Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 26 (Cyclotomic Fields)

Set $X=1+T$ and let $\phi :R\to R$ be the Frobenius substitution on cyclotomic power series, $\phi (g)(T)=g(X^p-1)$. It is injective: modulo $p$ the substitution is $g(T)\mapsto g(T^p)$, which is injective; if $\phi (g)=0$, this first gives $g\in pR$, and iterating gives $g\in {\bigcap }_r{p}^{r}R=0$.

We first prove that every formal power series has a unique expansionModulo $p$, the ring ${\mathbb{F}}_p[[T]]$ is free over ${\mathbb{F}}_p[[T^p]]$ with basis $1,T,\dots ,T^{p-1}$: group the coefficients by their exponents modulo $p$. Replacing this basis by $1,X,\dots ,X^{p-1}$ gives an invertible triangular change of basis with diagonal entries one. Lift the resulting coefficients to $R$, subtract their displayed expression from $f$, divide the remainder by $p$ and repeat. The p-adic limits of these corrections give the required $f_i$ because $R$ is p-adically complete. If an expression is zero, reduction modulo $p$ forces all $f_i$ to be divisible by $p$; repetition forces every $f_i$ to vanish. This proves both existence and uniqueness without dividing coefficients by $p$.

Define the Coleman trace operator by $\psi (f)=f_0$. To verify its averaging formula, work temporarily over ${\mathbb{Z}}_p[{\mu }_p][[T]]$, interpreting the substitutions in the $(\xi -1,T)$-adic topology. The argument $\xi X-1$ is topologically nilpotent there, so substituting into an arbitrary formal power series is legitimate. MoreoverThe sum of ${\xi }^i$ over the p-th roots of unity is $p$ at $i=0$ and zero at $1\le i<p$. ThereforeThe right side belongs to $R$, so the apparent denominator causes no integrality problem. Injectivity of $\phi$ proves that this is the unique map $\psi$ with the required identity. The construction also proves $\psi (\phi (g)f)=g\psi (f)$ and $\psi (\phi (g))=g$.

Write a bar for reduction modulo $p$. For $n\ge 1$,The binomial theorem in characteristic $p$ givessince $\left(\genfrac{}{}{0ex}{}{p-1}{i}\right)\equiv (-1{)}^i(\mathrm{mod}p)$ and $p$ is odd. Taking the zeroth basis coefficient yieldsThe endpoint case $n=1$ includes the constant term $T^0=1$.

Half exponents in an odd-order root of unity group mean multiplication by the inverse of two modulo its order. This uniquely defines ${\zeta }_n^{a/2}$ inside $K_n$, and the definitions are compatible as $n$ varies. Put $d_a(z)=z^{-a/2}(1-z^a)$ on these roots. The symmetric cyclotomic unit is $c_n(a,b)=d_a({\zeta }_n)/d_b({\zeta }_n)$. Complex conjugation replaces each $d_a$ by $-d_a$, so the ratio is real. It is a global unit: choosing positive representatives for $a,b$ modulo $p^{n+1}$ gives geometric-series expressions for $(1-{\zeta }_n^a)/(1-{\zeta }_n)$ and its inverse, and similarly for $b$. The remaining factor is a root of unity.

The conjugates of ${\zeta }_n$ over $K_{n-1}$ are $\xi {\zeta }_n$ with $\xi \in {\mu }_p$. Because $p\nmid a$, multiplication by $a$ permutes ${\mu }_p$, andAlso ${\prod }_{\xi \in {\mu }_p}\xi =1$ for odd $p$, so $N_{n,n-1}({\zeta }_n)={\zeta }_{n-1}$. Applying the compatible half power proves $N_{n,n-1}(d_a({\zeta }_n))=d_a({\zeta }_{n-1})$. Dividing the analogous identities for $a,b$ proves

Embed the fields into the p-adic cyclotomic tower. Its interpolating Coleman power series isHere $(b-a)/2\in {\mathbb{Z}}_p$, and binomial expansion defines the half power. Both numerator and denominator vanish to order one at $T=0$, with coefficients $-a$ and $-b$, so cancellation gives an integral unit with $f_{a,b}(0)=a/b$. This applies also to negative integers $a,b$. Evaluation at ${\zeta }_n-1$ gives $c_n(a,b)$; the same root-product calculation shows $\mathcal{N}{f}_{a,b}=f_{a,b}$ for the Coleman norm operator.

