On the main conjecture of Iwasawa theory for certain non-cyclotomic Z p -extensions
aa r X i v : . [ m a t h . N T ] F e b ON THE MAIN CONJECTURE OF IWASAWA THEORY FORCERTAIN NON-CYCLOTOMIC Z p -EXTENSIONS YUKAKO KEZUKA
Abstract.
Let K = Q ( √− q ) , where q is any prime number congruent to modulo , with ring of integers O and Hilbert class field H . Suppose p ∤ [ H : K ] is a prime number which splits in K , say p O = pp ∗ . Let H ∞ = HK ∞ where K ∞ is the unique Z p -extension of K unramified outside p . Write M ( H ∞ ) for the maximal abelian p -extension of H ∞ unramified outside the primesabove p , and set X ( H ∞ ) = Gal( M ( H ∞ ) /H ∞ ) . In this paper, we establishthe main conjecture of Iwasawa theory for the Iwasawa module X ( H ∞ ) . Asa consequence, we have that if X ( H ∞ ) = 0 , the relevant L -values are p -adicunits. In addition, the main conjecture for X ( H ∞ ) has implications toward (a)the BSD Conjecture for a class of CM elliptic curves; (b) weak p -adic Leopoldtconjecture. Introduction
The study of the main conjectures of Iwasawa theory, which relates the p -adic L -functions to the characteristic power series of certain Iwasawa modules, has beenvery influential in the development of modern number theory, and has been appliedto a wide circle of problems in which values of L -functions play a key role. Itprovided, for example, one of the most fruitful approaches to understanding theconjecture of Birch and Swinnerton-Dyer. The strongest general result known inthis direction so far is for elliptic curves E defined over an imaginary quadraticfield K with complex multiplication by the ring of integers of K . For these curves,Rubin showed [16, Theorem 11.1], by first proving appropriate main conjectures,that if L ( E/K, = 0 , then the p -part of the Birch–Swinnerton-Dyer conjectureholds for all p not dividing the number of roots of unity in K . In particular, theprime p = 2 is always omitted, and in many ways this is the most interestingprime, because the product of Tamagawa factors which appears in the formulaof the Birch–Swinnerton-Dyer conjecture can be divisible by a large power of .However, the arguments used in the proof seem very difficult to extend to coverthis case, except when E has potential ordinary reduction at the primes above .In this paper, we study a family of quadratic twists of elliptic curves with complexmultiplication which are no longer defined over K , but over the Hilbert class field H of K . We formulate and prove certain main conjectures at primes including p = 2 .Let K = Q ( √− q ) , where q is a prime congruent to modulo . Then thediscriminant of K is equal to − q , so the class number h of K is odd by genustheory. Let O denote the ring of integers of K , and let H be the Hilbert class fieldof K . Let p denote a prime such that ( p, q ) = 1 and p splits in K , say p O = pp ∗ .Write H ∞ = HK ∞ where K ∞ is the unique Z p -extension of K unramified outside This work is partially supported by the SFB 1085 “Higher invariants” at the University ofRegensburg, funded by the Deutsche Forschungsgemeinschaft (DFG). p , given by global class field theory. Let G = Gal( H ∞ /K ) and G = Gal( H ∞ /K ∞ ) ,and assume ( p, h ) = 1 . Denote by M ( H ∞ ) the maximal abelian p -extension of H ∞ unramified outside the primes of H ∞ above p , and write X ( H ∞ ) = Gal( M ( H ∞ ) /H ∞ ) . Then X ( H ∞ ) is a finitely generated torsion module over the Iwasawa algebra Z p [[ G ]] . Let I be the ring of integers of the completion of the maximal unram-ified extension of K p . We briefly recall the structure theorem for finitely gener-ated torsion modules over the Iwasawa algebra Λ I ( G ) = I [[ G ]] of G with coeffi-cients in I . Given a finitely generated torsion Λ I ( G ) -module M and χ ∈ G ∗ :=Hom( G, C × p ) , write M χ for the largest submodule of M on which G acts via χ .Since p ∤ G ) by assumption, any Λ I ( G ) -module decomposes into the direct sumof its χ -components. Furthermore, Λ I ( G ) χ is (non-canonically) isomorphic to thering I [[ T ]] of formal power series in indeterminate T with coefficients I . Thus, thewell-known structure theorem for finitely generated torsion I [[ T ]] -modules easilyimplies that there exist elements f , . . . , f r of Λ I ( G ) and pseudo-isomorphisms ⊕ rj =1 Λ I ( G ) / ( f i ) → M and M → ⊕ rj =1 Λ I ( G ) / ( f i ) . The ideal ( Q ri =1 f i )Λ I ( G ) is an invariant of M called the characteristic ideal of M , and is denoted by char( M ) . Furthermore, for every χ , we will denote by char ( M χ ) ⊂ Λ I ( G ) χ the characteristic ideal of the Λ I ( G ) χ -module M χ . Similarly,given a finitely generated torsion Z p [[ G ]] -module X , we will denote by char ( X ) and char ( X χ ) the characteristic ideals of X ˆ ⊗ Z p I and ( X ˆ ⊗ Z p I ) χ , respectively.In this paper, we shall prove the main conjecture of Iwasawa theory for theextension H ∞ /H . We remark that the case p = 2 is specifically excluded from[8, Chapter III] where de Shalit formulates the main conjecture and gives someevidence in favour of it. We construct in Section 3.1 an I -valued pseudo-measure ν p on G which interpolates relevant Hecke L -values. For a precise statement ofthis result, see Theorem 3.11. Define ϕ = I I ( G ) ν p , where I I ( G ) denotes theaugmentation ideal of Λ I ( G ) . We can now state the main result of this paper. Theorem 1 (Theorem 3.19, Main Conjecture for H ∞ /H ) . For every χ ∈ G ∗ , wehave char ( X ( H ∞ ) χ ) = ϕ χ . We end this section by outlining the proof and applications of Theorem 3.19. Theproof closely follows the methods of Rubin [16]. Later, Gonzalez-Avilés extendedRubin’s methods to study the case p = 2 [11, Theorem 3.5.1] using quadratic twistsof the elliptic curve X (49) which is defined over Q and has complex multiplicationby the ring of integers of Q ( √− . In the proof, it is vitally important that isa potentially ordinary prime for all quadratic twists of X (49) , since splits in Q ( √− . Unfortunately, there are no other elliptic curves with complex multipli-cation defined over Q for which is a potentially ordinary prime, since Q ( √− is the only imaginary quadratic field with class number one in which splits. Wewill thus extend their ideas by looking at a family of elliptic curves which are nolonger defined over Q or K . In [12], Gross proved the existence of an elliptic curve A ( q ) defined over the field J of index in H with complex multiplication by O and minimal discriminant − q . In the case q = 7 , we have A (7) = X (49) . Let E be any quadratic twist of A ( q ) by a quadratic extension of the form H ( √ λ ) /H of discriminant prime to q , λ ∈ K × . Let p be a prime such that E has good N THE MAIN CONJECTURE OF IWASAWA THEORY 3 reduction at all places of H above p , and p splits in K , say p O = pp ∗ . The theoryof complex multiplication then tells us that the reduction of E is of ordinary typeat all primes of H above p , a fact which will be important for the arguments ofIwasawa theory to follow. In particular, p = 2 satisfies these conditions, and wewill pay special attention to this case which is quite different in nature. Note thatthe ideals in O are in not in general principal since the class number of K is notequal to if q > . Given a non-zero ideal a of O , define E a = ∩ α ∈ a E α , where E α = ker (cid:16) E ( ¯ F ) α −→ E ( ¯ F )) (cid:17) . Let F n = H ( E p n ) , and set F ∞ = H ( E p ∞ ) , H = Gal( F ∞ /H ) . Let O p be the ring of integers of K p = Q p . We make an observation that in thecase q > , we do not have H = K , so that the formal group b E of E at a place v of H where E has good reduction is not in general a Lubin–Tate group of E over H v . We overcome this difficulty by an argument which involves relative Lubin–Tategroups introduced by de Shalit [8, Chapter I §1], showing that there is a canonicalisomorphism χ p : H → O × p given by the action of H on E p ∞ , and H = ∆ × Γ , where ∆ is cyclic of order p − or , if p > or p = 2 , and Γ is isomorphic to O p .Write ¯ E H ∞ and U H ∞ for the groups of global units and principal semi-local unitsdefined in Section 3.3. Let A ( H ∞ ) be the projective limit of the p -primary part ofthe ideal class group of H n = F n ∩ H ∞ with respect to the norm maps, and let ¯ C H ∞ be the group of elliptic units defined in Section 3.2. Global class field theoryprovides an exact sequence of Z p [[ G ]] -modules(1) → ¯ E H ∞ / ¯ C H ∞ → U H ∞ / ¯ C H ∞ → X ( H ∞ ) → A ( H ∞ ) → . We prove in Chapter 5 that char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) = ϕ χ for every χ ∈ G ∗ . In Chapter 4, we construct an Euler system of the elliptic units ¯ C H ∞ . Note that in [8, Chapter II], de Shalit sketches rather elaborate proofs of thefunctional equations that are satisfied by the elliptic function Θ (see [8, II.2.3]) inorder to study the properties of the elliptic units, referring back to arguments ofRobert and Kubert–Lang. In particular, in [8, Chapter III] de Shalit takes a throot of Θ and shows a norm compatibility relation satisfied by the elliptic units,modulo roots of unity. He then uses that w p , the number of roots of unity in K congruent to modulo p , is equal to for p > . In this paper, we will deal with a th root of Θ from the beginning, and all the functional equations will be proveddirectly by simple, purely algebraic arguments with rational functions on E , in thespirit of the Appendix of [5].We then use a variant of Čebotarev’s theorem and induction to establish a divisi-bility relation between the characteristic ideal of (cid:0) ¯ E H ∞ / ¯ C H ∞ (cid:1) χ and that of A ( H ∞ ) χ in Z p [[Γ]] . Since the characteristic ideals of a Γ -module behave well under extensionof scalars, this implies the following divisibility relation in Λ I (Γ) = I [[Γ]] : Theorem 2 (Theorem 4.18) . (1) If p is an odd prime, we have char ( X ( H ∞ ) χ ) | char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) . YUKAKO KEZUKA (2) If p = 2 , we have char ( X ( H ∞ ) χ ) | π k char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) for some integer k > , where π is a uniformiser of I . The uniformizer π in part (ii) of the above theorem appears due to the fact thatthe restriction map in Lemma 4.7 is not necessarily injective for p = 2 .In Chapter 5, we finish the proof of the main conjecture by showing that X ( H ∞ ) and U H ∞ / ¯ C H ∞ have the same Iwasawa invariants. We first follow the paper ofCoates and Wiles [7] to compute the Iwasawa invariants of X ( H ∞ ) , and then com-pute the Iwasawa invariants of U H ∞ / ¯ C H ∞ using the analytic class number formulaand Kronecker’s second limit formula.As a corollary, Theorem 3.19 implies that X ( H ∞ ) = 0 if and only if ϕ is aunit. It is known that (see [3, Section 5]), when p = 2 , X ( H ∞ ) = 0 for all primes q ≡ with q < and q = 431 . Thus Theorem 3.11 gives us informationon the p -adic valuation of L ( ψ kE/H ,k )Ω ∞ ( E/H ) k for positive integers k with k ≡ ,where ψ E/H denote the Grössencharacter of
E/H and Ω ∞ ( E/H ) denotes the com-plex period. In particular, it tells us in this case that ϕ is a unit if and only if ( χ p ( γ ) − L ( ψ E/H , / Ω ∞ ( E/H ) is a unit at p , where γ is a topological gener-ator of Γ . It follows that X ( H ∞ ) = 0 if and only if ord p (cid:18) L ( ψ E/H , ∞ ( E/H ) (cid:19) = − or − , if p > or p = 2 . This can be checked numerically in the case E = X (49) and p = 2 . Indeed, we can compute using the computer package Magma that L ( ψ E/H , / Ω ∞ ( E/H ) = , and we have ord p (cid:0) χ p ( γ ) − (cid:1) = 3 since γ is a topo-logical generator of Γ ≃ O p . We remark also that the main conjecture for H ∞ /H is an important step to proving the main conjecture for F ∞ /F , which canbe used to study the p -part of the Birch–Swinnerton-Dyer Conjecture for E/H .Furthermore, for p = 2 , the construction of the p -adic L -function in Chapter 3 andthe computation of the Iwasawa invariants in Chapter 5 of this paper are crucialfor the proof in [3] that X ( H ∞ ) is a finitely generated Z p -module (the case p > was proven independently by Gillard [9, Theorem 3.4] and Schneps [17, TheoremIV]). This can be applied to prove the weak p -adic Leopoldt conjecture for certainnon-abelian extensions. Recall that the Leopoldt conjecture holds for a numberfield F if the Leopoldt defect of F vanishes, or equivalently, if the p -adic regula-tor R p ( F ) of F (see (25) for a definition with respect to p ) does not vanish. Forabelian extensions of K , this follows from Baker’s theorem [1] on linear forms in the p -adic logarithms of algebraic numbers, which was proven by Brumer [2, Theorem1]. Now, for p = 2 , it can be shown by an argument involving Nakayama’s lemmathat X ( F ∞ ) is a finitely generated Z -module for any quadratic extension F of H and F ∞ = F H ∞ . This allows us to prove the weak p -adic Leopoldt conjecturefor a class of non-abelian extensions F ∞ /K , which asserts that the p -adic defect ofLeopoldt is always bounded as one goes up the Z p -extension F ∞ / F , or equivalently,that X ( F ∞ ) is Z p [[Γ]] -torsion (see [4, Lemma 14]). As a consequence, we obtainthat E ( H ∞ ) and E ( F ∞ ) modulo torsion are finitely generated abelian groups. Thisis discussed further in [3]. N THE MAIN CONJECTURE OF IWASAWA THEORY 5 The Gross Curves
