Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 26 (Introduction to Iwasawa Theory)

Put $C={\bigcup }_{r\ge 1}\mathbb{Q}({\mu }_{p^r})$, where ${\mu }_{p^r}$ is the group of $p^r$th roots of unity. The compatible Galois groups of the cyclotomic fields giveFor odd $p$, the Teichmüller character givesThe p-adic logarithm identifies $1+p{\mathbb{Z}}_p$ with $p{\mathbb{Z}}_p$, which is topologically isomorphic to ${\mathbb{Z}}_p$. More explicitly, $a\mapsto (1+p{)}^a$ is an isomorphism from the additive p-adic integers onto $1+p{\mathbb{Z}}_p$: the logarithm of $1+p$ has valuation one and generates $p{\mathbb{Z}}_p$. Take the fixed field of ${\mu }_{p-1}$ in $C$.

For $p=2$, use insteadThe p-adic logarithm identifies the second factor with $4{\mathbb{Z}}_2$, and $a\mapsto 5^a$ identifies it with ${\mathbb{Z}}_2$. The fixed field of $\{\pm 1\}$ in $C$ is the union of the maximal real subfields of the cyclotomic fields of $2$-power conductor. Thus in both cases the cyclotomic Zp-extension exists:where ${\Delta }_p={\mu }_{p-1}$ for odd $p$ and ${\Delta }_2=\{\pm 1\}$.

The closed subgroup $p^n{\mathbb{Z}}_p$ is the unique subgroup of index $p^n$, so the Galois correspondence supplies the unique degree-$p^n$ field $\mathbb{Q}[p^n]$. In the odd case it is the ${\Delta }_p$-fixed field in $\mathbb{Q}({\mu }_{p^{n+1}})$; in the even case it is the real subfield of $\mathbb{Q}({\mu }_{2^{n+2}})$. These finite layers are all totally real number fields.

Now set $A=\mathbb{Q}[p^n]$, $B=\mathbb{Q}[p^m]$, and let $H_A$ be the ordinary Hilbert class field of $A$. Thus $H_A/A$ is abelian, unramified at finite primes, and split at every real place, with degree $h(A)$. If $m=n$ there is nothing to prove. Otherwise the unique prime over $p$ is totally ramified in $B/A$, by the permitted total-ramification fact and multiplicativity of the ramification index. Any nontrivial intermediate field of $B/A$ is ramified at that prime. HenceIt follows that the two extensions are linearly disjoint field extensions, andUnramified extensions remain unramified after base change. The real places also remain split: $H_A$ and $B$ are totally real, so their compositum is totally real. Therefore $B{H}_A/B$ lies inside the ordinary Hilbert class field $H_B/B$. Its degree divides $[H_B:B]=h(B)$, provingThis is class number divisibility with split real places; using a class-field statement restricted to imaginary fields would unnecessarily omit the present real layers.

Write $\Gamma =\mathrm{Gal}(K_{\infty }/K)\cong {\mathbb{Z}}_p$. Its only finite subgroup is trivial. In particular, a real place cannot acquire complex inertia in this Zp-extension: the possible nontrivial inertia at a real place has order two. Thus every infinite place splits in the tower.

Suppose that no finite prime ramified. Every finite layer would then be an abelian everywhere unramified extension of number fields, including splitting at real places. Every layer would lie in the ordinary Hilbert class field of $K$, a finite extension. Their degrees are unbounded, a contradiction. Therefore some finite prime must ramify.

Let $v$ have residue characteristic $\ell \ne p$. Apply local class field theory to the corresponding decomposition group, a closed subgroup of $\Gamma$. The image of the unit group ${\mathcal{O}}_{K_v}^{\times }$ is the inertia group. Its first principal-unit subgroup is a pro-l group, so its continuous image in the pro-p group $\Gamma$ is trivial. The residue-unit quotient is the finite group ${\mathbb{F}}_{q_v}^{\times }$. Its image is finite, and is therefore also trivial in the torsion-free group ${\mathbb{Z}}_p$. The entire inertia group is trivial. HenceEquivalently, the tame ramification relation with a Frobenius element would force a tame inertia generator to satisfy ${\tau }^{q_v-1}=1$, which is impossible nontrivially in ${\mathbb{Z}}_p$.

