7→ (w0, w1, ...wn−1) where, for each i, wi = ap

n

0 + pa pn−1

1 + ... + p n_a

n. This construction clearly depends on p, but in the following the choice of the prime will always be evident.

Remark 1.1. If p is invertible in A, the previous map is a bijection and gives us an isomorphism between Wn(A) and An.

We now define the ring of Witt vectors of A to be the inverse limit of the Wn(A) with the maps

fn: Wn(A) → Wn−1(A) (a0, a1, ..., an) 7→ (a0, a1, ..., an−1) We denote this ring by W (A). We have a natural map

W (A) → AN

(a0, a1, ..., an, ...) → (w0, w1, ..., wn, ...)

As in the case of Wn(A), if p is invertible in A we get an isomorphism of rings between W (A) and AN_{, but this is false in general. For A = F}_p _{we obtain an easy} example of this construction, since W (Fp) = Zp.

We have a canonical way to lift elements from A to W (A), called the Teichmuller lift: to a ∈ A we associate [a] = (a, 0, 0, ...) ∈ W (A). We can define in W (A) a sequence of ideals Vm(A) = {(a0, a1, ..., an, ...)|ai = 0 for all i < m}. Clearly Vm+1(A) ⊂ Vm(A) and V1(A) is maximal; we identify the rings Wm(A) with the

(5)

2. EXTENSIONS OF NUMBER FIELDS AND p-ADIC FIELDS 3

quotients W (A)/Vm(A). We consider W (A) as a topological ring with the p-adic topology. If A is a ring of characteristic p we have an inclusion p.W (A) ⊂ V1(A); moreover, the p-adic topology and the one defined by the maximal ideal V1coincide, and W (A) is separated and complete with respect with this topology.

We suppose from ow on the characteristic of A equal to p. We can also lift the Frobenius homomorphism, sending x to xp_{, to a homomorphism of W (A), by} sending (a0, a1, a2, ...) to (a p 0, a p 1, a p

2, ...). We recall that A is perfect if and only if its Frobenius map is an automorphism. The following propositions show why we it is preferable to work with a perfect field A:

Proposition 1.1. If A is a perfect ring, we have pnW (A) = Vn(A) for all n. The projection on the first coordinate gives an isomorphism W (A)/V1(A) ∼= A. If A is a field, W (A) is a discrete valuation ring with residue field A.

Proposition 1.2. The ring W (A) is Noetherian if and only if the ring A is perfect.

2. Extensions of number fields and p-adic fields

We fix a finite place v on a number field K, corresponding to a prime p in the ring of integers OK; if v lies above p ∈ Q we will denote kvk = pk the cardinal of the residue field of K in v. By choosing an extension of v to ¯v on ¯K corresponding to a prime q in O_K¯ lying over p, we get a tower of fields

K ⊂ KD⊂ KE⊂ ¯K

where KD is the maximal extension of K in ¯K such that q ∩ KD is inert and unramified over p, while KE is the maximal extension of K such that q ∩ KE is unramified over p. We call Dv = Gal( ¯K/KD) ’the’ decomposition group at v and Iv= Gal(K/KE) ’the’ inertia group at v; these groups depend on the choice of ¯v, but groups associated to different choices are conjugated. We have furthermore an identification between the absolute Galois group of the completion Kvof K at v and the decomposition group Dv, which we obtain by choosing an embedding of K in ¯Kv (which amounts to choosing a place ¯v extending v to ¯K) and restricting the elements of Gal( ¯Kv/Kv) to the image of ¯K. We observe that, thanks to this identification, a representation of the group GK induces by composition a representation of the group GKv:

ρv: GKv ∼= Dv,→ GK → GL(V )

We have a similar but more subtle situation when considering a representation of the group GK,S, with S a finite set of finite places of K. When fixing a place v of K not belonging to S, we have again a map from GKv to GK,S (induced by an

embedding KS → Kv) which allows us to obtain, by composition, a representation of GKv by one of GK,S, but it is false in general that this map is an injection.

However, it has been recently shown by G. Chenevier and L. Clozel [CC09] that, for K = Q, the map Gal(Qp/Q) → GK,S is an injection if S contains at least two places; this is the same as stating that each embedding of QS in ¯Qp is dense.

From the description of the absolute Galois group of Kv as the decomposition group of a place ¯v over v, we get the tower of extensions of Kv:

Kv⊂ Kv,ur⊂ ¯Kv

where Kv,ur is the maximal unramified extension of Kv in ¯Kv and the fixed field of the inertia group Iv. The quotient group Gal(Kv,ur/K) is isomorphic to the

(6)

absolute Galois group of the residue field Fkvk = Fpk of Kv (hence isomorphic to ˆZ). Since Gal(F¯p/Fp) is topologically generated by the arithmetic Frobenius automorphism φp sending x ∈ ¯Fp to xp

k

, the group Gal(Kv,ur/Kv) is generated by its only element σp inducing φp on the residue field. We call σp the arithmetic Frobenius at v, and its inverse σ−1

p the geometric Frobenius at v.

The extension Kv,ur can be obtained by adding to Kv the pn-th roots of 1 for any n, and p will be a uniformizer in Kv,ur (a generator of the maximal ideal in OKv,ur). Given any algebraic extension k

0 _{of the residue field k of K}

v, there is a unique unramified extension K0 of Kv having k0 as residue field; K0 will be isomorphic to FracW (k0), W (k0) being the ring of Witt vectors of k0.

We can further distinguish between ’tame’ and ’wild’ inertia: we define the tamely ramified extension Kv,tr of Kv as the one obtained by adding to Kv,ur all the n-th roots of p with n prime to p. The Galois group of Kv,trover Kv,urwill be a profinite group of order prime to p (tame inertia group), while the Galois group of

¯

Kv over Kv,tr will be a pro-p-group (projective limit of p-groups). More precisely, we have an injective morphism of GKv-representations:

κ : Gal(Kv,tr/Kv,ur) ∼

→Y

l6=p Zl(1)

where the action of GKv on the left is the adjoint one. This is an isomorphism if

and only if Kvis unramified over Qp. The action of GKv on these groups factorises

through the quotient GKv/Iv = Gal(Kv,ur/K) = Gal(¯k/k) since the action of the

inertia group is trivial; GKv/Ivis generated (as a pro-cyclic group) by the Frobenius

σv, so the GKv/Iv compatibility is equvalent to the relation

˜

σvs˜σv−1= s p

for any s ∈ Gal(Kv,tr/Kv,ur) and a lift ˜σv of the Frobenius automorphism. If χ : Gal(Kur/K) → ˆZ is the map sending the Frobenius to 1 (thus giving the isomor-phism between these two groups) we can restate the Gal(Kv,ur/K)-compatibility as

sgs−1 = gχ(s) for every s ∈ Gal(Kv,tr/Kv) and g ∈ Gal(Kv,tr/Kv).

We can explicitate the previous isomorphism: if n is prime to p, ζn is an n-th root of p and µn the group of primitive n-th roots of 1, we get an isomorphism

Gal(Kv,ur(ζn)/Kv,ur) ∼= µn∼= (Z/nZ) by sending g to gζn

ζn ∈ µn; this is a GKv-equivariant isomorphism, with the adjoint

action on the left-hand side. Since we obtain Kv,tr by adding to Kv,ur all the n-th roots of p with n prime to p, by taking the projective limit in n of the prevoius maps we get the desidered morphism, called the Kummer map κ. If Kv is unramified over Qp, no root of p already belongs to K, so this map is an isomorphism.

By composition of κ with the projection on the l-th component of the image, we get the l-th Kummer map

κl: Gal(Kv,tr/Kv,ur) → Zl(1) κlgives an injective morphism

(7)

where Kv,tr,l is the ’l-part’ of Kv,tr, which is the tamely ramified extension of Kv obtained by adding to Kv,ur the lk-th roots of p for every k. This morphism is compatible with the action of Gal(Kv,tr,l/Kv); it is an isomorphism if and only if Kv has no l-ramification over Qp.

The Kummer map appears in an important result by Grothendieck; we first give two simple definitions:

Definition 2.1. We call an element A of GLn(F ) unipotent if A − 1 is nilpo-tent. A representation ρ : G → GLn(F ) of a profinite group G is unipotent if ρ(g) is unipotent for every g ∈ G; it is quasi-unipotent if there is an open subgroup U of G such that ρ|U is unipotent.

Remark 2.1. A ∈ GLn(F ) is unipotent if and only if all of its eigenvalues are 1.