For clarity, the normalization of the higher logarithmic derivative isIn the formal coordinate $T=e^z-1$, $D=d/dz$ andOnly the derivative of the logarithm is needed, so no choice of its constant term is involved. Differentiating gives the regular-at-zero expressionUsing the Bernoulli numbers defined by $z/(e^z-1)={\sum }_{r\ge 0}{B}_r{z}^r/r!$, with $B_1=-1/2$, givesThe two poles cancel. Taking $k-1$ further derivatives and evaluating at zero therefore yields all the requested values:In particular ${\delta }_1=0$, ${\delta }_2=(a^2-b^2)/12$, and ${\delta }_4=-(a^4-b^4)/120$. The odd case includes $k=1$; substituting $B_1$ into the even-index formula would incorrectly lose the symmetric normalization. All the displayed values lie in ${\mathbb{Z}}_p$, directly from the integral Coleman power series definition, even when the rational Bernoulli expression has an apparent p-adic denominator. For even $k\ge 2$ the same answer is $-(a^k-b^k)\zeta (1-k)$.

The relevant notion is a root-of-unity Euler system, which contains auxiliary-prime data as well as the p-power tower. Fix a finite set $S$ of primes containing two but not $p$, and let $W_S$ consist of all roots of unity whose orders are prime to every member of $S$. It is a map $\Phi :W_S\to {\overline{\mathbb{Q}}}^{\times }$ satisfying these three axioms:

The first axiom puts $\Phi (\zeta )$ in the maximal real subfield of $\mathbb{Q}(\zeta )$. For $\zeta ={\zeta }_n$, the second axiom is exactly norm compatibility from $K_n^{+}$ to $K_{n-1}^{+}$ for $n\ge 1$. Auxiliary roots with prime orders different from $p$ are essential: mere norm compatibility of a sequence in the tower is not a verification of all three axioms.

The real cyclotomic unit group at a p-power conductor is generated by $-1$ and the symmetric cyclotomic units $c_n(a,b)$ with $p\nmid ab$. Galois conjugates are included, since a conjugate of $c_n(a,b)$ is $c_n(ua,ub)$ for an integer $u$ prime to $p$. Thus a prescribed unit can be expressed aswhere $d_a(z)=z^{-a/2}-z^{a/2}$, with half exponents interpreted in the odd-order root group. Choose $S$ to contain two and every prime divisor of the $a_j$. For $z\ne 1$ in $W_S$ putEach factor at $z\ne 1$ is nonzero because raising $z$ to $a_j$ preserves its order. At $z=1$ each $d_{a_j}$ has a simple zero with first coefficient $-a_j$; the total exponent zero cancels the zeros and signs, giving the stated removable value. Consequently $\Phi ({\zeta }_n)=\epsilon$.

We verify the axioms individually. First, $d_a(z^{-1})=-d_a(z)$; the product's sign is $(-1{)}^{\sum e_j}=1$, so $\Phi$ is even. The unique half-root operation commutes with every field automorphism, all exponents are integers and $\eta$ is rational. Hence $\Phi (\sigma z)=\sigma \Phi (z)$, including at $z=1$.

Second, $q\notin S$ is odd and does not divide any $a_j$. For a variable $x$, choose half powers consistently and useHere $\rho \mapsto {\rho }^a$ permutes ${\mu }_q$, and the product of its half powers is one. Raising these identities to $e_j$ and multiplying gives the distribution identity, since ${\eta }^q=\eta$. If $x=1$ or $x^q=1$, take the removable values after cancellation; the rational-function identity remains valid there. Thus the distribution axiom holds also at the special roots where the uncancelled factors vanish.

Third, fix $q\notin S$, a root $z$ of order prime to $q$, and a prime $\mathfrak{q}$ above $q$. Every $\rho \in {\mu }_q$ reduces to one modulo $\mathfrak{q}$. If $z\ne 1$, its reduction retains its exact order: prime-to-q roots have distinct reductions because their order polynomial is separable in characteristic $q$. Therefore $1-z^{a_j}$ is a local unit, and all the factors and inverse factors are regular at that reduction. Reduction of the expression for $\Phi (\rho z)$ then gives $\Phi (z)$, since the half roots reduce compatibly. This proves the congruence and local integrality for nontrivial $z$.

