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 M∞/L∞/F∞ and M∞/L∞/F∞, and we give their Galois groups the following names

Y∞ := G(L∞/F∞) = limnG(Ln/Fn)  X∞ := G(M∞/F∞) = limnG(Mn/Fn).

Let Y∞, X∞ be defined similarly. That these extensions are Galois follows because F∞ 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 L∞,M∞ (similarly with everything curly). ∼ χ= ∗ Now, let G = Gal(F∞/Q) → Zp, G = Gal(F∞/Q), and Gn,Gn the obvious quotients. We form the Iwasawa algebra as the completed Zp- 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 X∞ coming from their being situated over the of F∞ (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 → X∞ → G(M∞/Q) → G → 0. It is not hard to show that if one picks γ ∈ G, lifts it to any element of G(M∞/Q) and then uses it to act by conjugation on X∞, this action is independent of lift. Moreover, the pro finite structures of X∞ and Λ(G) are compatible and pro-p, so this action extends and one can show.

Proposition 1.1. The profinite groups Y∞ and X∞ have a canonical structure as topo- logical Λ(G)-modules. Similarly, Y∞ and X∞ have a canonical structure as topological Λ(G)-modules. In fact, all the modules in sight are finitely generated over Λ(G), and X∞, Y∞,Y∞ 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 L∞ = F∞. 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 ∈ Λ(H) as a continuous Zp-valued measure on the group H in the following sense. Writing λ = P [g] ∈ [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]∈H/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 ∈ Λ(H). Since the Riemann zeta function ζ has a pole at 1, we must define a pseudo-measure to be a quotient ψ = λ1/λ2 where λ2 is not a zero divisor with the property that for any g ∈ H, (g − 1)ψ ∈ 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 fdψ := fd(g − 1)ψ. 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 X∞ as

ChG(X∞) = 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 , we get C∞ ⊂ U∞. 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 U∞/C∞. 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 =∼ U∞/C∞ → Λ(G)/ζpI(G). 1 To prove this, we exploit Coleman’s trick of identifying (un) ∈ U∞ 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 X∞. Let E∞ 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 ∼ U∞/E∞ = Gal(M∞/L∞) (the latter viewed as a submodule of X∞). By Galois theory, we deduce the exact sequence of Λ(G)-modules, each of which is finitely generated torsion 1 1 1 1 0 → E∞/C∞ → U∞/C∞ → X∞ → Y∞. By multiplicativity of the characteristic ideal and Iwasawa’s theorem, the main conjec- ture follows immediately from the following result. Theorem 1.6 (Rubin et al). We have an equality of characteristic ideals 1 1 ChG(E∞/C∞) = ChG(Y∞). To prove this, one works at a finite level and uses Kolyvagin’s Euler system game to prove a divisibility relation in one direction, and the equality is then deduced from the analytic class number formula. We remark2 that Vandiver’s conjecture would imply both modules are trivial, but that this conjecture seems totally out of reach at present so we really need Rubin’s argument and this is the real heart of the proof.

2Apologising to Rong whose talk I trolled by noting he seemed to be constructing elements in a trivial group. CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 5

1.4. Cryptic remarks on applications. We should probably convince ourselves at the very least that Herbrand-Ribet is an immediate consequence of the main conjecture as we stated it. Recall that this is a conjecture to do with the action of µp−1 on the odd part of the class group of Q(ζp). Unfortunately, one needs a somewhat elaborate Kummer theory argument (about five lectures of Coates’ Michaelmas 2011 Part III course) to show that − A∞ := colim Gal(Ln/Fn) is related to X∞ in the following manner (where hats denote Pontryagin and twists are Tate twists in the obvious sense). Theorem 1.7 (Delicate Kummer theory). There is a canonical identification of Λ(G)- modules ∼ ˆ− X∞ = A∞(1). The presence of the Tate twist explains certain annoying odd/even confusion that can arise, but unwrapping this statement and the statement of the main conjecture and deduc- ing the Herbrand-Ribet theorem is now an exercise.

2. Cyclotomic Units The key ‘motivic’ input that makes the whole game work is a certain group explicit units in Fn and Fn called cyclotomic units. In this section we mainly follow Washington’s book. n+1 Fix (switching notation slightly) a compatible system of p th roots of unity (ζn) so that F = (ζ ). We define D ⊂ O∗ to be the intersection of the subgroup generated by n Q n n Fn {ζ , 1 − ζa} inside F ∗ with O∗ . Let D = D ∩ O∗ be the real cyclotomic units. We n n n Fn n n Fn will for simplicity of notation and consistency with the literature henceforth write En,En for the full groups of units in these fields. 2.1. Basic properties of cyclotomic units. It will be useful to have an explicit gener- ating set for Dn.