K is now a p-adic field, k its residue field and l a prime different from p. Definition 2.2. A representation ρ : GK → GLn(Ql) is called semi-stable if its restriction to the inertia subgroup I = Gal(Ks/Kur) is unipotent. We say that ρ is potentially semi-stable if its restriction to IK is quasi-unipotent. Equivalently, ρ is potentially semi-stable if there exists a finite extension L of K such that the induced representation of GL is unipotent.

Theorem 2.1 (Grothendieck’s monodromy theorem). Let K be a p-adic field and ρ : GK → GL(V ) ∼= GLn(Ql) a Ql-representation of GK, ρ is potentially semi-stable. Equivalently, there exists a unique nilpotent endomorphism of V such that

ρ(σ) = exp(κl(σ)N ) ∀σ ∈ J where κl is the Kummer map associated to l as defined above.

Before giving a proof of the theorem, let’s remark why this is significant. In the setting of the Fontaine-Mazur conjecture we will consider representations of G_Q in GL2(Qp). As we saw earlier, such a representations induces representations of GQl

for every prime l; we will be interested in knowing wether a certain condition of ’potential semi-stability’ is satisfied for each of these local representations. For l 6= p (’l-adic representation’) the condition is the one given above, so Grothendieck’s theorem guarantees that it is always satisfied. This allows us to focus only on what happens for l = p (’p-adic representation’): in order to study this case we will have to develop quite a vast machinery, which will be the object of the next sections.

Proof. Since the image of ρ lies in the pro-group (projective limit of l-groups) GLn(Ql), ρ must factorise through the l-part Gal(Ktr,l/K) of IK. It is then sufficient to show that the quotient representation of Gal(Ktr,l/K), which we call again ρ, is quasi-unipotent. Indeed, thanks to the trivial surjection IK → Gal(Ktr,l/K), the preimage of an open subgroup of Gal(Ktr,l/K) on which ρ acts unipotently gives an open subgroup of IK on which ρ acts unipotently.

We call χl the character GK → Z∗l giving the action of Gal(Kur) on l∞-roots of unity); χl factorises through the pro-l-part of Gal(Kur/K), giving an injective map Gal(K(µl∞/K) → Z∗_l. We call again χ_l the induced map Gal(K_tr,l/K) →

ˆ

Z. The Gal(Kur/K) = Gal(¯k/k) compatibility of the l-th Kummer isomorphism Gal(Ktr,l/Kur)

κl

→ Zl(1) gives the relation sgs−1 = gχl(s)for every g ∈ Gal(Ktr,l/Kur) and s ∈ Gal(Ktr,l/K). By applying ρ to the previous equality we get that for all

(8)

s the matrices ρ(g) and ρ(g)χl(s) _{are conjugate through ρ(s). This means that,}

if λ is an eigenvalue of ρ(g) (in an extension of Qp), λχl(s) is also an eigenvalue of ρ(s), since it is an eigenvalue of its conjugate ρ(g)χl(s)_{. We thus get a group}

E = {λχ(s)}s∈Gal(Ktr,l/K) of eigenvalues of ρ(g), which must be finite. If we show

that the image of χlis an infinite subgroup of Zl(1), in order for E to be finite we must have λχl(s)_{= 1 for some s ∈ Gal(K}

tr,l/K), so that λ is a root of 1. But, the group µl∞(K) of l∞roots of 1 in K being finite (since isomorphic to µ_l∞(k)), the

group Gal(K(µ_l∞_{( ¯}_K))/K) is infinite; since χlis injective over this group, its image is infinite too.

Since we proved that every eigenvalue of ρ(g) is a root of 1, there exists an inte-ger m such that all eigenvalues of ρ(g)m_{are 1, so that ρ(g}m_{) = ρ(g)}m_{is unipotent.} Clearly this implies that the whole closure of the subgroup of Gal(Ktr,l/K) ∼= Zl(1) generated by gm

acts unipotently. Since the closure of a subgroup of Zlgenerated by one element is open, we obtain an open subset of Gal(Ktr,l/K) acting unipotently. To put ρ in the exponential form described by the theorem, we just take its logarithm τ (s) = log ρ(s), which we are allowed to do since the image of ρ lies in GLn(Qp). Let’s now take the restriction of ρ to a pro-cyclic open subgroup U (with generator g) on which it is unipotent (it exists thanks to the proof of the first part) and N ∈ Mn(Ql) such that τ (g) = κl(g)N (N exists because κl(g) 6= 0, otherwise κl would be zero on an open subgroup of IK, absurd because it is injective on its l-part). N is nilpotent, since its eigenvalues are the logarithms of those of ρ, so they are all zero. If gm is an element of U , we have τ (gm) = mτ (g) = mκl(g)N = κl(gm)N , so that τ (s) = κl(s)N for all s of the form gm, hence for all s ∈ U by

continuity. This proves the theorem.

Remark 2.2. Grothendieck’s theorem is true under the weaker hypothesis that K is a local field such that no finite extension of K contains the group µl∞(K) of

the l∞roots of unity in ¯K: this is indeed sufficient for the group Gal(K(µl∞( ¯K)/K)

to be infinite.

Remark 2.3. The nilpotent endomorphism appearing in the theorem satisfies the very important relation

ρ(g)N ρ(g)−1= qχ(g)N

for all g ∈ GK. In particular, for g equal to the arithmetic Frobenius φ of K we get ρ(φ)N ρ(φ)−1 = qN , or ρ(φ)N = qρ(φ)N . We will have in the p-adic case an analogue of N satisfying the same relation.

The main application of Grothendieck’s monodromy theorem lies in that it allows to define from an l-adic representation of GK a representation of other very important groups associated to K, the Weil group and the Weil-Deligne group which we now define. Recall that we have an exact sequence

1 → IK → GK→ Gk → 1

where Gk is topologically generated by the Frobenius automorphism φ. We want now to trivialise the topology on the unramified part:

Definition 2.3. We define the Weil group of K as the subgroup of GK con-sisting of the elements whose projections in Gk are of the form φmfor some m ∈ Z; we denote it WK.

(9)

Thanks to the compatibility relation sgs−1 = gχ(s)

for g ∈ IK and s ∈ GK, with χ(φm) = m, WKcan be identified with the semi-direct product

IKo Z

with m ∈ Z acting by conjugation on IK as φmzφ−m = qmz, q being the cardinal of the residue field k.

It is often more interesting to work with the Weil group rather than the com-plete Galois group. The reason for this is that it is useful to consider complex representations rather than just l-adic (or p-adic) ones, but the complex (conti-nous) representations of GKare uninteresting: because of its rich topology, they all have finite image. On the other side the complex representations of WK present a greater variety because we have the discrete topology on the unramified part rather than a profinite one.

We now construct the Weil-Deligne group by taking a semi-direct product with the additive group Ga of Qp:

Definition 2.4. The Weil-Deligne group of K is given by the semidirect prod-uct

Gao WK with w ∈ WK acting on x ∈ Ga as

wxw−1= q−χ(w)x

We define representations of WK as an usual representation with an additional condition:

Definition 2.5. If K is a p-adic field and E a field of characteristic 0, an E-representation of the Weil group WK is a finite dimensional E-vector space V with a morphism

ρ : WK → GL(V )

such that ρ is trivial on U o Z for an open subgroup U of IK.

Remark 2.4. We say that a representation is finite on a profinite group G if it has finite image: this is equivalent to saying that it is trivial on an open subgroup of G.

Since W DK = Gao WK, to give an E-representation of W DK is the same as to give E-representations ρ and τ of WK and Ga satisfying

ρ(w)τ (x)ρ(w)−1= q−χ(w)ρ(x)

We see by the previous relation the representation τ is actually determined by its values on the subgroup Z and by the representation ρ of WK, so giving τ amounts to giving τ (1) ∈ GL(V ) (indeed, this gives τ (m) for every m ∈ Z, the previous formula extends τ to Q and by continuity we get it on Ga). This justifies the following definition, where we require ρ to be a Weil representation as defined above and τ (1) to be of the form exp(N ) with N nilpotent:

Definition 2.6. A representation of the Weil-Deligne group W DK is a pair (ρ, N ) consisting of:

(10)

• a representation of WK

ρ : WK → GL(V )

in the preceding sense, which means trivial on an open subgroup of IK (finite on IK); • a nilpotent endomorphism N : V → V such that N ◦ ρ(w) = qχ(w)ρ(w) ◦ N for any w ∈ WK.

