CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE TOM LOVERING Abstract. In these notes, we follow the proof in [1] of the main conjecture of Iwasawa theory making heavy use of the Euler system of cyclotomic units. On the one hand, using the local theory of Coleman series and ideas of Iwasawa one obtains a connection with the p-adic zeta function. On the other hand by CFT and Rubin's refinement of the ideas of Kolyvagin (and the analytic class number formula) one obtains a connection with the tower of ideal class groups. The author is not an expert on this subject and these notes were written quickly, so beware errors. Contents 1. Basic setup and overview of the proof 1 1.1. Iwasawa's basic setup 1 1.2. Measures and the p-adic zeta function 3 1.3. Outline of proof 4 1.4. Cryptic remarks on applications 5 2. Cyclotomic Units 5 2.1. Basic properties of cyclotomic units 5 2.2. Analytic class number formula 6 3. Coleman's Theory and the Coates-Wiles map 7 3.1. Units and power series 7 3.2. Logarithmic differentiation and an exact sequence 8 3.3. The Coates-Wiles map and measures 9 3.4. Cyclotomic units and the p-adic zeta function 11 3.5. Proof of Iwasawa's theorem 12 4. Euler Systems and the completion of the proof 13 References 13 1. Basic setup and overview of the proof 1.1. Iwasawa's basic setup. Fix p an odd prime. We will study the cyclotomic tower + Fn := Q(ζpn+1 ) and the tower of real subfields Fn = Q(ζpn+1 ) . Over each of these one can construct maximal unramified abelian p-extensions Ln;Ln respectively, and maximal unramified-away-from-p abelian p-extensions Mn;Mn. Taking the obvious colimits, we 1 2 TOM LOVERING 1 obtain possibly infinite extensions M1=L1=F1 and M1=L1=F1, and we give their Galois groups the following names Y1 := G(L1=F1) = limnG(Ln=Fn) X1 := G(M1=F1) = limnG(Mn=Fn): Let Y1; X1 be defined similarly. That these extensions are Galois follows because F1 is Galois over Q, so taking an automorphism of Q¯ will act on them, and preserve the property of being an unramified (or unramified-outside-p) abelian p-extension, from which it is easy to deduce that such an automorphism also acts on L1;M1 (similarly with everything curly). ∼ χ= ∗ Now, let G = Gal(F1=Q) ! Zp, G = Gal(F1=Q), and Gn;Gn the obvious quotients. We form the Iwasawa algebra as the completed Zp-group ring Λ(G) := lim Zp[Gn]; Λ(G) = lim Zp(Gn): Of course one can make such a construction for any other pro finite abelian group, and later we will need a canonical map called the Mahler transform which makes the identification =∼ M : Λ(Zp) ! Zp[[T ]]: The point of this ring is it allows us to keep track of extra structures inside the Galois groups like X1 coming from their being situated over the Galois group of F1 (e.g. recall that the Herbrand-Ribet theorem gives information about the tame character acting on the class group). Indeed, note the short exact sequence (from Galois theory) 0 ! X1 ! G(M1=Q) ! G ! 0: It is not hard to show that if one picks γ 2 G, lifts it to any element of G(M1=Q) and then uses it to act by conjugation on X1, this action is independent of lift. Moreover, the pro finite structures of X1 and Λ(G) are compatible and pro-p, so this action extends and one can show. Proposition 1.1. The profinite groups Y1 and X1 have a canonical structure as topo- logical Λ(G)-modules. Similarly, Y1 and X1 have a canonical structure as topological Λ(G)-modules. In fact, all the modules in sight are finitely generated over Λ(G), and X1; Y1;Y1 are torsion. This latter fact is particularly handy, because one can (and Bourbaki does) prove a structure theorem for finitely generated torsion modules over such rings (they unfortunately aren't PIDs, but have some similar properties). Theorem 1.2 (Structure theory of finitely generated torsion Iwasawa modules). Let X be a finitely generated torsion Iwasawa module. Then it sits in an exact sequence n 0 ! ⊕i=1Λ(G)=fiΛ(G) ! X ! D ! 0 1 + Although we note that Vandiver's conjecture that p doesn't divide the class group of Q(ζp) is equivalent to the assertion that L1 = F1. However, if these extensions aren't trivial they are infinite by a theorem of Iwasawa. CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 3 where D has finite cardinality, and the characteristic ideal ChG(X) := f1f2:::fnΛ(G) is independent of this presentation and multiplicative in exact sequences. The main conjecture, as we will state it, gives a precise prediction for this ideal. 