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Put $X=1+T$ and $Y=X^p-1$, so that $\varphi (f)=f(Y)$. The essential integrality statement for the Frobenius substitution on cyclotomic power series is the decomposition into a finite free module:To prove it, reduce modulo $p$. Since $Y\equiv T^p$, grouping the coefficients of a formal power series according to their exponents modulo $p$ gives the unique expression ${\sum }_{j=0}^{p-1}{T}^j{h}_j(T^p)$. The change from $1,T,\dots ,T^{p-1}$ to $1,X,\dots ,X^{p-1}$ is a triangular matrix with diagonal entries $1$, so this is also a basis over ${\mathbb{F}}_p[[T^p]]$. Lift these $p$ coefficient series to $R$, subtract their contribution from $f$, divide the remaining error by $p$, and repeat. The sums of the successive lifts converge in the finer of the topologies on integral formal power series, because $R$ is complete in that topology. This proves existence of the decomposition over $R$. A relation among its summands reduces to the zero relation modulo $p$, forcing each coefficient series to be divisible by $p$. Repetition forces divisibility by every power of $p$, hence every coefficient series is zero. This proves uniqueness and, in particular, injectivity of $\varphi$.

For $f={\sum }_{i=0}^{p-1}{X}^i\varphi (f_i)$, defineThis is a ${\mathbb{Z}}_p$-linear map. To calculate the average over roots of unity, work over $\mathcal{O}={\mathbb{Z}}_p[{\zeta }_p]$. The substitutions $T\mapsto \zeta X-1$ are legitimate: their constant terms are topologically nilpotent, so the defining coefficient sums converge in $\mathcal{O}$. Since $(\zeta X{)}^p=X^p$, each $\varphi (f_i)$ is unchanged by the substitution. The sums of the powers of the roots of unity are $p$ for $i=0$ and zero for $1\le i<p$. ConsequentlyThus the average is integral and belongs to $\varphi (R)$, and the constructed Coleman trace operator has the required property. Injectivity of $\varphi$ forces uniqueness of any map satisfying that property. Finally, the decomposition of $\varphi (g)$ has $f_0=g$ and all other coefficients zero, giving $S\circ \varphi ={\mathrm{id}}_R$.

We will also use the trace of multiplication by $h$ on this finite free module. After extending scalars to split the conjugates, that trace is the sum just computed. Hence

Let $M_f$ be the matrix of multiplication by $f$ in the basis from Question 1. Define the Coleman norm operator by the norm for a finite free ring extension:The entries lie in $\varphi (R)$, and $\varphi :R\to \varphi (R)$ is an isomorphism, so this determines $Nf$ uniquely. If $f$ is a unit, $M_f$ is invertible and its determinant is a unit. Also $M_{fg}=M_f{M}_g$, giving $N(fg)=NfNg$. Splitting the conjugates, or diagonalizing multiplication after adjoining the roots of unity and inverting $p$, gives the equivalent formulaIn particular, this definition has the required target $R^{\times }$, rather than merely a ring with extra roots of unity in its coefficients.

For the requested ring congruence, observe that the reduction of $\varphi$ is the injective map $\overline{g}(T)\mapsto \overline{g}(T^p)$. If $\varphi (g)\in p^{k}R$, its reduction first gives $g=p{g}_1$. Then $\varphi (g_1)\in p^{k-1}R$; repeat to obtain $g\in p^{k}R$. Applying this to $g=f-1$ provesThis congruence reflection for Frobenius substitution is valid for all $f\in R$, without assuming $f$ is a unit.

A further property needed for the fixed points is Coleman norm contraction. If $g=1+p^{k}h$ with $k\ge 1$, expand the determinant:Here $e_j(M_h)\in \varphi (R)$ is the coefficient of degree $j$ in $\det (I+t{M}_h)$. The trace formula from Question 1 makes the linear term divisible by $p^{k+1}$; every later term is divisible by $p^{2k}$ and $2k\ge k+1$. Reflecting the resulting ring congruence through $\varphi$ givesIf $Nf=f$ and $f\in 1+pR$, iteration gives $f\in 1+p^{k}R$ for every $k$. Since ${\bigcap }_k{p}^{k}R=0$, $f=1$.