The reason to define Weil and Weil-Deligne representations this way is again the fact that we want to consider complex representations of these groups. Given a Weil representation over an E-vector space with E l-adic, we can obtain a complex representation simply by choosing an immersion E → C and composing:

ρ : WK → GLn(E) → GLn(C)

The representation obtained this way is continuous, since it is trivial on a subgroup of the form U o Z with U ⊂ IK open, hence factors through the quotient IK/U o Z whose topology is discrete (IK/U is finite). The same reasoning shows that a Weil-Deligne representation over E l-adic gives a complex Weil-Weil-Deligne representation.

We can now state this immediate corollary of the Grothendieck’s monodromy theorem:

Corollary 2.1. If K is a p-adic field and ρ : GK → GLn(E) is an l-adic representation of GK, we obtain a Weil E-representation by defining

ρ0: WK→ GLn(E) w 7→ ρ(w) exp(−κl(w)N )

where N is the nilpotent endomorphism associated to ρ. By the previous discus-sion ρ0 induces also a complex representation of WK. Moreover, the couple (ρ0, N ) defines a Weil-Deligne E-representation, hence also a complex one.

We conclude this section by giving an important definition:

Definition 2.7. If K is a number field, a representation V of GK or GK,S (for S as usual) is said to be unramified if the image of the inertia group at v is trivial.

3. The representations coming from modular forms

In this section we define modular forms as a special kind of holomorphic functions on the Poincar´e half-plane (the set of complex numbers with positive imaginary part); we then state the existence and the main properties of the Ga-lois representaton attached to a modular form, which plays a central role in the Fontaine-Mazur conjecture. We will not include the explicit construction of this representation.

(11)

3. THE REPRESENTATIONS COMING FROM MODULAR FORMS 9

3.1. Definition of modular forms. We give here a summary of the basic definitions, as found for example in [Ti00] or [Se77]; for a more detailed approach see [Bu98]. If C is as usual the complex plane, we call H ⊂ C the Poincar´e open half-plane defined by {z ∈ C|=z > 0}. For any fixed integer k, the group GL2R+ of real 2 × 2 matrices with positive determinant acts on H by the rule:

σ = a b c d : z 7→ σ.z =az + b cz + d

We observe that scalar matrices act trivially. We use the action of GL2(R)+ to define for any k ∈ Z an action of the same group on the space of holomorphic functions on H. If f is such a function and k ∈ Z is fixed, we put:

f 7→ f |kσ

f |kσ(z) = det(σ)k/2(cz + d)−kf (σ.z)

In order to define modular forms we will look at the action, with the above law, of the subgroup SL2(Z) ⊂ GL2(R)+ of 2 × 2 integer matrices with determinant 1. More precisely we will consider the action of certain subgroups of finite index of SL2(Z), its congruence subgroups:

Definition 3.1. We define the subgroup Γ(N ) ⊂ SL2(Z) as the kernel of the natural projection SL2(Z) → SL2(Z/N Z), so we have:

Γ(N ) = 1 b c 1 b, c ≡ 0( mod N )

We call a subgroup Γ of SL2(Z) a congruence subgroup if there exists N such that Γ(N ) ⊂ Γ.

Remark 3.1. We have Γ1 = SL2(Z). Every Γ(N ) is a normal subgroup of finite index of Γ(1), so any congruence subgroup of SL2(Z) is of finite index too.

Before working on functions, we have a better look at the properties of the action of a congruence subgroup on H. In proving the following proposition it can be useful to know that the group SL2(Z) is generated by the elements

S = 1 0 0 1 ; T = 1 1 0 1 .

Proposition 3.1. The action of SL2(Z) on H:

(1) is not faithful, but the induced action of PSL2(Z) = SL2(Z)/{±1} is; (2) is discontinous.

We get trivially from the proposition that

Proposition 3.2. If Γ is a congruence subgroup of SL2(Z), its action on H: (1) is faithful if and only if −1 /∈ Γ, and if −1 ∈ Γ the induced action of

Γ/{±1} is faithful; (2) is discontinous.

(12)

Since the action of a congruence subgroup Γ is discontinous, we can consider a fundamental domain for its action (for instance, a fundamental domain for SL2(Z) is {z ∈ H}). In fact we can do much more than this: we can give the quotient space Γ\H a structure of open Riemann surface (complex variety of dimension 1) and add to it some points to get a compact Riemann surface. We resume without proof the central results in this direction:

Definition 3.2. If Γ is a congruence subgroup, we call cusps the equivalence classes in P1(Q) = Q ∪ ∞ with respect to the action of Γ; we denote by H∗ the union H ∪ Q ∪ ∞.

The set C of cusps of Γ is finite, since SL2(Z) acts transitively on P1(Q) and Γ is a subgroup of finite order.

Theorem 3.1. The complex structure on H induces the structure of an open Riemann surface on the quotient Γ\H. There exists a complex structure on Γ\H∗= Γ\H∪C such that this becomes a compact Riemann surface (it is the compactification of Γ\H).

We usually denote Y (Γ) the open Riemann surface (’modular curve’) associated to Γ and X(Γ) its compactification; we use the notations Y (N ) and X(N ) when Γ = Γ(N ).

Let’s now focus on the action of a congruence subgroup Γ on functions H → C. Fix Γ and an integer k. Let’s take f a holomorphic function on H such that f |kσ = f for any σ ∈ Γ. By definition of congruence, we can find N such that Γ(N ) ⊂ Γ, so 1 N 0 1 belongs to Γ. Since f |k 1 N 0 1 (z) = f (z + N ), f must be periodic of period N , so we can write it as sum of a Fourier series:

f (z) =X n∈Z

ane2πinz/N

We will put q := e2πiz/N _{to simplify the notation. We observe that, if f is} Γ-invariant and σ ∈ SL2(Z), f0 = f |kσ is σ−1Γσ-invariant. Since σ−1Γσ contains either 1 N 0 1 or −1 N 0 −1

, we have f0(z) = ±f0(z + N ) for all z ∈ H, so f |kσ is always periodic of period 2N and has a Fourier expansion

f0(z) = f |kσ(z) = X

n∈Z

aσne2πinz/N

Fixed Γ, we define a modular form as a Γ-invariant function f such that the Fourier expansions of all f |kσ for σ ∈ SL2(Z) behave sufficiently well:

Definition 3.3. If Γ is a congruence subgroup, k ∈ Z, and f is a holomorphic function H → C invariant with respect to the degree k action of Γ, we say that f is:

• holomorphic at infinity if, for every σ ∈ SL2(Z), aσn= 0 for all n < 0 • vanishing at infinity if, for every σ ∈ SL2(Z), aσn= 0 for all n ≤ 0. In the first case we say that f is a modular form with respect to Γ and k, in the second case we say that it is a cusp form. We call Mk(Γ) the space of modular forms and Sk(Γ) ⊂ Mk(Γ) the subspace of cusp forms.

(13)

Remark 3.2. The reason for the expression ’at infinity’ is that we are asking for the Fourier series to converge (or to be zero in the vanishing case) when q goes to zero, which means that z goes to −∞. Clearly, it is sufficient to check the properties required in the theorem for a set of representatives of the classes in SL2(Z)/Γ. Such classes are in bijection with the set C of cusps of Γ: we associate to each cusp the class of elements (or one of its representatives) sending it to the point ∞ (which is always a cusp). Evaluating f on a representative for each class means exactly looking at the behavior of f at the corresponding cusp (we know by definition that it is holomorphic elsewhere).

We have the following result:

Theorem 3.2. For every congruence subgroup Γ and every integer k, Mk(Γ) is a finite dimensional C-vector space. If k ≤ 0, we have Mk(Γ) = 0.

Thanks to the fact that the action of Γ is discontinous, we have a way to define a product on the space Sk(Γ) of its cusp forms, the Petersson inner product. We define it first for Γ = SL2(Z). It is easy to see that the function

f (z)g(z)(=z)k

is invariant under the transformation z 7→ σ.z for any σ ∈ SL2(Z). This means that the following integral on the quotient SL2(Z)\H is well defined (it can be calculated on any fundamental domain):

hf, gi = Z

Γ\H

f (z)g(z)(=z)kdxdy y2

The convergence of this integral is assured by the fact that f and g are cusp forms; we can see this by writing them as sums of Fourier series. It is then immediate to see that previous expression gives a positive definite Hermitian product.

We define now the product for two cusp forms with respect to a congruence subgroup Γ: in order to do this, we just take N such that Γ(N ) ⊂ Γ and we add a normalising factor: hf, gi = 1 [SL2(Z) : Γ(N )] Z Γ(N )\H f (z)g(z)(=z)kdxdy y2

The normalising factor makes the product independent of N , and moreover it allows us to define the product on the space lim

−→ΓSk(Γ), which means we can take the product of cusp forms relative to two different subgroups (we just need to choose N such that Γ(N ) is contained in both). We check immediately that the product on Sk(Γ) is again Hermitian and positive definite.