Proposition 2.1 (Basis for the cyclotomic units). Dn is generated by −1 and the elements ζ−a/2 − ζa/2 u (a) = n n n −1/2 1/2 ζn − ζn where 1 < a < p2/2 and p 6 |a.

Qn ak mk Proof. Suppose we have a η k=1(1 − ζn ) . Observe that by the two identities −a −a a 1 − ζn = −ζn (1 − ζn) and pa a a p−1 a 1 − ζn = (1 − ζn)(1 − ζ0ζn) ... (1 − ζ0 ζn) 2 we may assume 1 ≤ ak < p /2 and p 6 |a. By taking p-adic valuation and using that it is P a unit, we also deduce that k mk = 0, so it is harmless to divide each of the factors by (1 − ζn). Finally, because we wish to end up with a real number, we deduce that there are two possibilities for η (differing by −1) resulting in the elements claimed. It is also clear that the elements listed above are indeed units.  6 TOM LOVERING

This result can be restated in the following useful form (recalling that the automorphisms −1 a −a of Fn take ζn + ζn 7→ ζn + ζn where a is prime to p). 2 Corollary 2.2. Let 1 < a < p /2 and p 6 |a. Then Dn is generated by −1 and the Galois conjugates of un(a).

Lemma 2.3 (Norm compatibility). If Nn : Fn → Fn−1 is the obvious norm map, Nn(un(a)) = un−1(a).

Proof. This is immediate from multiplicativity of Nn, that Nn(ζn) = ζn−1 and p−1 Y k Nn(1 − ζn) = (1 − ζ0 ζn) = 1 − ζn−1. k=0  2.2. Analytic class number formula. Recall the classical analytic class number formula

Theorem 2.4 (Class number formula). Let F be a number field, r1 real places and r2 complex places, with Dedekind zeta function ζF , regulator RF , class number hF and dis- criminant D. r r 2 1 (2π) 2 hK RK lims→1(s − 1)ζK (s) = . |µ(K)|p|D| n Applying this to F = Fn, and setting m = (p − 1)p /2 = [Fn : Q] we get (where the product is over characters of Gal(Fn/Q) each of which is visibly a Dirichlet character) n−1 Y 2 hF RF L(1, χ) = √ n n . χ6=1 D

Theorem 2.5 (Index of cyclotomic units). The cyclotomic units Dn have finite index in En and more precisely

[En : Dn] = hFn . Proof. Previously we wrote down an explicit generating set of the correct cardinality for Dn, and one can use this compute its regulator explicitly, obtaining after a little pain the identity pn+1−1 Y 1 X R(D ) = ± χ(a)log|1 − ζa|. n 2 n χ6=1 a=1 But another explicit computation using the Gauss sums τ(χ) shows that pn+1−1 −1 X a −τ(χ)L(1, χ ) = χ(a)log|1 − ζn|. a=1

A further explicit computation (using the obvious basis for the ring of integers of Fn) p Q gives the identity |D| = χ τ(χ). Putting all these ingredients together with the class number formula, we deduce that

R(Dn) = RFn hFn . CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 7

The desired result now follows immediately from the definition of a regulator and stan- dard commutative algebra. 

3. Coleman’s Theory and the Coates-Wiles map In this section we develop Coleman’s theory and the theory of the Coates-Wiles “higher logarithmic derivative” map, with a view to constructing the p-adic zeta function naturally from cyclotomic units and in such a context as to make Iwasawa’s theorem very natural. We super-vaguely remark that Coleman’s series are supposed to be linked with the theory of norm fields and Fontaine’s theory of (φ, Γ)-modules, and that I think people are able to play the following type of game in more general situations using this language, extracting a p-adic L-function as a purely p-adic local object. Given such a thing encodes important global data, the author is naively extremely impressed and inspired to do more p-adic Hodge theory.

3.1. Units and power series. We study the (totally ramified) local p-cyclotomic towers + Kn = Qp(ζpn+1 ) , Kn = Qp(ζpn+1 ), take colimits to get K∞, K∞. We (abusing notation) fix bases (ζn) for both the global and local p-cyclotomic characters, which determines restriction maps (if you like) identifying the Galois groups over Qp with G and G. We also manufacture some norm-compatible uniformisers $n = ζn − 1. ∗ Suppose we have a unit u ∈ Un ⊂ Kn. We could “expand it $n-adically” and get a power ∗ series f ∈ Zp[[T ]] representing it in the sense that f($n) = u. However, since v(πn) will be much less than v(p), these expansions are far from being unique. The miracle observed by Coleman is that if one considers a norm-compatible family of units u = (un) ∈ U∞ = limnUn, and tries to represent them all by the same power series, this is always possible and can be done uniquely. Since it will appear everywhere, we will let this power series ring be denoted R = Zp[[T ]].

