Serre s theorem on Galois representations attached to elliptic curves

Transcription

1 Università degi Studi di Roma Tor Vergata Facotà di Scienze Matematiche, Fisiche e Naturai Tesi di Laurea Speciaistica in Matematica 14 Lugio 2010 Serre s theorem on Gaois representations attached to eiptic curves Candidato: Vaerio Dose Matricoa Reatore: Prof. René Schoof Correatore: Prof. Fabio Gavarini Anno Accademico 2009/2010

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3 Serre s theorem on Gaois representations attached to eiptic curves Vaerio Dose Juy 14, 2010 Introduction Gaois representations are in genera vector spaces, with the action of the Gaois group of some extension of fieds. This kind of objects arise naturay in agebra and number theory. The simpest exampe is probaby the group of p-th roots of unity µ p for a fied K, which is a vector space of dimension one over the finite fied F p, with the action of the Gaois group Ga(K(µ p )/K). A richer situation comes from the study of eiptic curves defined over number fieds. Eiptic curves are a particuar type of agebraic curves which have the pecuiarity that the points with coordinates in some fied form a group with some composition aw. If the fied K of definition of an eiptic curve E has characteristic 0 (for exampe a number fied, which is a finite extension of Q), we have that the points P on E such that pp = P P =(neutra eement), form a group that has naturay the structure of vector space of dimension two over F p. Furthermore, on the set of these points there is an action of the Gaois group Ga(K/K) where K is an agebraic cosure of K. The study of Gaois representations attached to eiptic curves has been mainy focused on determining this action. Serre s theorem, appeared in [12] in 1972, is one of the few genera resuts of this type. It states that in many situations, the Gaois group acts as the whoe group of invertibe transformations of the reative vector space. In this thesis we wi provide a proof of Serre s theorem, as we as some exampe regarding particuar cases that arise in the proof. In section 0 there is an expanation of the statement of the theorem, and a brief ook at the ingredients that are used to prove it. The actua proof is contained in sections 1-4. The resuts are organized amost in the same way as in [12] and the proofs are given with some more detai. In particuar, in section 1 there is a study of certain subgroups of GL 2 (F p ) that are important for the proof 1

4 of Serre s theorem, in section 2 we concentrate on the action of the inertia subgroup of the Gaois group, in section 3 there is the anaysis of certain abeian representations, and finay section 4 contains the concusion of the proof. Section 5 contains a brief expanation on how we can generate eiptic curves with desired Gaois representation expoiting the moduar description of isomorphism casses of eiptic curves. Moreover, in this ast section there are some expicit cacuations of certain quadratic extensions of fieds that come out in specia cases. These cacuations were made with the hep of the software PARI/GP Cacuator ([7]). Acknowedgments. Besides the peope that have aready been mentioned on the cover, I woud ike to thank Burcu Baran for her advice. 2

5 Contents 0 Preiminaries Unramified extensions, Frobenius cass and Čebotarev s theorem Loca/Goba cass fied theory Idee group Loca cass fied theory Goba cass fied theory Agebraic groups Agebraic groups of mutipicative type and tori Construction of an extension of an agebraic group Linear representations of agebraic groups Eiptic curves Reduction of eiptic curves Gaois representations attached to eiptic curves Subgroups of GL 2 (F p ) Cartan subgroups Normaizers of Cartan subgroups Bore subgroups Subgroups of order prime to p Subgroups containing Cartan subgroups Representation of inertia group Projective structure of I t Representation of I in characteristic p Characters of I t Action of I on µ p Action of I on p-torsion points of an eiptic curve Abeian representations of Gaois group Abeian -adic representation of Gaois group with vaues in S m Characters of S m Representations of S m Reating characters of S m to -adic abeian representations of Gaois group Reating representations of S m to -adic abeian representations of Gaois group Appication to some Gaois representations attached to eiptic curves

6 4 End of the proof of Serre s theorem ϕ (G) contained in the normaizer N of a Cartan subgroup C and not in C ϕ (G) contained in a Bore subgroup or a Cartan subgroup Concusion of the proof Some cacuations in the case of eiptic curves with ϕ 11 (G) contained in the normaizer of a non-spit Cartan subgroup The moduar context X ns(11) References 64 4

7 0 Preiminaries 0.1 Unramified extensions, Frobenius cass and Čebotarev s theorem Here K is a number fied and Σ K is the set of the vauations/paces of K given by the prime ideas of its ring of integers O K. Let L be a finite Gaois extension of K with Gaois group G = Ga(L/K). Take v Σ K and w Σ L an extension of v to L. This means that if w is given by the prime idea q O L (w(x) = ord q (x)) and v by p O K then q p. A prime ideas q containing p are then conjugate under G ([8], p. 20, Proposition 19), and we say simiary that a the paces w dividing v are conjugate under G. There is a surjective homomorphism between the decomposition group D w = D q = {s G s(q) = q} and the Gaois group of the residue extension Ga ( O L/q/O K /p) ([8], p. 21). The kerne of this homomorphism is the inertia group I w = I q = {s G s(x) x mod q}. We have aso that D w is the Gaois group Ga(L w /K v ) where L w and K v are the competions of L and K with respect to the vauations w and v ([8], p. 31, Coroary 4), and this Gaois group contains I w. We say that the extension w of v is unramified if the inertia group I w is trivia. Furthermore, any set on which G acts is said to be unramified at w if the action of the subgroup I w G is trivia. Since a the paces of L dividing v are conjugate under G, a Gaois extension is unramified at a pace dividing v if and ony if it is in any other pace dividing v. Hence we can use the expression that such extension is unramified at v Σ K, or equivaenty that v does not ramify in such extension. Simiary, a group homomorphism ρ : G X is said to be unramified at v if ρ(i w ) = 1 for a w Σ L dividing v. If the extension L K is unramified at w, the preceding homomorphism D w Ga ( O L/q/O K /p) is actuay an isomorphism, thus there exists an eement in D w G which corresponds to the automorphism x x #{O K/p} in Ga ( O L/q/O K /p). This eement is caed the Frobenius eement F w of w. Since a the paces of L dividing v are conjugate under G, we have that a the Frobenius eements associated to a the paces of L dividing v form a conjugacy cass in Ga(L/K) that we ca the Frobenius conjugacy cass F v of v. If L is an abeian extension of K, the Frobenius cass of a pace v is reduced to one eement. For infinite Gaois extensions L of K, not necessariy unramified, we say that the Frobenius conjugacy cass F v of a pace v Σ K is the subset of Ga(L/K) formed by the automorphisms of L that are in the decomposition group of some extension of v to L, and when restricted to finite unramified Gaois extensions, they are equa to an eement of the respective Frobenius 5