1.2. Measures and the p-adic zeta function. Let H be a profinite abelian group. One can view λ 2 Λ(H) as a continuous Zp-valued measure on the group H in the following sense. Writing λ = P [g] 2 [H=U] for the projection of λ onto U, we define the U λU [g] Zp measure of an open subset S ⊂ H as X µλ(S) = limU λU[g] [g]2H=U:[g]\S6=; where this limit exists because S is a union of cosets of sufficiently small U. We do not give any more details here, but a reader wishing to know more can see for example [1, 3.2]. R We just note that one is able to define integrals H fdλ for each λ 2 Λ(H). Since the Riemann zeta function ζ has a pole at 1, we must define a pseudo-measure to be a quotient = λ1/λ2 where λ2 is not a zero divisor with the property that for any g 2 H, (g − 1) 2 Λ(H). Given a pseudo-measure and any nonconstant function f : H ! Cp we can still form the integral by choosing g such that f(g) 6= f(1) and setting Z 1 Z fd := fd(g − 1) : H f(g) − f(1) H One thing we shall soon prove using cyclotomic units (although there are plenty of other cool proofs) is the following fact. We always let the cyclotomic character be denoted =∼ ∗ χ : G ! Zp: Theorem 1.3 (Existence of Leopoldt-Kubota p-adic Zeta Function). There exists a pseu- domeasure ζp on G with the property that for any positive even k, Z k k−1 χ(g) dζp = (1 − p )ζ(1 − k): G Moreover, this integral is zero at k > 0 odd, and the above property characterises ζp uniquely. Now, ζp is only a pseudo-measure, so does not lie in the Iwasawa algebra, but letting I(G) be the augmentation ideal (kernel of the `degree' map Λ(G) ! Zp) it follows from the definition of a pseudo-measure that ζpI(G) is an ideal of Λ(G). Also, the vanishing for k odd allows us to view ζp as an element of Λ(G) and get an ideal ζpI(G) similarly. We can now finally state the main conjecture. Theorem 1.4 (Iwasawa Main Conjecture). We can identify the characteristic ideal of X1 as ChG(X1) = ζpI(G): 4 TOM LOVERING 1.3. Outline of proof. To prove this using the Euler system of cyclotomic units, we proceed as follows. Firstly, we shall study the tower of cyclotomic units Dn via its closure + 1 1 Cn inside the group of units Un of Kn = Qp(ζpn+1 ) . Passing to Cn ⊂ Un, those which 1 1 are congruent to 1 modulo $n = ζpn+1 − 1, and taking an inverse limit, we get C1 ⊂ U1. One can also identify the local Galois groups with the corresponding global Galois groups, and thus obtain an Iwasawa module structure on these objects, and in particular on their 1 1 quotient U1=C1. The miracle that probably led Iwasawa to make his conjecture in the first place is the following. Theorem 1.5 (Iwasawa). We have a canonical isomorphism of Λ(G)-modules 1 1 =∼ U1=C1 ! Λ(G)/ζpI(G): 1 To prove this, we exploit Coleman's trick of identifying (un) 2 U1 with power series in Zp[[T ]], which itself is related to the Iwasawa algebra via the Mahler transform in such a way that integrals of the cyclotomic character can be computed by taking logarithmic derivatives. Then a miraculous calculation involving Coleman's series associated to a tower of cyclotomic units (T + 1)−a=2 − (T + 1)a=2 f(T ) = (T + 1)−1=2 − (T + 1)1=2 recovers the Bernoulli numbers and proves both the existence of ζp and Iwasawa's theorem in one elegant swoop. We do this in some detail in section 3. Until this point we have used no class field theory and mainly local arguments at p. 1 However, we must now do so to relate these units to the Galois group X1. Let E1 be the obvious limit of closures of groups of global units. An elementary exercise in assembling class field theory and taking limits gives us a canonical Λ(G) isomorphism 1 1 ∼ U1=E1 = Gal(M1=L1) (the latter viewed as a submodule of X1). By Galois theory, we deduce the exact sequence of Λ(G)-modules, each of which is finitely generated torsion 1 1 1 1 0 ! E1=C1 ! U1=C1 ! X1 ! Y1: By multiplicativity of the characteristic ideal and Iwasawa's theorem, the main conjec- ture follows immediately from the following result.