To identify all fixed points, we also need $Nf\equiv f(\mathrm{mod}pR)$ for every unit $f$. In characteristic $p$, the extension ${\mathbb{F}}_p[[T]]/{\mathbb{F}}_p[[T^p]]$ has degree $p$ and its norm for a finite free ring extension is $\overline{f}\mapsto {\overline{f}}^p$. Indeed, after passing to fraction fields it is a purely inseparable field extension of degree $p$, so multiplication has only the eigenvalue $\overline{f}$, with multiplicity $p$, after splitting. Its determinant is ${\overline{f}}^p$. Thereforeand congruence reflection for Frobenius substitution gives $Nf\equiv f(\mathrm{mod}pR)$.

Now choose any lift $f_0\in R^{\times }$ of a given $a\in A^{\times }$ and put $f_n=N^n{f}_0$. Multiplicativity givesThus $(f_n)$ is a Cauchy sequence for the finer of the topologies on integral formal power series and converges to a unit $w(a)$ reducing to $a$. The Coleman norm operator is continuous: multiplication matrices and their determinants are continuous, and the finite-free coordinate decomposition and its inverse preserve the $p$-adic filtration defining the finer of the topologies on integral formal power series. Hence $Nw(a)=w(a)$. If two fixed units reduce to $a$, their quotient is fixed and belongs to $1+pR$, so it is $1$ by the preceding argument. This proves uniqueness, independence of the initial lift, and multiplicativity of the Coleman norm-fixed lift. In particular,Every unit $f\in R$ consequently has a unique factorization $f=w(\overline{f})v$ with $v\in 1+pR$, giving the canonical Coleman decomposition of power-series units:The final assertion as printed is false. The original PDF identifies a ring with a product of unit groups. With multiplication, $A$ has a zero and is not a group. With addition, it has exponent $p$, whereas $W\times A^{\times }$ has a nonidentity element of order $2$: $-1\in W$ because $p$ is odd and $N(-1)=(-1{)}^p=-1$. Thus neither interpretation makes the printed assertion true. The two boxed isomorphisms above are the valid natural conclusions; they do not require guessing which symbols the author intended to replace.

The preceding constructions lead to the Iwasawa theorem on local cyclotomic units. Its significance is that an explicitly calculated local unit quotient is governed by a p-adic L-function; class field theory then suggests that the same analytic object should govern global arithmetic. The latter prediction is the Iwasawa main conjecture, rather than a consequence of the local calculation alone.

Fix compatible roots of unity ${\zeta }_{p^n}$ and put $K_n={\mathbb{Q}}_p({\zeta }_{p^n})$, ${\pi }_n={\zeta }_{p^n}-1$. Let $V_{\infty }$ be the norm limit of the local principal units. Take its even part under complex conjugation to obtain $U_{\infty }^1$, the local principal-unit limit in the real tower. Let $C_{\infty }^1$ be the closed subgroup generated by norm-compatible real cyclotomic units, with their finite-order residue factors removed. WriteHere the last map is augmentation. Normalize the p-adic zeta pseudomeasure ${\zeta }_p$ byThe expression $x^k$ is well defined on $G$ for even $k$. A pseudomeasure lies in the total fraction ring of the Iwasawa algebra; the pole on its trivial finite-character component is removed by the augmentation ideal, so $I(G){\zeta }_p\subset {\Lambda }_G$. In this normalization the local theorem isThe following steps outline its proof and explain where the earlier questions enter.

First encode local units by Coleman power series. Evaluation at ${\pi }_n$ is surjective from $R$ to ${\mathcal{O}}_{K_n}$, and the product defining the Coleman norm operator givesThus norm-fixed series give norm-compatible sequences of local units. Conversely, lift $u_m$ from such a sequence to $f^{(m)}\in R^{\times }$ and take $g_m=N^{m-r_m}{f}^{(m)}$, where $r_m=\lfloor m/2\rfloor$. This has $g_m({\pi }_{r_m})=u_{r_m}$. The estimates in Question 2 give $N{g}_m/g_m\in 1+p^{m-r_m+1}R$. Consequently for fixed $n\le r_m$, the series $N^{r_m-n}{g}_m$ differs from $g_m$ modulo $p^{m-r_m+1}$ and its value at ${\pi }_n$ is $u_n$. The compactness of $R$ in the coarser of the topologies on integral formal power series supplies a convergent subsequence. Continuity of the Coleman norm operator and of evaluation shows that its limit is norm-fixed and takes every value $u_n$. It is a unit because the original principal units have residue $1$. Uniqueness follows from the Weierstrass preparation theorem: a nonzero series in $R$ has only finitely many zeros in the open unit disk, whereas the distinct ${\pi }_n$ give infinitely many interpolation points. This constructs the Coleman power series $f_u$.