8 conjugacy cass F v. If K ab is the maxima abeian extension of K, we ca the Frobenius cass of v in Ga(K ab /K), with abuse of anguage, the Frobenius eement F v Ga(K ab /K). Let A be a subset of Σ K. We say that the set A has density d if im n #{v A N(v) n} #{v Σ K N(v) n} = d where N(v) = #{O K /p} is the norm of the pace v and p is the prime idea of O K giving v. The foowing is Čebotarev s density theorem Theorem. Let L be a finite Gaois extension of K. Let C Ga(L/K) = G be a subset of the Gaois group stabe under conjugation. Let A C be the set of paces v Σ K such that v is unramified in L and the Frobenius cass F v is contained in C. Then the set A C Σ K has density #C #G. [11], p. I-7, Theorem. Aso [6], p.169, Theorem 10. Coroary. Let L be a Gaois extension of K (not necessariy finite), which is unramified outside a finite set of paces. Then the Frobenius eements of the unramified paces of L are dense in Ga(L/K). [11], p. I-8, Coroary Loca/Goba cass fied theory Idee group Let K be a number fied with ring of integers O K, Σ be the set of discrete vauations of K given by the prime ideas of O K and Σ the vauations of K given by the embeddings of K in R, C. For a v Σ Σ et K v be the competion of K with respect to the vauation v, and for a v Σ, et O v be the ring of integers of K v. Now we can define the idee group I { I = (x v ) v } Kv x v Ov for amost a v Σ v Σ Σ This is a topoogica group with the topoogy generated by the product topoogy and the subset Kv Ov v Σ v Σ 6

9 K itsef is a subgroup of I through the incusion K I x (x) v Σ Σ and we can define the idee cass group C = I/K. We have the foowing exact sequence 0 K I C 0 and C is aso a topoogica group with the quotient topoogy Loca cass fied theory Let K be a compete fied with respect to a discrete vauation. Loca cass fied theory aows us to describe the abeian extensions of K in a separabe cosure K s just through K itsef. The main theorem of interest to us is the foowing Theorem. Let L K s be an abeian extension of K. Then there exists an isomorphism K /N(L ) = Ga(L/K) where N : L K is the norm homomorphism. [8], p.196, Coroary. This isomorphism is caed the reciprocity isomorphism and by composition with the projection onto the quotient it becomes a surjective homomorphism ω L : K Ga(L/K) which is caed the reciprocity homomorphism. Let L be a Gaois extension of K such that K L L. Then for a x K we have that ω L (x) is the restriction of ω L (x) to L ([8], p. 197, Proposition 12). Thus we can actuay define reciprocity homomorphisms for infinite abeian extensions of K. In particuar, taking K ab the maxima abeian extension of K we obtain a surjective homomorphism K = Ga(Kab /K) which maps the group of units OK Ga(K ab /K) ([8], p. 198, Coroary). of K onto the inertia subgroup of 7

10 0.2.3 Goba cass fied theory Let K be a number fied and Σ, Σ its vauations as in Let K v be the competion of K with respect to v, and L a finite abeian extension of K in an agebraic cosure K. Then for a v Σ oca cass fied theory provides us with reciprocity homomorphisms f v : K v D v where D v Ga(L/K) is the decomposition group of any extension of v to L (it does not depend on the extension because L K is abeian). Furthermore, even if v Σ (K v = R, C) we have obvious reciprocity homomorphisms that are either trivia or as foows f v : R R /N(C ) = Ga(C/R) Ga(L/K) For any idee a I it is f v (a v ) = 1 for amost a v Σ (using [8], p. 197, Proposition 13 with the facts that a v Ov for amost a v Σ and L K is unramified at amost a v Σ). Taking the product of the f v s we obtain a homomorphism f : I K Kv Ga(L/K) v Σ Σ where I K is the idee group of K as in We have the foowing theorem Theorem. The map f is surjective, and its kerne is generated by K and N(I L ), where N : I L I K is the homomorphism induced on idees by the norm homomorphism from L w to K v, when w is an extension to L of a vauation v of K. [8], p. 221, Theorem. Proof in [2] This is one of the form in which Artin reciprocity aw is known. Since f is the product of the oca homomorphisms f v, we have that it maps the group of units O v K v onto the inertia group I v D v. Furthermore, if L K is unramified, any eement x I K with a components equa to 1 except the v-th x v that is a uniformizer of K v, is mapped by f onto the Frobenius eement F v Ga(L/K). By passing to infinite abeian extensions, as in the case of oca reciprocity, we get an isomorphism C/D = Ga(Kab /K) where K ab is the maxima abeian extension of K in K and D is the connected component of the identity in C = I/K. The idees x I with a components equa to 1 except the v-th that is a uniformizer of K v are mapped onto the Frobenius eement F v Ga(K ab /K). 8

11 0.3 Agebraic groups In this section, K is a fied of characteristic 0. An agebraic group (or group scheme) G over K is a functor from the category of K-agebras to the category of groups, which is aso represented by a particuar finitey generated K-agebra Λ = K[x 1,..., x n ]/I. This means that for any K-agebra A we have G(A) = Hom K (Λ, A) In this way we can equip G(K) with a Zariski topoogy, precisey the one induced by the agebraic set in A n K defined by the idea I. Actuay, the representing agebra Λ has further structure, that in some way transate in the context of agebras, the group structure of the functor Hom K (Λ, ). The resuting object is caed an Hopf agebra ([15], p. 7, 1.4). A homomorphisms between two agebraic groups G, F, is a natura transformation G F of functors. In particuar we have a group homomorphism G(A) F (A) for each K-agebra A. Yoneda s emma ([15], p. 6, Theorem) tes us that this kind of morphisms between representabe functors correspond to Hopf K-agebras morphisms Λ F Λ G between the agebras representing F and G. Indeed, we have the foowing theorem Theorem. Agebraic groups over K correspond to Hopf agebras over K [15], p. 9, Theorem. Notice that if we have an agebraic group G over K we can aso consider it as an agebraic group defined over an extension L of K, just by taking the restriction of the functor G to the category of L-agebras ([15], p. 11, 1.6). We wi indicate the resuting agebraic group with G L. The L-agebra representing G L is Λ L where Λ is the K-agebra representing G. In particuar if Λ = K[x 1,..., x n ]/I, then Λ L = L[x 1,..., x n ]/I and we have G(K) G(L) = G L (L). We can aso do the converse. Let H be an agebraic group over an extension L of K. Then we can obtain an agebraic group F over K defining F (A) = H(A K L) for any K-agebra A. The exampes of agebraic groups that we wi need are these (the definition is given for any K-agebra A) 9

