l-adic Representations Attached to Elliptic Curves

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Preliminaries

In this chapter we present briefly some basic results that we recall during the discussion of the work. We will introduce the notion of algebraic group in a more generic form such as an affine algebraic group scheme. We are in-terested in a particular algebraic groups: Tori. We will treat their property in §0.1.1. In §0.1.2we will focus our attention on semi-simple representa-tions of a commutative algebraic group and we will introduce the characters attached to a such representation. We will conclude our extremely concise introduction to algebraic group, with a construction of a particular push-out. In the next section we will give the essential result on local and global class field theory. They will be fundamental in chapter2 and chapter3. In particular we are interested in the ray class group attached to a some modulus m. The exact sequence associated to a ray class group together with the push-out above are the starting point of chapter3. In §0.3we will define what is an `-adic representation and its properties. Finally in the last section we will present elliptic curves and their connection with local fields. We will conclude the chapter defining the `-adic representations attached to an elliptic curve and we will lay the foundations for chapter 4.

0.1 Algebraic Group

Through this section K is a field with char (K) = 0.

Let F be a functor from K-algebras category to sets category or group category. Suppose exist a K-algebra A such that F is natural isomorphic to the functor HomK(A, −). Then we call F representable functor and we say that A represents F . The relationship between a representable functor and its representing algebra is explained by the following

Theorem 0.1.1 (Yoneda’s lemma [18], p. 6, Theorem). . Let F and G be representable functors, and let A e B be the respective representing algebras. Natural maps from F to G correspond to K-algebras homomorphisms from B to A.

Definition 0.1.1. An algebraic group (or affine algebraic group scheme) G over K is a representable group-functor from K-algebras cat-egory, with representing finitely generated K-algebra A = K[X1, . . . , Xn]/I.

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Algebraic Group

Equivalently we can define an algebraic group as a representable set-functor from K-algebras represented by a finitely generated K-algebra, together with three natural maps :

m : G × G −→ G u : {e} −→ G inv : G −→ G

where {e} is the terminal object of K-algebras category. This maps are such that the following diagrams commute

G × G × G G × G G × G G id × m m × id m m (associtivity) {e} × G G × G G G u × id ' m id (left unit) G G × G {e} G id m u (left inverse).

By Yoneda’s lemma we know that the natural maps m,u,inv are in corres-pondence with K-algebra homomorphisms of A, in particular such homo-morphisms put on A an Hopf algebra structure.

Theorem 0.1.2 ([18], p. 9, Theorem). There is a correspondence between algebraic groups over K and Hopf algebras over K finitely generated as K-algebras.

We give some examples :

i) The addictive group Ga: taken R a K-algebra, Ga(R) is the set R equipped with its own addictive structure. The representing algebra is simply K[X].

ii) The multiplicative group GL1, we use the symbol Gm: it is the algeb-raic group that to each K-algebra R associate Gm(R) the multiplicat-ive group of R. The K-algebra that represents it is

A = K[X, Y ]

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Algebraic Group

iii) GLn: GLn(R) = {invertible matrix with coefficinet in R}. It is rep-resented by the algebra

K[X11, . . . , Xnn, Y ] (det(Xij)Y − 1)

.

iv) Roots of unity µn: is the algebraic group that associate to R it n-th roots of unity: µn(R) = {r ∈ R | rn = 1}. the representing algebra is K[X]/(Xn− 1).

Consider a field extension L/K, and take G an algebraic group over K represented by the algebra A. Then we can construct an algebraic group GL over L by restriction of G to the algebras category. Indeed each L-algebra is, in a natural way, a K-L-algebra and therefore make sense G(R). So we have obtained a group functor from L-algebras category. For each L-algebra R0 we have the isomorphism HomL(A ⊗KL, R0) ' HomK(A, R0), then A0 = A ⊗KL represents GL. Moreover if A = K[X1, . . . , Xn]/I is the representing algebra of G then A0 is finitely generated L-algebra, indeed

A ⊗KL =

K[X1, . . . , Xn]

I ⊗KL ' L[X1, . . . , Xn]/I. Thus GL is an algebraic group over L.

If L/K is a finite extension the converse is true. Given an algebraic group G over L we define the functor (Res)L/K(G) that to each K-algebra R associate the group G(R⊗KL), then (Res)L/K(G) is an algebraic group ([11], p. 60, §5), and we call it Weil restriction of G obtained by restriction of scalar.

Definition 0.1.2. An homomorphism between algebraic group is a natural map F −→ G such that each map F (R) −→ G(R) is a group homomorph-ism.

Given an algebraic group G consider homomorphisms f : G −→ Gm. We call such f characters of G. For each f ∈ X(G), Yoneda’s lemma says that correspond an Hopf algebra map

ψf : K[X, 1/X] −→ A

Then ψf(X) must be an invertible element in A. In particular to ψf(X) correspond certain elements in the Hopf algebra called group-like. More precisely the characters of an algebraic group G represented by A correspond to group-like elements in A ([18], p. 14, Theorem).

Let A = K[X1, . . . , Xn]/I be the representing algebra of G. The ideal I define an algebraic set in AnK, hence induce a Zariski topology over the latter. From definition of G we have G(K) ' HomK(A, K), then we can equip G(K) with the topology introduced above. In particular we have that Ga, Gm, GLncorrespond to Ga(K), Gm(K), GLn(K) ([18], p. 32, §4.5, Corollary). One of the most important result is the following

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Algebraic Group

Algebraic Groups of Multiplicative type

Theorem 0.1.3. ([18], p. 25, §3.4, Theorem) Every algebraic group over K is isomorphic to a Zariski-closed subgroup of GLn.

0.1.1 Algebraic Groups of Multiplicative type

Given a finite abelian group H, let K[H] be the group algebra. Then K[H] is a finitely generate K-algebra, and we make this a Hopf algebra by making the elements of H group-like. The algebraic group G represented by K[H] is called diagonalizable. A characterization of diagonalizable algebraic groups is they are a finite product of copies of Gm and µn([18], p. 15, Theorem or [11], p. 219, Theorem 4.2).

We say that an algebraic group G is of multiplicative type if G_K is diagonalizable. This implies that for every K-algebra R we have

G_K(R) ' G_mK(R) × · · · × G_mK(R) × µn(R) × · · · × µn(R).

Algebraic groups of multiplicative type such that in the previous expression involve only copies of Gmare called Tori. If in the product we have d copies of G_mK we say that T is a torus of dimension d.

Remark 0.1.1. Let L/K be a finite field extension. Taken Gm over L con-sider T = (Res_L/K)(Gm), the algebraic group over K obtained by restriction of scalars. For every K-algebra R we have for definition T (R) = (R ⊗KL)∗ the invertible elements of (R ⊗KL). In particular T (K) = L∗. If [L : K] = d then T is a torus of dimension d. Indeed let {σ1, . . . , σd} be the embeddings of L into K. For every K-algebra R we have the isomorphism:

σ : T_K(R) =(R ⊗KL)∗ −→ GmK(R) × · · · × GmK(R) (r ⊗ x) 7−→ (σ1(x)r, . . . , σd(x)r). Hence T_K = G_mK× · · · × G_mK as wanted.

Let G be of multiplicative type with A = K[X1, . . . , Xn]/I the repres-enting algebra. Let X be the set of group-like elements of A ⊗KK. As we have seen before we can identify X with X(G) = Hom_K(G_K, G_mK). Taken the Galois groupG = Gal(K/K), we have a natural action of G on X since

X ⊂ A ⊗ K ' K[X1, . . . , Xn]/I

and G sends group-like elements in group-like elements. Thus X(G) come equipped with a G -module structure. We call X(G) character group of G. In particular this action is continuous and an important result is Theorem 0.1.4 ([18], p. 55, §7.3, Theorem). Taking character groups yields an anti-equivalence between algebraic group of multiplicative type and abelian groups on which G acts continuously.

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Algebraic Group

Linear Representations of a Commutative Algebraic Groups

0.1.2 Linear Representations of a Commutative Algebraic Groups

Let V be K-vector space of dimension n. A linear representations of a commutative algebraic group G over K on V , is an homomorphism of algebraic groups

ρ : G −→ GLV

where for all K-algebras GLV(R) = AutR(V ⊗KR) ' GLn(R). Since G is commutative a semi-simple representation is a representation that diagon-alize over K. Thus the image of G_K under a semi-simple representation is isomorphic to an algebraic subgroup of G_mK× · · · × G_mK× µn× · · · × µn. Note that if G is of multiplicative type every representation is semi-simple. To a semi-simple representation ρ we associate its trace:

θρ= X

nχ(ρ)χ

where nχ(ρ) ∈ Z is the multiplicity of χ ∈ X(G) that occur in the repres-entation over K. Let RepK(G) be the set of isomorphism classes of semi-simple linear representations of G. Fixed an extension L of K we have a map η : RepK(G) → RepL(GL). In particular this map in injective. Indeed suppose there exist two distinct representations (V, ρ) and (V0, ρ0) such that

GL GLV,L

GLV0_,L '

Called A, AV, AV0the representing algebra of G, GL_V, GL_V0, Yoneda’s lemma says that we have the commutative diagram:

A ⊗KL AV ⊗KL

AV0⊗_KL '

But the homomorphisms in this diagram are induced from those between A, AV, AV0. Therefore A AV AV0 ' −Y oneda−−−−→ G GLV GLV0 ρ ρ0 '

Hence we obtain the absurd (V, ρ) ' (V0, ρ0). The elements of RepL(GL) that belong to the image of η are called definible over K.