Take q to be any prime number with q ≡ . Let K = Q ( √− q ) , and fixan embedding K ֒ → C . Let E be an elliptic curve over C with End C ( E ) = O , thering of integers of K . Since K has prime discriminant, the class number, which wedenote by h , is odd. In the case q = 7 , we can take E to be any quadratic twist ofthe elliptic curve A = X (49) with equation A : y + xy = x − x − x − . This is the only family of quadratic twists of elliptic curves with complex multipli-cation defined over Q for which is a potentially ordinary prime, since q = 7 is theonly case in which K has class number one. In general, the theory of complex mul-tiplication tells us that the modular invariant j ( O ) is a real number which satisfiesan irreducible equation of degree h over K , and the Hilbert class field H of K isgiven by H = K ( j ( O )) . Furthermore, given a rational prime p , E has potentiallygood ordinary reduction at all primes of H above p if and only if p splits in K .From now on, let p be a prime number such that E has good ordinary reductionat all primes of H above p , and p splits in K , say p O = pp ∗ . We also define J = Q ( j ( O )) , which satisfies [ H : J ] = 2 . Then for any prime number q with q ≡ , Gross showed in [12, Theorem 12.2.1] that there exists an ellipticcurve A ( q ) which is defined over J with End H ( E ) = O . In the simplest case q = 7 ,we have A (7) = X (49) . This is done by constructing a Grössencharacter ψ q of H .Let a be an integral ideal of H . Define ψ q to be the unique Grössencharacter withconductor ( √− q ) such that, if a is an integral ideal of H with ( a , q ) = 1 , then ψ q ( a ) = α, where α is the unique generator of the principal ideal N H/K ( a ) which is a squarein O / √− q O . In particular, we have σ ( ψ q ) = ψ q for all σ ∈ Gal( H/ Q ) . Thisdefines an isogeny class of elliptic curves defined over H with Grössencharacter ψ q , j -invariant equal to j ( O ) and complex multiplication by O . Gross showed thatwe can pick out in this isogeny class a unique elliptic curve A ( q ) defined over J with Grössencharacter ψ A ( q ) /H = ψ q such that End H ( A ( q )) = O , j ( A ( q )) = j ( O ) and the minimal discriminant ideal is equal to ( − q ) . In addition, Gross found anexplicit equation for A ( q ) over J . We will show this via a slightly different method.Let us consider a generalised Weierstrass equation of A ( q ) of the form y + a xy + a y = x + a x + a x + a with a i ∈ H . Let ∆( A ( q )) denote the discriminant for this equation. We claim thatwe can have a i ∈ J with ∆( A ( q )) = − q . Given an integral ideal a of O , let σ a denote the image of a via the Artin isomorphism from the ideal class group of K to G = Gal( H/K ) , and let λ ( a ) denote the unique isogeny from A ( q ) to B = A ( q ) σ a of degree N a defined over H , characterised by λ ( a )( u ) = σ a ( u ) for any u ∈ A ( q )[ c ] with ( c , a ) = 1 . Let x ′ , y ′ be the coordinates of any generalisedWeierstrass equation for B , and let ∆( B ) be the discriminant of this equation. Wewrite ω A ( q ) = dx y + a x + a , ω B = dx ′ y ′ + a ′ x ′ + a ′ YUKAKO KEZUKA for the Néron differentials. Then we see that the value Λ( a ) ∈ H × defined by λ ( a ) ∗ ( ω B ) = Λ( a ) ω A ( q ) is such that ∆( B )Λ( a ) is independent of the choice of Weierstrass equation for B . Further, it is shown in [5, Appendix, Theorem 8] that there exists a unique c A ( q ) ( a ) ∈ H × such that c A ( q ) ( a ) gives a canonical th root in H of ∆( A ( q )) deg λ ( a ) ∆( B )Λ( a ) = ∆( A ( q )) N a − Λ( a ) . Taking appropriate values for a , we see in particular that ∆( A ( q )) has a th rootin H . By the definition of j (see [12, §1]), j ( A ( q )) has a cube root in H and j ( A ( q )) − has a square root in H . Note that the only roots of unity in H are ± , so j ( A ( q )) in fact has a cube root in J . Now we have the following. Proposition 2.1.
The curve A ( q ) has a model over Jy = x + mq · x − nq · where (2) m = j ( A ( q )) and n = j ( A ( q )) − − q , with discriminant equal to − q . Here, we take the positive square root for n .Proof. The arguments above show that m ∈ J , and n ∈ H . But j ( A ( q )) − and − q are both negative, so n ∈ J as well. An easy computation then showsthat indeed the curve defined by equation (2) has discriminant − q and j –invariantequal to j ( A ( q )) . Now, [13, Proposition 3.5] shows that there is an isomorphismover J from this curve to A ( q ) . (cid:3) The coefficients of the (2) are integral in J , except perhaps at and . It is notknown in general how to write a global minimal equation for A ( q ) over J explicitlyfor q > , although Gross has shown that it exists over J (see [13, Proposition 3.2]).Using a classical -descent, Gross showed that, for all q ≡ , we have [12,Theorem 22.4.1]: A ( q )( J ) = Z / Z , A ( q )( H ) = Z / Z × Z / Z , X ( A ( q ) /J )(2) G = 0 . There is one additional property of the curves A ( q ) which is important in carryingout arguments of Iwasawa theory for them. Let A ( q ) tor denote the torsion subgroupof A ( q )( J ) . By the theory of complex multiplication, H ( A ( q ) tor ) is an abelianextension of H . In fact, more is true since ψ A ( q ) /H satisfies ψ A ( q ) /H = ϕ A ( q ) ◦ N H/K where ϕ A ( q ) is a Grössencharacter of K with conductor ( √− q ) , and a theorem ofShimura [19, Theorem 7.44 ] states that the existence of such a ϕ A ( q ) is equivalentto H ( A ( q ) tor ) being an abelian extension of K .In what follows, we assume E is a quadratic twist of A ( q ) by a quadratic ex-tension of H of the form H ( √ λ ) , where λ is some non-zero element of K and thediscriminant of H ( √ λ ) /H is prime to q . Thus, in particular, E has good ordinaryreduction at the primes of H above . Proposition 2.2.
We have ψ E/H = ϕ K ◦ N H/K , where ϕ K is a Grössencharacter of K . N THE MAIN CONJECTURE OF IWASAWA THEORY 7
Proof.
We have remarked that ψ A ( q ) /H = ϕ A ( q ) ◦ N H/K . Now, E is a twist of A ( q ) by a quadratic extension M of H which we assumed to be of the form HM where M is a quadratic extension of K . Let χ M (resp. χ M ) be the quadratic character of H (resp. K ) defining M (resp M ). Then we have χ M = χ M ◦ N H/K by class fieldtheory. Now, since M /H has discriminant prime to p , we have ψ E/H = ψ A ( q ) /H χ M .It follows that we can take ϕ K = ϕ A ( q ) χ M . (cid:3) Applying [19, Theorem 7.44], we immediately obtain that the field H ( E tor ) isabelian over K . It also follows that E is isogeneous over H to all of its conjugatesunder G , since we have ψ E/H = ψ E σ /H , that is, E and E σ have isomorphic Ga-lois representations on their Tate modules, and so, they are isogenous over H byFaltings’ theorem.Given an ideal b of O prime to the conductor g of the Grössencharacter ϕ K , let σ b be the Artin symbol of b for H/K . Let λ E ( b ) : E → E σ b denote the unique H -isogeny whose kernel is E b , obtained by restricting the Serre–Tate character of the abelian variety B/K [18, Theorem 10], which is the restrictionof scalars of E from H to K . If v is any place of H , we write H v for the completionof H at v , and write O v for its ring of integers. In the case q = 7 , we have H = K ,so that for every place v of H where E has good reduction, the formal group b E of E at v is a Lubin–Tate group of E over H v . However, if q > , this is no longertrue because ψ E/H ( v ) will no longer be a local parameter of H v in general. We firstbriefly discuss how one handles this situation.Let v be any place of H lying above a prime w of K such that E has goodreduction at v , and let σ v ∈ G be the Frobenius at v . Let λ E ( v ) : E → E σ v denotethe unique isogeny induced by the isogeny λ E ( w ) . We remark that the isogeny λ E ( v ) is defined by the same formulae which define the isogeny λ ( v ) : A ( q ) → A ( q ) σ v . To see this, recall the notations in the proof of Proposition 2.2 and let τ be the nontrivial element of Gal( M /H ) . Then E ( H ) is isomorphic to the − eigenspace for the action of Gal( M /H ) on A ( q )( M ) , that is, the points on A ( q )( M ) on which τ acts as − . But we have λ ( v )( − P ) = − λ ( v )( P ) since isogeny preservesthe group law, and also we clearly have χ M ( τ ) = − . Hence λ ( v ) is independentof twist by χ M . This induces a homomorphism b λ E ( v ) : b E → b E σ v , of formal groups of the curves E and E σ v at v , defined over the ring of integers O v of H v . Thus, we can view b λ E ( v ) as an element of O v [[ t ]] satisfying(3) b λ E ( v )( t ) ≡ Λ( v ) t mod degree , b λ E ( v )( t ) ≡ t q w mod v, where Λ( v ) is an element of O v and q w denotes the cardinality of the residue fieldof the restriction w of v to K . Now, we can apply σ iv for i = 1 , . . . , f v , where f v denotes the residue degree of v in H/K , to λ E ( v ) and b λ E ( v ) . Then we see that N H v /K w Λ( v ) = ψ E/H ( v ) , since Q f v i =1 σ iv λ E ( v ) is the unique element of End H ( E ) = O which reduces modulo v to the Frobenius endomorphism at v . Thus b E is not itself a Lubin–Tate group,but b E together with the homomorphism b λ E ( v ) : b E → b E σ v is a relative Lubin–Tategroup, which was studied by de Shalit in [8, I §1]. The theory of Lubin–Tate groups YUKAKO KEZUKA generalises to relative Lubin–Tate groups, and in particular, we have the following([8, Proposition I.1.8]):
Theorem 2.3.
Let v be any place of H where E has good reduction, and let w be its restriction to K . Then for any n > , the extension H v ( E w n ) /H v is totallyramified, and its Galois group is isomorphic to ( O /w n ) × . We define F n = H ( E p n ) , and F = F or F , according as p = 2 or p > . Set F ∞ = H ( E p ∞ ) , H = Gal( F ∞ /H ) . Then by Theorem 2.3, we have a character χ p : H → O × p = Z × p giving the action of H on E p ∞ , which is an isomorphism. We write H = ∆ × Γ , where ∆ = Gal ( F/H ) is cyclic of order or p − if p = 2 or p > and Γ = Gal ( F ∞ /F ) is isomorphic to Z p . 3. Construction of the p -adic L -functions Construction of the p -adic L -function for H ∞ /H . Let K ∞ denote the unique Z p -extension of K unramified outside p , and let H ∞ denote the composite field HK ∞ . Then H ∞ is a subfield of F ∞ such that H ∞ /H is a Z p -extension, and it is clear that H ∞ = F ∆ ∞ . We now construct the p -adic L -function which interpolates the values of the L ( ψ kE/H , k ) for integers k > with k ≡ . This gives rise to the p -adic L -function for H ∞ /H , and it isan essential ingredient for the main conjecture related to the Birch–Swinnerton-Dyer conjecture for E/H at p = 2 . The construction of the p -adic L -functioninterpolating the values at k ≡ can be found in [14]. We will followthe ideas in [6] which deals with the case p > . The case p = 2 cannot be foundin literature.Write x , y for the coordinates of E/H . We fix a generalised global minimalWeierstrass equation for E over H , which exists by [13, Proposition 3.2], to be(4) y + a xy + a y = x + a x + a x + a . Recall that G denotes the Galois group of H over K . Then applying σ ∈ G to (4)gives a generalised global minimal Weierstrass equation for E σ /H . Let ω σ be theNéron differential on E σ , and note that the discriminant of this equation ∆( E σ ) is equal to (∆( E )) σ = ∆( E ) . Let L (resp. L σ ) be the period lattice of the Nerondifferential on our global minimal Weierstrass equation for E (resp. E σ ). Thenthere exists Ω ∞ ∈ C × such that L = Ω ∞ O . The uniformisation Φ : C /L ∼ −→ E ( C ) is accomplished through Φ( z, L ) = (cid:18) ℘ ( z, L ) − (( a ) + 4 a )12 , (cid:18) ℘ ′ ( z, L ) − a (cid:18) ℘ ( z, L ) − (( a ) + 4 a )12 (cid:19) − a (cid:19)(cid:19) . Given a principal ideal a = ( α ) with α ∈ O and ( a , f ) = 1 , define R a ( P ) = c E ( a ) Y U ( x ( P ) − x ( U )) − , where U runs over any set of representatives of E a \{O} modulo {± } , and c E ( a ) isan element of H whose th power is equal to ∆( E ) N a − / Λ( a ) , where Λ( a ) ∈ H × satisfies λ E ( a ) ∗ ( ω σ a ) = Λ( a ) ω. N THE MAIN CONJECTURE OF IWASAWA THEORY 9
Thus R a ( P ) is a rational function on E with coefficients in H . Let us write P forthe generic point on E σ with coordinates ( x, y ) . Applying σ ∈ G to the coefficientsof R a ( P ) , we obtain a rational function R σ a ( P ) on the curve E σ /H . For ease ofnotation, we will work with E but the arguments are identical if we replace this by E σ and consider rational functions on E σ over H . Proposition 3.1.