12 the mutipicative group G m : G m (A) = A = {a A a is invertibe}. Its representing agebra is K[x, y]/(xy 1). the n-th roots of unity µ n : µ n (A) = {a A a n = 1}. Its representing agebra is K[x]/(x n 1). the n-dimensiona inear group GL n : GL n (A) ={invertibe matrices with entries in A}. Its representing agebra is K[a 11,..., a nn, b]/(det((a ij ))b 1) Given an agebraic group G, we ca its group of characters the group X(G) = Hom K (G K, G m K ). The eements ϕ X(G) correspond to morphisms of Hopf agebras over K f ϕ : K[x, y]/(xy 1) Λ where Λ = Λ K = K[x 1,..., x n ]/I is the agebra representing G K and Λ = K[x 1,..., x n ]/I is the agebra representing G. Then f ϕ (x) is an invertibe eement of Λ. We see that characters of G correspond to certain eements of Λ that are caed group-ike eements of Λ ([15], p. 14, Theorem). The Gaois group Ga(K/K) acts naturay on Λ K[x 1,..., x n ]/I, and such action maps group-ike eements in group-ike eements, so that X(G) becomes a Ga(K/K)-modue Agebraic groups of mutipicative type and tori We have the foowing theorem Theorem. Every agebraic group over a fied is isomorphic to a Zariskicosed subgroup of some GL n. [15], p. 25, 3.4, Theorem. We can then consider any agebraic group as a group of matrices. An agebraic group G over K is caed of mutipicative type if over an agebraic cosure K of K its eements (as matrices) can be simutaneousy diagonaized. This impies ([15], p. 55, 7.2) that over K, G becomes a direct product of a finite number of groups ike G m and µ n. This means that for a K-agebras A we have G(A) = G K (A) = G m K (A) G mk (A) µ n 1 (A) µ nk (A) If the product is extended ony to groups ike G m, then we say that G is a torus. Every agebraic group G of mutipicative type is characterized by its character group. We have the foowing theorem 10

13 Theorem. Taking character groups yieds an anti-equivaence of categories between agebraic groups of mutipicative type over K and finitey generated abeian groups on which Ga(K/K) acts continuousy. [15], p. 55, 7.3, Theorem Construction of an extension of an agebraic group Let A be a commutative agebraic group over K and et 0 Y 1 Y 2 Y 3 0 be an exact sequence of abeian groups with Y 3 finite. Suppose we have a group homomorphism π : Y 1 A(K) We want to construct another commutative agebraic group B, a homomorphism of agebraic groups g : A B and a homomorphism of groups ɛ : Y 2 B(K), such that B is a pushout over Y 1 of A and Y 2, where Y 1 and Y 2 are considered as constant agebraic groups. This means that the diagram Y 1 π A(K) g Y 2 ɛ B(K) commutes and B has the universa property which is summarized in the foowing figure Y 1 Y 2 π A(K) g ɛ B(K) ɛ g f B B (K) (Suppose that there exist another agebraic group B, a homomorphism of agebraic groups g : A B and a homomorphism of groups ɛ : Y 2 B (K) such that the diagram among Y 1, Y 2, A(K), B (K) commutes. Then there exists a unique morphism f B : B B such that g = f B g and ɛ = f B ɛ.). In particuar, this property heps to give a different description of the character group of B which is X(B) = Hom K (B K, G m K ). In fact if we take B = G m, the universa property shows that characters in X(B) correspond 11

14 to pairs (ϕ, f), with ϕ X(A), f Hom K (Y 2, K ) and ϕ(π(y)) = f(y) for a y Y 1. The universa property aso impies the uniqueness up to isomorphism of such B, if it exists. We wi now provide a construction of B. For each y Y 3 choose an eement y Y 2 such that the image of y in Y 3 is y. If y, y Y 3 we have y + y = y + y + c(y, y ) where c(y, y ) is the image of an eement of Y 1 in Y 2. We now define B as the disjoint union of copies A y of A for a y Y 3. The group aw on B is given by the maps A y A y A y+y (a, b) ab [ π(c(y, y )) ] for a y, y Y 3. The compositions on the right-hand side are given by the group aw on A. We can aso define the maps g and ɛ in the foowing way g ɛ A A 0 B Y 2 B(K) a a [π(c(0, 0))] 1 y + z π(z) A y (K) B(K) for a a A, y Y 3 and z Y 1. Direct check shows that the defined B is a pushout over Y 1 of A and Y 2 (with f B (a) = g (a)ɛ (y) for a a A y B). We notice that for a extensions L of K we have an exact sequence and a commutative diagram 0 A(L) B(L) Y 3 0 A y (L) {y} 0 Y 1 Y 2 Y 3 0 π 0 A(L) g ɛ id B(L) Y Linear representations of agebraic groups Let H be a commutative agebraic group over K and G = Ga(K/K), where K is an agebraic cosure of K. A inear representation of H of degree d is a morphism of agebraic groups φ : H GL V 12

15 where V is a vector space over K of dimension d and GL V (A) = Aut A (V A) = GL d (A) for a K-agebras A. A inear representation of the abeian group H is said to be semi-simpe if it becomes diagona over K. This means that the image of H K in GL V K is isomorphic to an agebraic subgroup of G m K G mk, where the product has d factors. In this case the representation is actuay given over K through d eements of the character group X(H) = Hom K (H K, G m K ). Note that if H is an agebraic group of mutipicative type, then a its inear representations are semi-simpe, because H K woud be aready isomorphic to a subgroup of some product G m K G mk. To a semi-simpe representation φ of H we can associate its trace θ φ = n χ (φ)χ which is the forma sum of the eements χ X(H) that gives the representation φ over K. Let Rep K (H) be the set of isomorphism casses of inear representations of H that are aso semi-simpe. Then for every extension L of K we have a map between sets S : Rep K (H) Rep L (H L ) which is defined through extension of scaar (H GL V maps in H L GL V L ). This map is injective, in fact if we have commutative diagrams H L GL V L = GL V L Λ L Λ V L = Λ V L where Λ, Λ V, Λ V are respectivey the K-agebras representing H, GL V, GL V and the diagram on the right is obtained by the one on the eft by Yoneda s emma. Thus we obtain the desired commutative diagrams H GL V = GL V Λ Λ V = Λ V remembering that the maps Λ V L Λ L, Λ V L Λ L come from maps Λ V Λ, Λ V Λ, that correspond to inear representations of H over K. We say that a inear representation of H L can be defined over K if it is in the image of S. The next proposition gives an equivaent condition to this property. 13