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Algebraic Group

Linear Representations of a Commutative Algebraic Groups

Proposition 0.1.1. The map ρ → θρ defines a bijection between RepK(G) and the set of elements θ =P n_χ_{χ ∈ Z[X(G)] such that}

i ) θ is invariant by G = Gal(K/K). That is: for all σ ∈ G , χ ∈ X(G) we have nχ= nσ(χ).

ii ) nχ ≥ 0 for all χ ∈ X(G).

Proof. At the end of the last paragraph, we have seen the identification between X(G) and X, the set of group-like elements in A ⊗ K. Moreover theG -module structure on X(G) is given through the natural action of G on X. Let Gχ ⊂G be the subgroup that fix χ, and consider first the case where θ has the form

θ = X

σ∈G \Gχ

σ(χ). (1)

That is the sum extended to all distinct conjugate of χ. For Galois corres-pondence we have Kχ ⊂ K the fixed field of Gχ. Then Kχ is the smallest sub-extension of K in K such that the corresponding group-like element of χ belong to A ⊗K Kχ, where A is the representing algebra of G. This in-duce an homomorphism Kχ[X, 1/X] −→ A ⊗ Kχ, and by Yoneda’s lemma, GKχ −→ GmKχ. Thus we can think χ as 1-dimensional linear representation of HKχ. Taking restriction of scalars of GKχ we have, for all K-algebra R, the homomorphism

ψ : GKχ(R ⊗ Kχ) −→ GmKχ(R ⊗ Kχ) = (R ⊗ Kχ) ∗

.

As R-module R ⊗ Kχ has d = [Kχ : K] generators, and multiplication for invertible elements can be expressed as a matrix of GLd(R). The inclusion G(R) ,→ GKχ(R ⊗ Kχ) allow us to have a linear representation of dimension d = [Kχ : K]

ρ : G −→ GLd.

Let {σ1, . . . , σd} the embeddings of Kχ in K. Extension of scalars applied to GmKχ give us, for every K-algebra R, the isomorphism

σ : GmKχ(R ⊗ Kχ) =(R ⊗ Kχ)

∗_{−→ G}

mK(R) × · · · × GmK(R) (r ⊗ x) 7−→ (σ1(x)r, . . . , σd(x)r).

That is (GmKχ)K ' GmK× · · · × GmK as algebraic groups. Therefore the representation obtained from ρ by scalar extension to K is isomorphic to the diagonal representation

σ ◦ ψ : G_K−→ G_mK× · · · × G_mK g 7−→ (σ1(χ)g, . . . , σd(χ)g)

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Algebraic Group Push-out of Algebraic Groups

Clearly we have θρ = θ. The surjectivity now follow from the fact that every element in Z[X(G)], such that condition i) and ii) are satisfied, can be written as sum of (1). The injecitivity follow from the well-known result that representations with same characters are isomorphic.

Corollario 0.1.1. In order that ρ ∈ RepL(GL) can be defined over K it is necessary and sufficient that θρ∈ A ⊗KL belongs to A.

0.1.3 Push-out of Algebraic Groups

Let Y1, Y2, Y3 be abelian groups with Y3 finite. Suppose we have a short exact-sequence:

0 Y1 Y2 Y3 0 . (2)

Let G be a commutative algebraic group over K, and let ε : Y1−→ G(K) be a group homomorphism. Then we can construct the push-out of G over Y1 and Y2, i.e. there exists an unique (up to isomorphisms) algebraic group F over K, together with an homomorphism of algebraic groups ϕ : G −→ F and a group homomorphism ψ : Y2−→ F (K) such that:

i) the diagram Y1 Y2 G(K) F (K) ε ψ ϕ (∗) is commutative.

ii) If F0 is another algebraic group such that (∗) commute, then exist an homomorphism of algebraic group ϑ : F −→ F0 such that the following diagram Y1, Y2 G(K) F (K) F0 ε ψ ψ0 ϕ ϕ0 ϑ (3) commute.

Note that the last condition implies that F is unique up to isomorph-isms. We have to show the existence. For each y ∈ Y3 let αy ∈ Y2 be a representative of y. Fixed y, y0 ∈ Y3 we have αy+ αy0 = α_y+y0+ c(y, y0) with

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Ideles and Adeles Classes

c(y, y0) the image of an element of Y1. More precisely c is a 2 − cocycle de-fining the extension (2). Let F be the disjoint union of copies of G indexed on Y3. The group law on F is define by

πy,y0 :G_y× G0_y −→ G_y+y0

(gy, gy0) 7→ g_y+ g_y0 + ε(c(y, y0))

The maps ϕ : G −→ F and ψ : Y2 −→ F (K) are defined in the following way

ϕ : G −→ G0⊂ F ψ : Y2−→ F (K) g 7−→ g − ε(c(0, 0)) αy+ z 7−→ ε(z) A direct calculation gives the universal property of F .

Remark 0.1.2. Let L be an extension of K then we have the exact sequence 0 −→ G(L) −→ F (L) −→ Y3 −→ 0

and the commutative diagram

0 Y1 Y2 Y3 0

0 G(L) F (L) Y3 0

Remark 0.1.3. If we apply the construction to Gm we obtain an algebraic group F of multiplicative type. We know from (0.1.4) that such groups are completely identified by their character groups. In particular the universal property of F give a description of X(F ): it is the the set of the pairs (φ, χ), with φ : Y2 −→ K

∗

an homomorphism and χ ∈ X(G) is such that φ(y1) = χ(y1) for all y1 ∈ Y1.

0.2 Ideles and Adeles Classes

Through this section K is a number field. We define the set P

K as the set of finite places (or not-archimedean) of K. The set of infinite places (or archimedean) is indicated withP∞

K. The symbol P

K denotes the union of P

K and P∞

K. For v ∈ P

K let Kv the completion of K with respect to v. We know by Ostrowski theorem that

i) v ∈ P

K is equivalent to | · |p for an unique prime ideal p in OK, the ring of integers of K;

ii) v ∈P∞

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Ideles and Adeles Classes

A non-archimedean place v is called complex or real place if the corres-ponding σ is a real or complex embedding. Note that from ii) if v ∈P∞ K then Kv is equal to R or C. The group of units in Kv is denoted by Uv. For every p ∈ OK the quotient OK/p is a finite field. Taken v ∈ P_K, let O_v := {x ∈ Kv|v(x) ≤ 1} be the ring of integers of Kv, that is the closure of O_K in Kv. Let pv be the closure of p in Ov, the unique maximal ideal. We have OK/p ' Ov/pv, then from the finiteness of the first field we have Kv is a locally compact field, with respect to the topology induced from v ([16], p. 27, Proposition 1). Thus Ov is an open compact subgroup of the locally compact multiplicative group K+, and the same holds for Uv and K∗. Then we define the adele group of K,denoted AK, the restricted direct product of K+ with respect to Ov. When we consider the restricted product of K∗ with respect to Uv we call it the idele group of K

I =n(av) ∈ Y

K_v∗

av∈ Uv for almost all v o

with the product taken over all valuation v. From definition we note that I ⊂ A_K, in particular the idele group

I come equipped with a topology. First of all consider S a finite set of places containing P∞

K. We define the subgroup of idele group I_S = Y

v∈S

K_v∗ ×Y v /∈S

Uv

with the product topology. Note that the first factor is a finite product of locally compact spaces, hence compact. The second factor is product of compact spaces and so by Tychonoff is compact. Thus IS is locally compact. Then we define the topology on I decreeing that for each S the subgroup I_S is open. A basis for this topology consists of sets of the form Q

vAv where Av is open in Kv∗ and Av = Uv for all but finitely many v. From the considerations above, the idele group with this topology is a locally compact topological group. Since a non-zero element of K is a unit for all but finitely v ∈P

K then we have the inclusion

ı : K∗ I (4)

x (xv)

where xv = x for all v. The quotient C = I/K∗ is called the idele class group of K and we have the exact sequence of group:

1 K∗ ı I π C 1 .

Let I be the group of fractional ideal of K. We know that every element of I factors uniquely as finite product of powers of prime ideals. Then we

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Ideles and Adeles Classes Ray Class group modulo m

can think I as the free abelian group generated by the prime ideals of OK. We have the surjective homomorphism

ϕ : I I (av) 7→ Y v∈P K pordp(av) v

where ordp(av) is the integer rv such that av = πvrvu with u ∈ Uv and πv the uniformizer of Kv. The kernel of ϕ is clearly IP∞

K and the image of K ∗ under ϕ ◦ ı is the group of principal ideals P of I.

Hence we have

I K∗IP∞

K ' C with C the ideal class group.