We will be interested in the case of forms with respect to a congruence subgroup Γ belonging to a particular family of congruence subgroups indexed by the positive integers. For N ∈ Z+_{,we put}

Γ1(N ) = 1 b c 1 ∈ SL2(Z) c = 0( mod N )

and we look at this as a normal subgroup of Γ0(N ) = a b c d ∈ SL2(Z) c = 0( mod N )

For any fixed N we consider the space of modular forms with respect to Γ1(N ); we call N the level of such a form. It proves useful to look at the action of Γ0(N )

(14)

on these functions. The quotient Γ0(N )/Γ1(N ) is isomorphic to (Z/N Z)∗; we see that an ismorphism is given by projecting onto the right-down coordinate. Hence, given a character : (Z/N Z)∗→ C∗_{, we can define the subspace of M}

k(Γ1(N )) on which Γ0(N )/Γ1(N ) acts through :

Definition 3.4. With the above identification Γ0(N )/Γ1(N ) ∼= (Z/N Z)∗ and a character of this group, we denote with Mk(Γ0(N ), ) the subspace of Mk(Γ1(N )) of functions on which Γ0(N )/Γ1(N ) acts as

d.f = (d)f

Equivalently, Mk(Γ0(N ), ) is the subspace of modular forms for Γ1(N ) on which Γ0(N ) acts as:

f |k a b 0 d = (d)f

When varies over all characters of (Z/N Z)∗, the previous subspaces give a decomposition of Mk(Γ0(N )):

Proposition 3.3. If (Z/N Z)\∗ _{is the dual group of (Z/N Z)}∗_{, we have a direct} sum decomposition:

Mk(Γ1(N )) = M

∈ \(Z/N Z)∗

Mk(Γ0(N ), )

We denote Sk(Γ0(N ), ) the subspace of cusp forms in Mk(Γ0(N ), ); we have a decomposition for the space Sk(Γ1(N )) analogous to the one above:

Sk(Γ1(N )) = M

∈ \(Z/N Z)∗

Sk(Γ0(N ), )

Moreover, the spaces Sk(Γ0(N ), 1) and Sk(Γ0(N ), 2) are orthogonal with re-spect to the Petersson product if 16= 2.

Remark 3.3. The character and the degree k have the same parity, in the sense that

(−1) = (−1)k

The degree k and the character of a modular form f ∈ Mk(Γ0(N ), ) will appear in a significant way in the Galois representation associated to f .

3.2. Hecke operators and newforms. Let’s fix N , k, . We define, for every prime p, a linear operator acting on the space Mk(Γ0(N ), ):

Definition 3.5. Let f be a form in Mk(Γ0(N ), ), with Fourier expansion f = ∞ X n=0 anqn If p - N , we define an operator Tp on Mk(Γ0(N ), ) by f |Tp= ∞ X n=0 anpqn+ (p) ∞ X n=0 anqnp If p|N , we define an operator Up on the same space by

f |Up= ∞ X

n=0 anpqn

(15)

The operators defined above are called Hecke operators.

It is easy to check that the Hecke operators are well defined, in the sense that the image of a form of type (k, ) is a form of the same type. Furthermore, the subspace Sk(Γ0(N ), ) of cusp forms is invariant with respect to all the Hecke operators. We will focus from now on on their action on this space and we will look for Hecke eigenforms, where

Definition 3.6. A Hecke eigenform is a cusp form which is an eigenfunction for every Hecke operator. We denote λf,p ∈ C its eigenvalue with respect to the Hecke operator associated to the prime p.

It is easy to show from our definition of the Hecke operators that:

Proposition 3.4. If f is a Hecke eigenform, P∞n=0 is its Fourier expansion, we have:

• a16= 0

• λf,p= a−11 ap for every prime p

Furthermore, if a1 = 1, the coefficients an are multiplicative in the sense that if m and n are coprime amn= aman.

We call the eigenform f normalised if a1 = 1. We have an important result regarding the eigenvalues of a Hecke eigenform (two different proofs can be found in [DS74] and in [Se77]):

Theorem 3.3. Let’s fix N , k, . If Ep is the set of eigenvalues of Tp (or Up) acting on the space Sk(Γ0(N ), k, ) and E =S_pEp, then E is contained in the ring of integers of a number field (finite extension of Q).

We give now a further decomposition of the spaces Sk(Γ0(N ), ). We want to distinguish which forms in this space come from forms belonging to Sk(Γ0(N0), ) for some N0 dividing N , and which are ’new’ (and we will call these newforms and we will see that the space they span has a basis of eigenforms). This makes sense because, if N0|N , we have an inclusion Γi(N ) ⊂ Γi(N0) for i = 1, 2, so from a form f ∈ Sk(Γ1(N0), ) and an integer d|N/N0 we can obtain a form in Sk(Γ1(N ), ) defined by:

z 7→ f (dz)

The forms in obtained this way from forms in Sk(Γ0(N0), ) for some N0|N span a subspace of Sk(Γ0(N )) which we denote S_k+(Γ0(N ), ). We have a canonical way to define a complement to this subspace:

Definition 3.7. We denote Sk+(Γ(N ), ) the orthogonal to S −

k(Γ(N ), ) in Sk(Γ(N ), ) with respect to the Petersson inner product. We call the elements of S_k+(Γ0(N ), ) newforms.

We state this important result, known as the ’multiplicity one theorem’: Theorem 3.4. The space S+k(Γ0, ) has a basis consisting of Hecke eigenforms. We can take the elements of the base to be normalized. If f ∈ S_k+(Γ0(N ), ) and f0 ∈ S_k+(Γ0(N ), ) are two normalised newforms which are also Hecke eigenforms having the same eigenvalue with respect to Tpfor almost all p, we must have N = N0 and f = f0.

(16)

3.3. The Galois representation attached to a cuspidal newform. We state now the main result concerning the construction of Galois representations associated to certain modular forms.

Theorem 3.5. Let f ∈ Sk(Γ0(N ), ) be a Hecke eigenform. Let E be the number field generated over Q by its eigenvalues. Let v be a finite place of E of residual characteristic p. There exists a 2-dimensional Ev-vector space Vf,v and a representation

ρf,v: Gal( ¯Q/Q) → GL(Vf,p) such that

(1) ρ is unramified outside the set S of primes in Q dividing N p (which means that the image of an inertia subgroup over a prime q /∈ S is trivial); (2) if Frobq is a geometric Frobenius at a prime q in Q not dividing N p, the

characteristic polynomial of its action on Vf,v is:

det(I − ρ(Frobq)X, Vf,v) = 1 − λf,qX + (q)qk−1X2

Remark 3.4. The two properties above are independent of the choice of an inertia subgroup or a Frobenius in q, since all inertia subgroups in q are conjugates and the same is true for all Frobenius.

The construction of the Galois representation associated ot a cusp eigenform is split in three cases depending on the degree k: k = 1, k = 2, k > 2. The construction for k = 2 was the the first to be achieved, by Eichler and Shimura [Sh70]; it was generalized to k ≥ 2 by Deligne in [De71], while the case k = 1 was treated by Deligne and Serre and can be found in [DS74].

The representation of G_Q attached to a modular form induces local represen-tations through the inclusions G_Q_l ⊂ G_Q: we will be interested in these, specially in the case l = p.

4. Fontaine’s rings of periods

In this section we will define some classes of representations, associated to certain Qp-algebras called Fontaine’s rings of periods, on which we will focus our attention in what follows. We will give the constuctions without proving every proposition; a summary of the subject can be found in [Fo93], while the complete proofs are in [FO13].

We fix from now on a p-adic field K; we will denote by K0the maximal subex-tension of K/Qp unramified over Qp and C the completion of an algebraic closure

¯

K with respect to a valuation v extending the normalized one on K. The natural action of GK on ¯K extends by continuity to C.

From now on, let B be a topological Qp-algebra on which GK acts continously; the action of GK on B naturally extends to an action on F racB. We give the following:

Definition 4.1. A B-representation V of GKis trivial if there exists a B-basis of V consisting of GK-invariant elements. A Qp-representation V is B-admissible if B ⊗_QpV is trivial as a B-representation.