At $z=1$, write $\rho =1+T$. The cancelled expressionis a unit formal power series over ${\mathbb{Z}}_q$, because every $a_j$ and two are q-adic units. Its constant term is $\eta {\prod }_j{a}_j^{e_j}$. Evaluating at $T=\rho -1$ proves the remaining congruence modulo every prime over $q$. This also covers $\rho =1$. Hence every real cyclotomic unit at any layer is a value of an Cyclotomic Euler system satisfying all three axioms.

For completeness, at a nontrivial root $z$ every ratio $d_a(z)/d_b(z)$ is a global unit. After removing a root-of-unity factor, both it and its inverse are geometric sums because $a,b$ are invertible modulo the order of $z$. Thus the values used at nontrivial roots really are units, not just nonzero algebraic numbers. Values at one may be rational S-units, which is compatible with the definition.

The auxiliary norm factor can also be read off explicitly. If $z$ has order $m>2$ and $q\nmid m$, then for $\rho \ne 1$ of order $q$,Here the arithmetic Frobenius sends $z$ to $z^q$. This convention makes the sign of the Euler factor explicit, while the case of a repeated prime $p$ gives the exact tower norm relation already proved in Question 2.

Write $F_n=K_n^{+}$, $F_{\infty }={\bigcup }_n{F}_n$, $G=\mathrm{Gal}(F_{\infty }/\mathbb{Q})={\Delta }^{+}\times \Gamma$, where $|{\Delta }^{+}|=(p-1)/2$ and $\Gamma \simeq {\mathbb{Z}}_p$. Set $A={\mathbb{Z}}_p[[G]]$. Choosing a topological generator $\gamma$ of $\Gamma$ identifies its Iwasawa algebra with $\Lambda ={\mathbb{Z}}_p[[S]]$, $S=\gamma -1$; this group variable is distinct from the local-unit interpolation variable $T$. The Teichmüller character decomposes $A$ into a product of copies of $\Lambda$, one for each character of ${\Delta }^{+}$, equivalently each even Teichmüller character of $\mathrm{Gal}(K_0/\mathbb{Q})$. All characteristic-ideal statements below are componentwise in this product.

Let $U_n^1$ be the local principal units at the unique prime above $p$. Let $E_n^1$ and $C_n^1$ be respectively the p-adic closures of global units and cyclotomic units inside $U_n^1$, and take their inverse limits under the field norms. Define $X_{\infty }$ as the Galois group of the maximal abelian pro-p extension of $F_{\infty }$ unramified away from $p$, and $Y_{\infty }$ as that of the maximal everywhere unramified extension. This is the real tower: its prime-ramified module is torsion, by the Leopoldt theorem for abelian number fields. It must not be confused with the positive-rank prime-ramified module of the full complex cyclotomic tower. The unramified Iwasawa torsion theorem gives torsion for $Y_{\infty }$.

A convenient formulation of the Iwasawa main conjecture uses the p-adic zeta pseudomeasure ${\zeta }_p$ on $G$. With the sign conventionits product with the augmentation ideal $I(G)$ is integral and principal. The pole on the trivial-character component is cancelled by that ideal; simply writing an integral quotient by ${\zeta }_p$ would be incorrect. The main assertion in this convention isWe explain the local calculation, the remaining global obstruction, and how the cyclotomic Euler system removes it.

The local calculation is the Iwasawa theorem on local cyclotomic units:One can see why this theorem has exactly this analytic term. For a norm-fixed Coleman power series $f$, its corrected logarithm isThe ratio lies in $1+pR$, so the expression is integral. The norm identity puts its Amice transform in the kernel of the Coleman trace operator, hence gives a measure on ${\mathbb{Z}}_p^{\times }$. Its kth moment is $(1-p^{k-1}){\delta }_k(f)$. The full local exact sequence has p-power roots of unity at its kernel and cokernel. Passing to the real, or even, part removes both, since complex conjugation acts as minus one on their Tate module and $p$ is odd. The corrected logarithm therefore identifies $U_{\infty }^1$ with $A$.

Question 2 now shows that the image of $c(a,b)$ is $([b]-[a]){\zeta }_p$: its even moments are $(1-p^{k-1})(a^k-b^k)B_k/k$. A choice of integer $e$ generating ${\mathbb{Z}}_p^{\times }$ topologically gives a norm-compatible generator $c(e,1)$, after multiplication by its constant Teichmuller lift to make it principal. That constant has zero logarithmic derivatives. Its conjugates generate the closed cyclotomic-unit module, and $[e]-[1]$ generates $I(G)$. This yields the displayed local quotient and its characteristic ideal. The generation statement is an important ingredient; the mere calculation of a few moments would not prove the theorem.