Theorem 3.1 (Coleman). Let u = (un) be a norm-compatible system of units. Then there ∗ exists a unique power series fu(T ) ∈ R with the property that for each n, fu($n) = un. Proof. The uniqueness is immediate from the Weierstrass preparation theorem, which im- plies that whenever an element of R has infinitely many zeroes it is in fact equal to zero. The existence part is interesting but quite computational, so we will outline the main steps. The operators we define will also be important and continue to feature in the rest of this section. (1) Firstly, one defines an operator φ : f(T ) 7→ f((1 + T )p − 1).

One observes this is injective and has the property that φ(f)($n) = f($n−1). (2) Next, one defines “norm” and “trace” operators N , ψ characterised by the proper- ties: Y φ ◦ N (f)(T ) = f(η(1 + T ) − 1),

η∈µp 8 TOM LOVERING

1 X φ ◦ ψ(f)(T ) = f(η(1 + T ) − 1). p η∈µp (3) Check that for all k, φ(f)(T ) ≡ 1 mod pkR implies f(T ) ≡ 1 mod pkR. (4) Use this to prove the “norm contraction property” that whenever s ≥ t ≥ 0, and f ∈ R∗, we have N sf ≡ N tf mod pt+1. k (5) Check that NKn/Kn−k f($n) = N (f)($n−k). With each of these in place we finish off as follows. Suppose u = (un) is a norm- ∗ compatible system of units. Pick for each n any fn ∈ R such that fn($n) = un. n ∗ Key trick: Set gn = N f2n. Since R is compact we can find a convergent subsequence ∗ gni , converging to some g ∈ R . Claim: We may take fu = g. That is, we are claiming g($n) = un for all n. To prove this it suffices to check the numbers gm($n) → un. Suppose m ≥ n. Then noting that property (4) implies m 2m−n m+1 gm($n) = N f2m($n) ≡ N f2m($n)(mod p R), and property (5) implies 2m−n N f2m($n) = NF2m/Fn f2m($2m) = NK2m/Kn u2m = un, we deduce the required result. 

We can make some immediate useful remarks. Firstly, observe that N (fu) = fu. Indeed, it is clear from (5) and the uniqueness statement. Also observe that whenever f ∈ R∗ and N (f) = f, we have that each un = f($n) is a unit, and un−1 = f($n−1) = φ(f($n)) = ∗ φ(N ($n)) = NKn/Kn−1 (un). So letting W be the subgroup of R such that N f = f, we have an isomorphism of groups ∗ U∞ 3 u 7→ fu ∈ W ⊂ R . So we have really embedded our systems of units inside a power series ring. In fact we can let G act on R in the obvious way (1 + T ) 7→ (1 + T )χ(g), and Coleman’s map becomes G-equivariant, and even a map of Λ(G)-modules. 3.2. Logarithmic differentiation and an exact sequence. We continue to develop these ideas, the eventual aim being to get a well-understood map from U∞ into the group Rψ=0 of elements in R with trace zero, which we will later identify with the Iwasawa algebra ∗ of Zp and use to deduce Iwasawa’s theorem. There are quite a few steps along the way, so we proceed mostly without proofs, referring the reader to [1, §2]. Define the logarithmic differentiation map ∆ : R∗ → R by f 0(T ) f(T ) 7→ (1 + T ) . f(T ) It is easy to check that this maps W into the additive group Rψ=1 of elements f such that ψ(f) = f (this equation once φ is applied is exactly (1 + T ) times the logarithmic CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 9 derivative of the equation φN f = φf). It is somewhat harder (one studies reduction mod p) to check that this map is in fact surjective, giving the following exact sequence. Proposition 3.2 (Exact sequence 1). We have the short exact sequence ψ=1 0 → µp−1 → W → R → 0. The following is rather easier. Note that Rψ=0 is the additive group of trace free elements of R. Proposition 3.3 (Exact sequence 2). Let θ : R → R be 1 − φ. Then it sits in an exact sequence ψ=1 ψ=0 ev0 0 → Zp → R → R → Zp → 0. We now wish to write the composite map θ ◦ ∆ in a slightly different way, undoing the “differentiation” but retaining the “logarithmic” and “kill φ-invariants” parts. Let D : f(T ) 7→ (1 + T )f 0(T ). We seek L : W → Rψ=0 such that the following diagram commutes