0.2.1 Ray Class group modulo m

A modulus is a formal product m =Q

vvmv where v are places of K, and mv are all but finitely equal to 0, and satisfy

a) mv = 0 if v is a complex place; b) mv ∈ {0, 1} if v is a real place; c) mv ≥ 0 if v is a finite place of K. The finite set S ⊂P

K that contains all finite place v such that mv > 0, is the support of m. To every m we associate the group Um=QvUm,v, where:

i) Um,v= {u ∈ Uv| v(1 − u) ≥ mv} if v ∈ S; ii) Um,v= R∗>0 if mv = 1 and v is a real place; iii) Um,v= Uv if mv = 0 and v is a finite place,

iv) Um,v= Kv∗ if mv = 0 and v ∈P∞_K.

By definition Um is an open subgroup of I. Let Im be the quotient I/Um and Cm the quotient C/π(Um) that, for third theorem of isomorphism, is isomorphic to I/K∗Um. Then we have the short exact sequence of abelian group

1 K∗/Em Im Cm 1 (5)

where Em= {x ∈ O∗K| ı(x) ∈ Um}. We call Cm the ray class group mod-ulo m.

Let S be a finite set of prime ideals and consider IS the free abelian group generated by the prime ideals not in S. Taken a modulus m we denote with S(m) the finite set of prime ideals corresponding to the support

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`-adic Representations of Number Fields

of m. We define PS(m) ⊂ IS(m) _{the subgroup of principal ideals (a) such} that σ(a) ∈ R>0 for all real embedding and a ≡ 1 mod. m. We call ideal class group modulo m the quotient Cm = IS(m)/PS(m) and we have the short exact sequence of abelian group

1 PS(m) IS(m) Cm 1.

We have the isomorphism Cm ' Cm ([12], p. 365, Proposition 1.9). In par-ticular as Cm is finite ([10], p. 146, Theorem 1.7) the isomorphism above implies that the ray class group modulo m is a finite abelian group.

0.3 `-adic Representations of Number Fields

In this section we adopt the same notation as in §0.2. Let L/K be a finite Galois extension and G = Gal(L/K). Let v ∈ ΣK and p ∈ OK the asso-ciated prime ideal to v. Consider w ∈ ΣL with P ∈ OL the corresponding prime ideal, we say that w extends v to L iff P | p. The Galois group acts commuting the prime ideals of OL that lives above p. Consider Dw the subgroup of G that fix w

Dw:= {σ ∈G | σ(w) = w}

called the decomposition group of w. Let lw, kv be the residue field of Lw and Kv respectively, as we have seen in §0.2 they are finite. So the extension lw, kv is of Galois. Every σ ∈ Dw induce σ ∈ Gal(lw/kv), we have an homomorphism from Dw to Gal(lw/kv) whose kernel Iw is the inertia group of w. This homomorphism is surjective ([9], p. 101, Corollary 1), hence Dw/Iw ' Gal(lw/kv). In particular Dw/Iw is a finite cyclic group generated by the Frobenius element Fw, where Fw(λ) = (λ)||p|| for all λ ∈ lw with ||p|| = card(kv). We say that w (resp. v) is unramified if Iw = {1}. Since only a finite number of prime ideal of OK ramified in OL, then all but a finite many v ∈ ΣK are unramified.

If L is an arbitrary algebraic extension of Q we define ΣLas the projective limit of the set ΣLn where Ln ranges over all finite sub-extension of L/Q. Then if L/K is an arbitrary Galois extension of K, for each w ∈ ΣL, we can define Dw, Iw, Fw as before. Let v ∈ ΣK be an unramified place of K and let w be a place of L that extend v. Then for every σ ∈ G \ Dw the valuation σ(w) of L extends v and we cane easily seen that Fσ(w)= σFwσ−1. We denote with Fv the conjugacy class of Fw inG = Gal(L/K).

Let P be a subset of ΣK. For each integer n we define an(P ) = card({v ∈ P | ||pv|| ≤ n}). Let ζ be a real number, then P has density ζ if

lim n→∞

an(P ) an(ΣK)

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`-adic Representations of Number Fields

Theorem 0.3.1 (Tchebotarev’s density theorem [12], p. 545, Theorem 13.4; or [8], p. 169, Theorem 10). Let L be a finite Galois extension of K, with Galois group G . let X ⊂ G be stable by conjugation. Let PX be the set of places of v ∈ ΣK, unramified in L, with the Frobenius class Fv contained in X. Then PX has density equal to card(X)/card(G ). In particular for every σ ∈ G , there exists infinitely many unramified places w ∈ ΣL such that Fw= σ.

We have an important corollary

Corollario 0.3.1 ([4], p. 300 ,Corollary 4.10). Let L/K be an arbitrary Galois extension, which is unramified outsied a finite set S. then The Frobenius elements of the unramified places of L are dense in Gal(L/K).

Let ` be a prime number and consider V a finite dimensional Q` vector space. If dim_Q_`V = n then Aut(V ) ' GLn(Q`). In particular GLn(Q`) is an `-adic Lie group ([14], ch. 3). Let K be an algebraic closure of K and G = Gal(K/K) then we give the follow definition:

Definition 0.3.1. An `-adic representation ofG (or by abuse of notation of K) is a continuous homomorphism ρ :G −→ Aut(V ). We say that ρ is unramified at v ∈ ΣK if ρ(Iw) = 1 for any w ∈ Σ_K extending v.

By Galois correspondence consider L the fixed field of H = ker(ρ) in particular L/K is a Galois extension with Galois group G /H . Since G /H ,→ Aut(V ) we have ρ(Iw) = {1} ⇔ Iw = {1} that is ρ is unramified at v if and only if v is unramified in L/K. If ρ is unramified at v then its restriction to Dw factors through Dw/Iw for any w extending v, so ρ(Fw) ∈ Aut(V ) is defined. The element ρ(Fw) is called the Frobenius of w in the representation ρ, and it is denoted Fw,ρ. The conjugacy class of Fw,ρ in Aut(V ) depends only on v and it is denoted by Fv,ρ and the polynomial det(1 − Fv,ρT ) is denoted Pv,ρ(T ).

Definition 0.3.2. The `-adic represenation ρ is said to be rational if there exists a finite subset S of ΣK such that

i) ρ is unramified at all v ∈ ΣK\ S,

ii) if v /∈ S, the coefficients of Pv,ρ(T ) belong to Q

Suppose we have ρ0 an `0-adic representation of K, with `0 6= ` a prime number. If the characteristic polynomials of the Frobenius elements for ρ and ρ0 are the same for almost all v ∈ ΣK we say that ρ0 and ρ are compatible. More precisely if there exists a finite subset S ⊂ ΣK such that for all v ∈ ΣK\ S holds ρ0 and ρ are unramified and Pv,ρ0(T ) = P_v,ρ(T ), then we say that this representations are compatible. Let P be the set of prime numbers

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Elliptic Curves

Definition 0.3.3. Let P be the set of prime numbers. For each ` ∈ P let ρ` be a rational `-adic representation of K. We call (ρ`)`∈P a compatible system of representations if ρ`0, ρ_` are compatible for any two prime `, `0. We says that (ρ`)`∈P is a strictly compatible system if there exists a finite subset S ⊂ ΣK such that

i) ρ` is unramified at v and Pv,ρ`(T ) has rational coefficients for all v /∈ S ∪ S`, with S` := {v ∈ ΣK| pvdivides `}

ii) Pv,ρ`(T ) = Pv,ρ`0(T ) for all v /∈ S ∪ S`∪ S`0.

We denote with Pv(T ) the common value of the polynomials in ii). Further-more when a system is strictly compatible, there is a smallest set S having properties i, ii, we call S the exceptional set of the system.

0.4 Elliptic Curves

Let K be a field. An elliptic curves is a smooth curve over K of genus 1 with a point O, called basepoint, with coordinates in K. Every elliptic curve can be written, in the projective plan, as a plane cubic with O the point at infinity ([17], p. 59, Proposition 3.1). Thus we have a cubic polynomial attached to an elliptic curve, and in affine coordinates the polynomial has the form

y2+ a1xy + a3y = x3+ a2x2+ a4x + a6

with a1, a2, a3, a4, a6 ∈ K. We call it the Weierstrass equation of E. A such equation, not need attached to an elliptic curve, come together with two important element the discriminant ∆ ∈ K and the j-invariant j ∈ K. If ∆ 6= 0 we say that the curve defined by the equation is non-singular or smooth. The name j-invariant is justified from the result: two elliptic curves are isomorphic over an algebraic closure K of K if and only if they have the same j-invariant ([17], p. 45, Proposition 1.4). When char(K) 6= 2, 3 the Weierstrass equation has the more simple form

y2 = x3+ Ax + B

with A, B combinations of a1, a2, a3, a4, a6, hence A, B ∈ K. For this equa-tion the discriminant and the j-invariant can be expressed via this formulas

∆ = −16(4A2_{+ 27B}3₎ _{j = −1728}(4A) 3

∆ .

The points of an elliptic curve E defined over a field K have a group structure with respect to a composition law ⊕ : E × E −→ E, where ⊕ is an element of K(E) the function field of E over K and the neutral element with respect this law is O.

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Elliptic Curves Reduction of Elliptic Curves

For each m ∈ Z consider [m] : E −→ E the multiplication by m map [m]P = P ⊕ · · · ⊕ P

| {z }

m terms

We denote with E[m] the kernel of this map, it is the subgroup whose elements are the points of E of order m, we call it the subgroup of m-torsion of E. If m is coprime with the characteristic of K, or char(K) = 0 then E[m] ' Z/mZ × Z/mZ, if ` = char(K) we have for all strictly positive integer n: E[`n] = {O} or E[`n] = Z/`nZ and only one of this two case occur ([17], p. 86, Corollary 6.4).