Next turn its multiplicative information into an integral p-adic measure. For a norm-fixed unit $f$, use the Frobenius-corrected logarithmThe ratio belongs to $1+pR$ by reduction modulo $p$, and the logarithm divided by $p$ is integral because $p$ is odd. Equivalently $\mathcal{L}(f)=(1-\varphi /p)\log f$, interpreting the constant coefficient with the p-adic logarithm on units. The averaging identity for the logarithm is understood on the open $p$-adic unit disk, where the translated logarithmic series converge; the corrected result itself lies in $R$. The logarithmic product formula and Question 1 giveUnder the Amice transform, the root-of-unity average projects a p-adic measure onto its restriction to $p{\mathbb{Z}}_p$: the averaged factor $(1+T{)}^x$ survives exactly for $x\in p{\mathbb{Z}}_p$. Consequently $S\mathcal{L}(f)=0$ says that $\lambda (\mathcal{L}(f))$ is supported on ${\mathbb{Z}}_p^{\times }$. This gives a Galois group-equivariant homomorphism from local principal units to the Iwasawa algebra of ${\mathbb{Z}}_p^{\times }$.

The integral lifting step in this proof is the Coleman unit-measure exact sequenceHere $\mathrm{Col}(u)=\lambda (\mathcal{L}(f_u))$. The last map is surjective, since $m_1({\delta }_1)=1$, and it has the cyclotomic action on its target. Its composition with $\mathrm{Col}$ is zero: $D\varphi =p\varphi D$ implies $D\mathcal{L}(f)(0)=0$. The kernel on the unit side consists of norm-compatible $p$-power roots of unity. To see the shape of the kernel, write $z=\log (1+T)$; the equation $\mathcal{L}(f)=0$ becomes $h(pz)=ph(z)$ for $h(z)=\log f(e^z-1)$, so only its linear term can survive. Integral solutions are $f=(1+T{)}^c$, $c\in {\mathbb{Z}}_p$, after excluding finite residue factors by the principal unit condition. For the image, use the following explicit integral lifting of the corrected cyclotomic logarithm.

Take a p-adic measure $\mu$ on the units with first moment zero, and write $g$ for its Amice transform. Then $Sg=0$ and $g^{\prime }(0)=0$. Put $H=\exp (pg)\in 1+pR$ and solve $f^p=H\varphi (f)$ with $f(0)\in 1+p{\mathbb{Z}}_p$. The constant equation is solved by $f_0=\exp (p{g}_0/(p-1))$. In degree one the terms involving $f_1$ cancel, and the remaining condition is $H_1=0$, exactly $g_1=0$; choose any $f_1\in {\mathbb{Z}}_p$. For $n\ge 2$, after fixing the lower coefficients, the coefficient multiplying the unknown $f_n$ isThe other terms are divisible by $p$: for the already integral truncated series, $f^p\equiv \varphi (f)(\mathrm{mod}p)$, while $H\equiv 1(\mathrm{mod}p)$. Division therefore gives an integral $f_n$, since $f_0$ and $1-p^{n-1}$ are units. Recursion constructs $f\in R^{\times }$ with $\mathcal{L}(f)=g$. Now $pSg=\log (Nf/f)=0$, and $Nf/f\in 1+pR$ by Question 2. The p-adic logarithm is injective there, so $Nf=f$. This supplies the desired Coleman power series of a norm-compatible sequence of local units. The freely chosen linear coefficient accounts precisely for the kernel $(1+T{)}^c$. It proves the integral surjectivity step, including the integrality that rational inversion of $1-\varphi /p$ would leave unproved.

Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. As $p$ is odd, taking the even part is exact and removes both end terms. It follows that $\mathrm{Col}:U_{\infty }^1\simeq {\Lambda }_G$. The integral exact sequence is essential here: the support calculation alone would not prove this isomorphism.