We want to rewrite the admissibility condition in order to better suit our needs; to this purpose we give another definition:

(17)

4. FONTAINE’S RINGS OF PERIODS 15

(1) B is a domain (2) BGK _{= (FracB)}GK

(3) If b ∈ B is such that g.b ∈ Qpb ∀g ∈ G, then b is invertible in B. Remark 4.1. These three conditions are always satisfied if B is a field. In order to study the admissibility condition, let’s consider the space of GK invariant elements of the representation B ⊗_Q_pV : we set

DB(V ) := (V ⊗QpB)

GK

DB(V ) is clearly a K-vector space. We have a natural map αV : B ⊗BGK (B ⊗_QpV )

GK _{→ B ⊗}

QpV

which is B-linear and GK-equivariant. The utility of the regularity condition lies in the following two theorems:

Theorem 4.1. If B is (Qp, GK)-regular, for any Qp-representation of GK the map αV is injective and we have

V is B-admissible ⇐⇒ αV is an isomorphism

Proof. We can suppose B is a field (hence BGK _{too), since, if C = FracB, by}

regularity we have BGK _{= C}GK _{and injections}

B ⊗BGDB(V ) ,→ B ⊗BGDC(V ) ,→ C ⊗BGDC(V ) B ⊗_Q_pV ,→ C ⊗_Q_pV

so the injectivity of C ⊗BGDC(V ) → C ⊗QpV implies that of B ⊗BG DB(V ) →

B ⊗_Q_pV .

We see by the definition of the map αV that it is injective if and only if any ele-ments b1, ...bnof DB(V ) linearly indipendent over BG are still linearly independent over B. Let’s suppose this is false and let’s take a counterexample with n minimal, so that there exist λ1, ..., λn∈ B such that

n X

i=1

λibi= 0

We show that we can take these coefficients to be in BG_{. By rescalng we can} suppose λn= −1, so that bn =P

n−1

i=0. Since bn∈ DB(V ) is G invariant, we get n−1 X i=0 λibi= xn= g(xn) = n−1 X i=0 g(λi)bi from which n−1 X i=0 (g(λi) − λi)bi= 0

By minimality of n, we must have g(λi) = λi for every i and every g ∈ GK, so that λi∈ BGK for every i.

The admissibility condition is equivalent to the existence of a basis of B ⊗_Q_pV consisting of invariant elements (which means they belong to DB(V )); this is clearly equivalent to the surjectivity of αV. Since αV is always injective, this is equivalent

(18)

We see immediately that αV is an isomorphism if and only if we have the dimensional equality

dimBGD_B(V ) = dim_Q_pV

The injectivity of αV tells us that (≤) is always true for B regular. In the cases we will consider the field BG _{will be K or K}

0.

Since a morphism of Qp-representations f : V → W induces a map of B-representations B ⊗ f : B ⊗_Q_pV → B ⊗ W by B ⊗ f (b ⊗ v) = (b ⊗ f (w)), which restricts to a map of B-modules DB(f ) : DB(v) → DB(W ), it is easy to check that V 7→ DB(V ) defines a functor from Rep_Q_p(GK) to the category of K-vector spaces. By tensor functor we mean a functor compatible with tensor products and duals and sending Qp to K. We have then:

Theorem 4.2. If B is (Qp, GK)-regular, we denote by RepB_Q_p(GK) the full subcategory of Rep_Q_p(GK) whose objects are the B-admissible GK-representations. The restriction of DB to RepB_Q_p(GK) defines then an exact and faithful tensor functor.

In the following paragraphs we will choose as B various algebras, called Fontaine’s rings of periods in honor of the first one to introduce them; the study of the cor-responding conditions of B-admissibility is an essential step in the study of p-adic representations.

4.1. Hodge-Tate representations. Let’s take C as before, with the natural action of GK; for each i ∈ Z, we can consider C(i), the i-th Tate twist of C. We use these to build an interesting C-algebra:

Definition 4.3. We define BHT =L_i∈ZC(i)

If t is a generator of Zp(1), so that for g ∈ GKg.t = χ(g)t with χ the cyclotomc character, we can also write Zp(i) = Zpti and

BHT = C(t, 1/t)

The completion of BHT with respect to its natural valuation isB¯HT = C((t)). We verify that

Proposition 4.1. BHT is (Qp, GK)-regular.

We cite without proof this result of Sen, which we will need to prove the previous regularity (recall that IK is the inertia subgroup of GK):

Theorem 4.3. If η : GK → C∗ is a character we denote by C(η) the corre-sponding one dimensional C-representation of GK; we have C(η)GK ∼= K if η(IK) is finite and C(η)GK _{= 0 otherwise.}

Proof. (of the proposition) BHT is clearly a domain. Since K ⊂ BGHTK ⊂ F racBGK

HT ⊂ ¯B GK

HT, it suffices to show that ¯B GK

HT = K. Let’s suppose b = P

i∈Zbiti is GK invariant. Since g(ti) = χi(g)ti ∈ Qpti, in order to have

g(b) =X i∈Z g(bi)χi(g)ti= X i∈Z biti= b

(19)

we must have g(bi)χi(g)ti = citi for every i ∈ Z, which means each biti must be invariant. Thanks to Sen’s theorem, CGK _{= K and C(i)}GK _{= 0 for i 6= 0, so our}

result follows.

It remains to proof that g.b ∈ Qpb for all g implies b ∈ BHT∗ . Let’s define η : GK→ C∗ such that g.b = η(g)b. By writing b asP_i∈Zbitiand using the definition of η, we get g(bi)χi(g) = η(g)bi. If bi 6= 0, the one dimensional representation Cbi= C(ηχ−i) has a GK-stable Qp-subspace, Qpbi. By Sen’s theorem, this means that ηχ−i(IK) must be finite (for all i such that bi 6= 0). If i0 is a value of i staisfying this condition, no twist of C(ηχ−i0_{) by χ}j _{can satisfy it, so we can have}

at most one value of i such that bi 6= 0. This means b is of the form bi0t

i0_{, thus}

invertible in BHT.

This result allows us to give this:

Definition 4.4. A p-adic representation of GK is called a Hodge-Tate repre-sentation if it is BHT-admissible.

We put DHT(V ) = DBHT = (B ⊗QpV )

GK ₌L(C(i) ⊗

QpV )

GK_{. We have a}

natural grading on DHT, given by

griDHT = (C(i) ⊗QpV )

GK

We then define:

Definition 4.5. If V is a Hodge-Tate p-adic representation of GK and i is an integer, the i-th Hodge-Tate number hi(V ) of V is dim_Qpgr

i_D

HT(V ). We call Hodge-Tate weights of V the integers i ∈ Z such that hi(V ) 6= 0; the value of hi(V ) is the multiplicity of the weight i.

If V has dimension n we clearly haveP

i∈Zhi= n.

Example 4.1. For the one-dimensional representation Qp(1) given by the cy-clotomic character we have h−1= 1 and h(j) = 0 for all j 6= −1. More generally, the representation Qp(i) has Hodge-Tate weight −i (with multiplicity 1).

4.2. De Rham, crystalline and semi-stable representations. We will now construct a tower of (Qp, GK)-regular C-algebras

Bcris⊂ Bst⊂ BdR

giving the corresponding conditions of admissibility on representations, which we will call respectively semi-stable, crystalline and de Rham. The algebra BHT can’t be inserted in the tower, but we will see that BHT =L griBdRfor the grading on BdR.

We will construct first BdR, beginning with this general construction:

Definition 4.6. Given a ring A of characteristic p, we define the ring R(A) as the projective limit of N copies of A, (Ai)i∈N with Ai = A, with the maps φi: Ai+1→ Ai given by the Frobenius automorphism: φi(a) = ap.

The ring R can be viewed as a sort of ’perfectisation’ of A: it is always perfect and it is equal to A if A is perfect. We want to work with a perfect ring in order for its ring of Witt vectors to be Noetherian.

We will now concern ourselves with the case in which A is the residue field ¯

(20)

(we want to work with C rather than ¯K because of its completeness). It follows from the definition that we can identify R with the set

{(a(r)) ∈ OC/pOC such that (a(r+1))p= a(r)}

but we also get an dentification by simply taking the x(i) _{to be in O}

C, rather than OC/pOC, and we denote them in this case by xi. We obtain a valuation v on R by choosing v(x) = v(x0). The following proposition summarizes the basic properties of the ring R:

Proposition 4.2. R is perfect of characteristic p and complete with the given valuation; its residue field is ¯k. We have a section ¯k → R gven by:

a 7→ [ap−n]_n∈N

where, for x ∈ ¯k, [x] is the Teichmuller representative of a in W (¯k) = OKnr.