16 Proposition. The map φ θ φ induce a bijection between Rep K (H) and the set of eements θ = n χ (φ)χ Z[X(H)] satisfying θ is invariant for G = Ga(K, K) (n χ = n s(χ) for a s G, χ X(H)). n χ 0 for a χ X(H). Proof. We first show that the image of such map actuay respects the two hypothesis. The second is obvious. For the first consider the map α : Z[X(H)] Λ K = K[x 1,..., x n ]/I nχ χ n χ χ where Λ is the representing agebra of H and χ is the eement in Λ K corresponding to χ. Then θ φ is G-invariant if and ony if its image by α is an eement of Λ. To prove this, note that for a h = (h 1,..., h n ) H(K) we have θ φ (h) = α(θ φ )(h) and θ φ (h) K because φ is a representation over K. Now et {e α } be a K-basis of K with e α0 = 1 for some α 0. Then we get α(θ φ ) = λ α e α, λ α Λ, and since α(θ φ )(h) K for a h H(K) it is α λ α = 0 for a α α 0 and α(θ φ ) = λ α0 Λ. The injectivity of φ θ φ is guaranteed by the injectivity of S (with L = K) and by the inear independence of characters ([15], p. 15, Lemma). Indeed, two semi-simpe representations of H with the same trace have the same diagona form over K. Now, take χ X(H), and et G χ be the subgroup of G that fixes χ. Let θ = s G/G χ s(χ) so that the sum is extended to a the different conjugates of χ. We have seen that the eements of X(H) correspond to certain invertibe eements of Λ K, and the action of G on X(H) is given through this correspondence. Then we have that the fied K χ fixed by G χ is the smaest extension K K χ K such that the eement of Λ K corresponding to χ is actuay an invertibe eement of Λ K χ, which is the agebra representing H Kχ. This provides us (by Yoneda s emma) with a morphism H Kχ G mkχ which we can consider as a inear representation of degree 1. Then, using restriction of scaars we can define a morphism H(A) H Kχ (A K χ ) ϕ G mkχ (A K χ ) = (A K χ ) for a K-agebras A. If we write K χ as a K-vector space of dimension [K χ : K] in some basis, we get that A K χ is an A-modue generated by 14

17 [K χ : K] eements. Thus, mutipication by an invertibe eement in A K χ can be represented by an eement of GL [Kχ:K](A) and in this way we obtain a inear representation of degree [K χ : K] φ : H GL [Kχ:K] To cacuate the trace of this representation we appy extension of scaars to ϕ : H Kχ ( K χ ) G mkχ ( K χ ). For any K-agebra A we have an isomorphism σ : G mkχ (A K χ ) = (A K χ ) = Gm K (A) G mk (A) a x (σ 1 (x)a,..., σ [Kχ:K](x)a) where σ 1,..., σ [Kχ:K] are the embeddings of K χ in K. These isomorphisms define an isomorphism of agebraic groups between G mkχ ( K χ ) K and G m K G mk where the products has [K χ : K] factors. Hence the representation constructed from φ by scaar extension to K is isomorphic to the diagona representation σ ϕ : H K G m K G mk h (σ 1 (χ)(h),..., σ [Kχ:K](χ)(h)) where σ 1,..., σ [Kχ:K] act on χ through their action on the eement of Λ K χ corresponding to χ. Now it is cear that θ φ = θ. The proposition foows from the fact that any eement of Z[X(H)] which respects the hypothesis is the sum of eements ike θ (sum of a the different conjugates of some character). Coroary. Let H be a commutative agebraic group over K. Let ψ Rep L (H L ) for some extension L of K. Then ψ can be defined over K if and ony if α(θ ψ ) is an eement of Λ. 0.4 Eiptic curves Let K be a fied. In this section E is a projective agebraic curve over K, smooth, of genus 1 and with at east one point O with coordinates in K. Such curve is caed an eiptic curve. For an eiptic curve, it exists a cubic mode in the projective pane, such that the point O is the ony point at infinity of the curve ([14], p. 59, Proposition 3.1). The equation obtained have in the affine pane the form y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 15

18 with a 1, a 2, a 3, a 4, a 6 K. An equation such as this is caed a Weierstrass equation for E. If the characteristic of K is not 2 or 3, we can actuay reach an equation for E of the simper form y 2 = x 3 + Ax + B with A, B K (for this, the next and a the basic properties of the Weierstrass equation see [14], III.1). To a Weierstrass equation over K, not necessariy of an eiptic curve, is associated a quantity K that is caed the discriminant of the equation. If 0 then the Weierstrass equation defines an eiptic curve (smooth). In this case we can associate to the equation another quantity j K which is caed the j invariant (or moduar invariant). The Weierstrass equations of two isomorphic eiptic curves have the same j-invariant, thus we can tak of the j invariant of an eiptic curve E. Conversey, two eiptic curves with the same moduar invariant are isomorphic over an agebraic cosure K of K. The points of an eiptic curve E over K form an abeian group with composition aw E E E given by rationa functions with coefficients in K. This group is caed the Morde-Wei group of E, and the neutra eement is the point O. Then, for any number n N we can define the subgroup E n of the n-torsion points on E(K), which are those points P with coordinates in K such that [n](p ) = P P = O, where is the composition on E and it is repeated n times. If the characteristic of K does not divide n we have that E n = Z/nZ Z/nZ but otherwise, this is never true. In particuar if p is the characteristic of K, it is E p = Z/pZ or Ep = {O} ([14], p. 86, Coroary 6.4). Consider now a prime number different from the characteristic of K. As said above we have E n = Z/ n Z Z/ n Z for a n N. Furthermore, we have group homomorphisms E n+1 [] E n and we can thus take the projective imit of these groups T = im n E n which is caed the Tate modue of E. This is a Z -modue of rank 2, where Z is the ring of -adic numbers Z = im n Z/ n Z. 16