Let ` be a prime number, the map [`] is a transition map from E[`n+1] to E[`n] thus we can take the projective limit and we obtain the Tate module

T`(E) = lim_←− n

E[`n]. Since for all n each E[`n_{] is a Z/`}n

Z-module then T` is a Z`-module. As for the m-torsion groups we can describe the structure of the Tate module: T` ' Z`× Z` if ` 6= char(K) otherwise if ` = char(K) > 0 we have T` ' {0} or T` ' Z`.

Let E1 and E2 two elliptic curves defined over K. An isogeny from E1 to E2 is a rational morphism ϕ : E1 −→ E2 such that the basepoint of E1 is mapped in the basepoint of E2. The set of all isogeny from E to E is denoted by End(E). Take f, g ∈ End(E) we define (f + g) as (f + g)(P ) = f (P ) ⊕ g(P ), and since ⊕ is a rational map then the sum of isogeny just defined is an isogeny. We can define a product taking the composition of two isogeny. Therefore End(V ) is a ring. Note that for all m ∈ Z the map [m] is an isogeny: it is a rational map and [m]O = O. So Z ,→ End(E) and if hold the isomorphism End(E) ' Z we say that E is without complex multiplication.

0.4.1 Reduction of Elliptic Curves

Let K be a field complete with respect to a discrete valuation v. Let E be an elliptic curve defined over K. We may suppose that the coefficients of the Weierstrass equation belong to OK the valuation ring of K. This implies that v(∆) ≥ 0, and since v is discrete there exists a Weierstrass equation with coefficients in OK such that minimizes v(∆). We call such equation the minimal Weierstrass equation of E.

Let m be the unique maximal ideal of OK, and let k = OK/m be the residue field. The ideal m is a principal ideal, we call π a uniformizer of OK if it is a generator of m. We call the projection ˜· : OK −→ k = OK/πOK the reduction modulo π. Chosen a minimal Weierstrass equation for E we can reduce modulo π its coefficients and obtain one for a curve eE, possibly singular, over k. We have a reduction map from E(K) to eE(k) that maps

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Elliptic Curves

The `-adic Representation attached to Elliptic Curves

P = [x, y, z] to eP = [x,e ey,ez]. If v(∆) = 0 we have e∆ 6= 0, so eE is not singular and we say that E is a good reduction. If v(∆) > 0 then e∆ = 0 so eE is singular, moreover if eE has a node we say that E has a bad reduction of multiplicative type. Finally if v(∆) > 0 and eE has a cusp we say that E has a bad reduction of additive type. However there exists a finite extension K0 of K such that E has only or a good reduction or a bad reduction of multiplicative type ([17], p. 197, Proposition 5.4).

Note that if E has a good reduction then the reduction map is surject-ive ([17], p. 188, Proposition 2.1). Furthermore if m is a integer prime to char(k) then we have the isomorphism E[m](K) ' eE[m](k) ([17], p. 192, Proposition 3.1)

Let K be a number field and consider E an elliptic curve over K. Let v a finite place of K then we can consider E defined over the completion Kv of K with respect to v. Taking a minimal Weierstrass equation for E over Kv we denote with eEv the reduced curve over the residue field kv of Kv. We say that E has a good (resp. bad) reduction at v if E has a good (resp. bad) reduction when we see it as elliptic curve over Kv. Moreover take any Weierstrass equation for E over K with coefficients a1, a2, a3, a4, a6 ∈ K as in the preceding paragraph, and let ∆ be its discriminant. So for all but finitely many finite place v of K we have

v(ai) ≥ 0 for i = 1, . . . , 6 and v(∆) = 0.

For all such v the given equation is already a minimal Weierstrass equation and in particular the reduce curve eEv is not singular. This implies that the set Sbad of the finite places of K such that E has a bad reduction is finite. The next theorem due to Shafarevich allow us to show that we have only a finite number of elliptic curves (up to isomoprhisms) that are K-isogenous to a given E defined over a number field K.

Theorem 0.4.1 ([17], p. 293, theorem 6.1). Let S be a finite set of places of K containing Σ∞_K. Then there are only a finite number (up to isomoprhisms) of elliptic curves defined over K, that have a good reduction at every place outside S.

Since isogenous curves have the same set of bad reduction places ([17], p. 202, Corollary 7.2) we can conclude

Proposition 0.4.1. Let E be an elliptic curve over K. Then there are only finitely many elliptic curves, up to isomorphisms, which are K-isogenous to E.

0.4.2 The `-adic Representation attached to Elliptic Curves

We have seen that the set E[m] of points of E of order m is an abstract group, however E[m] come equipped with another algebraic structure. Let

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Elliptic Curves

K be an algebraic closure of K with Galois groupG . We have a continuous action ofG on E[m]. Indeed, for all σ ∈ G and P ∈ E[m] we have

[m]σ(P ) = σ([m])σ(P ) = σ([m]P ) = σ(O) = O

where the second equality holds because the coefficients of the rational map [m] are in K. Thus E[m] is a G -module. If we suppose that m is prime to char(K), or char(K) = 0, we have the continuous homomorphism ρ : Gal(K/K) −→ Aut(E[m]) = GL2(Z/mZ). Consider E[`n], ` a prime number, the action of G on those groups commute with the map [`]. So we have a continuous action of G on the Tate module T`. If we choose a Z`-basis for T`(E) we have the continuous homomorphism

ρ :G −→ Aut(T`(E)) ' GL2(Z`)

We want representations with value in some Q`-vector space, then we define V`(E) = T`(E) ⊗Z`Q` and G acts in a natural way on V`(E).

Definition 0.4.1. The `-adic representation attached to E is the con-tinuous homomorphism

ρ`:G = Gal(K/K) −→ Aut(V`(E)) ' GL2(Q`).

Let K be a field as in §0.4.1and k its residue field. Let K be an algebraic closure of K with Galois groupG . Let w ∈ Σ_Kan extension of v and Iw ⊂G the corresponding inertia group. Consider E an elliptic curve defined over K then we have just seen thatG acts on the groups of m-torsion, m ∈ Z. If Iw acts trivially on E[m] (resp. T`(E)) we say that E[m] (resp. T`(E)) is unramified at v (that is ρ(Iw) = {1} in Aut(E[m]) or Aut(T`(E)). The next theorem is called the criterion of N´eron-Ogg-Shafarevich, and gives the relation between the good reduction of E and when E[m] and T`(E) are unramified at v:

Theorem 0.4.2 ([17], p. 201, Theorem 7.1). Let E an elliptic curve over the field K. Then the following are equivalent

a) E has a good reduction,

b) E[m] is unramified at v for all integers m ≥ 1 prime to char(k), c) T`(E) is unramified at v for all prime numbers ` prime to char(k), d) E[m] is unramified at v for infinitely many integers m ≥ 1 prime to

char(k).

Consider an elliptic curve E defined over a number field K. At the end of §0.4.1 we have shown that the set Sbad containing the finite places v of K such that E has a bad reduction is finite. Let S` be the set of finite

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Elliptic Curves

places that not divide a fixed prime number `, so by criterion of N´ eron-Ogg-Shafarevich for all finite places v /∈ S_bad∪ S_` we have that ρ` is unramified at v. Moreover if E has a good reduction at v and v - `, then we have the isomorphism E[`](K) ' eE[`](kv), where kv is the residue field of the completion of K with respect to v. So the action of ρ`(Fw) on E[`](K) correspond to the Frobenius endomorphism φ on eE[`](kv). This implies

det(1 − Fv,ρ`) = det(1 − ρ`(Fw))

= 1 − T r(ρ`(Fw))T + det(ρ`(Fw))T2 = 1 − T r(φ)T + det(φ)T2.

But 1 − T r(φ)T + det(φ)T2 is a polynomial with integers coefficient ([17], p. 142, Theorem 2.3.1) and it not depends from `, hence (ρ`) forms a strictly compatible system of representations of G .

The following theorem is called the Irreducibility theorem

Theorem 0.4.3. Let E be an elliptic curve over a number field K without complex multiplication (cf. §0.4). Then

a) the representation ρ` :G −→ Aut(V`) is irreducible for all primes `, b) the representation ϕ` : G −→ Aut(E[`]) is irreducible for almost all

primes `.

We need the following lemma

Lemma 0.4.1. Under the same hypothesis of the irreducibility theorem, if there exists E1, E2 elliptic curves defined over K such that

ϑ : E1−→ E ψ : E2−→ E

are K-isogenies with non-isomorphic cyclic kernel, then E1, E2 are non-isomorphic over K.

Proof. Let n1, n2 be the order of ker(ϑ), ker(ψ) respectively. Assume E1 σ ' E2 over K. Take the dual isogeny ˆϑ : E −→ E1, it has ker( ˆϑ) of order n1 ([17], p. 81, Theorem 6.1). Composing the isogenies we obtain

E −→ Eϑˆ ₁−→ Eσ ₂ −→ E.ψ

In particular ψ ◦ σ ◦ ˆϑ is an endomorphism of E. Since EndK(E) = Z it must be [a] for some a ∈ Z. However ker([a]) = E[a] = Z/aZ × Z/aZ since char(K) = 0. Thus n1, n2 must be divide a and a2 = n1n2, that implies the contradiction a = n1 = n2.