The field FracR is a complete nonarchimedean perfect field of characteristic p; fur-thermore it is algebraically closed.

We recall that, given any ring A, we have a homomorphism ρ : W (A) → AN (a0, a1, ..., an, ...) 7→ (w0, w1, ..., wn, ...) with wi(a) = a pi 0 + pa pi−1 1 + ... + p i_a

i. We choose A = OCand we use the wi to define a map W (R) → OC. We proceed in two steps:

(i) We observe that, thanks to the identificaton of R with {(ar) ∈ OC/pOC such that ap_r+1= r}, we get natural maps

W (R) → Wn(OC/pOC)

(a0, a1, ..., an, ...) 7→ ((a0)n, (a1)n, ..., (an−1)n)

Since this maps are compatible with the morphisms fn : Wn(OC/pOC) → Wn−1(OC/pOC) given by fn(x0, ..., xn−1) = (x

p 0, ..., x

p

n−2), we get a map from W (R) to proj lim Wn(OC/pOC) = Wn(OC).

(ii) For n fixed, wn is by definition a map Wn+1(OC) → OC; by composing with the projection OC→ OC/pnOC we get a map wn0 : Wn+1(OC) → OC/pnOC. Since the kernel of the projection map Wn+1(OC) → Wn+1(OC/pOC) is contained in the kernel of w_n0, we get a map θn : Wn+1(OC/pOC) → OC/pnOC. Thanks to the commutativity of the following diagram, we obtain an induced map on the inverse limits: Wn+1(OC/pOC) OC/pnOC Wn(OC/pOC) OC/pn−1OC fn θn+1 θn

where fn is the map defined above. By the first step, we can identify W (R) to proj lim Wn+1(OC/pOC), thus getting a map

(21)

More explicitly, we have θ(x0, x1, ..., xn, ...) =P ∞ n=0p

n_x(n)

n ; we observe that θ is a continous homomorphism of W (OC/pOC)-algebras. We have moreover:

Proposition 4.3. The map θ is surjective. Its kernel is principal, generated by the element ξ = ω + p, where ω is any element of R satisfying ω(0) _{= −p; we} also haveT∞

n=0(ξ) n _{= 0.}

We observe that θ can be extended to a morphism of FracW (OC/pOC)(= W (OC/pOC)

h 1 p i

)-algebras, again surjective and continous with kernel (ξ). We are now ready to define the algebra BdR:

Definition 4.7. We define the ring BdR+ as B_dR+ = proj lim W (R) 1

p

/(ker θ)n= proj lim W (R) 1 p

/(ξn) and BdR as its field of fractions

BdR = FracBdR+ = B + dR 1 ξ

It is possible to deduce from our construction that BdRonly depends on C = ˆK,¯ not on K itself. We call a Qp-representation of GK a De Rham representation if it is BdR-admissible.

The definition of BdR as inverse limit of quotients by a principal ideal gives us both a valuation on BdR, defined by v(ξ) = 1, and a filtration, defined as

FiliBdR = B+dRξ i

We recall that a filtration on a K-vector space V is a sequence (FiliK)_i∈Zsuch that Fili+1K ⊂ FiliK, FiliK is equal to V for i << 0 and to 0 for i >> 0. The filtered vector spaces form a category with the morphisms V → W being the K-linear ones such that the image of FiliV lies in FiliW . It is moreover a Tannakian category with the filtrations on the tensor product V ⊗ W and dual V∗given by:

Fili(V ⊗ W ) = M i1+i2=i

Fili1_{v ⊗ Fil}i2_W

FiliV∗= {f : V → K such that f (x) = 0 ∀x ∈ Fil−i+1V } We denote DdR(V ) := DBdR(V ) = (BdR ⊗QpV ) GK_{. The filtration on B} dR carries on to a filtration on DdR(V ): FiliDdR(V ) = (FiliBdR⊗QpV ) GK

Since BdR is a field, it is (Qp, GK)-regular and, as we have seen earlier, the functor DdR is a faithful exact tensor functor from the category of De Rham Qp -representations of GK to the one of K-vector space; moreover, thanks to the pre-vious remark, we can consider DdR as a functor to the category FilK of filtered K-vector spaces.

We want to relate the rings BdRand BHT and, as a result, De Rham representa-tions to Hodge-Tate ones. We write, for a filtered algebra B, gri_{B = Fil}i_B/Fili+1_B and grB =L

i∈Zgr

(22)

Proposition 4.4. We have grBdR= M i∈Z B_dR+ (i)/ξB_dR+ (i) =M i∈Z C(i) = BHT

It follows from this that a De Rham Qp-representation of GK is also Hodge-Tate. The converse is false.

Proof. We prove the result by finding a generator of the maximal ideal in B_dR+ on which GK acts via the cyclotomic character. Let’s choose ∈ R such that (0) _{= 1 and}(1) _{6= 1 (the idea behind this choice is that generates U}1

R = {x ∈ R|v(x − 1) ≥ 0} as a pro-cyclic group). The element [] − 1 = (, 0, ...) − 1 ∈ W (R) belongs to ker(θ), since its image in OC is (0)− 1 = 0. We are then allowed to define the ’logarithm’ of [] as the sum in B_dR+ of the series

∞ X n=1 (−1)n([] − 1) n n

and we call t := log[]. It can be shown by an explicit calculation that [] − 1 belongs to Fil1B_dR+ but not to Fil2B_dR+ ; it follows immediately that t satisfies the same conditions and is thus a generator of Fil1BdR, so that FiliBdR= BdRti. If we look at the action of GK on t, we get

g.t = g. log([]) = log(g.[]) = log([]χ(g)) = χ(g) log[] = χ(g)t so that B_dR+ ti _{= B}+

dR(i). We can finally write:

grBdR = M i∈Z B+_dRti/B_dR+ ti+1 =M i∈Z B_dR+ (i)/tB_dR+ (i) =M i∈Z C(i) = BHT since B_dR+ /tB_dR+ is the residue field C of B_dR+ with respect to the usual valuation.

Let’s now take a p-adic representation V of GK. Thanks to the first part of the proof, we can write an exact sequence

0 → Fili(BdR) → Fili+1BdR→ C(i)

By tensoring with V and taking the GK invariants, we get an immersion griDdR(V ) ,→ (C(i) ⊗QpV )

GK

By summing over i ∈ Z we finally get an immersionL

i∈ZDdR(V ) ,→ DHT(V ), from which the inequality dimKDdR(V ) ≤ dimKDHT(V ). Since V is De Rham, we have dimKDdR(V ) = dim_Qp(V ): together with the previous relation, this gives

dimKDHT = dimQpV , so that V is Hodge-Tate.

Example 4.2. As an example of a Hodge-Tate representation which is not De Rham, we can take V such that the sequence of G_Q_p-representations

0 → Qp→ V → Qp(1) → 0

is exact and non-split; V exists because Ext1_(Qp(1), Qp) = H1(Qp, Qp(−1)) = Qp. It can be shown that such a V is Hodge-Tate but not De-Rham.

We work now towards the definition of the rings Bcris and Bst: Definition 4.8. We denote:

(23)

(1) A0

cris the W (R)-submodule of W (R) h

1 p i

obtained by adding to W (R) all the elements of the form ωm(a) := a

m

m! for a ∈ ker θ and m ∈ Z + _{(it is} called the divided power envelope of W (R) with respect to ker θ).

(2) Acris the projective limit proj lim A0cris/p n_A0

cris (3) B_cris+ the module Acris

h 1 p i

We observe that in the construction of A0

cris it suffices to add the elements of the form ωn(ξ), since ker θ = (ξ). The modules above are all rings, thanks to the relation ωm(a)ωn(a) = m+n_n ωm+n(a).

Remark 4.2. We can extend the homomorphism θ : W (R) → OC to A0crisand Acris, we get a map ¯θ : W (R)

θ

→ OC → OC/pOC; it can be shown that the ideal ker ¯θ = (ker θ, p) is a divided power ideal of Acris, meaning that ωn(a) ∈ Acris for every a ∈ ker ¯θ. We will need this result later.