Let b be an integral ideal of K with ( b , a ) = 1 . Then we have R σ b a ( λ E ( b )( P )) = Y R ∈ E b R a ( P ⊕ R ) . Proof.
Recall that the kernel of λ E ( b ) is E b , and λ E ( b ) is injective on E a since ( b , a ) = 1 . Hence, the left hand side and the right hand side of the above equationhave the same divisor, and R σ b a ( λ E ( b )( P )) Q R ∈ E b R a ( P ⊕ R ) is a non-zero element of H . It can be shown, thanks to the unique scaling factor c E ( a ) in our definition of the rational functions, that this constant is equal to .See [5, Appendix, Theorem 4] for details. (cid:3) Let k be a positive even integer, so that the conductor of ϕ kK is (1) . We write P n for a primitive p n -division point of E . Note that R a ( P ) has a zero of order N a − at P = O , and R a ( P n ) is not a unit. To get rid of this zero at P = O , define theindex set (5) I = { ( a i , n i ) , i = 1 , . . . , r, a i = ( α i ) ⊂ O , ( a i , p ) = 1 , n i ∈ Z with r X i =1 n i (N a i −
1) = 0 } . Given D = ( a i , n i ) ∈ I , define R D ( P ) = r Y i =1 R a i ( P ) n i . Then R D ( P ) has no zero at P = O , and R D ( P n ) is a unit, as we will see in Corollary3.14.Define G k ( L ) = P w ∈ L \{ } w k for k > , G ( L ) = lim s → P w ∈ L \{ } w − | w | − s and G ( L ) = 0 . Proposition 3.2.
Let s be an integral ideal of K prime to f such that σ s = σ .Then for any D = ( a i , n i ) ∈ I and k > an even integer, we have (cid:18) ddz (cid:19) k log R σ D (Φ( z, L σ )) | z =0 = r X i =1 − n i ( k − ϕ kK ( s )Λ( s ) k Ω k ∞ (cid:0) N a i − α ki (cid:1) L ( ϕ kK , σ, k )) . Proof.
Let L = Z ω + Z ω be a complex lattice, whose basis is ordered so that ω /ω belongs to the upper half plane. We will modify the Weierstrass σ -functionslightly, and define Θ( z, L ) = exp (cid:26) − G ( L ) z (cid:27) σ ( z, L ) . Recall that for any integer k > , we can define the Kronecker–Eisenstein series H k ( z, s, L ) = X w ∈ L ( z + w ) k | z + w | s , where the sum in taken over all w ∈ L , except − z if z ∈ L . This series converges for Re(s) > k2 + 1 , and it has analytic continuation to the whole complex s -plane. Thenon-holomorphic Eisenstein series E ∗ k ( z, L ) is defined by E ∗ k ( z, L ) := H k ( z, k, L ) .Furthermore, it is well-known that (see [10, Corollary 1.7]) for any z ∈ C \ L , wehave(6) ddz log Θ( z + z , L ) = z A ( L ) − + ∞ X k =1 ( − k − E ∗ k ( z , L ) z k − , where A ( L ) = ω ω − ω ω πi . By [10, Theorem 1.9], for any principal integral ideal a = ( α ) with ( a , f ) = 1 , we have Θ ( z, L σ ) N a Θ ( z, α − L σ ) = Y w ∈ α − L σ /L σ w =0 ( ℘ ( z, L σ ) − ℘ ( w, L σ )) − , so we can write R σ D (Φ( z, L σ )) = r Y i =1 (cid:18) c E ( a i ) Θ ( z, L σ ) N a i Θ ( z, α − i L σ ) (cid:19) n i . Now, (6) gives ddz log Θ( z, L σ ) = ∞ X k =1 ( − k − G k ( L σ ) z k − and G k ( L σ ) = 0 for k odd. Therefore, ddz log R σ D (Φ( z, L σ )) = r X i =1 X k > k even − n i (N a i − α ki ) G k ( L σ ) z k − by the homogeneity of G k . Let b be an ideal of K . Setting k = s , g = (1) and ρ = Ω ∞ in [10, Proposition 5.5], we see that the partial Hecke L -function L ( ϕ kK , σ b , k ) is identically equal to G k ( L σ b ) = ϕ kK ( b )Λ( b ) k Ω k ∞ L ( ϕ kK , σ b , k ) . Hence, setting b = s and noting ϕ kK ( a i ) = α ki ( k is even), we obtain (N a i − α ki ) G k ( L σ ) = ϕ kK ( s )Λ( s ) k Ω k ∞ (cid:0) N a i − α ki (cid:1) L ( ϕ kK , σ, k ) . This completes the proof of the proposition. (cid:3)
Define Ψ D ( P ) = R D ( P ) N p R σ p D ( λ E ( p )( P )) , so that we have(7) Y R ∈ E p Ψ D ( P ⊕ R ) = 1 . N THE MAIN CONJECTURE OF IWASAWA THEORY 11
Now, we fix an embedding i v : K → K p , and we let v denote the prime of H above p determined by i v . Let t = − xy be a parameter for this formal group. Let A D ( t ) be the development as a power series in t of the rational function Ψ D ( P ) .We will show that A D ( t ) ∈ P O P [[ t ]] , and so C D ( t ) = p log A D ( t ) ∈ O P [[ t ]] . Lemma 3.3.
Let B D ( t ) denote the t -expansions of R D ( P ) . Then B D ( t ) is a unitin O v [[ t ]] .Proof. Let U be any non-zero element of E a i . Let P denote any prime of H ( U ) above v , and let O P be the ring of integers of the completion of H ( U ) at P . Notethat x ( U ) is integral at P because ( a i , p ) = 1 . Thus we see that the coefficientsof the t -series expansion of x ( P ) − x ( U ) all belong to O P . Moreover, R D ( P ) isholomorphic at t = 0 , and so there are no negative powers of t in its t -seriesexpansion. It remains to show that the constant term of the t -series expansion of R D ( P ) is a unit. This follows from Corollary 3.14, where we show that R D ( P n ) isa global unit for a point P n of E with ord P ( P n ) < . (cid:3) From this we obtain
Corollary 3.4.
Let A D ( t ) denote the t -expansion of Ψ D ( P ) . Then A D ( t ) belongsto m v [[ t ]] , where m v denotes the maximal ideal of O v .Proof. Write B D ( t ) = ∞ P n =0 a n t . Thus, by the previous lemma, a n ∈ O v for all n > and a ∈ O × v . Now, A D ( t ) = B D ( t ) p B σ p D ( b λ E ( v )( t ) ) , where b λ E ( v ) : b E → b E σ p isa homomorphism of formal groups of the curves E and E σ p at v induced by λ E ,defined over O v of H v . Then (3) gives us b λ E ( v )( t ) ≡ t p mod v , and we see that B σ p D (cid:16)b λ E ( v )( t ) (cid:17) = ∞ X n =0 a σ v n ( b λ E ( v )( t )) n ≡ ∞ X n =0 a pn t pn mod v. On the other hand, B D ( t ) p ≡ ∞ X n =0 ( a n t n ) p ≡ ∞ X n =0 a pn t pn mod v so A D ( t ) ≡ v , as required. (cid:3) Lemma 3.5.
Define C D ( t ) by C D ( t ) = 1 p log A D ( t ) . Then C D ( t ) ∈ O v [[ t ]] , and (8) X ω ∈D σ,p C D ( t [+] ω ) = 0 , where D σ,p denotes the group of p -division points on the formal group b E at a place v of H lying above p and [+] denotes the group law on b E . This group can be identifiedwith E p .Proof. We have C D ( t ) = 1 p X n > ( − n − ( A D ( t ) − n n .2 YUKAKO KEZUKA
Define C D ( t ) by C D ( t ) = 1 p log A D ( t ) . Then C D ( t ) ∈ O v [[ t ]] , and (8) X ω ∈D σ,p C D ( t [+] ω ) = 0 , where D σ,p denotes the group of p -division points on the formal group b E at a place v of H lying above p and [+] denotes the group law on b E . This group can be identifiedwith E p .Proof. We have C D ( t ) = 1 p X n > ( − n − ( A D ( t ) − n n .2 YUKAKO KEZUKA The first claim is now clear from the previous lemma as n > ord v b ( n ) + 1 for n > .The final equation then follows from (7). (cid:3) Let I be the ring of integers of the completion of the maximal unramifiedextension K ur p of K p . By [8, Proposition 1.6], we have an isomorphism δ σ,v : b G m ∼ −→ b E σ defined over I , where b G m denotes the formal multiplicative group and b E σ denotesthe formal group of E σ at v . Let J σ D ( W ) = C D ◦ δ σ,v ( W ) ∈ I [[ W ]] , and let µ D ,σ be the I -valued measure on Z p determined by J σ D ( W ) , that is,(9) J σ D ( W ) = Z Z p (1 + W ) x dµ D ,σ ( x ) . We claim that the measure µ D ,σ is supported on Z × p . Indeed, let Λ I ( Z p ) (resp. Λ I ( Z × p ) ) be the ring of I -valued measures on Z p (resp. Z × p ). Then we have aninclusion ι : Λ I ( Z × p ) ֒ → Λ I ( Z p ) given by extending the measures on Z p to Z × p byzero. Given a measure µ in Λ I ( Z p ) , write f µ ( W ) ∈ I [[ W ]] for the correspondingpower series given by the isomorphism Λ I ( Z p ) ∼ = I [[ W ]] . Then it is well-known(see [8, I.3.3] for more details) that µ belongs to ι (cid:0) Λ I ( Z × p ) (cid:1) if and only if f µ satisfiesthe equation X ζ ∈ µ p f µ ( ζ (1 + W ) −
1) = 0 . It follows from (8) that this is satisfied by J σ D . Writing also µ D ,σ for the corre-sponding measure in Λ I ( H ) , we have(10) Z H χ k p dµ D ,σ = Z Z p x k dµ D ,σ = D k J σ D ( W ) | W =0 , where D = (1 + W ) ddW . We have an isomorphism b G m ∼ −→ b G a given by W e z − ,hence we see immediately that D = ddz . Moreover, we have δ σ,v ( W ) = Ω σ,v W + · · · with Ω σ,v ∈ I × , so(11) D k J σ D ( W ) | W =0 = (cid:18) ddz (cid:19) k ( J σ D ( e z − | z =0 = 1N p Ω kσ,v (cid:18) ddz (cid:19) k log Ψ σ D (Φ( z, L σ )) | z =0 . Lemma 3.6.
We have Ω σ,v = Λ( s )Ω v , where Ω v ∈ I × is the coefficient of W inthe formal power series t = δ v ( W ) , and δ v : b G m ∼ −→ b E is an isomorphism definedover I .Proof. We have λ E ( s ) ∗ ( ω σ ) = Λ( s ) ω by definition, so that λ E ( s ) (Φ( z, L )) =Φ(Λ( s ) z, L σ ) . Hence, writing exp( z, L σ ) for the formal power series in z obtainedby expressing t = − x/y in terms of z using the isomorphism Φ( z, L σ ) for E σ , wealso have λ E (exp( z, L )) = exp(Λ( s ) z, L σ )) . Now, regarding z as the parameter ofthe formal additive group, exp( z, L σ ) is the exponential map of b E σ . It then followsby the uniqueness of the exponential maps for the formal groups that δ σ,v ( e z/ Ω σ,v −
1) = exp( z, L σ ) . On the other hand, we have δ σ,v = b λ E ( s ) ◦ δ v ( W ) , where b λ E ( s ) : b E → b E σ is theisomorphism over H v of formal groups induced by λ E ( s ) . Hence we have δ σ,v ( e z −
1) = exp(Λ( s )Ω v z, L σ ) . N THE MAIN CONJECTURE OF IWASAWA THEORY 13
The assertion follows by comparing the coefficients of z in the above equations. (cid:3) Lemma 3.7.