19 Let now (E, O), (E, O ) be two eiptic curves over K with their respective point with coordinates in K. An isogeny from E to E is an agebraic morphism f : E E with coefficients in K, such that f(o) = O. We have that every isogeny is a homomorphism between the Morde-Wei groups of E and E ([14], p. 71, Theorem 4.8). The group of endomorphisms End(E) of an eiptic curve E is the group formed by isogenies E E with composition given by the equation (f + g)(p ) = f(p ) g(p ). The mutipication [n] by a number n N is obviousy an eement of End(E), and hence the group of endomorphisms aways contains Z. If End(E) = Z we say that the eiptic curve E is without compex mutipication, otherwise if End(E) is bigger, we say that E has compex mutipication Reduction of eiptic curves Here K is a number fied, or a compete fied with respect to a discrete vauation, and O K is its ring of integers. Let E be an eiptic curve over K. If v is a discrete vauation of K, we can obtain a Weierstrass equation for E with a the coefficients in O K and with minima vauation v( ) ([14], p. 186, Proposition 1.3). Then we can appy reduction moduo the maxima idea p = {x O K v(x) > 0}. We obtain an equation with coefficients in the residue fied k = O K /p, which is a fied of characteristic p p. This equation defines a curve Ẽ over k which can be singuar and therefore is not necessariy an eiptic curve. We have a map π : E Ẽ (t : x : y) (t mod p : x mod p : y mod p) where we took projective coordinates for the points in E and Ẽ. If / p, this means that the curve Ẽ is not singuar, hence it is an eiptic curve over k and we say that E has good reduction at v (or p). In this case π is actuay a surjective homomorphism between the Morde-Wei groups of E and Ẽ ([14], p. 188, Proposition 2.1). If p does not divide n the map E n Ẽn is an isomorphism ([14], p. 192, Proposition 3.1) and the modue of n-torsion points E n is unramified over K (if L is the extension constructed from K adjoining the coordinates of the points in E n, then L K is an unramified Gaois extension) ([14], p. 195, Proposition 4.1). If instead p, then Ẽ is singuar. We say then that E has bad reduction at v. Let j be the moduar invariant of E. If v(j) = v( ) < 0 then we say that the reduction at v is of mutipicative type, otherwise we say it is of additive type. If a curve has mutipicative reduction at every pace of bad reduction, we say that the eiptic curve E is semistabe. A eiptic curves over K become semistabe over a finite extension of K ([14], 17

20 p. 197, Proposition 5.4). The foowing is known as Šafarevič s theorem. It gives information about eiptic curves over a number fied with the same paces of bad reduction. Theorem. Let K be a number fied and S Σ K a finite subset of paces of K. Then there are ony a finite number of eiptic curves over K up to isomorphism that have good reduction at a paces outside S. [14], p. 293, Theorem 6.1. Aso [11], p. IV-7, Theorem. Coroary. Let E be an eiptic curve over a number fied K. There are ony a finite number of eiptic curves E over K up to isomorphisms, such that there exists a non trivia isogeny between E and E. [14], p. 294, Coroary 6.2. Aso [11], p. IV-7, Coroary Gaois representations attached to eiptic curves Let K be a number fied and G = Ga(K/K). Let E be an eiptic curve over K and P E for a prime number. Of course we have [](P ) = O. Let σ G. The map [] has coordinates in K so it commutes with σ. Thus we have [](σ(p )) = σ([](p )) = σ(o) = O. Therefore σ(p ) E. This means that the Gaois group G acts on the modue E, which is a vector space of dimension 2 over the finite fied F. Thus, we have a representation ϕ : Ga(K/K) Aut(E ) = GL 2 (F ) for any prime number. These are caed Gaois representations of K attached to E. In the same way we can obtain homomorphisms Ga(K/K) Aut(E n) = GL 2 (Z/ n Z) for a n N. Hence, we take the projective imit of the groups Aut(E n) = GL 2 (Z/ n Z) with respect to n (with natura transition maps) to get Ga(K/K) Aut(T ) = GL 2 (Z ) where T is the Tate modue of E associated to the prime number, and Z is the ring of -adic numbers. Now, tensoring with the -adic fied Q, which is the fraction fied of Z, we obtain a representation ρ : Ga(K/K) Aut(V ) = GL 2 (Q ) 18

21 where V = T Q is a vector space over Q of dimension 2. This is caed the -adic Gaois representation of K attached to E. Note now that the group End(E) acts on the torsion points of E (by the equivaence [n]f(p ) = f([n]p )), so that aso End(E) has representations in E, V for any prime number. Moreover, the morphisms in End(E) have coefficients in K, hence the action of G = Ga(K/K) on E, V, commutes with the action of End(E) on the same vector spaces. It is natura to ask if the image of the Gaois representation ϕ is the biggest possibe, that is the biggest subgroup of Aut(E ) = GL 2 (F ) commuting with the image of End(E). If E is without compex mutipication, the image of the representation ( ) of End(E) in Aut(E ) λ 0 = GL 2 (F ) consists ony of the scaar matrices, 0 λ λ F, in fact we ask if the foowing equaity hods ϕ (G) = Aut(E ) = GL 2 (F ) Serre s theorem comes in this framework, and states that the answer is yes for amost a prime numbers. The first step to understand the Gaois representations attached to E is to study their irreducibiity (absence of proper stabe subspaces). We have the foowing theorem Theorem. Let E be an eiptic curve without compex mutipication. Then the representation ρ prime numbers. : Ga(K/K) Aut(V ) is irreducibe for a the representation ϕ : Ga(K/K) Aut(E ) is irreducibe for amost a prime numbers. Lemma. Let E be an eiptic curve over K without compex mutipication. Let E, E be eiptic curves over K such that there exist isogenies E E, E E with kernes that are non isomorphic finite cycic groups. Then E and E are not isomorphic Proof. Let n, n be the orders of the kernes of E E and E E. Suppose E, E isomorphic. It exists an isogeny E E that has cycic kerne of order n ([14], p. 81, Theorem 6.1). Then we can make the composition E E = E E 19