It remains to calculate the image of the cyclotomic units. For $a,b\in {\mathbb{Z}}_p^{\times }$, consider the symmetric cyclotomic unit seriesThe quotient has an invertible constant term $a/b$, because both numerator and denominator have a simple zero at $T=0$. The factor $X^{(b-a)/2}$ makes the expression unchanged under $X\mapsto X^{-1}$. The product over the roots of unity gives $N{f}_{a,b}=f_{a,b}$; one may verify this first for integer $a,b$ prime to $p$ and then extend by continuity. Dividing by the norm-compatible Teichmuller representative makes the series correspond to principal units and does not change its logarithm. The generating function for the Bernoulli numbers, with $z=\log X$, givesThe symmetric factor cancels the linear term. Applying $1-\varphi /p$, and then the moment formula of Question 3, yieldsfor even $k\ge 2$. These moments determine an even p-adic measure on the units: polynomials are dense by the Mahler theorem, even projection preserves density, and multiplication by $x^2$ is invertible on the units, so the positive even-degree polynomials are dense there as well. ThusThe closed ideal generated by all differences $[b]-[a]$ is $I(G)$. The closed subgroup generated by the corresponding cyclotomic units, including its Galois group translates, therefore has image exactly $I(G){\zeta }_p$. Taking the quotient of the isomorphism $U_{\infty }^1\simeq {\Lambda }_G$ proves the boxed local theorem.

Finally introduce global arithmetic. Let ${\overline{E}}_{\infty }^1$ be the closure of norm-compatible global units in the real tower, let $X_{\infty }^{+}$ be its p-ramified Iwasawa module, and let $Y_{\infty }^{+}$ be its unramified Iwasawa module. With compatible unit and splitting conventions, class field theory gives the class-field unit sequence; quotienting its unit terms by $C_{\infty }^1$ givesIn the cyclotomic tower the prime above $p$ is principal, so the unramified class-field convention also makes it split. On the torsion character components, multiplicativity of characteristic ideals impliesThe local theorem has already calculated the first factor analytically. Equality of the global unit and class-group factors would therefore identify the characteristic ideal of $X_{\infty }^{+}$ with the same ideal of the p-adic L-function. That is the content predicted by the Iwasawa main conjecture. More explicitly, for a nontrivial even finite character $\chi$, the character component of $I(G)$ is the whole component and the prediction isHere ${\Lambda }_{\chi }$ is the power-series algebra of the pro-$p$ part with coefficients in the character-value ring. The trivial component requires the augmentation correction for the zeta pole. Equivalently the characteristic ideals of ${\overline{E}}_{\infty }^1/C_{\infty }^1$ and $Y_{\infty }^{+}$ agree componentwise. A formulation using odd class-group characters must include the appropriate cyclotomic twist and inversion, as in Kummer reflection in Iwasawa theory. The motivation is an analytic formula for local units together with a global class-field exact sequence; the global equality is the additional arithmetic assertion.

The related class-number growth theorem illustrates why passage to an Iwasawa module is powerful. For the cyclotomic ${\mathbb{Z}}_p$-extension, let $p^{e_n}$ be the $p$-part of the class number at level $n$. The unramified Iwasawa torsion theorem, the Iwasawa module structure theorem and the class-group control relations give, for sufficiently large $n$,Here $\mu ,\lambda$ are the Iwasawa invariants. To see the two growth terms, use a topological generator $\gamma$ and, after a fixed sufficiently high level $n_0$, the norm polynomial ${\nu }_{n,n_0}=({\gamma }^{p^n}-1)/({\gamma }^{p^{n_0}}-1)$. An elementary summand $\Lambda /(p^a)$ contributes $a(p^n-p^{n_0})$ to the length of its quotient by this polynomial. A summand associated with a distinguished polynomial of degree $d$ contributes $dn$ plus a constant: at each root $\alpha$ the valuation of the norm polynomial is $n$ plus a fixed constant for large $n$. If $1+\alpha$ is a root of unity already killed at level $n_0$, its value is instead exactly $p^{n-n_0}$ and gives the same linear contribution. The finite errors in the structure and control maps stabilize and contribute to $\nu$. Thus module structure explains growth, while the Iwasawa main conjecture goes further by identifying the characteristic series of an Iwasawa module themselves with analytic ones.