We make another very important remark:

Remark 4.3. Since the p-adic topology is separated on W (R) h 1 p i , hence on A0

cris, the map A0cris → Acris is an injection. Moreover, we obtain by continuity inclusions Acris,→ Bcris+ ,→ B

+

dR. We can describe more explicitly these rings as: Acris= ( X m∈Z+ amωm(ξ) with am∈ W (R), an→ 0 for n → ∞ ) B_cris+ = ( X m∈Z+ amωm(ξ) with am∈ W (R) 1 p , an→ 0 for n → ∞ )

Since A0_cris and B_cris+ are stable for the action of GK on BdR+ , we can consider them as GK-modules. We finally define:

Definition 4.9. Bcris is the ring Bcris1_t, where t = log[] ∈ B_dR+ as before. Remark 4.4. It can be shown that t belongs to Acris and tp−1 to pAcris, so that Bcris is also equal to Acris

h 1 p i

.

We recall that we have a Frobenius map φ on W (R)h1_pi; we can check that φ(A0

cris) = A0cris, so that the restriction of φ gives us a Frobenius map on A0cris; by continuity, this map extends to one on Acris and Bcris+ . We observe that

φ(t) = φ(log[]) = log([]p) = p log([]) = pt so we can put φ 1_t = _pt1 to get a Frobenius map on Bcris.

In order to define Bstwe will need to construct a logarithm map over (F racR)∗, valued in BdR. We note UR1 = {x ∈ R|v(x − 1) ≥ 1} ⊂ U

1

R= {x ∈ R|v(x − 1) ≥ 0}. We trace the steps of this construction:

(i) We begin by defining the logaritm on U1

R: we take log x to be the sum of the usual series

∞ X n=0 (−1)n+1([x] − 1) n n ∈ Acris We are allowed to do this since, being x in U1

R, ¯θ([x] − 1) = 0, where ¯θ is the map defined in a previous remark, so that by the same remark ωn([x] − 1) belongs

(24)

to Acris, so the series converges because its general term is (n − 1)!ωn([x] − 1) and (n − 1)! → 0 for n → ∞.

(ii) We will extend the logarithm map in the next steps in such a way to satisfy the usual property log(xy) = log x + log y. We begin by defining it for x ∈ U_R+: for any such x there exists m big enough such that xpm ∈ U1

R, so we put log x = 1

pmlog x pm

(iii) For an arbitrary x ∈ R∗, we can write x = ax0 with a ∈ k∗ and x0∈ U_R+, so we can choose log x = log x0.

(iv) Let’s now take x ∈ (FracR)∗; if v(x) = r_s, the element y = _ωxsr (where ω

is the usual element of R∗ with ω(0)_{= −p) belongs to R}∗_{. We want now to define} the logarithm of ω in order to take

log x = log y + r log ω s

If θ is the usual map from W (R)h1_pito C, we have θ_−pω , so we are allowed to sum the logarithmic series for _−pω and we put

log(ω) = log( ω −p) = ∞ X n=0 (−1)n−1 h ω −p− 1 i n ∈ B + dR This completes our definition of the map log : (FracR)∗ _{→ B}+

dR. We observe that the kernel of this map is just ¯k∗; moreover, if U is the image of the restricted map log : U_R+→ B+

cris, the image of the whole of (FracR)∗ through the logarithmic map is just U ⊕ Qplog ω. We then define:

Definition 4.10. Bst:= Bcris[log ω]

Remark 4.5. Because of the choice we made for log ω, the immersion Bst→ BdR is non-canonical.

Bst is clearly stable by the action of GK. It can also be shown that log ω is transcendental over the fraction field of Bcris, so that the map Bcris[x] → Bst sending x to log ω is an isomorphism.

We can extend the Frobenius map to Bst by putting φ(log ω) = p log ω We define another important operator on Bst:

Definition 4.11. We call monodromy operator the map N : Bst→ Bst X n∈N bn(log ω)n7→ − X n∈N nbn(log ω)n−1

The operator N is the only Bcris-derivation of Bst such that N (ω) = −1. The letter N with which we denote it is due to the fact that, as we will see, N will induce a nilpotent operator on modules associated to finite dimensional semi-stable representations.

Remark 4.6. Thanks to the transcendence of log ω on Bcris, Bcris coincides with the kernel of the monodromy operator on Bst.

(25)

By an explicit calculation we can see that φ and N satisfy the following key properties:

Proposition 4.5. (i) N commutes with the action of GK (ii) If φ is the Frobenius map on Bst, N φ = pφN

We also observe that the filtration on BdR induces one on Bcris and Bst. We denote Ccrisand Cst the respective fraction fields of Bcrisand Bst. Having constructed Bcris and Bst, we are now ready to define the respective admissible representations; as usual, we first need to show that:

Proposition 4.6. Bcris and Bstare (Qp, GK)-regular; more precisely, we have (recall that K0 is the maximal subfield of K unramified over Qp):

(i) Bcris and Bst are domains (ii) BG0 cris= B G0 st = C GK cris= C GK st = K0

(iii) if x ∈ Bcris (resp. Bst) and g.x ∈ Qpx for every g ∈ G, then b ∈ Bcris∗ (resp. Bst∗)

Proof. Bcris and Bst are domains since subrings of the field BdR. We have the inclusions K0 ⊂ (Bcris+ )

GK _{⊂ B}GK cris ⊂ B GK st ⊂ C GK st ⊂ B GK dR = K, where the first inclusion follows from

(B_cris+ )GK _{⊃ (W (R)} 1 p )GK _{= W (R)}GK 1 p = W (OK/pOK) 1 p = K0 Point (ii) now follows from the following lemma:

Lemma 4.1. The map K ⊗K0Bst→ BdR is injective.

Proof. (of the lemma) Since [K : K0] is finite and log ω is transcendental over Ccris, it is transcendental also over K ⊗K0Ccris, so that K ⊗K0Bst= K ⊗K0

(Bcris[log ω]) = (K ⊗K0Bcris)[log ω]. Since the map K ⊗K0Bcris→ BdRis injective,

(K ⊗K0Bcris)[log ω] → BdR is, too.

We prove now (iii). Since ¯k is the residue field of R, we have an inclusion W (¯k) ⊂ W (R), from which W (¯k)1 p ⊂ W (R) 1 p. We denote W (¯k) 1 p ⊂ C by P and its algebraic closure in C by ¯P : P and ¯P are fields and we have inclusions P ⊂ Bcris and ¯P ,→ B_dR+ . Moreover, ¯P ∩ Bcris = P . Indeed, if it were ¯P ∩ Bcris = Q with Q a nontrivial extension of P , we would have P ⊂ L ⊂ Q for a finite (nontrivial) extension L of P , so that (FracQ)GL _{= L. However the unramified subfield of L}

must be P , so by point (2) we get BGL

st = P , absurd.

We take now an element b ∈ BdR satisfying the condition in (iii) and we show that it belongs to ¯P . Up to multiplication by ti _{for a suitable i, we can suppose} that b belongs to B_dR+ − Fil1BdR. Let’s consider ¯b := θ(b) ∈ C: Qp¯b is GK-stable, hence isomorphic to Qp(η) for some character η. Sen’s theorem tells us that η(IK) must be finite and that ¯b must belong to ¯P ⊂ B_dR+ . The difference b − ¯b must then lie in FiliBdR− Fili+1BdR for some i ≥ 1; clearly, Qp(b − ¯b) is still GK-stable and isomorphic to Qp(η) (for the same η as before). However, t−i(b − ¯b) belongs to B_dR+ − Fil1BdR and GK acts on Qpt−ib through the character χ−iη (where χ is the cyclotomic character): since i ≥ 1, this character is not finite on IK, hence again by Sen’s theorem the only chance for Qpt−i(b − ¯b) (and equivalently for Qp(b − ¯b)) to be GK-stable is that b − ¯b = 0, which means b ∈ ¯P . If now b ∈ Bst, by what we just proved we must have b ∈ ¯P ∩ Bst= P . Since P is a field contained in Bcris, b must be invertible in Bcris (hence also in Bst).

(26)

We then give the following:

Definition 4.12. If V is a Qp representation V of GK, we call it crystalline if it is Bcris-admissible and semi-stable if it is Bst-admissible.

Since the fixed field of Bcrisand Bst is K0, we will have natural maps: αcris: Bcris⊗K0Dcris(V ) ,→ Bcris⊗Qp(V )

αst: Bst⊗K0Dst(V ) ,→ Bst⊗Qp(V )

Thanks to the (Qp, GK)-regularity of the respective rings, we have the equiva-lences:

V is crystalline ⇐⇒ αcris is an isomorphism ⇐⇒ dimK0Dcris(V ) = dimQpV

V is semi-stable ⇐⇒ αst is an isomorphism ⇐⇒ dimK0Dst(V ) = dimQpV

We now have four classes of representations: Hodge-Tate, De Rham, semi-stable and crystalline. We have already shown that De Rham implies Hodge Tate; we can now complete the chain of containments by showing that:

Proposition 4.7. A crystalline representation is semi-stable and a semi-stable one is De Rham.