22 which is an eement of End(E), and since E is without compex mutipication, it has to be a mutipication [a] by a number a N. We have then an exact sequence 0 Z/n Z Z/aZ Z/aZ Z/n Z 0 hence n, n must divide a and n n = a 2. This impies a = n = n, that is impossibe. We can now prove the theorem. Proof. We begin with the first assertion. It is enough to show that there is not any subspace of V = T Q of dimension 1 that is stabe for Ga(K/K). Suppose there is one and ca X its intersection with the Tate modue T. X is a free Z modue of rank 1, so et x be a generator and et n 0 be the smaer n N such that the image of x in E n = T / n T is not zero. Then for a n > n 0 the image X (n) of X in E n is a cycic submodue of E n of order n n 0. Furthermore, X (n) is stabe for Ga(K/K) by construction. Thus, there exists an eiptic curve E (n) and an isogeny E E (n) such that X (n) = Z/ n Z is the kerne of this isogeny ([14], p. 74, Proposition 4.12). The above emma shows that any two curves in the set {E (n) n > n 0 } are not isomorphic. This contradicts the coroary to Šafarevič s theorem in The second assertion can be proved in a very simiar way (see [11], p.iv- 9, Theorem, (b) and aso [14], p.294, Coroary 6.3). 1 Subgroups of GL 2 (F p ) In this section p is a prime number and V is a vector space of dimension 2 over the fied F p = Z/pZ. We have GL(V ) = GL 2 (F p ). We wi now study some of the subgroups of GL(V ). 1.1 Cartan subgroups Given two different ines in V, if C is the subgroup in GL(V ) of a the eements for which those ines are stabe, we ca such group a spit Cartan subgroup. ( ) Choosing a convenient basis of V, C can be written in the 0 form, hence it is abeian of order (p 1) 0 2. Its image in PGL(V ) = GL(V )/F p is a cycic group of order p 1. Let k End(V ) be a fied with p 2 eements. We ca the subgroup k of GL(V ) a non-spit Cartan subgroup. It is a cycic group of order p 2 1 and its image in PGL(V ) is aso cycic of order p

23 The determinant homomorphism det : C F p is surjective. It s cear when C is spit. If C is non-spit, we can think of it as the matrices representing mutipication by invertibe eements when we write F p 2 in some base as a vector space over F p. Then just note that the extension F p 2 of F p is obtained (when p 2) adjoining to F p a square root of an eement generating F p. We note that if the discriminant = Tr(s) 2 4det(s) of the characteristic poynomia of s GL(V ) is zero, then s is either a mutipication by a scaar and ( s is ) in a the Cartan subgroups ( of GL(V ) ), or s can be written in the form λ and we have s λi = which is a noninvertibe eement in 0 λ 0 0 End(V ) different from 0, so s is in none of the Cartan subgroups of GL(V ). Instead, if 0 and p 2, then s is contained in exacty one Cartan subgroup. In fact, if is a square in F p, it foows that s can be diagonaized and so it is contained in a spit Cartan subgroup. Otherwise, if is not a square, then s generates a non-spit Cartan subgroup of matrices that diagonaize over F p 2 in the same base as s. Finay, observe that if a subgroup H of PGL(V ) is the image of any Cartan subgroup of GL(V ), then there is no other Cartan subgroup of GL(V ) whose image in PGL(V ) is equa to H. 1.2 Normaizers of Cartan subgroups Suppose p 2. Let C be a Cartan subgroup of GL(V ) and k = F p C End(V ) the subagebra generated by C. k is isomorphic to F p F p or F p 2. In both cases #Aut(k) = 2. In fact if C is spit then k = F p F p and the ony nontrivia automorphism is the one doing (a, b) (b, a). Instead if k = F p 2 (C non-spit) we have Aut(k) = Ga(F p 2/F p ) = Z/2Z. Let now N be the normaizer of C in s GL(V ). There is the exact sequence 0 C N Aut(k) 0 s (x sxs 1 ) ( ) 0 so C has index 2 in N. The eements in N C are those of the form 0 if C is spit, or they are those such that s(λx) = λ p s(x) (λ k = F p 2, x V ) if C is non-spit. ( ) 2 0 b We observe that s N C impies s 2 F p. In fact = a 0 ( ) ab 0, that expains the caim when C is spit. When C is non-spit, we 0 ab 21

24 note that the eements of C are the matrices representing the mutipication by an invertibe eement in F p 2 when we write it (in some base) as a vector space over F p. Then aso the automorphism σ : x x p of F p 2 is F p -inear and it is represented by a matrix. Now, the eements of N C are those of the form σx with x C and we have (σx) 2 = σxσx = σx p+1 = x p2 +p = x p+1 which is aways an eement λ F p (such that λ p = λ). Thus, if we ca C 1 and N 1 the images of C and N in PGL(V ), N 1 is then the normaizer of C 1 in PGL(V ) and the eements in N 1 C 1 are a of order 2. Proposition 1.1. Suppose p 5. Let C be a Cartan subgroup of GL(V ) and N be its normaizer. Let C be another ( Cartan ) subgroup of GL(V ) or 0 a subgroup that can be written in the form, with C 0 1 N. Then we have C = C or C C. Proof. Let C 1, N 1 and C 1 be the images of C, N and C in PGL(V ). We know that C 1 is cycic of order p ± 1 > 2. If s is a generator of C 1 then s C 1, otherwise s wi be in N 1 C 1 where a eements are of order 2. Moreover, the images in PGL(V ) of two different Cartan subgroups have trivia intersection, so being s C 1 C 1 we have C 1 = C 1 and the proposition foows. 1.3 Bore subgroups Given one ine in V, if B is the subgroup of GL(V ) of a the eements for which that ine is stabe, we ca such group a Bore subgroup. ( Choosing ) a convenient basis, the eements of B can be written in the form and 0 so B is of order p(p 1) 2. The foowing proposition gives a condition for a subgroup of GL(V ) to be contained in a Bore subgroup. Proposition 1.2. Let G be a subgroup of GL(V ) of order divisibe by p. Then, either G contains SL(V ) or G is contained in a Bore subgroup of GL(V ). Proof. ( ) The eements of GL(V ) of order p can a be written in the form 1 1. In fact an eement x of order p satisfies the poynomia condition 0 1 x p 1 = 0, that, as we are in characteristic p, is (x 1) p = 0. So, the eigenvaues of x are both equa to 1. Furthermore, x cannot be the identity, hence it can be written in the showed form in some basis and it has exacty one fixed ine. If a eements of G of order p fix the same ine, then that 22