For the final assertion, the cyclotomic Zp-extension of $K$ is $K\mathbb{Q}[p^{\infty }]$. Fix $v\mid p$ and put $E=K_v$, a finite extension of ${\mathbb{Q}}_p$. Let $C_n$ be the completion of $\mathbb{Q}[p^n]$ at its unique prime over $p$. This is a totally ramified extension of ${\mathbb{Q}}_p$ of degree $p^n$. In a compatible local algebraic closure,Here the inequality follows from the tower through $C_n$. The right side is unbounded. These composita occur among the completions of the cyclotomic tower over $K$, so $v$ has unbounded ramification index, in particular nontrivial inertia group. ThusThis argument allows an arbitrary finite intersection between $K$ and the rational cyclotomic tower; it does not assume disjointness.

Let $\mathcal{G}=\mathrm{Gal}(M_{\infty }/F)$. The Galois group sequence isFor $g\in \Gamma$ and $x\in X$, choose a lift $\tilde{g}\in \mathcal{G}$ and defineThis is independent of the lift because $X$ is abelian. Write $X$ additively. As an abelian pro-p group, it is a compact ${\mathbb{Z}}_p$-module, with $ax$ defined by the limit of integer powers, and the conjugation action is continuous and ${\mathbb{Z}}_p$-linear. It extends to the Iwasawa algebraIndeed, use the finite quotients of $X$ obtained by intersecting it with open normal subgroups of $\mathcal{G}$; on each quotient the action factors through one of these finite group rings. This defines the natural compact Galois module structure, even when $X$ is not finitely generated. Choosing $\gamma$ identifies $\Lambda$ with ${\mathbb{Z}}_p[[T]]$ through $T=\gamma -1$.

The subgroup $(\gamma -1)X$ is closed: it is the image of the compact group $X$ under a continuous map. Every element $(\gamma -1)x$ is a commutator of a lift of $\gamma$ with $x$. ConsequentlyConversely, in the quotient by $(\gamma -1)X$, the lift of $\gamma$ centralizes the image of $X$. Its powers are dense in the procyclic subgroup they generate. Since these powers map densely onto $\Gamma$, compactness shows that $\mathcal{G}$ is generated by $X$ and that procyclic subgroup. The quotient is therefore abelian. HenceOne can also see this by choosing a lift of $\gamma$. Since $\mathcal{G}$ is a pro-p group, the closed subgroup generated by this lift is ${\mathbb{Z}}_p$, and its map to $\Gamma$ is an isomorphism. This gives a semidirect product splitting.

The maximal abelian subextension $M/F$ is the fixed field of the closed commutator subgroup. Because $F_{\infty }/F$ is abelian, $F_{\infty }\subseteq M$. Restriction of the Galois correspondence to $X$ now gives the coinvariant formulaThe closed image above is important: the formula is a quotient of compact groups, not a quotient by an unclosed abstract subgroup. This is abelianization over a Zp-extension.

Let $F_n=\mathbb{Q}({\mu }_{p^{n+1}})$ and $F_0=\mathbb{Q}({\mu }_p)$. The cyclotomic character identifiesChoose $\gamma$ with cyclotomic value $u=1+p$, and putThe Iwasawa algebra of $G$ is $\Lambda [\Delta ]$. Conjugation, independent of lift as in the preceding solution, makes $X$ a compact Galois module over this ring. It is the p-ramified Iwasawa module, not the unramified class-group module.

Because $p-1$ is a unit in ${\mathbb{Z}}_p$, the Teichmüller character $\omega$ gives orthogonal character idempotentsThey sum to one, so $X={⨁}_{\chi }{X}_{\chi }$, where $X_{\chi }=e_{\chi }X$. Each is a module over the one-variable ring $\Lambda$. Complex conjugation is the element $-1$ of $\Delta$; characters with $\chi (-1)=1$ are even, and those with $\chi (-1)=-1$ are odd.

The basic Iwasawa-module rank theorem in this tower givesIn particular,Thus it would be incorrect to describe all of $X$ as a torsion module.