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Elliptic Curves

Proof. a) If EndK(E) = Z it is enough show there is no one-dimension Q` -vector space of V` stable under the action ofG = Gal(K/K). Suppose there were one, its intersection X with the Tate module would be a submodule of T`(E) = Z`× Z`. In particular X and T`(E)/X are both free Z`-module of rank 1. For n ≥ 0 let X(n) be the image of X in E[`n] = T`(E)/`nT`(E). This is a cyclic submodule of E[`n] of order `nstable under the action ofG . In particular it corresponds to a finite subgroup of E, therefore there exists an elliptic curve E(n) such that E −→ E(n) is an isogeny with kernel X(n) ([17], p. 74, Proposition 4.12). The above lemma proves that the curves E(n), n ≥ 0 are pairwise non-isomorphic contradicting proposition0.4.1.

b) If E[`] is not irreducible, there exists a Galois submodule X`of E that is a F`-vector space of dimension 1. So X`is cyclic of order ` and in the same way as above we have an isogeny E −→ E/X` with cyclic kernel of order `. The above lemma proves that the curves wich correspond to different values of ` are non-isomorphic, and this contradict again proposition0.4.1.

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Chapter 1

Subgroup of GL

₂

_(F

_p

)

Through this chapter V denote a Fp-vector space of dimension 2. Then GLV, the automorphism group of V , is isomorphic to GL2(Fp). We want to study its subgroups.

1.1 Cartan subgroups

Let D1, D2 be two different lines such that V = D1⊕ D2. Then we can consider the subgroup C ⊂ GL2(Fp) that fix D1 and D2

C := {s ∈ G2| sD1 = D1 and sD2 = D2} .

Fixed a basis of V formed by a vector of D1and one of D2 we can express all element s ∈ C as a diagonal matrix∗ 0

0 ∗

. In particular C is an abelian group of order (p − 1)2, and if p 6= 2 then define uniquely {D1, D2}. We call C the split Cartan subgroup of GL2(Fp).

Let C1 be the subgroup of C whose elements are those that acts on D1 trivially. Then every element s ∈ C1 is a matrix with entries

1 0 0 ∗

, in particular C1 is a cyclic group of order p − 1 and we call it a split half Cartan subgroup. Consider C = C0 · F∗p the group generated by a split half Cartan subgroup C0 and the homotheties, then C is the unique Cartan subgroup containing C0. The image of C in PGLV = GLV/F∗p it is the same as the image of C0, so it is a cyclic group of order p − 1. We call it the split Cartan subgroup of PGLV. We know by primitive element theorem that Fp2 ' F_p[α] with α solution of a monic irreducible polynomial of degree 2 with coefficients in Fp. A basis for Fp[α] over Fp is {1, α}. Then multiplication by a non-zero element x + αy ∈ Fp[α]∗can be expressed as a matrix x α

2_y

y x

with entries in Fp. The determinant of the matrix associated to an element x + αy ∈ Fp[α]∗ coincide with the norm of x + αy

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Cartan subgroups

relatively at the field extension Fp[α]/Fp, in particular it is non-zero. So this matrices belong to GL2(Fp) and form a subgroup. A subgroup of GL2(Fp) that arise in this way is called non-split Cartan subgroup. Note that it is a cyclic group of order p2 − 1 and when we consider its image in PGLV then it is cyclic of order p + 1 and we call it non-split Cartan subgroups of PGLV.

If p 6= 2 and s ∈ GLV then the characteristic polynomial of s is fs = X2 − Tr(s)X + det(s) with discriminant ∆s = Tr(s)2 − 4det(s). If ∆s 6= 0 then s belong to a unique Cartan subgroup. Indeed, if ∆s is a square in Fp then s is diagonalizable and has two distinct eigenvalues over Fp, in particular it is not a multiplication by a non-zero scalar of Fp. So s belongs to a unique split Cartan subgroup because the intersection of two of them is only F∗p. Otherwise if ∆s is not a square in Fp then s represent the multiplication by a non-zero element of Fp2, thus generates a non-split Cartan subgroup.

Let k = FpC be the algebra of End(V ) generated by a Cartan sub-group C and suppose that p 6= 2 if it is split. Then k is a commutative semi-simple algebra of rank 2 whose multiplicative group is C. Indeed, if C is split, k is the sub-algebra of all diagonal matrix in End(V ) and k ' Fp×Fp, otherwise k ' Fp2 if C is not-split. We have only two automorphisms of k. Indeed we have only two automorphism on Fp× Fp: the identity and that defined by (x, y) 7→ (y, x). While on Fp2 the unique non-trivial automorph-ism is the Frobenius automorphautomorph-ism σ : x 7→ xp.

We denote with N the normalizer of C in GLV. Each element s ∈ N induce an automorphism ϕs of k defined as

x 7→ ϕs(x) = sxs−1

for all x ∈ k. If the corresponding automorphism to s is the identity, then s commute with all elements of k so s belongs to k by the double centralizer theorem ([6], p. 115, Theorem 2.43). But s is a unit of k, hence it belongs to C. We have deduced that the kernel of the map N −→ Aut(k) is C. Furthermore since |Aut(k)| = 2 we have that C has index 2 in N .

Assume that C is split, an easy calculation gives the following description of N N := s ∈ GLV s =∗ 0 0 ∗ or s =0 ∗ ∗ 0 . (1.1)

The elements s ∈ N \ C are such that sD1 = D2 and sD2 = D1. On the other hand if C is non-split each s ∈ N \ C acts on ax, a ∈ k and x ∈ V , as

s(ax) = aps(x). (1.2)

However the description of N for C non-split it is not direct as before. To find the structure of N as subgroup of GL2(Fp) we proceed in the following way. The Frobenius automorphism of Fp2 is an invertible F_p-linear map. If

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Subgroups with Order Divisible by p.

we fix a basis of Fp2 over F_p, then σ is representable as a matrix of GL₂(F_p). We denote with SF the associated matrix of σ. Moreover it is an inner automorphism by the Skolem-Noether theorem ([10], p. 126, Theorem 2.10). Fix any s ∈ N , we have seen it induces an automorphism ϕs on k. In particular suppose that ϕs is not trivial then it is induced by the Frobenius automorphism of Fp2. So ϕ_sacts on C as conjugation by S_F. Then we have sxs−1 = S_F−1xSF for all x ∈ C. Thus the element SF · s commute with all elements of C. However we have seen that C coincides with its centralizer in GL2(Fp) by the double centralizer theorem, and since s /∈ C then it is in the non-trivial coset of C in the subgroup < C, SF >.

But then

N =< C, SF > . (1.3)

Remark 1.1.1. If C is a split Cartan subgroup then every element s ∈ N \C has the form s =0 a

b 0

. Then, identifying F∗p with its image in GL2(Fp), s2₌ab 0

0 ab

∈ F∗

p. Assume now that C is non-split. We have shown that every element s /∈ C of the normalizer has the form s = S_F · c with c ∈ C. Then s2 = (SF · c)2 = cp+1 that is an element of F∗p.

Consider now, N1, C1 the images in PGLV of N and C respectively. The group N1 is the normalizer of C1 in PGLV. Actually we can say more about the structure of N1. Indeed by the remark above, and the structures (1.1),(1.3) of N in each respectively case of C, show that N1 is a dihedral group with C1 its cyclic group. In particular every element in N1\ C1 has order 2.

The next proposition underlying the importance of the normalizer N . Proposition 1.1.1. Let C be a Carta subgroup of GL(V ) and N its nor-malizer. Let C0 be a Cartan subgroup (resp. a split half Cartan subgroup) of GL(V ) contained in N . Assume p ≥ 5 if C0 is split and p ≥ 3 otherwise. Then C0 = C (resp. C0 ⊂ C).

Proof. Let C1, N1 and C10 be the images in PGLV of C, N and C0. under our hypothesis C₁0 is a cyclic group of order p ± 1 > 2. Let s be a generator of C0. Note that s is forced to belong to C1. If it is not then s ∈ N1\ C1 and we have seen that each element there must have order 2 that clearly is not the case of s. Since two Cartan subgroup have trivial intersection we have C₁0 = C1. But then C0F∗p= C and we are done.

1.2 Subgroups with Order Divisible by p.

Let D be a line of V . The subgroup of GLV defined as

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Soubgroups of Order Prime to p

B := {s ∈ GL_V | sD = D}

it is called Borel subgroup. Every element of B is a matrix of the form ∗ ∗

0 ∗

, thus it has order p(p − 1)2 and D is the unique line fixed by B. If a Cartan subgroup is contained in a Borel subgroup then it is split or a split half Cartan subgroup, and in the first case D is one of the two lines attached to it.

Proposition 1.2.1. Let G be a subgroup of GLV of order ph whit h some positive integer. Then either G contains SLV or G is contained in a Borel subgroup of GLV.

Proof. The matrix 1 1 0 1

has order p and generates in GLV a p-Sylow subgroup P :=1 1 0 1 := s ∈ GLV s =1 ∗ 0 1 .