For an even integer k > , we have Λ( s ) − k Ω v − k Z H χ k p dµ D ,σ = r X i =1 − n i ( k − ϕ kK ( s )Λ( s ) − k Ω − k ∞ c k ( a i ) (cid:18) L ( ϕ kK , σ, k ) − ϕ kK ( p )N p L ( ϕ kK , σσ p , k ) (cid:19) , where c k ( a i ) = N a i − α ki .Proof. We have λ E ( p )Φ( z, L σ ) = Φ(Λ( p ) σ z, L σσ p ) and Λ( sp ) = Λ( s )Λ( p ) σ , so (cid:18) ddz (cid:19) k log R σσ p D ( λ E ( p )Φ( z, L σ )) | z =0 = r X i =1 − n i ( k − ϕ kK ( sp )Λ( s ) k Ω k ∞ c k ( a i ) L ( ϕ kK , σσ p , k ) . Therefore, (cid:18) ddz (cid:19) k log Ψ σ D (Φ( z, L σ )) | z =0 = (12) r X i =1 − n i ( k − ϕ kK ( s )N p Λ( s ) k Ω k ∞ c k ( a i ) (cid:18) L ( ϕ kK , σ, k ) − ϕ kK ( p )N p L ( ϕ kK , σσ p , k ) (cid:19) . Combining (10), (11) and (12), the proof of the proposition is complete. (cid:3)
Define G ∗ = Hom( G, C × p ) , where G = Gal( H/K ) as before. For each χ ∈ G ∗ , put D D ( χ, k ) = X σ ∈ G χ ( σ ) ϕ K ( s ) − k Z H χ k p dµ D ,σ . We conclude immediately that D D ( χ, k ) = c k ( D )( k − X σ ∈ G χ ( σ ) L ( ϕ kK , σ, k ) Λ( s ) k Ω kv Λ( s ) k Ω k ∞ ! (cid:18) − ϕ kK ( p ) χ − ( σ p )N p (cid:19) , where c k ( D ) = P ri =1 − n i c k ( a i ) . Let C denote a set of integral ideals representingof the ideal class group of K with ( c , pf ) = 1 for any c ∈ C , and set Ω ∞ ( E/H ) = Q c ∈C Λ( c )Ω ∞ and Ω p ( E/H ) = Q c ∈C Λ( c )Ω v . Recalling L ( ψ kE/H , k ) = Y χ ∈ G ∗ X σ ∈ G χ ( σ ) L ( ϕ kK , σ, k ) and the factorisation of primes of K in H given by class field theory, we obtain thefollowing lemmas: Lemma 3.8.
For any even integer k > , we have Y χ ∈ G ∗ D D ( χ, k ) = c k ( D ) h (( k − h Ω p ( E/H ) k Ω ∞ ( E/H ) − k L ( ψ kE/H , k ) · Y w | p − ψ kE/H ( w )N w ! . Lemma 3.9.
There exists a measure ν D in Λ I ( G ) such that for all k > , k ≡ , we have Ω p ( E/H ) − k Z G χ k p dν D = c k ( D ) h (( k − h Ω ∞ ( E/H ) − k L ( ψ kE/H , k ) · Y w | p − ψ kE/H ( w )N w ! . Note that, since k ≡ , we have χ k p ( τ ) = 1 for any τ ∈ ∆ . Hence, wecan naturally consider ν D as an element of Λ I ( G ) . The only dependence of ν D on D occurs in the factor c k ( D ) h . We claim that we can remove this factor and obtaina pseudo-measure which is independent of D . Lemma 3.10.
There exists an element D in the index set I defined in (5) and θ D ∈ Λ I ( G ) such that θ D | Γ generates the augmentation ideal of Λ I (Γ) ⊂ Λ I ( G ) and Z G χ k p dθ D = c k ( D ) h for all k > .Proof. Choose α ∈ O so that α ≡ p m +1 , α ≡ p m mod p ∗ m +1 where m = 1 or according as p > or p = 2 , and define a = ( α ) . Take a = a , a = a , n = 1 , n = − . Then ( { a , a } , { n , n } ) ∈ I . Write σ a for the Artin symbol ( a , H ∞ /K ) of a for H ∞ /K . Note that ( a , H/K ) = 1 since a is principal, so thatwe can consider σ a as an element of Γ . We will show that the measure θ D = − (N a − σ a − (N a − σ a )) = σ a − σ a , has the desired property. Indeed, we have χ k p ( θ D ) = c k ( D ) h , so it remains to showthat θ D | Γ generates the augmentation ideal of I [[Γ]] . In order to do this, let us fixa topological generator of γ of Γ , and write σ a | Γ = γ a , σ a | Γ = γ b where a, b ∈ Z p .It suffices to show that θ D | Γ = (1 − γ ) · u for u ∈ Z p [[Γ]] × . Now, we have Γ ≃ Z p and p m log : 1 + p m Z p → Z p sending p m x p m ∞ P i =1 ( − i − (cid:16) p m xi (cid:17) i is anisomorphism. Hence p m +1 | α − implies a ≡ p , and α ∗ generates p m O p so b p . Now, σ a | Γ − σ a | Γ = γ a − γ b = γ a (1 − γ b − a ) , where clearly γ a is a unit, and also b − a p so − γ b − a is a product of (1 − γ ) and a unit, as required. (cid:3) We define(13) ν p = ν D /θ D . This is a pseudo-measure, since (1 − γ ) · θ D is a unit by the proof of Lemma 3.10.The following is an immediate consequence of Lemma 3.9 and Lemma 3.10. Theorem 3.11.
There exists a unique element ν p belonging to the quotient field Λ I ( G ) such that, for all integers k > with k ≡ , we have Ω p ( E/H ) − k Z G χ k p dν p = (( k − h Ω ∞ ( E/H ) − k L ( ψ kE/H , k ) Y v ∈ P − ψ kE/H ( v )N v ! . Furthermore, the denominator of ν p is given by γ − , so that ( γ − ν p ∈ Λ I ( G ) . If we in addition assume ( p, h ) = 1 , the idempotents e χ := G ) P g ∈ G χ − ( g ) g corresponding to any χ ∈ G ∗ lie inside Λ I ( G ) , and thus we can decompose ν p as asum of elements in e χ Λ I (Γ) . Given χ ∈ G ∗ , let ν χ p ∈ Λ I (Γ) denote the χ -part of ν p in the decomposition. Then we have shown that ν χ p ∈ Λ I (Γ) for every χ = 1 ,and ( γ − ν χ p ∈ Λ I (Γ) for χ = 1 . Thus, identifying Λ I (Γ) with I [[ T ]] via the mapsending γ to T , we have ν χ p ∈ I [[ T ]] /T when χ is trivial. The pseudo-measure ν p will be used for the main conjecture for H ∞ /H . N THE MAIN CONJECTURE OF IWASAWA THEORY 15
Elliptic Units.
In this section, we will use the rational function R D to generate elliptic units.For n > and σ ∈ G , let P ( σ ) n be a primitive p n -division point on E σ satisfying λ E σ ( p ) P ( σ ) n = P ( σσ p ) n − , where σ p is the Artin symbol of p for H/K . Note that wecan write P ( σ ) n = Φ( ρ, L σ ) for some ρ ∈ p − n L σ \ L σ . Given an integral ideal b of K prime to a i and p , the image of P n under the Artin symbol of b for H ( E p n ) /K is λ E ( b )( P n ) , so a choice of P ( σ b ) n for the Artin symbol σ b of b for H/K is given by P ( σ b ) n = Φ(Λ( b ) ρ, L σ b ) , which is a point on E σ b .It can be shown that R a ( P n ) ∈ K ( p n ) , where K ( p n ) denotes the ray class fieldof K modulo p n (see [14, Theorem 4.3.1]). Proposition 3.12.
For any integer m > , we have N F m /F m − R a ( P m ) = R σ p a ( P ( σ p ) m − ) , where σ p = ( p , H/K ) denotes the Artin symbol of p for the extension H/K .Proof.
Write Φ( v, L ) = P m . The conjugates Φ( v, L ) τ of Φ( v, L ) as τ runs over Gal( F mh /F ( m − h ) are Φ( v + u, L ) for Φ( u, L ) ∈ E p . Hence N m,n R a (Φ( v, L )) = Y u ∈ p − L/L R a (Φ( v + u, L )) . But by Proposition 3.1, the right hand side is equal to R σ p a ( λ E ( p )(Φ( v, L ))) = R σ p a (cid:0) Φ(Λ( p ) v, L σ p ) (cid:1) , and Φ(Λ( p ) v, L σ p ) is a primitive p m − torsion point of E σ p .Hence Φ(Λ( p ) v, L σ p ) = P ( σ p ) m − by our choice of p -power torsion points. (cid:3) Let L be an arbitrary finite extension of K . We say that a ∈ L is a universalnorm from L ( E p ∞ ) if it is a norm from L ( E p n ) for every n > . The following iswell-known (see [5, Lemma 5]). Lemma 3.13.
Let L be a finite extension of K , and a ∈ L × a universal norm from L ( E p ∞ ) . Then every prime which divides a lies above p . Corollary 3.14. R D ( P n ) are global units.Proof. We note that if D = ( a i , n i ) , then R a i ( P n ) is a unit outside p again byLemma 3.13 because R a i ( P m ) ( m = 1 , , . . . ) is norm compatible in the tower F ∞ over F by Corollary 3.12. If P | p is a prime of F n , we have ord P ( x ( P n )) < but ord P ( x ( U )) > for any U ∈ E a i \{O} , giving ord P ( x ( P n ) − x ( U )) = ord P ( x ( P n )) .Recalling that ord P ( c E ( a i )) = 0 , we have ord P ( R a i ( P n )) = (N a i −
1) ord P ( x ( P n )) ,because ( E a i \{O} ) / {± } has order (N a i − . Hence ord P ( R D ( P n )) = 12 ord P ( x ( P n )) X i n i (N a i −
1) = 0 , since P i n i (N a i −
1) = 0 by the definition of D . It follows that R D ( P n ) is a unit. (cid:3) Let H n = F n ∩ H ∞ . For n > e with e = 0 or according as p is odd or even,we have [ H n : H ] = p n − − e . Note in particular that H n = H for n < e .Furthermore, for each n > , the classical theory of complex multiplication shows that F n = H ( E p n ) contains the field K ( p n ) , the ray class field of K modulo p n .Then if p = 2 , we have H n = K ( p n ) and H ∞ = K ( p ∞ ) = [ n K ( p n ) is a Z p -extension of H . We identify Γ with Gal( H ∞ /H ) . If p > , [ K ( p n ) : H ] = p n − ( p − and K ( p n ) strictly contains H n .Let U H n denote the group of semi-local units of H n ⊗ K K p = ⊕ P | p H n, P which arecongruent to modulo the primes above p . We denote by U H ∞ the projective limitof the groups U H n with respect to the norm maps. Similarly denote by U F n and U F ∞ the corresponding objects for F n and F ∞ . Let R σ D ( P ( σ ) n ) = N K ( p n ) /H n R σ D ( P ( σ ) n ) .In particular, R σ D ( P ( σ ) n ) = R σ D ( P ( σ ) n ) if p = 2 . Definition 3.15. (1) Define the group C H n to be the group generated by R σ D ( P ( σ ) n ) for all σ ∈ G , as D runs over the index set I .(2) We let ¯ C H n denote the closure of C H n in U H n , and define the group of ellipticunits ¯ C H ∞ = lim ←− ¯ C H n ⊂ U H ∞ where the inverse limit is taken with respect to the norm maps. Note that C H n is stable under the action of Gal( H n /K ) , and does not dependon the choice of P n . Note also that the roots of unity in H n are just {± } .Given u = ( u n ) ∈ U F ∞ , let g u ( W ) ∈ O F ⊗ O O p [[ W ]] denote the Coleman powerseries of u (see [8, Theorem I.2.2] for more details), where O F denotes the ring ofintegers of F . We write f log g u ( W ) = log g u ( W ) − p X ω ∈D σ, p log g u ( W [+] ω ) , where we recall that D σ, p = b E σ p can be identified with E σ p . It is well-known [8,Lemma I.3.3] that f log g u ( W ) has integral coefficients. Define i : U F ∞ → Λ I (Gal( F ∞ /K )) by u µ u := Y χ ∈ G ∗ X σ ∈ G χ ( σ ) ϕ K ( s ) − k µ u ,σ , where µ u ,σ is the measure satisfying(14) f log g u ◦ δ σ,v ( W ) = Z H (1 + W ) χ p ( τ ) dµ u ,σ ( τ ) . This induces an injective pseudo-isomorphism ([8, Proposition III.1.3])(15) i : U F ∞ ˆ ⊗ Z p I → Λ I (Gal( F ∞ /K )) . Let u D = ( R σ D ( P ( σ ) n )) . Then by construction, f log g u D = C σ D where C σ D is defined inLemma 3.5, and thus i ( u D ) = ν D = Q χ ∈ G ∗ P σ ∈ G χ ( σ ) ϕ K ( s ) − k µ D ,σ . N THE MAIN CONJECTURE OF IWASAWA THEORY 17
Statement of the Main Conjecture for H ∞ /H . From now on, we always assume that ( p, h ) = 1 , where h denotes the class numberof K . This implies that H ∞ /H is totally ramified at all primes above p , since K ∞ /K is totally ramified at all primes above p .Denote by M ( H ∞ ) the maximal abelian p -extension of H ∞ unramified outsidethe primes of H ∞ above p , and write X ( H ∞ ) = Gal( M ( H ∞ ) /H ∞ ) . For every n > , let E H n be the group of global units of H n , and let ¯ E H n bethe closure of E H n ∩ U H n in U H n in the p -adic topology. Then we define ¯ E H ∞ =lim ←− ¯ E H n , where the inverse limits are taken with respect to the norm maps. Astandard result from global class field theory says that the Artin map induces a Gal( H n /K ) -isomorphism U H n / ¯ E H n ≃ Gal( M ( H n ) /L ( H n )) , where M ( H n ) is themaximal abelian p -extension of H n unramified outside of the primes of H n above p , and L ( H n ) is the maximal unramified abelian p -extension of H n . Hence, writing X ( H n ) = Gal( M ( H n ) /H n ) , we have an exact sequence, and taking the projectivelimit over n , we obtain(16) → U H ∞ / ¯ E H ∞ → X ( H ∞ ) → Gal( L ( H ∞ ) /H ∞ ) → , where L ( H ∞ ) = ∪ n > L ( H n ) is the maximal unramified abelian p -extension of H ∞ .Let A ( H n ) denote the p -primary part of the ideal class group of H n , and let A ( H ∞ ) denote the inductive limit of A ( H n ) taken with respect to the naturalmaps coming from the inclusion of fields. Class field theory identifies A ( H ∞ ) with Gal( L ( H ∞ ) /H ∞ ) . Thus we obtain the fundamental exact sequence needed for theproof of the main conjecture:(17) → ¯ E H ∞ / ¯ C H ∞ → U H ∞ / ¯ C H ∞ → X ( H ∞ ) → A ( H ∞ ) → . Recall that G = Gal( H ∞ /K ) . Then we have G = G × Γ so that characters of G can naturally be considered as characters of G . Lemma 3.16.