W −−−−→L Rψ=0     y∆ yD Rψ=1 −−−−→θ Rψ=0. Formally integrating θ ◦ ∆ appropriately, one is led to define 1  f(T )p  Lf(T ) = log . p φ(f)(T ) One can splice exact sequences 1 and 2, and use this commutative square to get a new exact sequence. We let A denote the collection of power series of the form η(1 + T )a where p−1 ∼ a ∈ Zp, η = 1, so abstractly as a multiplicative group A = µp−1 × Zp. Theorem 3.4 (Fundamental exact sequence). We have the following exact sequence

L ψ=0 ev0◦D 0 → A → W → R → Zp → 0. 3.3. The Coates-Wiles map and measures. We now relate the Iwasawa algebra to R via the Mahler transform. Firstly, let us consider the Iwasawa algebra Λ(Zp). The Mahler transform associates to λ ∈ Λ(Zp) the power series ∞ ! X Z x M(λ) := dλ T n. n n=0 Zp

For example, one can check that the topological generator 1Zp of Zp is sent to 1 + T . Proposition 3.5. This map defines a ring isomorphism =∼ M : Λ(Zp) → R. 10 TOM LOVERING

A crucial feature of this transform is that the integrals of the functions xk with respect to M−1(g) can be computed explicitly by repeated logarithmic differentiation as follows. Proposition 3.6. For any g ∈ R, we can integrate polynomials using Z k −1 k x d(M (g)) = (D g(T ))T =0. Zp ∗ One can also view Zp ⊂ Zp and use this to get a slightly odd embedding of Iwasawa ∗ algebras Λ(Zp) ⊂ Λ(Zp) I will shy away from talking about, except to say that the image of this subring under M is precisely the ring Rψ=0. ∼ = ∗ Finally, we can use the cyclotomic character χ : G → Zp to obtain the following. ∼ Proposition 3.7. There is a canonical isomorphism M˜ : Λ(G) →= Rψ=0. With this identification, one can reinterpret the integration formula above to get Z Z k k −1 k χ(g) dλ = x d(M M˜ (λ)) = (D M˜ (λ))T =0. G Zp We can put these ideas together and get an elegant method to pass from units to measures on G, which can then be computed by a simple operation δk on fu. ψ=0 Given a unit, Coleman gives us fu ∈ W , and recall we then defined L(fu) ∈ R . Finally, we can take the inverse Mahler transform (via the cyclotomic character) to get an element of Λ(G). Corollary 3.8. There is a canonical “Coleman” map

Col : U∞ → Λ(G)

−1 taking u = (un) 7→ M˜ L(fu). To compute it as a measure we define the Coates-Wiles3 map   0  k−1 (1 + T )fu(T ) δk(u) = D . fu(T ) T =0 k This is not quite the same as D L(fu) (remember that L included some φ and power of p stuff), so the formula above picks up an extra factor and becomes

Proposition 3.9 (Computing measures coming from units). Suppose u ∈ U∞ is a unit, and k ≥ 0. Then Z k k−1 χ(g) dCol(u) = (1 − p )δk(u). G

3A version of this appeared first in their work on the conjecture of Birch and Swinnerton-Dyer for CM elliptic curves. CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 11

3.4. Cyclotomic units and the p-adic zeta function. The stage is now set for the first major miracle of these notes. Let us consider the cyclotomic units ζ−a/2 − ζa/2 u (a) = n n , n −1/2 1/2 ζn − ζn which define a norm-compatible family u ∈ U∞ (in fact inside U∞). With such an explicit formula, we can just write down Coleman’s power series4 (T + 1)−a/2 − (T + 1)a/2 f (T ) = . u (T + 1)−1/2 − (T + 1)1/2

At the moment this is a power series with Zp coefficients, but if we take a equal to some positive integer, of course this can also be viewed a series with Q coefficients, so we might hope to embed Q in C and evaluate the Coates-Wiles map (just defined in terms of formal operations on power series with no reference to the base ring) using techniques from complex analysis. z ezf 0(ez−1) Let us do this, and make the substitution T = e − 1, so set g(z) = f(ez−1) , the point being that k−1 δk(u) = g (0). As I teach my “Math 1b” students, to compute these, it will suffice to write down a Taylor series expansion for g(z) about 0. But letting h(z) = f(ez − 1), h0(z) = ezf 0(ez − 1), so we just need an expansion for the logarithmic derivative of e−az/2 − eaz/2 h(z) = . e−z/2 − ez/2 a+a−1 2 But we can use the identity a−a−1 = 1 + a2−1 to compute ! h0(z) 1 e−az/2 + eaz/2 e−z/2 + ez/2  a 1  = − a − = − − . h(z) 2 e−az/2 − eaz/2 e−z/2 − ez/2 e−az − 1 e−z − 1 Now recall that the Bernoulli numbers are defined by the identity

∞ 1 X Bk = tk−1. et − 1 k! k=0 So 0 ∞ ∞ k−1 h (z) X Bk X Bk z = − (a(−az)k−1 − (−z)k−1) = (−1)k(ak − 1) . h(z) k! k (k − 1)! k=0 k=0

Bk And finally recall that ζ(1 − k) = − k , and this vanishes for odd k, and we deduce the following proposition.