Since all p-Sylow subgroups are conjugate ([3], p. 99, Theorem 2.12.2) then each element x ∈ GLV of order p has matrix representation

1 1 0 1

. In particular x fix only one line Dx. If for all x ∈ G of order p we have Dx= D for a line D of V , then G fixes D, and therefore G is contained in a Borel subgroup. If this not occur then there exists two different lines Dx fixed by G. If we take them as coordinates axis then G contain two element x, y that have matrix representations

x =1 a 0 1 y =1 0 b 1 .

the proposition follow since SL2(Fp) is generated by the matrix of GL(Fp) of the form1 ∗ 0 1 and 1 0 ∗ 1 ([7], p. 537, Lemma 8.1).

1.3 Soubgroups of Order Prime to p

Lemma 1.3.1. Let R be discrete valuation ring with residue field k of posit-ive characteristic p. Consider H < PGLd(k) a finite group with order prime to p. Then there exists a finite subgroup bH < PGLd(R) such that its image under the natural map PGLd(R) −→ PGLd(k) is isomorphic to H.

Proof. First we lift H to a finitely generated subgroup U of GLd(k) such that U/(U ∩ k∗) = H. Then U ∩ k∗has finite index in U , so is a finitely generated abelian group, that implies U ∩ k∗ = Z × F with Z a finite group and F a free abelian group. Note that k∗ has no element of order p. Therefore the

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order of Z is not divisible by p, so U/F is finite with order not divisible by p. We can think U as k∗H, since both k∗ and H have no element of order p, we have that also U contains no elements of such order.

We will prove that for each i ∈ N, there is a subgroup Ui ≤ GLd(R/πiR), such that Ui ∼= Ui−1 under the natural map, and also

Ui∩ (R/πiR)∗∼= Ui−1∩ (R/πi−1R)∗

under the natural map. Indeed if there exists such groups we have a pro-jective system

· · · → U_i→ U_i−1→ · · · → U₁ = U

with all maps isomorphisms. Then the inverse limit bU is also isomorphic to U , and can be viewed as subgroup of GLd(R). Moreover, we also have

b

U ∩R∗ ∼= U ∩k∗ by the natural map, and thus bH := bU / bU ∩R∗∼= H naturally, which is what we want to prove.

The Ui’s are constructed inductively, with U1 = U already given. Sup-pose we have defined Ui. Let Gi+1 ≤ GLd(R/πi+1R) be a finitely gen-erated subgroup which maps onto Ui. We can choose Gi+1 so that Gi+1∩ (R/πi+1R)∗maps onto Ui∩(R/πiR)∗. In particular, Gi+1∩(R/πi+1R)∗ con-tains a free abelian group Fi+1 mapping onto the corresponding subgroup Fi ∼= F of Ui.

Set P = Gi+1 ∩ 1 + Md(πiR/πi+1R), so Gi+1/P ∼= Ui. Now, let A ∈ Md(πiR/πi+1R). Then A2 = 0 since ab = 0 for a, b ∈ πiR/πi+1R. Since p(R/πR) = 0 it follows pR ⊆ πR hence pπiR ⊆ πi+1R, that is, p(πiR/πi+1R) = 0. Therefore pA = 0. We then have

(I + A)p = p X i=0 p i Ai = I + pA = I.

Then P is finitely generated and thus a finite p-group.

We have Fi+1∩ P = 1, and we can apply the Schur-Zassenhaus theorem to the finite group Gi+1/Fi+1 to conclude that P Fi+1/Fi+1 has a complement Ui+1/Fi+1. Then Ui+1 has the desired properties.

Theorem 1.3.1. Let H be a finite subgroup of PGL2(Fp) with order prime to p. Assume that H it is not cyclic or dihedral. Then H is isomorphic to one of this group

i) the alternating group of 4 elements A4, ii) the symmetric group of 4 elements S4, iii) the alternating group of 5 elements A5.

In particular the order of H can be only 12, 24, 60 and its elements can be only of order 1, 2, 3, 4, or 5.

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Proof. The field Fpis the residue field of the complete discrete valuation ring Zp. By the previous lemma we have that H can be lifted to a subgroup bH of PGL2(Zp) isomorphic to H. In particular we can identify bH to its image in PGL2(Qp) under the natural map PGL2(Zp) ,→ PGL2(Qp) ,→ PGL2(Qp) remembering that Qp is the field of fractions of Zp. Now, Qp ' C because it is an algebraic closed field of characteristic zero, thus PGL2(Qp) ' PGL2(C). It is a known fact that the finite subgroups of PGL2(C) are only those of our thesis, and it follows from two results:

a) there is an isomorphism of abstract groups PSU2(C) ' SO3(R) ([5], pag. 41, Corollary 2.12.3), where PSU2(C) denote the image of the spe-cial unitary group SU2(C) ⊂ GL2(C) inside the quotient PGL2(C) = GL2(C)/C∗;

b) every finite group of PGL2(C) is conjugate to a subgroup of PSU2(C) ([5], p. 44, Corollary 2.13.2).

Now, since the finite subgroups of SO3(R) are only those in our thesis ([5], p. 46, Theorem 2.13.5) we are done.

We have an immediate corollary:

Corollario 1.3.1. If p = 2, 3 all finite subgroup of PGL2(Fp) of order prime to p are only cyclic or dihedral.

Remark 1.3.1. This theorem holds also if we replace Fp with a general field k. If char(k) = 0 then every finite subgroup of PGL2(k) can identify with a finite subgroup of PGL2(k) ' PGL2(C) and it works. If k has char(k) = p > 0 then we can associate to it a discrete valuation ring C such that its residue field is k and its field of fractions is a field of characteristic 0. A such ring exist and it is called Cohen ring. Then we can apply the same reasoning of the theorem and obtain what we want.

Now, let G be a finite group of GL(V ) with order prime to p, and let H be its image in PGL(V ). The theorem then says that we have only this possibilities:

i) H is cyclic. Take h that generates H, let g be an element of G that maps onto h. In particular the discriminant of its characteristic poly-nomial ∆ can be not 0 because the only matrices of GL(V ) with null discriminant are those of the form

α 0 0 α α 1 0 α

that in PGL(V ) are rispectively the identity and the elements of order p. Then g belongs to a unique Cartan subgroup (cf. §1.1). Hence H is contained in a unique Cartan subgroup of PGL(V ). Then G is contained in a unique Cartan subgroup of GL(V ).

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ii) H is dihedral, so it contains a non-trivial cyclic subgroup C0 of index 2. So, using the reasoning above, C0 is contained in a unique Cartan subgroup C of PGL(V ). Since H normalizes C0it normalizes C. Com-ing back to GL(V ) we obtain that that G is contained in a Normalizer N of a Cartan subgroup of GL(V ).

iii) H is isomorphic to A4, S4, A5 so its element have order 1, 2, 3, 4 or 5. We can now state and prove the main theorem of this chapter. It plays a crucial role in chapter 4.

Theorem 1.3.2. Let G be a subgroup of GL(V ) that contains a Cartan subgroup C (resp. a split half Cartan subgroup). Assume that p 6= 5 if C is split. Then only one of the following holds:

a) G = GL(V );

b) G is contained in a Borel subgroup;

c) G is contained in a Normalizer of a Cartan subgroup.

Proof. We have two cases: the order of G is divisible by p or it is prime to p. Assume the first. Proposition1.2.1 says that G is contained in a Borel subgroup or it contains SL(V ). However, since G contains C, the restriction to G of the determinant map det : GL(V ) −→ F∗p is surjective. Indeed if C is a split Cartan subgroup or a split half Cartan subgroup it is clear because in both case we have the matrices of the form 1 0

0 ∗

. If C is non-split then we have seen that every element of C represents multiplication by a non-zero element of Fp2 (cf. §1.1). In particular determinant map restricted on these matrices coincides with the norm map

N_F p2/Fp : F ∗ p2 −→ F∗_p x 7→ Y σ∈G σ(x)

where G = Gal(Fp2/F_p). However G is cyclic of order 2 generated by the Frobenius automorphism, so N_F

p2/Fp(x) = x · x

p _{= x}p+1_{. Both F}∗ p2, F∗_p are cyclic of order p2 − 1 and p − 1 respectively. In particular we have | ker(N_F p2/Fp)| = gcd(p 2_{− 1, p + 1) = p + 1 and} |Im(N_F p2/Fp)| = p2_{− 1} gcd(p2_{− 1, p + 1)} = p − 1

so it is surjective and this holds also for the determinant map restricted to C. We can now conclude that if G contains SL(V ) = ker(det) then it must be GL(V ). Let now G be a subgroup of GL(V ) of order prime to p.

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If its image H in P GL(V ) is cyclic or dihedral the theorem is proved. If p = 2, 3 by corollary1.3.1 we have done. Suppose then p ≥ 5, the image C1 in PGL(V ) of C is cyclic of order p ± 1 ≥ 6 because we suppose p ≥ 7 if C is split. But none of the groups A4, S4, A5 have elements of such order, so H is not isomorphic to any one of them and we have shown the theorem.