We have i ( ¯ C H ∞ ˆ ⊗ Z p I ) = I I ( G ) · ν p , where i denotes the map in (15) and I I ( G ) denotes the augmentation ideal of Λ I ( G ) .Proof. Recall that i ( u D ) = ν D = θ D ν p . Hence we just need to show that I I ( G )Λ I ( G ) is generated by θ D , D ∈ I . In Lemma 3.10, we have found D ∈ I such that θ D | Γ generates I I (Γ) . It follows that for every χ ∈ G ∗ , we have i (cid:0) ( ¯ C H ∞ ˆ ⊗ Z p I ) χ (cid:1) = ( I I ( G ) · ν p ) χ . The result now follows since we have an isomorphism I [[ G ]] ≃ I [[Γ]][ G ] and thedecomposition I I ( G ) = ⊕ χ ∈ G ∗ I I ( G ) χ , where I I ( G ) χ = e χ I I (Γ) and I I (Γ) isthe augmentation ideal of Λ I (Γ) , which is generated by γ − . (cid:3) Define ϕ = I I ( G ) ν p ⊂ Λ I ( G ) . The following is an immediate consequence ofthe last two results. Theorem 3.17.
We have an exact sequence of Λ I ( G ) -modules → (cid:0) U H ∞ / ¯ C H ∞ (cid:1) ˆ ⊗ Z p I → Λ I ( G ) / ϕ → D → , where D is finite. Given a finitely generated torsion Z p [[ G ]] -module X , recall that char ( X χ ) ⊂ Λ I ( G ) χ denotes the characteristic ideal of the Λ I ( G ) χ -module ( X ˆ ⊗ Z p I ) χ . Corollary 3.18.
For every χ ∈ G ∗ , we have char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) = ϕ χ . We are now ready to state the main conjecture for H ∞ /H whose proof is inChapter 5. Theorem 3.19 (Main Conjecture for H ∞ /H ) . For every χ ∈ G ∗ , we have char ( X ( H ∞ ) χ ) = ϕ χ . Before we move on, we will verify that Theorem 3.19 holds for p = 2 and E = X (49) , which is equal to the case E = A ( q ) with q = 7 . In this case, we have M ( H ∞ ) = H ∞ , because the maximal abelian extension of K in M ( H ∞ ) coincideswith the union ∪ n K ( p n ) of ray class fields K modulo p n . Thus X ( H ∞ ) = 0 , and itfollows that Theorem 3.19 holds if and only if ϕ is a unit. By Theorem 3.11, thisholds if and only if ( χ p ( γ ) − L ( ψ E/H , / Ω ∞ ( E/H ) is a unit at p . It is easy tocheck that this holds, as discussed already in the introduction.4. Euler Systems
Euler Systems of the Elliptic Units.
Let G n = Gal( H n /K ) and Λ n = Z p [ G n ] . We denote by Λ( G ) = Z p [[ G ]] = lim ←− Λ n theIwasawa algebra of G . In this section, we will treat the G -modules occurring in thefundamental exact sequence 17 as Λ( G ) -modules. In fact, they are finitely generatedand torsion as Λ( G ) -modules. Given a finitely generated torsion Λ( G ) -module X ,write char Λ ( X ) for the characteristic ideal of X given by the structure theoremfor finitely generated torsion Λ( G ) χ ≃ Z p [[Γ]] -modules, and char Λ ( X χ ) for thecharacteristic ideal of X χ as a Λ( G ) χ -module. This means that char Λ ( X ˆ ⊗ Z p I ) =char( X ) . The aim of this chapter is to define and study Euler systems of theelliptic units ¯ C H ∞ , defined in Chapter 3, for the tower H ∞ /H . The method ofEuler systems we follow is due to Rubin [16, Chapter 1]. When combined with anapplication of Čebotarev density theorem, the results in this chapter enable us toprove a divisibility relation analogous to [16, Theorem 8.3]: char Λ ( A ( H ∞ )) divides p k char Λ ( ¯ E H ∞ / ¯ C H ∞ ) , for an integer k > ( k = 0 when p > ).Fix an integer ℓ > . Let I ℓ be the set of squarefree ideals of O which aredivisible only by primes q of K such that(i) q splits completely in H n /K , and(ii) N q ≡ p ℓ + e , where e = 0 or according as p is odd or even.Recall that K ( q ) denotes the ray class field of K modulo q . In the followinglemma, we define the field H n ( q ) . N THE MAIN CONJECTURE OF IWASAWA THEORY 19
Lemma 4.1.
Given a prime q ∈ I ℓ , we have a unique (cyclic) extension H n ( q ) of H n of degree p ℓ inside H n K ( q ) . Furthermore, H n ( q ) /H n is totally ramified at theprimes above q , and unramified everywhere else.Proof. Since q is unramified in H n /K , we have K ( q ) ∩ H n = H ∩ H n = H . Hence,we have Gal( H n K ( q ) /H n ) = Gal( K ( q ) /H ) , which isomorphic to ( O / q O ) × / e µ K ) via the Artin map, where e µ K denotes theimage of µ K under reduction modulo q . Since ( q ,
2) = 1 , the reduction modulo q map is injective, and this is cyclic of order (N q − / ( µ K )) where µ K ) = 2 .Hence it has a unique subgroup of order p ℓ since N q ≡ p ℓ + e , where e = 0 or if p > or p = 2 . Furthermore, H n K ( q ) /H n is totally ramified at the primesabove q and unramified everywhere else, so the assertions of the lemma follow. (cid:3) If r = Q li q i ∈ I ℓ , we write H n ( r ) for the composite H n ( q ) · · · H n ( q l ) , and put H n ( O ) = H n . Definition 4.2.
Fix an integer ℓ > . An Euler system relative to ℓ is a collectionof global units α = { α σ ( n, r ) : n > e, r ∈ I ℓ , σ ∈ G } satisfying(i) α σ ( n, r ) is a global unit of H n ( r ) , (ii) If q is a prime such that rq ∈ I ℓ , then (18) N H n ( rq ) /H n ( r ) ( α σ ( n, rq )) = α σ ( n, r ) − Frob − q where Frob q is the Frobenius of q in Gal( H n ( rq ) /K ) .(iii) (19) N H n +1 ( r ) /H n ( r ) ( α σ ( n + 1 , r )) = α σσ p ( n, r ) , where σ p = ( p , H/K ) ∈ G . The elliptic units give rise to an Euler system. The following is [15, Corollary7.7].
Proposition 4.3.
Suppose m is an ideal of O prime to af , P ∈ E m is a primitive m -division point of E and r is a prime ideal of K dividing m , say m = m ′ r . Then N K ( m ) /K ( m ′ ) R a ( P ) = ( R σ r a ( λ E ( r )( P )) − Frob − r if r ∤ m ′ R σ r a ( λ E ( r )( P )) if r | m ′ .where Frob r denotes the Frobenius of r in Gal( K ( m ′ ) /K ) , and N K ( m ) /K ( m ′ ) denotesthe norm map from K ( m ) to K ( m ′ ) . Proposition 4.4.
For all positive integers m > n , we have N H m /H n ¯ C H m = ¯ C H n , where N H m /H n denotes the norm map from H m to H n .Proof. By Corollary 3.12, we have N H m /H n R D ( P m ) = R σ m − n p D ( P ( σ m − n p ) n ) . Hence wehave N H m /H n C H m = C H n . This completes the proof of the proposition. (cid:3)
Proposition 4.5. If u ∈ C H n , then there exists an Euler system α with α σ ( n,
1) = u .Proof. It suffices to consider the case u = R σ D ( P ( σ ) n ) . Given r ∈ I ℓ , define α σn ( r ) = R σ D (cid:16) λ E σ ( r ) − ( P ( σ ) n ) (cid:17) . Then clearly α σn (1) = u and α σn ( r ) is a global unit in H n ( r ) .Furthermore, if q is a prime in I ℓ and rq ∈ I ℓ , then σ q = 1 , so by Proposition 4.3we have N H n ( rq ) /H n ( r ) ( α σn ( rq )) = R σ D (cid:16) λ E σ ( r ) − ( P ( σ ) n ) (cid:17) − Frob − q = α σn ( r ) − Frob − q , and similarly N H n +1 ( r ) /H n ( r ) ( α σn +1 ( r )) = R σσ p D (cid:16) λ E σσ p ( r ) − ( P ( σσ p ) n ) (cid:17) = α σσ p n ( r ) . Therefore, defining α σ ( n, r ) = α σn ( r ) gives the result. (cid:3) An Application of the Čebotarev Density Theorem.
Let ℓ > be a fixed integer. For every prime q ∈ I ℓ , write G q = Gal( H n ( q ) /H n ) .Then G q is cyclic of order p ℓ so we fix a generator τ q . Fix n > e , and let I H n = ⊕ Q Z Q denote the group of fractional ideals of H n written additively, where the sum runsover the prime ideals of H n . For every prime q of K , let I q = ⊕ Q | q Z Q = Z [ G n ] Q . For y ∈ H × n let ( y ) q , [ y ] and [ y ] q be the projection of the principal ideal ( y ) in I q , I H n /p ℓ I H n and I q /p ℓ I q respectively. Note that [ y ] and [ y ] q are well-defined for y ∈ H × n / ( H × n ) p ℓ .Suppose now that Q is a prime of H n lying above a prime q ∈ I ℓ , and we let e Q be the prime of H n ( q ) above Q . Suppose x ∈ H n ( q ) × and ρ ∈ G q . Then x − ρ mod e Q ∈ ( O H n ( q ) / e Q ) × , where O H n ( q ) denotes the ring of integers of H n ( q ) .We let x − ρ mod Q denote the image of x − τ q in ( O H n / Q ) × , and write ( x − τ q ) /d for the unique d -th root of the image of x − τ q in ( O H n / Q ) × / (( O H n / Q ) × ) p ℓ , where d = (N q − /p ℓ . Then the map H n ( q ) → ( O H n / Q ) × / (( O H n / Q ) × ) p ℓ , x → ( x − τ q ) /d is surjective, with kernel { x ∈ H n ( q ) × : ord e Q ( x ) ≡ p ℓ } . Let w be the imageof x under this map. Then setting l Q : ( O H n / Q ) × / (( O H n / Q ) × ) p ℓ ∼ −→ Z /p ℓ Z , w → ord e Q ( x ) mod p ℓ gives an isomorphism. Now define a map ϕ q : ( O H n / q O H n ) × / (( O H n / q O H n ) × ) p ℓ → I q /p ℓ I q by ϕ q ( w ) = P Q | q l Q ( w ) Q , where we also write l Q for the map composed with thenatural projection ( O H n / q O H n ) × / (( O H n / q O H n ) × ) p ℓ → ( O H n / Q ) × / (( O H n / Q ) × ) p ℓ . N THE MAIN CONJECTURE OF IWASAWA THEORY 21
Proposition 4.6.