4 (T +1)−a/2−(T +1)a/2 ∗ More precisely, we should note that T ∈ R and multiply two series together to get the expression shown. 12 TOM LOVERING

Proposition 3.10. Let uc be the tower of cyclotomic units un(a). Then we have ( 0 if k odd, δk(uc) = ζ(1 − k)(ak − 1) if k even. Combining this with the results of the previous chapter, we deduce that the Coleman measure Col(uc) is very nearly the desired p-adic zeta function. We must clearly divide it a by the element σa − 1 in Λ(G) (where σa(ζn) = ζn) and the only thing to check is that this defines a pseudo-measure. Theorem 3.11 (Existence of the p-adic zeta function). The element ζ := Col(uc) is p σa−1 a pseudo-measure on G, independent of the choice of a, and satisfies the interpolation property Z ( k 0 if k > 0 odd, χ(g) dζp = G ζ(1 − k) if k > 0 even. Vanishing on odd powers of the cyclotomic character, this can also be viewed as a measure on G. Proof. Once we know it is a pseudo-measure the integral formula follows immediately from what we already know. For independence from the choice of a just do a calculation. Then use the idea of considering a = e a primitive root mod p and such that ep−1 6≡ 1(mod p2), so that σa is a topological generator of G, which combined with the fact that cyclotomic units are generated by their Galois conjugates, makes checking the above is a pseudo-measure an easy exercise.  3.5. Proof of Iwasawa’s theorem. In this section we finally prove Iwasawa’s theorem 1 1 about the Iwasawa module structure of U∞/C∞. Recall that Cn is the closure of the 1 cyclotomic units Dn in Un and then C∞ the collection of inverse systems of such elements congruent to 1 mod $n. We start with a lemma 2 p−1 Lemma 3.12. Pick e ∈ N a primitive root modulo p and such that p 6 |e − 1. Take η 5 1 the (p − 1)st such that ηe ≡ 1(mod p). Then ηu(e) = η(un(e)) generates C∞ as a Λ(G)-module.

Proof. This follows immediately from σa topologically generating Λ(G), and the fact that the cyclotomic units are generated over Z by their Galois conjugates.  Recall the fundamental exact sequence

L ψ=0 ev0◦D 0 → A → W → R → Zp → 0. Using the Coleman map and the Mahler transform on the middle terms, and tracing through a load of identifications, one can rewrite this as an exact sequence

L˜ 0 → µp−1 × Tp(µp∞ ) → U∞ → Λ(G) → Tp(µp∞ ) → 0.

5 We are working with a closure in a Zp-module here so such an action is possible. CYCLOTOMIC UNITS AND THE IWASAWA MAIN CONJECTURE 13

1 Pulling back along U∞ ⊂ U∞, we get an exact sequence of Λ(G)-modules (the things above weren’t even a Zp-module, having some prime to p torsion), but there has always been some continuous G-action around of which one should carefully keep track).

1 L˜ 0 → Tp(µp∞ ) → U∞ → Λ(G) → Tp(µp∞ ) → 0. Now let J be the subgroup of G of order 2 and take J-invariants everywhere. Since p > 2, this operation is exact (fun easy exercise: prove this in the most naive way possible), and we recover a canonical isomorphism ˜ 1 =∼ L : U∞ → Λ(G). But Iwasawa’s theorem is now almost immediate. We have written a canonical identifi- 1 ˜ cation of U∞ with Λ(G). We also know by the previous section that L(ηu(e)) = ηζp(σe −1). Finally since σe topologically generates the Galois group, we note that Λ(G)(σe −1) = I(G) ˜ 1 the augmentation ideal. So using the lemma, we observe that L(E∞) = ζpI(G) ⊂ Λ(G), proving Iwasawa’s theorem that ˜ 1 1 =∼ L : U∞/C∞ → Λ(G)/ζpI(G). 4. Euler Systems and the completion of the proof Hopefully to be written after discussions with Yihang, probably based on chapters 5 and 6 of Coates-Sujatha.

References [1] Coates-Sujatha Cyclotomic Fields and Zeta Values