25 ine is stabe for G. In fact, for a s G, and t of order p, we have that sts 1 is aso of order p. Furthermore, if t fixes the ine D then sts 1 fixes sd. Since a the eements of order p fix the same ine, it is sd = D and thus G is contained in a Bore subgroup of GL(V ). If instead, there are in G at east two eements of order p with different stabe ines, ( we) can( choose ) a basis such that G wi contain the eements of the form,. This impies that G contains SL(V ) ([5], p. 537, Lemma 8.1). 1.4 Subgroups of order prime to p Here is a proposition about such subgroups in PGL 2 (F p ). Proposition 1.3. Let H be a subgroup of PGL 2 (F p ) of order prime to p. If H is not cycic or dihedra, then it is isomorphic to one of the foowing group: A 4, the aternating group on 4 eements S 4, the symmetric group on 4 eements A 5, the aternating group on 5 eements In particuar the order of the eements in H can be ony 1, 2, 3, 4 or 5. Proof. Take S the set of eigenspaces (over an agebraic cosure of F p ) of the eements of H except the identity. Notice that a the eements in PGL 2 (F p ) that have equa eigenvaues, are the identity or eements of order p (see proof of Proposition 1.2), then a non trivia eements of h have two distinct eigenspaces over an agebraic cosure of F p. The group H acts on S. In fact if E S we have ge = E for some g H, then (hgh 1 )he = he for a h H. Let S E = {h H he = E} be the isotropy group of E for some E S. Notice that #S E = #S he for any h H (g hgh 1 is a bijection from S E in S he ). Let now O 1,..., O r be the orbits in S for the action of H. It is #O i = #H/#S E for a E O i, i = 1,..., r. Choose representatives E 1,..., E r of O 1,..., O r. We have 2#H 2 = #H #S E1 (#S E1 1) + + #H #S Er (#S Er 1) 23

26 where the two sides of the equivaence comes from two ways of counting pairs (h, E) where h H with h 1 and E is an eigenspace for h. We obtain then 2(1 1 #H ) = (1.1) #S E1 #S Er Note that #S Ei 2 for a i = 1,..., r because E i is an eigenspace for same non trivia eement of H. Then r 4 is impossibe, otherwise the right hand side of (1.1) wi be at east 2, which is impossibe because the eft hand side is obviousy smaer than 2. Aso r = 1 is impossibe for simiar reason. If r = 2, then it must be #S E1 = #S E2 = #H. In this case a the non trivia eements of H have the same two eigenspaces E 1, E 2. This means that H is contained in the image of a Cartan subgroup of GL 2 (F p ) and thus it is cycic (see 1.1). Now suppose r = 3 and #S E1 #S E2 #S E3. It cannot be #S E1 3 otherwise the right hand side of (1.1) wi be greater than 2, which is impossibe as aready noted. Then #S E1 = 2, #H is even and #S E3 #H/2. 1 If #S E3 = #H/2 then (1.1) become + 1 = 1, so #S E1 = #S E1 #S E2 #S E2 = 2. In this case S E3 has index 2 in H, hence it is norma in H (aso it is the kerne of the homomorphism H {permutation group of O 3 } = S 2 given by the action of H on S). Take two eements x, y GL 2 (F p ) whose image in PGL 2 (F p ) is contained in S E3. Then in same basis they are both superior trianguar( matrices. ) The commutator 1 xyx 1 y 1 can be written in the form so it is the identity 0 1 or an eement of order p. Hence xyx 1 y 1 is the identity because its image in PGL 2 (F p ) is in H which is a group of order prime to p. x and y commutes and they have the same two eigenspaces. We have then that S E3 is contained in the image of a Cartan subgroup of GL 2 (F p ) and H is contained in the image of the normaizer of such Cartan subgroup. Furthermore, S E3 is cycic and any eement in H S E3 has order 2 (see 1.2) hence H is dihedra. Suppose now #S E3 < #H/2 and thus #S E2 > 2. It cannot be #S E2 4, otherwise the right hand side of (1.1) woud be at east = 2 which is impossibe. Hence #S E2 = 3. With #S E1 = 2 and #S E2 = 3, that is the ony possibiity remaining, it cannot be #S E3 6, otherwise again the right hand side of (1.1) woud be greater than = 2. There remain ony the three cases #S E3 = 3, 4, 5. 24

27 If #S E1 = 2, #S E2 = 3, #S E3 = 3, then #H = 12. In this case we have #O 3 = 4. Then we can consider the homomorphism H {permutation group of O 3 } = S 4 given by the action of H on the set of eigenspaces S. This homomorphism is injective, otherwise there is a non trivia eement of H with 4 distinct eigenspaces, impossibe. Then, since #H = 12, it is H = A 4 If #S E1 = 2, #S E2 = 3, #S E3 = 4, then #H = 24. In this case we have #O 2 = 8. We have aready showed in the case #S E3 = #H/2 with the commutator argument, that if two eements have an eigenspace in common, then they have both eigenspaces in common. Then, if we define X = {(E, F ) = (F, E) E, F O 2, E F, he = E, hf = F for some h H} we have #X = 4 and we can consider the homomorphism H {permutation group of X} = S 4 given by the action of H on S. Suppose there exist a H with a 1 such that a is in the kerne of this homomorphism. Then given (E, F ) X we have either ae = E, af = F or ae = F, af = E. This impies a 2 = 1 and excudes the possibiity of ae = E, af = F for some (E, F ) X because the isotropy group of an eement in O 2 has order 3. Take now x S E = S F with x 1 for some pair (E, F ) X. x has order 3, then it acts triviay on (E, F ) and acts ike A 3 on the remaining three pairs of X. Hence ax is an eement of H of order 6, which is impossibe because no isotropy group has order greater than 4. The above homomorphism is injective, and since #H = 24, it is H = S 4 If #S E1 = 2, #S E2 = 3, #S E3 = 5, then #H = 60. From the equivaence #S E = #S he, we get that for a E S it is #S E = 2, 3, 5 that are a prime numbers. Then a non trivia eements of H have order 2, 3 or 5. Furthermore, two cycic subgroups of H of the same order, are conjugated under H, because they are the isotropy groups of two eigenspaces in the same orbit O i (ge = E (hgh 1 )he = he for a h H). A norma subgroup of H hence must contain a or none of the eements of H of a given order. The eements of order #S Ei (i = 1, 2, 3, #S Ei = 2, 3, 5) are a the non trivia eements of 25