Suppose α = { α σ ( n, r ) : n > e, r ∈ I ℓ , σ ∈ G } is an Eulersystem. Given σ ∈ Gal(
H/K ) , there exists a canonical map κ α : I ℓ → H × n / ( H × n ) p ℓ such that for every n > and r ∈ I ℓ we have κ α ( r ) = α σ ( n, r ) D a mod ( H n ( r ) × ) p ℓ ,where D a = Q q | a P p ℓ − i =0 iτ i q . Furthermore, if r ∈ I ℓ and q a prime of K , then(i) If q ∤ r then [ κ α ( r )] q = 0 .(ii) If q | r then [ κ α ( r )] q = ϕ q ( r / q ) .Proof. This follows easily from an alternative definition of Euler systems as a Galoisequivariant map which takes values in ∩ n, r H n ( r ) × . See [16, Proposition 2.2] and[16, Proposition 2.4] for details. (cid:3) Lemma 4.7.
Let res : H × n / ( H × n ) p ℓ → H n ( µ p ℓ + e ) × / ( H n ( µ p ℓ + e ) × ) p ℓ , be the natural map, where e = 0 or if p > or p = 2 . Then res is injective if p > , and if p = 2 .Proof. We have H × n / ( H × n ) p ℓ ≃ H ( H n /H n , µ p ℓ ) and H n ( µ p ℓ + e ) × / ( H n ( µ p ℓ + e ) × ) p ℓ ≃ H ( H n ( µ p ℓ + e ) /H n ( µ p ℓ + e ) , µ p ℓ ) by Hilbert’s Theorem 90. Hence ker(res) = H (Gal( H n ( µ p ℓ + e ) /H n ) , µ p ℓ ) . Also, H ∞ ∩ K ( µ p ∞ ) = K because p and p ∗ aretotally ramified in K ( µ p ∞ ) /K , but H ∞ /K is unramified outside p . It follows that H ∞ ∩ Q ( µ p ∞ ) = Q , and Gal( H n ( µ p ℓ + e ) /H n ) = ( Z /p ℓ + e ) × ≃ ∆ × Z /p ℓ − Z . Here, ∆ = Gal( H n ( µ p e ) /H n ) is cyclic of order p − or p if p is odd or even, and Gal( H n ( µ p ℓ + e ) /H n ( µ p e )) ≃ Z /p ℓ − Z . So if p > , Gal( H n ( µ p ℓ + e ) /H n ) is cyclicand we have ker(res) = 0 , as required. If p = 2 , taking the inflation-restrictionsequence gives → H (∆ , µ ) → ker(res) → H (Gal( H n ( µ ℓ +1 ) /H n ( µ )) µ ℓ ) , and H (∆ , µ ) = H (Gal( H n ( µ ℓ +1 ) /H n ( µ ) , µ ℓ ) = Z / Z . Hence | , and the result follows. (cid:3) We now employ the Čebotarev density theorem to obtain a prime of H n lyingabove that of I ℓ with properties desirable for the induction argument in Section4.3. Let I ( G ) denote the augmentation ideal of Λ( G ) . Theorem 4.8.
Suppose χ ∈ G ∗ . Let v ∈ (cid:16) H × n / ( H × n ) p ℓ (cid:17) χ and define V tobe the finite Λ n -submodule of ( H × n / ( H × n ) p ℓ ) χ generated by v . Finally, fix φ ∈ Hom Λ n ( V, Λ n /p ℓ Λ n ) , φ = 0 . Let c ∈ p e I ( G ) A ( H n ) χ , where e = 1 if p = 2 and e = 0 otherwise. Then there is a prime q ∈ I ℓ and a prime Q of H n above q suchthat(i) the ideal class of Q in A ( H n ) χ is equal to c ,(ii) [ v ] q = 0 and there exists r ∈ ( Z /p ℓ Z ) × such that ϕ q ( v ) = p e rφ ( v ) Q .Proof. Write H ′ n = H n ( µ p ℓ + e ) , and V res = V /V ∩ ker(res) , where res is the mapin Lemma 4.7. Note that V = V res if p = 2 by Lemma 4.7, and Kummer theorygives Gal( H ′ n ( v /p ℓ ) /H ′ n ) ≃ Hom( V res , µ p ℓ ) . Fix a primitive p ℓ -th root of unity2 YUKAKO KEZUKA
Suppose χ ∈ G ∗ . Let v ∈ (cid:16) H × n / ( H × n ) p ℓ (cid:17) χ and define V tobe the finite Λ n -submodule of ( H × n / ( H × n ) p ℓ ) χ generated by v . Finally, fix φ ∈ Hom Λ n ( V, Λ n /p ℓ Λ n ) , φ = 0 . Let c ∈ p e I ( G ) A ( H n ) χ , where e = 1 if p = 2 and e = 0 otherwise. Then there is a prime q ∈ I ℓ and a prime Q of H n above q suchthat(i) the ideal class of Q in A ( H n ) χ is equal to c ,(ii) [ v ] q = 0 and there exists r ∈ ( Z /p ℓ Z ) × such that ϕ q ( v ) = p e rφ ( v ) Q .Proof. Write H ′ n = H n ( µ p ℓ + e ) , and V res = V /V ∩ ker(res) , where res is the mapin Lemma 4.7. Note that V = V res if p = 2 by Lemma 4.7, and Kummer theorygives Gal( H ′ n ( v /p ℓ ) /H ′ n ) ≃ Hom( V res , µ p ℓ ) . Fix a primitive p ℓ -th root of unity2 YUKAKO KEZUKA ζ , and let ι : Λ n /p ℓ Λ n → µ p ℓ be the map sending P a σ σ mod p ℓ to ζ a . Define β := p e ( ι ◦ φ ) . Then by Lemma 4.7, we have β ∈ p e Hom( V res , µ p ℓ ) . Let b bethe element of p e Gal( H ′ n ( v /p ℓ ) /H ′ n ) corresponding to β via the Kummer map,so that β ( v ) = b ( v /pℓ ) v /pℓ . Let L n denote the unramified extension of H n such that A ( H n ) χ = Gal( L n /H n ) . Then we see that there exists a submodule W of V res suchthat Gal( L ′ n /L n ∩ H ′ n ) = Gal( L ′ n H ′ n /H ′ n ) = Hom( W, µ p ℓ ) , where L ′ n = L n ∩ H ′ n ( v /p ℓ ) . On the other hand, Gal( H ′ n /H n ) acts trivially on Gal( L ′ n H ′ n /H ′ n ) and µ p ∞ ( H n ) = µ , so that Hom( W, µ p ℓ ) = Hom( W, µ ) . There-fore, p e Gal( L ′ n /L n ∩ H ′ n ) = 0 , and b restricted to L ′ n is trivial. Furthermore, I ( G ) annihilates Gal( L n ∩ H ′ n /H n ) since H ′ n is abelian over H , so we can consider c asan element of p e Gal( L n /L ′ n ) . Hence we can choose ρ ∈ Gal( L n H ′ n ( v /p ℓ ) /H n ) suchthat ρ | L n = c and ρ | H ′ n ( v /pℓ ) = b . By the Čebotarev density theorem, we can picka prime Q of H n lying above a prime q ∈ I ℓ whose Frobenius is equal to ρ . Thenclass field theory identifies [ Q ] ∈ A ( H n ) χ with Frob Q ∈ Gal( L n /H n ) , so (i) follows.Now, [ v ] q = 0 because all primes lying above q are unramified in H ′ n ( v /p ℓ ) /H n ,and v is a p ℓ -th power in H ′ n ( v /p ℓ ) . Also, ord Q ( p e φ ( v ) Q ) = 0 ⇔ p e ( ι ◦ φ ( v )) = b (( v ) /p ℓ )( v ) /p ℓ = 1 ⇔ v is an p ℓ -th power modulo Q .On the other hand, ord Q ( ϕ q ( v )) = l Q ( v ) = 0 if and only if v is an p ℓ -th power mod-ulo Q . It follows that there exists r ∈ ( Z /p ℓ Z ) × with ord Q ( ϕ q ( v )) = r ord Q ( p e φ ( v ) Q ) ,and the map sending v to ϕ q ( v ) − p e rφ ( v ) Q gives rise to a G n -equivariant injec-tive homomorphism from V to ⊕ h ∈ G n h =1 ( Z /p ℓ Z ) Q h . But the latter has no non-zero G n -stable submodules. (cid:3) The Inductive Argument.
Let e = 0 or if p > or p = 2 . For n > e , let Γ n = Γ p n − − e . Define I ( H n ) tobe kernel of the restriction map Λ( G ) → Λ n , that is, the ideal of Λ( G ) generatedby { σ − σ ∈ Γ n } . Proposition 4.9. X ( H ∞ ) is a finitely generated torsion Λ( G ) -module, and it hasno non-zero finite submodule. Furthermore, X ( H ∞ ) /I ( H n ) X ( H ∞ ) is finite for any n .Proof. The first statement follows from [4, Lemma 13, Lemma 14]. Iwasawa theoryshows that I ( H n ) X ( H ∞ ) = Gal( M ( H ∞ ) /M ( H n )) , because M ( H n ) is the largestabelian extension of H n inside M ( H ∞ ) . Hence we have an exact sequence(20) → X ( H ∞ ) /I ( H n ) X ( H ∞ ) → X ( H n ) → Gal( H ∞ /H n ) → , where X ( H n ) = Gal( M ( H n ) /H n ) . Clearly the Z p -rank of Gal( H ∞ /H n ) is , andthe same is true for X ( H n ) since we have rank Z p ( X ( H n )) = rank Z p ( U H n / ¯ E H n ) byclass field theory, and the right hand side is equal to by the p -adic analogue ofLeopoldt’s conjecture which holds for abelian extensions of K . (cid:3) Lemma 4.10. char Λ ( A ( H ∞ )) is prime to I ( H n ) . N THE MAIN CONJECTURE OF IWASAWA THEORY 23
Proof. A ( H ∞ ) is a quotient of X ( H ∞ ) , so A ( H ∞ ) /I ( H n ) A ( H ∞ ) is a quotientof X ( H ∞ ) /I ( H n ) X ( H ∞ ) . Since the latter is finite by Proposition 4.9, we alsohave that A ( H ∞ ) /I ( H n ) A ( H ∞ ) is finite. Let f , . . . , f k ∈ Λ( G ) be such that char Λ ( A ( H ∞ )) is given by (cid:16)Q ki =1 f i (cid:17) Λ( G ) . It can then be shown using the snakelemma that Q ki =1 Λ( G ) / ( I ( H n ) + f i Λ( G )) is finite. The result now follows. (cid:3) Let π U : U H ∞ /I ( H n ) U H ∞ → U H n , π E : ¯ E H ∞ /I ( H n ) ¯ E H ∞ → ¯ E H n and π C :¯ C H ∞ /I ( H n ) ¯ C H ∞ → ¯ C H n denote the maps induced by the projection map, and let D p = Q P | p D P denotes the group generated by the decomposition groups D P of P in H ∞ /H . Write I ( D p ) for the ideal of Λ( G ) generated by { σ − σ ∈ D p } . Lemma 4.11. (i) I ( D p ) ker π U = I ( D p ) coker π U = 0 ,(ii) I ( D p ) ker π E = 0 ,(iii) There exists an ideal B of finite index in Λ( G ) such that I ( D p ) B coker π E = 0 . Proof.
For (i), see [16, Theorem 5.1]. By Proposition 4.9, X ( H ∞ ) Γ n is a finitesubmodule of X ( H ∞ ) , and therefore is equal to zero. It follows from (16) that (cid:0) U H ∞ / ¯ E H ∞ (cid:1) Γ n = 0 . Now [16, Theorem 7.6 (i)] easily applies to give an injec-tion ker π E → ker π U , so assertion (ii) follows from (i). To prove assertion (iii),apply the snake lemma to (16) and use the fact that X ( H ∞ ) Γ n = 0 to obtain A ( H ∞ ) Γ n ≃ ker π U/ E , where π U/ E : (cid:0) U H ∞ / ¯ E H ∞ (cid:1) /I ( H n ) (cid:0) U H ∞ / ¯ E H ∞ (cid:1) → U H n / ¯ E H n denotes the map induced by the projection map. Note that A ( H ∞ ) Γ n is finite,since A ( H ∞ ) /I ( H n ) A ( H ∞ ) is finite. Assertion (iii) now follows from (i) and (ii)on taking B to be the annihilator of the maximal finite submodule of A ( H ∞ ) in Λ( G ) . (cid:3) Lemma 4.12. rank Λ( G ) ( ¯ C H ∞ ) = 1 and coker( π C ) = ker( π C ) = 0 .Proof. By Lemma 3.16, there is a isomorphism of Λ( G ) -modules ¯ C H ∞ ≃ I ( G ) , where I ( G ) is the augmentation ideal of Λ( G ) , so the first statement follows onnoting that rank Λ( G ) (Λ( G ) /I ( G )) = rank Λ( G ) ( Z p ) = 0 . By Proposition 4.4, theprojection map π C is surjective, so coker π C = 0 . Now, the first statement ofthe theorem gives ¯ C H ∞ /I ( H n ) ¯ C H ∞ ≃ Λ n as Λ( G ) -modules. Furthermore, ¯ C H n isisomorphic to a submodule Y of finite index in Λ n . Define a map f : Λ n → Y sothat it commutes with the map π C . Then clearly ker π C ⊂ ker f and coker f is aquotient of coker π C , which is equal to zero. Thus ker f is finite, and hence equalto zero since Λ n has no non-zero finite submodules. The theorem now follows. (cid:3) In the next result, we obtain a map θ λ,n : ¯ E H n → Λ n which allows us to relate agenerator of char Λ ( ¯ E H ∞ / ¯ C H ∞ ) to the image under θ λ,n of an element of ¯ C H n . Corollary 4.13. char Λ ( ¯ E H ∞ / ¯ C H ∞ ) is prime to I ( H n ) . Furthermore, there existsan ideal B ⊂ Λ( G ) such that for every λ ∈ I ( G ) B , there is a map θ λ,n : ¯ E H n → Λ n satisfying λ char Λ ( ¯ E H ∞ / ¯ C H ∞ )Λ n ⊂ θ λ,n ( ¯ C H n ) . Proof.