Proof. The first implication follows immediately from the inclusion Dcris(V ) ⊂ Dst(V )

Indeed this inclusion together with the injectivity of αst gives the inequali-ties dimK0Dcris(V ) ≤ dimK0Dst(V ) ≤ dimQpV , so if V is crystalline, so that

dimK0Dcris(V ) = dimQp(V ), these must be equalities.

Even though Bst⊂ BdR, we don’t have an analogous inclusion between Dst(V ) and DdR(V ), since the fixed fields of Bst and BdR, K0 and K, are in general different. However, since [K : K0] is finite, we have an inclusion K ⊗K0Bst,→ BdR,

which gives us:

K ⊗K0Dst(V ) = K ⊗K0(Bst⊗QpV ) GK _{= (K ⊗} K0(Dst⊗QpV ) G K = = ((K ⊗K0Bst) ⊗QpV ) GK_{,→ D} dR(V ) from which

dimKK ⊗K0Dst(V ) = dimK0Dst(V ) ≤ dimKDdR(V ) ≤ dimQpV

If V is semi-stable we have dimK0Dst(V ) = dimQpV , so we have equalities in

the last line and V is De Rham.

Remark 4.7. From the previous proof we deduce that, if V is semi-stable, we have an isomorphism between K ⊗K0Bst and BdR.

We observe that all the structure we have on Bst (filtration, Frobenius map and monodromy operator) carries over to Dst(V ) for V a semi-stable represen-tation: this structure will play a central role in our further study of semi-stable representations. More precisely, if φ and N are as before, defined on Bst, we define them on Bst⊗QpV by putting:

φ(b ⊗ v) = φ(b) ⊗ v N (b ⊗ v) = N b ⊗ v

(27)

Since φ and N commute with the action of GK, Dst(V ) = (Bst⊗QpV )

GK _is

stable under such maps and we can define Frobenius and monodromy on Dst(V ) by restriction. In particular it is still true that φ is bijective and N φ = pφN .

All we said about φ and N is still true if V is an arbitrary representation, not necessarily semi-stable.

About the filtration, we can’t define it on Dst, but rather on DK(V ) = K ⊗K0

Dst(V ): since for V semi-stable this space is isomorphic to DdR(V ), we just take the filtration FiliDdR(V ) (in the case of an arbitrary representation, we have an immersion but not necessarily an isomorphism, so we must intersect the filtration with DK(V )).

We make the following very important remark:

Remark 4.8. Since Bcris is the kernel of the monodromy operator acting on Bst, a p-adic representation V of GK is crystalline if and only if it is semi-stable and the induced monodromy operator on Dst(V ) is constantly zero.

We can now give the definition we need in order to state the Fontaine-Mazur conjecture:

Definition 4.13. Let now V be a p-adic representation of GQp and F an

algebraic extension of Qp. We say that V is F -semi-stable (resp. crystalline) if it is semi-stable as a representation of GF; we say that V is potentially semi-stable (resp. crystalline) if it is F -semi-stable for a finite extension F of Qp.

If V is a representation of the absolute Galois group of a global field rather than a p-adic one, we can evaluate all the properties of representations by considering the representations induced by localisation:

Definition 4.14. If K is an algebraic extension of Q, we say that a p-adic representation of GK has the property P at a finite place v if the representation of GKv obtained by composition with the immersion GKv ,→ GK has the property P .

We can finally state the conjecture: Conjecture 1. Let

ρ : G_Q→ GL2( ¯Qp)

a 2-dimensional p-adic representation of G_Qwhich is irreducible, unramified outside a finite set of finite places of Q and not given by the Tate twist of an even repre-sentation of G_Q which factors through a finite quotient of G_Q. Then ρ is attached to a cuspidal newform f if and only if it is potentially semi-stable at p.

Actually, we have:

Proposition 4.8. If G is a profinite group, the image of a representation ρ : G → GLn( ¯Qp) is contained in a subgroup of the form GLn(E) for some finite extension E of Qp.

Proof. The union of the closed subgroups {GLn(E)}, where E varies over all p-adic fields, is the whole group GLn( ¯Qp). Hence the union of the closed subgroups {GLn(E) ∩ ρ(G)} is the whole ρ(G). Since G is profinite it is locally compact, hence its image is locally compact too and thus a Baire space. Since we wrote it as a union of closed subgroups, it must be contained in one of them, as desired.

(28)

Thanks to the previous result, the presence of ¯Qp in the conjecture just means that we are considering representations with values in GL2(E) for any p-adic field E. Indeed, as we saw earlier in this chapter, a representation coming from a modular form is of this kind.

It is known that representations coming from a modular form are odd. Since the conjecture predicts that even representations which are not twists of finite ones must be modular, in order for the conjecture to be true such representations must not exist. In particular, since a finite representation has Hodge-Tate weights (0, 0), all its twists have equal Hodge-Tate weights: the conjecture then predicts that there exists no even representation with distinct Hodge-Tate weights. For a proof of this (nontrivial) result under a few additional hypotheses, see [Cal11].

We make a final remark about ’potential’ properties. We defined these only in the cases we really need (potentially semi-stable or crystalline representations of Qp), but we can give an immediate generalisation:

Definition 4.15. If K is a p-adic field, K0 an algebraic extension of K and V a p-adic representation of GK, we say that V is F -semi-stable (resp. crystalline, de Rham, Hodge-Tate) if it is semi-stable (resp. crystalline, de Rham, Hodge-Tate) as a representation of GK0; we say that V is potentially semi-stable (resp. crystalline,

de Rham, Hodge-Tate) if it is K0_{-semi-stable (resp. crystalline, de Rham,} Hodge-Tate) for a finite extension K0 of K.

It is not difficult to prove that:

Proposition 4.9. A p-adic representation of GK is potentially Hodge-Tate (resp. de Rham) if and only if it is Hodge-Tate (resp. de Rham).

It follows immediately from the above proposition and the known implication semi-stable ⇒ de Rham that a potentially semi-stable representation of GK is de Rham. The converse is also true:

Theorem 4.4. A de Rham representation of GK is potentially semi-stable. This was known as the p-adic monodromy conjecture and was proved for the first time by Berger in 2002 [Ber02].

(29)

CHAPTER 2

Modules associated to potentially semi-stable

representations

We have seen in the previous chapter that, given a p-adic field K and a semi-stable p-adic representation V of GK, we can associate to ρ the K0-vector space defined by Dst(V ) (where K0 is the maximal unramified subextension of K/Qp). More precisely we saw that this correspondence gives an exact and faithful tensor functor from the category of semi-stable Qp-representations of GK to the one of finite-dimensional K0-vector spaces. In order to move towards a classification of semi-stable p-adic Galois representations, lying at the heart of the Fontaine-Mazur conjecture, our strategy (due essentially to Fontaine) will be to make use of the functor Dst to switch our study from representations to a suitable category whose objects are K0-vector spaces with additional structure (coming from the represen-tations), which we can understand and handle more easily than representations.

In this perspective, our first step will be to find a suitable category, whose ob-jects being K0-vector spaces, such that the functor Dst induces an equivalence be-tween Repst_Q_pGK and this new category. We will later apply our results to the study of potentially semi-stable representations (of G_Q_p) and focus on the 2-dimensional case, appearing in the conjecture, managing give an explicit and nearly complete description of the essential image of Dst in this setting, as done in [FM].

1. The category of (φ, N )-modules

Since the objects of VectK0 clearly do not carry enough structure to give

a significant correspondence with representations, we want to endow them with some structure which arises naturally when looking at modules of the form Dst(V ). Briefly, we want to define a category towards which the functor Dstbe fully faithful; to get an equivalence with Repst_Q

pGK we will then have to find the essential image

of Dst in this category. We have already done part of the work at the end of the first chapter, by observing that the structure on Bst (filtration, Frobenius map, monodromy operator) carries over to Dst(V ), so that Dst gives a functor from Repst_Q_pGK to the following category:

Definition 1.1. A filtered (φ, N )-module over K is a finite-dimensional K0 -vector space D together with:

• a bijective map φ : D → D which is semi-linear with respect to the action of the Frobenius automorphism σ of K0/Qp (Frobenius map)

• a K0-linear map N : D → D (monodromy operator), satisfying the rela-tion N φ = pφN

A look at the Fontaine-Mazur conjecture and some related results

Contents

Stating of the Fontaine-Mazur Conjecture

Modules associated to potentially semi-stable

representations