The class-field unit sequence gives the global comparisonMultiplicativity of characteristic ideals in exact sequences givesThus the local theorem proves the main conjecture precisely when one proves $\mathrm{char}(B)=\mathrm{char}(Y_{\infty })$. These terms measure the global-unit and ideal-class obstructions. Discarding them would silently impose a much stronger arithmetic hypothesis.

The first global step is the Euler-system divisibility for real cyclotomic class modules: a generator $f_Y$ of $\mathrm{char}(Y_{\infty })$ divides a generator $f_B$ of $\mathrm{char}(B)$ on every character component. Here is the mechanism. Choose auxiliary primes $q$ splitting completely in a finite layer and satisfying $q\equiv 1(\mathrm{mod}{p}^m)$. For a generator ${\sigma }_q$ of its auxiliary cyclic group, putApply products of these operators to the unit Cyclotomic Euler system of Question 3. The norm axiom and this group-ring identity make the derived elements invariant modulo $p^m$-th powers, so Kummer descent gives classes in $F_n^{\times }/(F_n^{\times }{)}^{p^m}$. The congruence axiom identifies the valuation at each newly introduced prime with a residue symbol of the preceding class; valuations away from the chosen auxiliary primes vanish modulo $p^m$. The Chebotarev density theorem supplies primes in prescribed ideal classes with the required residue-symbol behavior. Adding these primes successively forces the elementary divisors of the class group to divide the available cyclotomic-unit index. Taking the norm limit yields $f_Y\mid f_B$. Finite control defects do not survive localization at height-one primes; using two coprime annihilators of such a finite defect also shows that no hidden p-power factor is left. This is the substantive global argument furnished by the Cyclotomic Euler system, not a consequence of Iwasawa's local theorem alone.

For the reverse comparison use the cyclotomic unit index formula and norm descent, keeping invariants as well as coinvariants. At the base field put $A_0$ for its p-primary ideal class group, and let $N_{\infty }(E_0^1)$ be the universal norm of units in a Zp-extension. Class-field and unit descent, using total ramification and the principal prime over $p$, giveThese are the required norm-defect control statements. The cyclotomic unit index formula, with the p-adic closure justified by Leopoldt, gives $\#A_0=\#(E_0^1/C_0^1)$. Also $B^{\Gamma }=0$: its coinvariants are finite by the displayed control, so its invariants are finite by the Iwasawa module structure theorem, whereas $B$ is a submodule of the cyclic local quotient $H$, which has no nonzero finite submodule. One must retain $Y_{\infty }^{\Gamma }$; dropping it would incorrectly identify all global units with universal norms.

Define the finite Euler characteristic of a one-variable Iwasawa module by ${\chi }_{\Gamma }(M)=\#M_{\Gamma }/\#M^{\Gamma }$. The preceding identities now give an exact cancellation:For a torsion $\Lambda$-module with finite invariant and coinvariant groups, the structure theorem gives ${\chi }_{\Gamma }(M)=|f_M(0){|}_p^{-1}$. Write $f_{B,\chi }=f_{Y,\chi }{h}_{\chi }$ using the Euler-system divisibility. After forgetting ${\Delta }^{+}$, the characteristic series are the products of their character-component series. Equality of these Euler characteristic of a one-variable Iwasawa module values givesEvery summand is nonnegative, since $h_{\chi }\in {\mathbb{Z}}_p[[S]]$. Each is therefore zero. A formal power series with p-adic-unit constant term is a unit, so every $h_{\chi }$ is a unit. Hence $\mathrm{char}(B)=\mathrm{char}(Y_{\infty })$, and cancellation in the class-field exact sequence completes the boxed main conjecture.

The familiar reflected odd class-group formulation follows from Kummer reflection in Iwasawa theory, with the inversion of the group variable and the Tate twist both included. For example, for nontrivial even $\chi$, if $f_{\chi }((1+p{)}^s-1)=L_p(\chi ,s)$, then the p-ramified series is $g_{\chi }(S)=f_{\chi }((1+p)(1+S{)}^{-1}-1)$, and $g_{\chi }((1+p{)}^{1-s}-1)=L_p(\chi ,s)$. The proof above includes the exceptional trivial component through augmentation regularization. It assumes neither a vanishing real class group nor Vandiver's conjecture: the Euler-system divisibility and the norm-defect cancellation replace that extra hypothesis.