28 the isotropy groups of the eigenspaces in O i. They are in number 1 60 (#S Ei 1). So in H there are 15 eements of order 2, 20 2 #S Ei eements of order 3 and 24 eements of order 5. These cardinaities impy that H does not have any proper norma subgroups, H is simpe. Since the ony simpe group of order 60 is A 5, it is Proposition is proved. H = A 5 Let G be a subgroup of GL(V ) of order not divisibe by p, and et H be its image in PGL(V ). The previous proposition impies that we are in one of the foowing situations: H is cycic. Take a generator h of H. It is the image in PGL(V ) of some eement g G. Let be the discriminant of the characteristic poynomia of g. cannot be zero, otherwise h woud be the identity or an eement of order p. Instead, if 0 we know that g is contained in a unique Cartan subgroup of GL(V ) (see section 1.1). H is then contained in the image in PGL(V ) of a Cartan subgroup of GL(V ). This impies that aso G is contained in a Cartan subgroup of GL(V ). H is dihedra. Then H contains a cycic subgroup C of index 2. The previous case impies that C is contained in the image C in PGL(V ) of a Cartan subgroup of GL(V ). Since C is norma in H we have that H is contained in the normaizer of C C in PGL(V ). Then G is contained in the normaizer of a Cartan subgroup of GL(V ). H is isomorphic to A 4, S 4 or A 5. The eements of H are then of order 1, 2, 3, 4 or Subgroups containing Cartan subgroups The foowing proposition gives a condition for a subgroup of GL(V ) to be equa to the whoe GL(V ). Proposition 1.4. Suppose p > 5. Let G be a subgroup of GL(V ) containing ( a ) Cartan subgroup C or a subgroup C that can be written in the form 0. Then, we are in one of the foowing cases: 0 1 G = GL(V ) 26

29 G is contained in a Bore subgroup of GL(V ) G is contained in the normaizer of a Cartan subgroup of GL(V ). Proof. If the order of G is divisibe by p, then Proposition 1.2 impies that either G is contained in a Bore subgroup of GL(V ) or it contains SL(V ). In the atter case we have G = GL(V ). In fact, the image of C by the determinant map det : GL(V ) F p is surjective (see 1.1), then we can write each eement of GL(V ) as a product cs with c C and s SL(V ). If the order of G is not divisibe by p et H be the image of G in PGL(V ). Section 1.4 then tes us that it is enough to excude the case of H isomorphic to A 4, S 4, A 5. The image of C in PGL(V ) is cycic of order p ± 1 > 5, but A 4, S 4 and A 5 do not have eements of order greater than 5 hence the proposition foows. 2 Representation of inertia group In this section K is a fied of characteristic 0 that is compete with respect to a discrete vauation v and v(k ) = Z. Let O K be its ring of integers, p the maxima idea of O K and k = O K /p the residue fied. We suppose k finite of characteristic p > 0 and we write e = v(p) <. Let K be an agebraic cosure of K. The vauation v extends in a unique way to a vauation of K ([8], p. 28, Proposition 3). We have v(k ) = Q ([8], p. 29, Coroary 4) and the residue fied k of K is an agebraic cosure of k ([8], p. 54, Coroary 1). We have K K ur K t K where K ur is the maxima unramified extension of K contained in K and K t is the biggest tamey ramified extension of K ur in K, which means that K t is the composite of a the finite ramified extension of K ur with ramification index not divisibe by p. We write G = Ga(K/K), I = Ga(K/K ur ) and I p = Ga(K/K t ) respectivey the Gaois group, the inertia group and the inertia p-group of K. G has the structure of profinite group (projective imit of finite groups) through the equaity Ga(K/K) = im L Ga(L/K) where the imit is extended to a finite Gaois extensions L of K in K. In the same way, aso I is a profinite group, and I p is the biggest pro-p-group (projective imit of finite p-groups) contained in I. In fact, if we take a finite Gaois extension K of K ur in K not contained in K t, its degree must 27

30 be divisibe by a power p n > 1, because the ramification index divides the degree of the extension ([8], p. 29, Coroary 1). Now, Syow theorems impy that there is a finite Gaois extension L of K ur with K ur L K t and L = K K t. This shows that the projective component given by K in I p is a finite group of order p n. On the other hand any finite Gaois extension of K ur of degree a power of p, cannot have any intersection with K t, because such intersection woud not be tame. I p is a norma subgroup of I. In fact the restriction of s Ga(K/K) to a every finite Gaois extension of K preserves ramification indexes, then we have s(k t ) K t, that is s(k t ) = K t. Now just note that for any eement g I p, we have that sgs 1 fixes s(k t ) = K t. The quotient I t = I/I p = Ga(K t /K ur ) is a projective imit of finite groups of order prime to p. 2.1 Projective structure of I t Let d be a positive integer not divisibe by p and µ d the group of d-th roots of unity in K. Then µ d K ur, because the poynomia x d 1 does not have mutipe roots in characteristic p and so neither does the minima poynomia of an eement of µ d. This group thus become isomorphic to the group of d-th roots of unit in k taking reduction by maxima idea. If x is a uniformizer of K ur and K d = K ur (x 1 d ), the extension K d of K ur is tamey ramified of degree d. In fact, we have that the residue fied of K ur is the agebraic cosure k of k ([8], p. 54, Coroary 1), then we can sove in K ur the equation x d = u for a u OK ur (directy, writing u as a convergent series ([8], p. 32, Proposition 5)). We get that a the eements α of Kur with v(α) 0 mod d are in (Kur) d and so Kur/(K ur) d = Z/dZ is generated by the uniformizer x, by the exactness of 0 Our Kur v Z 0. Finay, Kummer theory ([2], Chap. III, 2, Lemma 2) impies that the ony cycic Gaois extension of K ur of degree d is K d = K ur (x 1 d ), which is tame because of degree prime to p. Its Gaois group is isomorphic to µ d, in fact we have the natura isomorphism θ d : Ga(K d /K ur ) µ d such that s(x 1 d ) = θd (s) x 1 d which does not depend on the choice of x or x 1 d. The fied K t is the composite of the K d s for a d not divisibe by p, because any Gaois extension of K ur of degree prime to p is union of cycic Gaois extension of order prime to p and we showed that the ony cycic Gaois extension of K ur of order d is K d. Hence I t = Ga(K t /K ur ) = im d Ga(K d /K ur ), and the isomorphisms θ d define an isomorphism θ : I t im d µ d 28