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Chapter 2

Tamely Inertia Group

Let K be a field complete with respect to a discrete valuation v and suppose that is normalized. Let OK be its valuation ring, m its maximal, ideal and k the residue field OK/m of characteristic p. We have that Z = v(K∗). We define the ramification index e of p as the integer e = v(p), 1 ≤ e ≤ ∞.

Let Ksbe a separable closure of K. The valuation v extends to a unique valuation Ks ([16], p. 29, Corollary 2), and the corresponding residue field is an algebraic closure k of k ([16], p. 54, Corollary 1). We have the following tower of extension:

K ⊂ Kur ⊂ Kt⊂ Ks

where Kur is the maximal unramified sub-extension of K in Ks, and Kt is the maximal tamely ramified sub-extension of K in Ks, that is the compositum of all ramified extension of K in Ks with ramification index prime to p = char(k).

Let

G = Gal(Ks/K), I = Gal(Ks/Kur), Ip = Gal(Ks/Kt). (2.1) We have thatG is the inverse limit of all Gal(L/K) where L ranges over all finite Galois sub-extension of K in Ks. Using the notation of ramification theory (cf. [16], chap IV) we say that G is the inverse limit of the G−1,L’s with L as above. So we have that I is the inverse limit of the G0,L’s and we call it inertia sub-group of G ([16], p. 63, Corollary). Finally Ip is the inverse limit of the G1,L’s, and since they are p-groups then also Ip is a p-group. In particular every G1,L is the maximal p-group of G0,L ([16], p. 67, Corollary 1), then Ip is the maximal pro-p-group contained in I. Moreover Ip is a normal subgroup of I ([16], p. 62, Proposition 1) and the quotient is a commutative pro-finite group with order prime to p ([16], p. 67, Corollary 1). We call It = I/Ip the tamely inertia subgroup of G (or of K).

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Structure of the Tamely Inertia Subgroup

2.1 Structure of the Tamely Inertia Subgroup

Let d ≥ 1 be an integer prime to p. Then polynomial xd− 1 is separable over k, so its roots belong to ks and by Hensel lemma they can be lifted to Kur. Let µd be the group of the d-th roots of unity of Kur. Let π be a uniformizer for Kur and consider the field Kd= Kur(π1/d). The polynomial x1/d − π is irreducible over K_ur by the generalized Eisenstein’s criterion. This implies that the extension Kd/Kur is tamely ramified, and of degree d. If s ∈ Gal(Kd/Kur) we have that s(π1/d) is a root of xd− π, hence there exists a unique d-th root of unity θd(s) such that

s(π1/d) = θd(s)π1/d. (2.2) The map θd: Gal(Kd/Kur) −→ µd is clearly an isomorphism.

Kt is the composite of Kd’s, with (d, p) = 1 ([2], p. 32, Corollary 1). We have then

It= Gal(Kt/Kur) = lim_←− d

Gal(Kd/Kur), in particular the homomorphisms θd defines the isomorphism

θ : It−→ lim_←− d

µd.

Note that since Itis isomorphic to a projective limit of cyclic group then it is abelian.

Assume now that the residue field k of K is Fq with q = pn. Let L be an abelian tamely ramified extension of K. By definition of Ktholds K ⊂ L ⊂ Kt, so the inertia sub-group I(L/K) ⊂ G(L/K) is a quotient of It. Then we have a surjective homomorphism α : It' lim_←−

d

µd−→ I(L/K). Since L/K is abelian, local class field theory provide us the reciprocity homomorphism ω : K∗ −→ Gal(L/K). In particular if U = O∗

K then ω(U ) = I(L/K) and ω(1 + m) = {1}. On the other hand U/(1 + m) = k∗, hence ω factor to this quotient and define a surjective homomorphism ω : k∗ −→ I(L/K). But k∗= F∗q= µq−1, then we can study the relation between α and ω.

Proposition 2.1.1. Let θq−1 be the homomorphism of It on k∗ = µq−1 defined in 2.2. For all s ∈ It we have

α(s) = ω ◦ θq−1(s−1).

Proof. Let d = q − 1. Let x be an uniformizer of K and consider Kd = Kur(x1/d). The homomorphism ω : k∗ −→ I(L/K) is surjective, so I(L/K) must be finite of order dividing d. Thus, by the structure of Kt, L ⊂ Kd.

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Characters of It

We can suppose L = Kd, by functoriality property of α and ω. If s ∈ It we have seen that θd acts as

s(x1/d) = θd(s)x1/d.

Let t ∈ k∗ and let ω(t) the corresponding element in I(L/K). Let (x, t)dbe the local symbol associated to x and t ([16], chap. XIV) then

ω(t)(x1/d) = (x, t)dx1/d.

But (x, t)d= t−v(x)= t−1([16], p. 211, Corollary), and if we take t = θd(s−1) we have ω(t)(x1/d) = θd(s)x1/d = s(x1/d). Since s and ω(t) are the identity on Kur then they are equal over Kd= L.

2.2 Characters of I

t

Let V be a vector space of dimension n over a field k1of characteristic p. Let ρ :G −→ GL(V ) be a continuous linear representation of G with value in V . With the next proposition we characterize the semi-simple representation of G .

Proposition 2.2.1. Let ρ be a semi-simple representation.

Then ρ(Ip) = {1}, where Ip is the inertia p-group introduced in 2.1.

Proof. It is enough prove this for simple representations. Assume the ρ is simple, and let V0 the subset of the elements of V stable under the action of ρ(Ip). We have seen that Ip is a p-group and every linear representation of a p-group contains the unit representation ([16], p. 139, Theorem 2), so V0 6= 0. Since Ip is a normal subgroup of G , for all g ∈ G , h ∈ Ip and x ∈ V0

ρ(g−1h g)(x) = x then

ρ(h)(ρ(g)(x)) = ρ(g)(x)

this implies ρ(g)(x) ∈ V0for all g ∈G , so V0= V and the proposition follow.

The proposition above implies that if ρ is semi-simple the action of I on V factorize trough the quotient It= I/Ip. The image of It by ρ is a cyclic group, hence abelian, with order prime to p. If k1is big enough, for example separable closed field, then ρ(It) is diagonalizable and we can say that the restriction of ρ to It is given by n characters ψi : It−→ k1∗, i = 1, . . . , n. Consider the group X = Hom(It, k∗s) of the continuous characters of Itwith value in k_s∗. For example the homomorphisms θd: It−→ Gal(Kd/Kur) ' µd introduced in §2.1 are elements of X. Moreover we can describe X using

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Characters of It

them. Let (Q/Z)0 the subset of the elements of Q/Z with order prime to p. Every α ∈ (Q/Z)0 has the form α = a/d, a, d ∈ Z and (d, p) = 1. Let χα be the a-th power of θd, note that since θmmd= θd then χα does not depend from the particular form a/d chosen for α.

Proposition 2.2.2. The map

ϑ : (Q/Z)0 −→ X α 7→ χα is an isomorphism.

Proof. The group (Q/Z)0 is the union of the subgroup (1/d Z)/Z with (d, p) = 1. We know that It is the inverse limit of µd, d as above, so X is the inverse limit of the groups Xd= Hom(µd, ks∗). The proposition is now proved because ϑ induces an isomorphism between (1/d Z)/Z and Xd.

Let n ≥ 1 be an integer, and q = pn. We define a fundamental char-acter of level n the composition of

θq−1 : It−→ µq−1= F∗q

with an automorphism of Fq. Thus it is a character of the form χ = θp_q−1i , i = 0, . . . , n − 1.

Let w be the unique extension to Ks of the valuation v of K, we have w(K_s∗) = Q ([16], p. 29, Corollary 4). Let M be the maximal ideal of the valuation ring OKs. For every α ∈ Q we define Mα:= {x ∈ Ks| w(x) ≥ α} and let M+

α be the elements of Mα such that we have a strict inequality. The quotient Vα = Mα/M+α is a k-vector space of dimension 1, where k is the residue field of Ks. To see this take b ∈ OKs such that v(b) = α. For every ¯x ∈ Vα, ¯x 6= 0, chosen a representative x ∈ mα, we have v(x) = α. By definition of valuation, v(xb−1) = 0 therefore xb−1 ∈ O∗_K

s. Hence there exists an a ∈ O∗_K

s such that x = ab. Thus ¯x = ˆa¯b ∈ k ¯b, where ¯b is the image of b in Vα and ˆa is the projection of a in k. Finally ¯b is linearly independent in the sense if ˆc¯b = ¯0 then v(cb) > α that implies v(c) > 0 so c ∈ M and ˆ

c = ˆ0. We have shown that Vα = h¯bi_k.

Now, G acts on the valuations extending v permuting them. However there is a unique extension w of v, so w(s(x)) = w(x) for all s ∈ G . In other terms the action of G does not affect the valuation. Therefore we have a natural action of G on Vα. Let s ∈ G , let σ be the image of s in Gk = Gal(k/k) = Gal(ks/k). The automorphism on Vα defined by s is σ-linear. In fact, for all λ ∈ k∗ we can lift the multiplication-by-λ map to a multiplication-by-a in Mα, where a ∈ O∗_K_s is such that a ≡ λ (mod M).