The first assertion follows from Lemma 4.12 and the snake lemma. Forthe second assertion, we can adopt the proof of [16, Corollary 7.10] on letting B = A A where A satisfies Lemma 4.11 (iii) and A is the annihilator of char Λ ( ¯ E H ∞ / ¯ C H ∞ ) /θ ( ¯ C H ∞ ) . (cid:3) Fix f , . . . , f k ∈ Λ( G ) so that char Λ ( A ( H ∞ )) is given by (cid:16)Q ki =1 f i (cid:17) Λ( G ) . Thenext lemma is [16, Proposition 6.5]. Lemma 4.14.
There exists an ideal B of finite index in Λ( G ) such that there existclasses c , . . . c k ∈ A ( H n ) satisfying B Ann( c i ) ⊂ f i Λ n for every i , where Ann( c i ) ⊂ Λ n is the annihilator of c i in A ( H n ) / ( c Λ n + · · · + c i − Λ n ) . Let ℓ > be a fixed integer. Given a prime Q of H n lying above q ∈ I ℓ , I q is afree Z [ G n ] -module of rank generated by Q , and we define v Q : H × n → Λ n by v Q ( w ) Q = ( w ) q , ¯ v Q : H × n / ( H × n ) p ℓ → Λ n /p ℓ Λ n by ¯ v Q ( w ) Q = [ w ] q The following lemma is a combination of [16, Lemma 8.2] and [11, Lemma 3.8.4]and will be used in the induction argument of Theorem 4.16.
Lemma 4.15.
Fix an integer ℓ > . Suppose χ ∈ G ∗ , v ∈ (cid:16) H × n / ( H × n ) p ℓ (cid:17) χ , q ∈ I ℓ is a prime, Q is a prime of H n lying above q , S is a set of primes of K not containing q , and f, λ , λ , λ ∈ Λ( G ) , with λ = 2 if p = 2 . Write B n for thesubgroup of A ( H n ) generated by the primes of H n lying above the primes in S , c forthe image of Q in A ( H n ) χ and V for the Λ n -submodule of H × n / ( H × n ) p ℓ generatedby v . Suppose also that we have(i) [ v ] r = 0 for a prime r of K not in S ∪ { q } ,(ii) the annihilator Ann( c ) ⊂ Λ χn of c in A ( H n ) χ /B χn satisfies λ Ann( c ) ⊂ f Λ χn ,(iii) A ( H n ) χ ) | p ℓ and ¯ v Q ( v ) divides ( p ℓ / A ( H n ) χ )) λ in Λ χn /p ℓ Λ χn , and(iv) f Λ( G ) is prime to I ( H n ) .Then there exists a G n -equivariant map φ : V → Λ n /p ℓ Λ n satisfying f φ ( v ) = λ λ λ ¯ v Q ( v ) . Finally, we make full use of the results from Chapter 4 and apply induction toestablish a divisibility relation between char Λ ( A ( H ∞ ) χ ) and char Λ ( ¯ E H ∞ / ¯ C H ∞ ) χ . Theorem 4.16.
Let k be the number of f i appearing in char Λ ( A ( H ∞ )) .(i) If p > and χ ∈ G ∗ , we have char Λ ( A ( H ∞ ) χ ) divides I ( D p ) k +2 char Λ ( ¯ E H ∞ / ¯ C H ∞ ) χ . (ii) If p = 2 and χ ∈ G ∗ , we have char Λ ( A ( H ∞ ) χ ) divides k +6 char Λ ( ¯ E H ∞ / ¯ C H ∞ ) χ . Proof.
Fix a generator β of char Λ ( ¯ E H ∞ / ¯ C H ∞ ) χ . Let B be an ideal of finite indexin Λ( G ) satisfying the conditions in Lemma 4.11 (iii) and Lemma 4.14. Take λ ∈ I ( D p ) B (or I ( G ) B if p = 2 ). The existence of B is clear from the proof of Lemma4.14. Pick ℓ > large enough so that we have(21) p ℓ λ Λ n ⊂ p n +4 ke λ k β ( A ( H n ) χ )))Λ n . Here, e = 1 or according as p = 2 or p = 2 . The choice of ℓ will be justifiedlater in the proof. Now, by Corollary 4.13, there exists θ λ,n : ¯ E H n → Λ n such that λ β ∈ θ λ,n ( ¯ C χH n ) . Thus, we may fix u ∈ ¯ C χH n with θ λ,n ( u ) = λ β , and also we fix N THE MAIN CONJECTURE OF IWASAWA THEORY 25 u ∈ C χH n with u ≡ u mod ( ¯ C χH n ) p ℓ . By Proposition 4.5, we have an Euler system α and σ ∈ G with α σ ( n,
1) = u . Let κ α be the map defined in Proposition 4.6,and let c , . . . , c k ∈ A ( H n ) be as given in Lemma 4.14. We will use induction toselect primes Q , . . . , Q k +1 of H n lying above primes q , . . . , q k +1 of K satisfying:(22) [ Q i ] = λ c χi in A ( H n ) χ , and q i ∈ I ℓ , (23) ¯ v Q ( κ α ( q ) χ ) = r p e λ β and f i − ¯ v Q i ( κ α ( a i ) χ ) = r i p e λ ¯ v Q i − ( κ α ( a i − ) χ ) , where a i = q · · · q i and r i ∈ ( Z /p ℓ Z ) × .For i = 1 , we take c = λ c χ ∈ p e I ( G ) A ( H n ) χ , W = (cid:16) ¯ E H n / ¯ E H n ∩ ( H × n ) ℓ (cid:17) χ , φ = p e θ λ,n and apply Theorem 4.8 and Proposition 4.6. Then we obtain a prime Q of H n lying above a prime q ∈ I ℓ such that [ Q ] = λ c χ in A ( H n ) χ and r ∈ ( Z /p ℓ Z ) × satisfying [( κ α ( q ) χ )] Q = ϕ q ( κ α (1) χ ) = ϕ q ( α σ ( n, r p e φ ( u ) Q = r p e θ λ,n ( u ) Q = r p e λ β Q . Thus we have ¯ v Q ( κ α ( q ) χ ) = r p e λ β by the definition of [ · ] Q .Now, let < i < k and suppose we have selected primes Q , . . . , Q i satisfying(22) and (23). We will define Q i +1 . Recall a i = Q j i q j . Let V i be the Λ n -submodule of H × n / ( H × n ) p ℓ generated by κ α ( a i ) χ . We will apply Lemma 4.15 with Q = Q i , v = κ α ( a i ) χ , λ = λ = λ and S = { q , . . . , q i − } . This is possible becauseconditions (i), (ii) and (iv) of Lemma 4.15 are satisfied thanks to Proposition 4.6,Lemma 4.14 and Lemma 4.10, and (iii) is satisfied because by (23), ¯ v Q i ( κ α ( a i ) χ ) divides p ie λ i β in Λ χn /p ℓ Λ χn , so by the choice of ℓ made in (21), ¯ v Q i ( κ α ( a i ) χ ) divides (cid:0) p ℓ / A ( H n ) χ ) (cid:1) λ in Λ χn /p ℓ Λ χn . Thus, we obtain a map φ i : V i → Λ n /p ℓ Λ n such that(24) f i φ i ( κ α ( a i ) χ ) = p e λ ¯ v Q i ( κ α ( a i ) χ ) . Now, applying Theorem 4.8 by setting V = V i , c = λ c χi +1 , φ = φ i , we obtain a prime q i +1 ∈ I ℓ and a prime Q i +1 of H n lying above it. Then (i) and (ii) of Theorem 4.8gives (22) for i + 1 . Furthermore, by Proposition 4.6 (ii) and Theorem 4.8 (ii), forsome r i +1 ∈ ( Z /p ℓ Z ) × we have f i [ κ α ( a i +1 ) χ ] Q i +1 = r i +1 p e f i φ i ( κ α ( a i ) χ ) Q i +1 = r i +1 p e λ ¯ v Q i ( κ α ( a i ) χ ) Q i +1 , where the last equation follows from (24). This proves (23) for i + 1 . Finally,combining (23) for i k + 1 gives k Y i =1 f i ¯ v Q k +1 ( κ α ( a k +1 ) χ ) = rp (4 k +4) e λ k +2 β in Λ n /p ℓ Λ n for some r ∈ ( Z /p ℓ Z ) × . It follows that char Λ ( A ( H ∞ )) = k Y i =1 f i divides p (4 k +4) e λ k +2 β Λ( G ) = p (4 k +4) e λ k +2 char Λ (cid:0) ¯ E H ∞ / ¯ C H ∞ (cid:1) . This holds for every λ ∈ I ( D p ) B (or I ( G ) B if p = 2 ), so in particular, holds for λ being the greatest common divisor λ of all elements in this ideal. If p > , wehave λ Λ( G ) ⊃ I ( D p ) and the divisibility in (i) follows. If p = 2 , it is easy to show that we have λ Λ( G ) = 2 I ( G ) . This concludes the proof of Theorem 4.16, because char Λ ( A ( H ∞ )) is prime to I ( G ) by Lemma 4.10. (cid:3) Corollary 4.17.
Let p > . Then char Λ ( A ( H ∞ )) divides char Λ ( ¯ E H ∞ / ¯ C H ∞ ) . Proof.
We have shown in Lemma 4.10 that char Λ ( A ( H ∞ )) is prime to I ( D p ) , so byTheorem 4.16, char Λ ( A ( H ∞ )) divides char Λ ( ¯ E H ∞ / ¯ C H ∞ ) . (cid:3) Recall that p ∤ [ H : K ] by assumption. Theorem 4.18.
We have char Λ ( X ( H ∞ )) = char Λ (cid:0) U H ∞ / ¯ C H ∞ (cid:1) if and only if char Λ ( A ( H ∞ )) = char Λ (cid:0) ¯ E H ∞ / ¯ C H ∞ (cid:1) , and char Λ ( X ( H ∞ )) | e (6 k +6) char Λ (cid:0) U H ∞ / ¯ C H ∞ (cid:1) . Proof.
Recall from (17) that we have an exact sequence → ¯ E H ∞ / ¯ C H ∞ → U H ∞ / ¯ C H ∞ → X ( H ∞ ) → A ( H ∞ ) → , and therefore char Λ ( A ( H ∞ ))char Λ (cid:0) U H ∞ / ¯ C H ∞ (cid:1) = char Λ ( X ( H ∞ ))char Λ (cid:0) ¯ E H ∞ / ¯ C H ∞ (cid:1) .The last assertion of the theorem follows from Theorem 4.16 and Corollary 4.17. (cid:3) Proof of the Main Conjecture for H ∞ /H The Iwasawa Invariants of X ( H ∞ ) . Recall that G ≃ G × Γ , where we identify G with Gal( H ∞ /K ∞ ) and Γ with Gal( K ∞ /K ) . Let us consider Λ I (Γ) as a Λ I ( G ) -module via χ ∈ G ∗ . Given afinitely generated torsion Λ I ( G ) -module M , recall that char( M ) ⊂ Λ I ( G ) de-notes the characteristic ideal of M . If X is a Λ( G ) -module and χ ∈ G ∗ , we write X χ for e χ ( X b ⊗ Z p I ) , where e χ is the idempotent corresponding to χ . This is justi-fied because the characteristic ideal of a Γ -module behaves well under extension of scalars. This comes from the fact that we can identify Λ I (Γ) with I [[ T ]] upon fixing a topological generator γ of Γ .Recall also that any f ( T ) ∈ I [[ T ]] can be written uniquely, by the p -adic Weier-strass preparation theorem, in the form f ( T ) = π m P ( T ) U ( T ) where π is a uniformiser of I , P ( T ) is a distinguished polynomial, that is, a monicpolynomial whose coefficients are divisible by π , and U ( T ) is a unit in I [[ T ]] . Let ǫ be the absolute ramification index of I . The invariants µ ( f ) = mǫ and λ ( f ) =deg P ( T ) are called the Iwasawa µ -invariant and λ -invariant of f , respectively. TheIwasawa invariants of Λ( G ) -modules are defined similarly, and if M = X b ⊗ Z p I isobtained from a Λ( G ) -module X by extension of scalars to I , the invariants of M coincide with those of X . Let f χ be a generator of char ( X ( H ∞ ) χ ) and let g χ be agenerator of char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) . We set f = Q f χ and g = Q g χ . By Theorem4.18, we have Theorem 5.1. f χ | π ek g χ for some integer k ≥ , e = 0 if p > and e = 1 if p = 2 . N THE MAIN CONJECTURE OF IWASAWA THEORY 27
In order to prove Theorem 3.19, we just need to show f χ and g χ define the sameideal, thanks to Corollary 3.18. Thus it remains to show that f and g the havethe same Iwasawa invariants. We shall compute them separately, and show thatthey are equal. First, we compute the invariants of X ( H ∞ ) using class field theory,and in Section 5.2 we compute the invariants of U H ∞ / ¯ C H ∞ using the analytic classnumber formula and Kronecker’s second limit formula.For the rest of the chapter, fix an integer n > e , where e = 0 or if p isodd or even. Recall from the proof of Proposition 4.9 that X ( H ∞ ) /I ( H n ) X ( H ∞ ) is equal to Gal( M ( H n ) /H ∞ ) , where M ( H n ) is the maximal abelian p -extension of H n which is unramified outside the primes of H n above p . Thus the asymptoticformula of Iwasawa [21, Theorem 13.13] gives Theorem 5.2.