Here is the class-field unit sequence explaining these ranks. Let $U_{\infty }$ be the inverse limit, under local norms, of the pro-$p$ completions of the unit groups at the unique prime over $p$ in $F_n$. Let ${\overline{E}}_{\infty }$ be the inverse limit of the closures of the global units in these local unit groups, and letunder ideal norms. The Artin reciprocity maps yieldThe final module is the unramified Iwasawa module and is finitely generated torsion, as proved in the other essay. Local-unit theory gives $U_{\infty }$ Iwasawa-module rank $p-1$, with rank one in each $\Delta$-character. The Leopoldt theorem for abelian number fields gives ${\overline{E}}_{\infty }$ rank $(p-1)/2$, with rank one in each even character and zero in each odd character. In odd characters the possible norm-compatible roots of unity contribute a rank-zero term, not a free summand. Subtracting ranks in the exact sequence gives the displayed result and also finite generation.

For any character, the Iwasawa module structure theorem describes $X_{\chi }$ up to pseudo-isomorphism:where the $f_k$ are irreducible distinguished polynomials and the kernel and cokernel of a pseudo-isomorphism are finite. For a torsion component, its characteristic ideal is generated byThe Iwasawa invariants are $\mu =\sum a_j$ and $\lambda =\sum b_k\mathrm{deg}{f}_k$. A characteristic ideal describes the elementary divisors only in aggregate; it is not an assertion that the module is cyclic or that an odd component is actually free.

The trivial character component $X_1$ is zero. Indeed, the maximal abelian pro-$p$ extension of $\mathbb{Q}$ unramified outside $p$ is precisely the rational cyclotomic Zp-extension, by the Kronecker–Weber theorem. Coprime descent using the character idempotent $e_1$, followed by the abelianization over a Zp-extension formula, identifies $(X_1{)}_{\Gamma }$ with the additional abelian pro-$p$ quotient over $\mathbb{Q}$. There is no such extension beyond the cyclotomic one, so $(X_1{)}_{\Gamma }=0$, and the Compact Nakayama lemma gives $X_1=0$.

The central arithmetic description concerns the nontrivial even characters. Let $g_{\chi }(T)$ be the integral Kubota-Leopoldt p-adic L-function power series with conventionFor positive integers $k$ of the matching parity, the interpolation iswhere the Dirichlet character in the Euler factor is its primitive associate, and $B_{k,\eta }$ is a generalized Bernoulli number. The Iwasawa main conjecture, a theorem here, saysThe trivial character is excluded from this displayed analytic normalization: its $p$-adic zeta function has a pole and must be treated separately.

The proof mechanism starts with norm-compatible cyclotomic units, for instance ratiosand their character projections. A Coleman power series encodes a norm-compatible local unit by a single power series. Applying its logarithmic derivative to these cyclotomic units produces the p-adic L-function above. If $C_{\infty }$ denotes their closed norm-limit module, this calculation determinesThe remaining global input isfor nontrivial even $\chi$. In the class-field unit sequence modulo $C_{\infty ,\chi }$, multiplicativity of characteristic ideals cancels these two terms and gives the boxed formula for $X_{\chi }$.

There are two established routes to this global input. The Mazur-Wiles theorem constructs suitable abelian extensions from Galois representations associated with modular forms and the Eisenstein ideal; their sizes supply the missing divisibility. The cyclotomic Euler system route uses norm relations for cyclotomic units at auxiliary primes and descent to bound the class-group module. The analytic class number formula and the cyclotomic-unit index formula then supply the equality of characteristic ideals. These are substantial arithmetic theorems, rather than consequences of the abstract structure theorem alone.

A useful further result is the Ferrero-Washington theorem: $\mu (Y)=0$ for this cyclotomic tower, more generally for cyclotomic towers of abelian number fields. Its proof rules out an identically zero reduction modulo $p$ of the relevant $p$-adic $L$-series, using distribution of $p$-adic digits. Together with the main conjecture, it gives $\mu (X_{\chi })=0$ for each nontrivial even $\chi$. Therefore these torsion components are finitely generated over ${\mathbb{Z}}_p$ up to finite error.

