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A cyclotomic Euler system must relate units at many auxiliary conductors, not just the successive layers of one p-power tower. A useful axiomatization is the root-of-unity Euler system. Fix a finite set of excluded primes $S$ containing $2$, and let $W_S$ be the roots of unity whose orders are prime to every prime in $S$. A system is a nonzero-valued function $\Phi :W_S\to {\overline{\mathbb{Q}}}^{\times }$ with three properties. It is even and Galois equivariant:It satisfies the distribution relation at every prime $\ell \notin S$:Finally, if $\ell \nmid \mathrm{ord}(\xi )$, it satisfies the local congruenceThe values in this congruence are required to be integral at the prime; in the unit example below they are local units. Evenness puts the values in the maximal real cyclotomic subfields. Equivariance identifies conjugates, distribution gives exact norm identities, and the congruence supplies information at newly introduced primes. These are the three distinct inputs used in the descent, rather than a single norm-coherence assumption.

Here is a concrete system extending the units in Q2. Fix a nonzero integer $a$ prime to $p$ and let $S$ contain $2$ and the prime divisors of $a$, but not $p$. PutEvery value is defined and nonzero: the excluded primes make $a$ invertible modulo the order of $\xi$, and that order is odd. The quotient is the same Laurent polynomial $b_a$ used in Q2. Replacing $\xi$ by ${\xi }^{-1}$ changes both numerator and denominator by a minus sign, so it is even; its rational coefficients give Galois equivariance.

For the distribution axiom, $\ell \notin S$ makes $\ell$ odd and prime to $a$. Both exponents $2$ and $2a$ permute ${\mu }_{\ell }$, and ${\prod }_{\rho \in {\mu }_{\ell }}\rho =1$. The calculation in Q2 therefore holds with $p$ replaced by $\ell$:Raising to $p-1$ proves the axiom. At removable points it holds by evaluating the regular Laurent polynomials, so it includes $\xi =1$ and ${\xi }^{\ell }=1$.

For the congruence axiom, a root $\rho$ of order $\ell$ reduces to one at every prime above $\ell$. The integral Laurent polynomial $b_a$ thus gives $b_a(\rho \xi )\equiv b_a(\xi )$. These values are local units when $\xi$ has order prime to $\ell$: its reduction retains that order, and, unless $\xi =1$, both $1-{\xi }^2$ and $1-{\xi }^{2a}$ remain nonzero; at $\xi =1$ the value is the unit $a$. The stated power preserves the congruence. Thus ${\Phi }_a$ satisfies all three Euler-system axioms, and its values at ${\zeta }_n$ are precisely $c_n(a)$. Products, Galois translates, suitable field norms and integral group-ring powers produce the other cyclotomic-unit systems needed in the theory.

The distribution relation explains the Euler factors. If $\ell \nmid m$, $\xi$ has order $m$ and $\rho$ is a primitive $\ell$th root, the conjugates over $\mathbb{Q}({\mu }_m)$ run through the nontrivial $\ell$th roots. Removing the single factor with $\rho =1$ from the distribution product givesWe use arithmetic Frobenius, so ${\mathrm{Fr}}_{\ell }(\xi )={\xi }^{\ell }$. This is the cyclotomic Euler norm factor at a new prime. At a higher power of a prime already dividing the conductor, all $\ell$ translates occur, and the ordinary successive-layer norm relation has no missing-factor quotient. A mere p-power compatible sequence therefore contains only part of the Euler-system information.

To explain the axiomatic method, fix a p-power modulus $P=p^r$ and auxiliary primes $\ell$ with $P\mid \ell -1$, split in the chosen real base field. For the cyclic auxiliary Galois group of order $\ell -1$, choose a generator ${\sigma }_{\ell }$ and defineSubtracting consecutive coefficients gives the concrete identityThis is the Kolyvagin derivative operator for a cyclic group. Apply products of these operators to Euler-system values at squarefree auxiliary conductors. The norm relations control the apparent failure of invariance modulo $P$; Kummer descent, using Hilbert 90 and retaining roots-of-unity corrections where present, turns the derivative data into classes over the base field. They have controlled valuations and are unramified away from the selected auxiliary primes and the excluded set. The local congruence relates the valuation component at a newly added prime to the finite residue-symbol component of the preceding class, with the sign and normalization determined by the chosen generator. This finite-to-singular relation is precisely the additional information not obtainable from norm compatibility alone.

The Chebotarev density theorem supplies auxiliary prime ideals in specified ideal classes while satisfying the splitting and congruence conditions. Reading the controlled valuations as principal-ideal relations successively bounds the p-primary elementary divisors of the ideal class group by the divisibility of the initial cyclotomic-unit class. On suitable nontrivial character components this gives the finite bound $\#A^{\chi }\le \#(E/C{)}^{\chi }$; passing through the norm tower yieldswhere divisibility is between characteristic generators, so the associated ideal containment is reversed. The Euler-system divisibility for real cyclotomic class modules requires the usual local conditions and control of finite errors; it is not a claim that arbitrary compatible units automatically annihilate every class group. The cyclotomic unit index formula, together with the global norm-defect comparison, gives the reverse equality. Q3's exact sequence then proves the main conjecture.

This account separates the theorem inputs from the explicit calculations: the distribution and congruence identities for cyclotomic units are verified above, and the derivative operator is computed directly; the Chebotarev/Kummer descent bounds and the global index comparison are the deeper arithmetic steps. Their usefulness is that the descent uses the axioms rather than a special closed formula for each individual auxiliary unit.

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.

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.