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Characters of It

In particular s(ax) = s(a)s(x) and s(a) ∈ O∗_K_s, so s(a) ≡ σ(λ) (mod M) and we are done. Moreover by uniqueness of w we deduce that G coincide with the decomposition group of w. Then G /I ' Gk, and from the con-siderations above I acts on Vα as a linear map. However Vα has dimension 1, the only possibility for I to acts on Vα is multiplication by a character ϕα : I −→ k

∗

. More precisely for all s ∈ I and x ∈ Vα the action of s is defined by the following formula

s(x) = ϕα(s)x. (2.3)

As k∗ does not contain non-trivial elements of order p, the character ϕα is trivial on Ip. So we can think it as a character of the tamely inertia subgroup It= I/Ip. It make sense ask if there is a connection between ϕα and one of the characters χα defined in2.2.2.

Proposition 2.2.3. The character ϕα given by the action of I on Vα is identical to a character χα.

Proof. Let α, β ∈ Q. The map

M_α× M_β −→ M_α+β (x , y) 7→ xy

induces an isomorphism Vα× Vβ−→V˜ α+β. Such isomorphism clearly com-mute with the action of G . This implies that ϕαϕβ = ϕα+β, i.e. the map α 7→ ϕα is an homomorphism.

let d ≥ 1 be an integer prime to p, and consider x a d-root of an uni-formizer of K. Then

w(x) = 1/d, s(x) = θd(x)x ∀s ∈ It.

Therefore ϕ1/d = θd = χ1/d. For all α ∈ Q there exists an integer q power of p, such that qα = a/d with a ∈ Z and d ≥ 1 is prime to p. Being the maps α 7→ ϕα, α 7→ χα homomorphisms we obtain

ϕq_α= ϕqα = ϕa/d = ϕ_1/da

= χ_1/da

= χa/d= χqα= χqα.

Finally the group of characters of It has no elements of p-torsion, hence it must be ϕα = χα.

Assume now K of characteristic 0. Let µp be the group of the p-th roots of unity contained in Ks. The Galois group G acts on µp, so we have an

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G -module attached to an elliptic curve with good reduction

homomorphism χ : G −→ Aut(µp) ' F∗p. The image of Ip trough χ is the identity (cf. Prop.2.2.1), hence there is an action of It on µp given by the character

χ : It−→ F∗p.

Proposition 2.2.4. Let e = w(p) be the absolute ramification index of K. The character χ is the e-th power of the fundamental character of level 1 θp−1.

Proof. Let α = e/(p − 1). Take ω a primitive p-th roots of unity, then p = u(1 − ω)ϕ(p) where u ∈ O_K∗

s and ϕ is the Euler’s totient function. This implies w(1−ω) = α and there is an injective homomorphism from µpto Mα. Thus passing to the quotient we have an injective homomorphism µp −→ Vα, and this map commute with the action of G . In particular commute with the action of It, and since it acts on Vα through the character θep−1 (see Prop.2.2.3) the proposition follow.

2.3 G -module attached to an elliptic curve with

good reduction

From now on, we assume the characteristic of K is zero, so K = Ks. Let E be an elliptic curve defined on K (cf. §0.4). Let E[p] be the p-torsion subgroup of E, that is an Fp-vector space of dimension 2. We have seen in §0.4.2that E[p] is aG -module, and we are interested in to know better this structure. By Weil pairing (see [17], Chapter III, § 8)V2

E[p] ' µp, so the determinant map of the representation ofG in E[p] is the same as a character of G −→ F∗_p = Aut(µp) given by the action of G on µp. Proposition2.2.4 shows that the restriction of this character on It is the e-th power of θp−1. Consider the reduction eE of E at v (cf. §0.4.1). By reduction map we obtain the follow exact sequence ([17], p. 188, Proposition 2.1)

0 −→ Xp−→ E[p] −→ eE[p] −→ 0.

For an elliptic curve E1 defined over a field of characteristic p we have ([17], p 145, Theorem 3.1)

a) or E1[pr] = {O} for all integers r ≥ 1; b) or E1[pr] = Z/prZ for all integers r ≥ 1.

So we say that E has a good reduction of height 2 if a holds, otherwise we say that E has a good reduction of height 1. We study the two cases separately.

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Good reduction of height 1

In this case we have Xp ' Z/pZ. The points in Xp are those mapped in [0, 1, 0], so their coordinates [x, y, z] satisfy w(x), w(z) > 0 and w(y) = 0. Since G doesn’t affect the valuation, Xp is stable under the action of the Galois group. If we chose a basis (e1, e2) of E[p] over Fp such that Xp = Fpe1, the image of G in Aut(E[p]) ' GL2(Fp) is contained in a Borel subgroup∗ ∗

0 ∗

(cf. §1.2). Moreover being Ip a pro-p-group its image in GL2(Fp) is contained in the group

1 ∗ 0 1

. The action of Iton Xpand eE[p] is given by multiplication of two charaters χx, χy respectively, both with value in F∗p. In particular the action of G on E[p] is given composing thee surjective homomorphism G −→ G /I ' Gal(k/k), with the natural action of Gal(k/k) on eE[p]. So we deduce χy = 1. Furthermore the determinant of the representation ofG on E[p] is χxχy, and for what we have seen at the beginning of this section we finally have χx= θp−1e .

Proposition 2.3.1. Assume e = v(p) = 1, the following hold

i ) the two characters giving the action of It on E[p] are 1 and the fun-damental character of level 1 θp−1,

ii ) chosen a basis (e1, e2) of E[p] over Fp such that e1 ∈ Xp, if Ip acts trivially on Ep then the image of I in GLE[p] is a cyclic group of order p − 1 whose matrix representation is∗ 0

0 1

,

iii ) if the action of Ip on E[p] is not trivial, the image of I in GLE[p] is a group of order p(p − 1) whose matrix representation is ∗ ∗

0 1

. Proof. The first assertion follow directly from what we have said before. Now, χx = θp−1 is a surjective homomorphism from It −→ F∗p then the image of I is a group of order a multiple of p − 1. Hence, being χy = 1 and the image ofG contained in a Borel subgroup, the image of I is contained in∗ ∗

0 1

. We have only two cases: or the image of I is the cyclic group of order p − 1 ∗ 0

0 1

and so Ip acts trivially, or the image of I is the whole group and so it has order p(p − 1) and Ip acts on E[p] by

1 ∗ 0 1

.

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Assume that e = 1. If (x, y) ∈ E(K) such that the corresponding reduced point is the basepoint [0, 1, 0], then x, y /∈ O_K and thus w(x), w(y) < 0. The minimal Weierstrass equation for E is of the form y2 = x3 + Ax + B with A, B ∈ O_K (cf. §0.4 and §0.4.1), hence 2w(y) = 3w(x) that means |w(y)| > |w(x)|. Then w(x/y) > 0 that is t = x/y ∈ M. The composition law of E can be written as a formal group law

F (t, t0) = t + t0− a₁tt0− a₂(t2t0+ tt02) + · · ·

whose coefficients live in OK ([17], chapter IV, § 1). Furthermore we can define the multiplication by p associated to F whose formal series has the property [p](t) = P

i≥1aiti = pt + (higher order terms) ([17], p. 122, Pro-position 2.3). Since eE[p] = 0 the height of F is 2 ([17], p. 134, Theorem 7.4), so the coefficients of the formal series defining [p] satisfy

ai≡ 0 (mod M) for i < p2 and ap2 ∈ M./ (2.4) The kernel of [p] is identified with E[p] that we remember is a Fp-vector space of dimension 2. Taken x ∈ M belonging to the kernel of [p], we have

p + a2x + · · · + ap2xp 2₋₁

+ . . . = 0.

By (2.4) if we comparing the valuations we obtain w(x) = 1/(p2− 1) and we let α = 1/(p2− 1). If x, y ∈ M are in ker([p]) then x, y ∈ Mα and by the definition of the formal group law

F (x, y) ≡ x + y (mod M+_α).

Thus the restriction of the projection Mα −→ Vα = Mα/M+α on ker([p]) is an injective homomorphism. This homomorphism commute with the action of the G so Ip acts trivially on ker([p]) because it acts trivially on Vα by proposition2.2.1. Identifying the kernel of [p] with its image in Vαwe obtain by proposition2.2.3that It acts on ker([p]) as

s(x) = θp2₋₁x. Being θp2₋₁ : I_t −→ F∗

p2 surjective we deduce that ker([p]) is stable under multiplication by elements of F∗_p2 so it is a sub-Fp2-vector space of V_α. Fur-thermore card(E[p]) = card(ker([p])) = p2 _{so its dimension is 1. We have} just shown the following proposition

Proposition 2.3.2. Assume e = 1, then

i ) E[p] is Fp2-vector space of dimension 1 such that the action of I_t on it is given by a fundamental character of level 2 θ_p2₋₁,

l-adic Representations Attached to Elliptic Curves

Contents

Preliminaries

0.1

Algebraic Group

0.2

Ideles and Adeles Classes

0.3

`-adic Representations of Number Fields

0.4

Elliptic Curves

Chapter 1

Subgroup of GL

2

(F

p

)

1.1

Cartan subgroups

1.2

Subgroups with Order Divisible by p.

1.3

Soubgroups of Order Prime to p

Chapter 2

Tamely Inertia Group

2.1

Structure of the Tamely Inertia Subgroup

2.2

Characters of I

2.3

G -module attached to an elliptic curve with

good reduction

₂

_(F

_p