Let f be a characteristic power series for X ( H ∞ ) as a Z p [[Γ]] -module. For sufficiently large n, we have ord p ( X ( H ∞ ) /I ( H n ) X ( H ∞ )) = µ ( f ) p n − − e + λ ( f )( n − − e ) + c, where µ ( f ) and λ ( f ) are the Iwasawa invariants of X ( H ∞ ) and c ∈ Z is independentof n . We will now compute the p -adic valuation of the index [ M ( H n ) : H ∞ ] using themethods of Coates and Wiles [7], and use it to find ord p ( X ( H ∞ ) /I ( H n ) X ( H ∞ )) .We note that p is assumed to be an odd prime number in [7], but it can be extendedto p = 2 in our case, using the fact that splits in K and ( p, h ) = 1 by assumption.Set [ H n : K ] = d , and recall that this is equal to p n − − e h . Let ξ , . . . ξ d denote thedistinct embeddings of H n into Q p extending the embeddings of K into Q p givenby p . Since H n is totally imaginary, rank Z ( E H n / ( E H n ) tor ) = d − . We pick a basis ǫ , . . . , ǫ d − for E H n / ( E H n ) tor , and put ǫ d = 1 + p or p if p is odd or even. Wedefine the p -adic regulator of Leopoldt:(25) R p ( H n ) = ( d log ǫ d ) − det (log( ξ i ( ǫ j ))) i,j d . For each n > e , let C H n denote the idele class group of H n , and put Y n = ∩ m > n N H m /H n C H m . We write Φ p = H n ⊗ K K p , and let P denote the set of primes of H n lying above p . For each P ∈ P , let U H n, P be the group of units in the completion of H n at P which are congruent to modulo P , and let t > be such that p − t O P ⊂ log U H n, P .The p -adic logarithm gives a homomorphism log : U H n, P → H n, P whose kernel hasorder w P = µ p ∞ ( H n, P ) . We write log U H n = Q P ∈ P log U H n, P , so that we have log : U H n → Φ p with kernel w p = Q P ∈ P w P . Then the arguments of [7, Lemma7] apply, and we have(26) ord p [ Y P ∈ P p − t O P : log U H n ] = ord p w p Y P ∈ P N P + td. Let V n = 1 + p n O p denote the local units of K p which are congruent to modulo p n , and define D n = V e ¯ E H n ⊂ U H n . Furthermore, let ∆ H n /K denote thediscriminant of H n /K , and pick a generator ∆ n of the ideal ∆ H n /K O p . Lemma 5.3. ord p ([log U H n : log D n ]) = ord p R p ( H n ) √ ∆ n w p Y P ∈ P N P − + n. Proof.
Using methods analogous to [7, Lemma 8], we can show that(27) ord p [ Y P ∈ P p − t O P : log D n ] = ord p (cid:18) R p ( H n ) √ ∆ n (cid:19) + td + n − − e + ord p (log( ǫ d )) . We have ord p (log( ǫ d )) = 1+ e , so the right hand side of (27) is equal to ord p (cid:16) R p ( H n ) √ ∆ n (cid:17) + td + n . The result now follows from (26). (cid:3) Corollary 5.4. ord p ([ U H n : D n ]) = ord p R p ( H n ) ω H n √ ∆ n Y P ∈ P N P − + n. Proof.
This is an immediate consequence of Lemma 5.3, obtained by applying thesnake lemma to the following commutative diagram D n U H n U H n /D n
00 log D n log U H n log U H n / log D n . log log with exact rows. (cid:3) Following the arguments of [7, Lemma 5], we see that Y n ∩ U H n = ker (cid:0) N Φ p /K p | U Hn (cid:1) . (28)We claim that ¯ E H n is contained in Y n ∩ U H n , and compute its index in the nextlemma. Indeed, if ξ ∈ E H n and p = 2 , then clearly N H n /K ( ξ ) = 1 . If p = 2 , we have N H n /K ( ξ ) ∈ {± } . But by local class field theory, we have N Φ p /K p ( U H n ) = V n , sofor n > e we must have N H n /K ( ξ ) = 1 . The claim now follows from (28). Lemma 5.5. [ Y n ∩ U H n : ¯ E H n ] = ord p R p ( H n ) ω H n √ ∆ n Y P ∈ P (1 − (N P ) − ) + n Proof.
By (28) and the definition of D n , we have ¯ E H n = ker (cid:0) N Φ p /K p | D n (cid:1) , and N Φ p /K p ( D n ) = ( V e ) d = ( V e ) p n − − e = V n . Hence, applying Lemma 28 again,we obtain a commutative diagram with exact rows E H n D n V n Y n ∩ U H n U H n V n . N Φ p /K p N Φ p /K p N THE MAIN CONJECTURE OF IWASAWA THEORY 29
Lemma 5.5 now follows from Lemma 28 on noting that ord p (cid:16)Q P ∈ P (1 − (N P ) − ) (cid:17) =ord p (cid:16)Q P ∈ P (N P ) − ) (cid:17) . (cid:3) Theorem 5.6.
Let M ( H n ) be the maximal abelian p -extension of H n which isunramified outside the primes in P , and write h H n for the class number of H n .Then ord p ([ M ( H n ) : H ∞ ]) = ord p h H n R p ( H n ) ω H n √ ∆ n Y P ∈ P (1 − (N P ) − ) + n. Proof.
Let L ( H n ) be the maximal unramified extension of H n in M ( H n ) . Thuswe may identify Gal( L ( H n ) /H n ) with A ( H n ) , the p -primary part of the ideal classgroup of H n . Class field theory gives an isomorphism Y n ∩ U H n / ¯ E H n ∼ = Gal( M ( H n ) /L ( H n ) H ∞ ) . Note that L ( H n ) ∩ H ∞ = H n because H ∞ /H n is totally ramified at the primes in P . Thus → Y n ∩ U H n / ¯ E H n → Gal( M ( H n ) /H ∞ ) → A ( H n ) → . The theorem now follows from Lemma 5.5 and the fact that ord p ( A ( H n ))) =ord p ( h H n ) . (cid:3) Corollary 5.7.
Let f be a characteristic power series for X ( H ∞ ) as a Γ -module.Then for sufficiently large n , µ ( f ) · p n − − e ( p −
1) + λ ( f ) = 1 + ord p h H n +1 R p ( H n +1 ) p ∆ n +1 / h H n R p ( H n ) √ ∆ n ! , where µ ( f ) and λ ( f ) are the Iwasawa invariants of X ( H ∞ ) .Proof. Noting that each P ∈ P has inertia degree zero in H n +1 /H n , it is clearfrom Theorem 5.6 that the right hand side of the above equation is equal to ord p ([ M ( H n +1 ) : H ∞ ] / [ M ( H n ) : H ∞ ]) . Recalling that Gal( M ( H n ) /H ∞ ) = X ( H ∞ ) /I ( H n ) X ( H ∞ ) , Theorem 5.2 gives ord p ([ M ( H n +1 ) : H ∞ ] / [ M ( H n ) : H ∞ ]) is equal to ( µ ( f ) p n − e + λ ( f )( n − e )+ c ) − ( µ ( f ) p n − − e + λ ( f )( n − − e )+ c ) = µ ( f ) p n − − e ( p − λ ( f ) . This completes the proof of the corollary. (cid:3)
The Iwasawa Invariants of the p -adic L -function. In this section, we will compute the Iwasawa invariants of U H ∞ / ¯ C H ∞ and showthat they coinside with those of X ( H ∞ ) computed in Corollary 5.7. We will followthe methods discussed in [8, Chapter III.2]. Again, the prime p is assumed to beodd in Chapter III of [8], but the methods still holds for p = 2 thanks to ourassumptions that p splits in K and p ∤ [ H : K ] . Fix a generator g χ ∈ Λ I (Γ) of char (cid:0) ( U H ∞ / ¯ C H ∞ ) χ (cid:1) , and set g = Q g χ . Let µ ( g ) and λ ( g ) denote the Iwasawainvariants of g . The following is [8, Lemma III.2.9]. Lemma 5.8.
Recall that Γ n = Γ p n − − e . For any character ρ of Γ of finite order,write l ( ρ ) = n − − e if ρ (Γ n ) = 1 but ρ (Γ n − ) = 1 . Then for n sufficiently large, ord p Y l ( ρ )= n − e ρ ( g ) = µ ( g ) · p n − − e ( p −
1) + λ ( g ) . Given a ramified character ε of G = G × Γ , write ε = χρ where χ is a characterof G and ρ is a character of Γ . Let f ε denote the conductor of ε , f ε = f ε ∩ Z , andlet B n be the collection of all ε with p n || f ε . Then Theorem 5.9.
For n sufficiently large, ord p Y l ( ρ )= n − e ρ ( g ) = 1 + ord p h H n +1 R p ( H n +1 ) p ∆ n +1 / h H n R p ( H n ) √ ∆ n ! . Proof.
Our arguments are analogous to Proposition III.2.10 and III.2.11 of [8]. Any ε ∈ B n can be written in the form ε = χρ where χ is a character of G and ρ is acharacter of Γ with l ( ρ ) = n . Define G ( ε ) as in [8, II.4.11] and S p ( ε ) be as definedin [8, III.2.10 (14), III.2.11 (14’)]. Recall that g χ is also a generator of ϕ χ byCorollary 3.18, where ϕ = I I ( G ) ν p and ν p satisfies Theorem 3.11. Then followingthe methods of [8, Theorem II.5.2] but using R D (Φ( z, L )) = Q ri =1 R a i (Φ( z, L )) n i constructed in Chapter 3 instead of the elliptic function Θ( z ; L, a i ) used in [8], andnoting R a i (Φ( z, L )) is a th root of Θ( z ; L, a i ) and that θ D ν p = ν D , we obtain ρ ( g χ ) = Z G χ χρdν χ p = (cid:26) G ( ε ) S p ( ε ) if χ is non-trivial ( ρ ( γ ) − G ( ε ) S p ( ε ) if χ = 1 ,where γ is a topological generator of Γ . Hence Q l ( ρ )= n − e ( ρ ( γ ) −
1) = Q ζ ∈ µ pn − e ( ζ − . Noting that ord p (cid:16)Q ζ ∈ µ pn − e ( ζ − (cid:17) ) = 1 , we obtain(29) ord p Y l ( ρ )= n − e ρ ( g ) = 1 + ord p Y ε ∈ B n +1 − e G ( ε ) S p ( ε ) . On the other hand, using the analytic class number formula and Kronecker’s secondlimit formula (see [20, §0.2.7, §I.2.2 and §IV.3.9 (6)]) for the fields H n +1 and H n gives:(30) ord p Y ε ∈ B n +1 − e G ( ε ) S p ( ε ) = ord p h H n +1 R p ( H n +1 ) p ∆ n +1 / h H n R p ( H n ) √ ∆ n ! Combining (29) and (30) completes the proof of Proposition 5.9. (cid:3)
Comparing Corollary 5.7, Lemma 5.8 and Theorem 5.9, we conclude that f and g have the same Iwasawa invariants. As discussed at the beginning of Section 5.1,this together with the divisibility relation obtained in Theorem 4.18 completes theproof of Theorem 3.19. N THE MAIN CONJECTURE OF IWASAWA THEORY 31 acknowledgement
This work is based on my PhD thesis at the University of Cambridge, and I wouldlike to express my sincere gratitude to my supervisor, John Coates, for patientlyadvising and encouraging me. His expertise and enthusiasm helped me overcomecountless difficulties over the course of my doctoral studies. I would also like toexpress my warm thanks to Junhwa Choi, Tom Fisher, Guido Kings, Jack Lamplughand Sarah Zerbes for giving me many helpful comments on this work. Finally, I amvery grateful to the referees for their very careful reading of the paper and for thevaluable feedback which helped improving the quality of this paper.
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