Finally, Kummer reflection in Iwasawa theory relates the even $p$-ramified module to an odd unramified component:where $(1)$ is the Tate twist by the cyclotomic character. The corresponding power-series substitution isThis explains why the $p$-ramified interpolation involves $u^{1-s}$ while the reflected class-group interpolation involves $u^s$. Both the involution and the twist matter. The rank theorem, the class-field sequence, the structure theorem and the explicit even-character characteristic ideals together give the known compact Galois module structure; none justifies replacing every odd component by a free module without further argument.

Let $F_n$ be the degree-$p^n$ layer, ${\Gamma }_n={\Gamma }^{p^n}$, andConjugation makes $Y$ a compact $\Lambda$-module. The goal is to show that it is finitely generated and has Iwasawa-module rank zero. This proof applies to every Zp-extension, not only a cyclotomic one, and makes no Leopoldt conjecture assumption.

By the ramification argument above, only primes over $p$ ramify in $F_{\infty }/F$, and at least one does. Every nonzero closed inertia group in $\Gamma \cong {\mathbb{Z}}_p$ is open. After replacing $F$ by a finite layer, every prime which ramifies is totally ramified in the remaining tower. The number $s\ge 1$ of such primes then stays constant. This replacement does not affect whether $Y$ is a torsion module: ${\mathbb{Z}}_p[[\Gamma ]]$ is finite free over the Iwasawa algebra of an open subgroup, and the two module ranks vanish together.

Let ${\mathcal{G}}_n=\mathrm{Gal}(L_{\infty }/F_n)$, and let $D_n$ be its maximal abelian quotient. The abelianization over a Zp-extension formula gives an exact sequenceChoose a lift of a generator of ${\Gamma }_n$ if a splitting is desired. We will bound the ${\mathbb{Z}}_p$-rank of the middle term independently of $n$.

At each of the $s$ ramified primes, inertia in $L_{\infty }/F_n$ maps isomorphically onto ${\Gamma }_n$: its kernel is inertia in the unramified extension $L_{\infty }/F_{\infty }$, hence trivial, and the map onto the totally ramified base inertia is surjective. Thus each image in $D_n$ is procyclic and has ${\mathbb{Z}}_p$-rank at most one. No other finite prime contributes inertia. Quotienting $D_n$ by the closed subgroup generated by these $s$ images gives an abelian extension of $F_n$ unramified at all finite primes. Its Galois group is finite by class field theory, using the ordinary or narrow ideal class group according to the treatment of real places. Any infinite-place inertia is finite and does not affect the rank bound. Therefore $D_n$ is finitely generated over ${\mathbb{Z}}_p$ andThe preceding exact sequence now yields the uniform boundIt does not assert that these coinvariant modules are always finite when several primes ramify.

At $n=0$, $Y/TY$ is finitely generated over ${\mathbb{Z}}_p$, so $Y/(p,T)Y$ is finite. Lift a finite basis of this quotient to $Y$. The Compact Nakayama lemma shows that these lifts generate $Y$ over $\Lambda$. Briefly, the quotient by their compact generated image satisfies $Q=(p,T)Q$. Every finite continuous $\Lambda$-quotient of $Q$ has nilpotent action by $(p,T)$: $p$ is nilpotent, and modulo $p$ a finite pro-$p$ action makes $\gamma -1$ nilpotent. Such a quotient must be zero. Finite quotients separate points of a compact pro-p group, so $Q=0$.

Suppose now that $Y$ had positive $\Lambda$-rank $r$. The Iwasawa module structure theorem provides a pseudo-isomorphism towith $E$ a finitely generated torsion module. Passing to ${\Gamma }_n$-coinvariant modules leaves a finite cokernel. Butis free of rank $p^n$ over ${\mathbb{Z}}_p$. Thus the coinvariant modules would have rank at least $r{p}^n$, contradicting the uniform bound $s-1$. Equivalently their ranks have asymptotic form $r{p}^n+O(1)$, with the torsion elementary divisors contributing only a bounded rank.

It follows that $r=0$. Since $\Lambda$ is an integral domain, rank zero means every element is killed by a nonzero scalar. ThereforeThis is the unramified Iwasawa torsion theorem. The crucial arithmetic inputs are finite class fields and the bounded number of ramified primes; neither vanishing of a $\mu$-invariant nor a cyclotomic main conjecture is required.