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For a height-one CM elliptic curve formal group and compatible ${\mathfrak{p}}^n$-division parameters $t_n$, an integral Coleman series interpolates a norm-compatible sequence of local units $u_n$. Its norm relation is multiplication of its translates by the $\mathfrak{p}$-torsion, with coefficient Frobenius where required by the unramified base. Coates–Wiles homomorphisms and the Frobenius-corrected logarithm turn this local series into critical values and measures.

Let $V_{\infty }$ be the norm limit of local principal units in the $p$-power cyclotomic field tower, for odd $p$. The Coleman power series $f_u$ gives the Frobenius-corrected logarithm $p^{-1}\log (f_u^p/\varphi f_u)$, whose inverse Amice transform is supported on ${\mathbb{Z}}_p^{\times }$. This is the middle map in the displayed exact sequence; the last map is the first moment $\mu \mapsto \int xd\mu$. Its kernel consists of norm-compatible $p$-power roots of unity. The logarithmic difference equation has one resonant degree, namely degree one in $\log (1+T)$, accounting for the last term. Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^{\times }/\{\pm 1\}]]$.

Let $R={\mathbb{Z}}_p[[T]]$, $p$ odd, and $\varphi (T)=(1+T{)}^p-1$. An element $g\in R$ is the Frobenius-corrected logarithm of a unit $f$ with $f(0)\in 1+p{\mathbb{Z}}_p$ if and only if $g^{\prime }(0)=0$. Necessity follows by differentiating at zero, using ${\varphi }^{\prime }(0)=p$. For sufficiency put $H=\exp (pg)\in 1+pR$ and solve $f^p=H\varphi (f)$. Choose $f_0=\exp (p{g}_0/(p-1))$. The linear equation reduces to $H_1=0$, so $f_1$ is free. In degree $n\ge 2$ the unknown $f_n$ has coefficient $p{f}_0^{p-1}(1-p^{n-1})$. The remaining terms are divisible by $p$ because reduction of any integral truncated $f$ satisfies $f^p\equiv \varphi (f)(\mathrm{mod}p)$. Division gives a unique integral $f_n$. Two lifts differ by $(1+T{)}^c$, $c\in {\mathbb{Z}}_p$, as follows from the logarithmic equation in the coordinate $z=\log (1+T)$. If also $Sg=0$, the Coleman trace operator gives $\log (Nf/f)=pSg=0$. As $Nf/f\in 1+pR$, injectivity of the p-adic logarithm implies $Nf=f$.

The Iwasawa main conjecture links two descriptions of cyclotomic arithmetic: Galois modules built from abelian extensions, and p-adic L-functions built by interpolation of special values. A precise statement must distinguish local units, global units and the permitted ramification of the extension.

Use global fields $F_n=\mathbb{Q}({\zeta }_n{)}^{+}$, distinct from the local fields $K_n$ in the preceding questions. Put $F_{\infty }={\bigcup }_n{F}_n$, $G=\mathrm{Gal}(F_{\infty }/\mathbb{Q})\cong {\mathbb{Z}}_p^{\times }/\{\pm 1\}$, and $\Omega ={\mathbb{Z}}_p[[G]]$. Its pro-p subgroup is $\Gamma \cong {\mathbb{Z}}_p$, and choosing a generator identifies its Iwasawa algebra with $\Lambda ={\mathbb{Z}}_p[[S]]$. The prime-to-p finite quotient has order $(p-1)/2$, so its character idempotents decompose $\Omega$ into one-variable power-series rings. Characteristic ideals below are understood componentwise in this decomposition.

Let $\mathcal{U}$ be the norm limit of local principal units in the completions of $F_n$ at $p$, $\mathcal{E}$ the norm limit of the closures of global principal units there, and $\mathcal{C}$ the closed module generated by compatible cyclotomic units. Thus $\mathcal{C}\subseteq \mathcal{E}\subseteq \mathcal{U}$. Let $X$ be the p-ramified Iwasawa module of $F_{\infty }$, and let $Y$ be its unramified Iwasawa module, equivalently the norm limit of the p-primary ideal class groups. In this real tower the relevant character components are finitely generated torsion modules; the abelian-field Leopoldt theorem for abelian number fields supplies the required rank comparison for $X$.

For a torsion one-variable Iwasawa module, the Iwasawa module structure theorem gives, up to finite kernel and cokernel, elementary factors $\Lambda /(p^{a_i})$ and $\Lambda /(q_j(S{)}^{b_j})$ with distinguished irreducible $q_j$. Its characteristic ideal is generated by $p^{\sum a_i}{\prod }_j{q}_j^{b_j}$; the generator is determined only up to a unit. This records all height-one elementary divisors, not just the order of a finite class group.

Let $I=\ker (\Omega \to {\mathbb{Z}}_p)$ be the augmentation ideal, and let ${\zeta }_p$ be the even p-adic zeta pseudomeasure, normalized byEven powers are well-defined on ${\mathbb{Z}}_p^{\times }/\{\pm 1\}$. The trivial-character pole means that ${\zeta }_p$ itself need not be integral; $I{\zeta }_p$ is an integral ideal. In this real, p-ramified convention the main conjecture isOn nontrivial even character components this is an equality with the ideal of a Kubota-Leopoldt p-adic L-function power series. If $\gamma$ corresponds to $u=1+p$ and $S=\gamma -1$, the corresponding interpolation convention is $g_{\chi }(u^{1-s}-1)=L_p(\chi ,s)$. Other formulations use odd components of the unramified module of the full complex tower; Kummer reflection in Iwasawa theory introduces the reflected character $\omega {\chi }^{-1}$ and the accompanying Tate twist/involution. These modules and power-series variables cannot be exchanged without those corrections. The main conjecture is a theorem, proved by the Mazur-Wiles theorem and also through the cyclotomic Euler-system method.

The Iwasawa theorem on local cyclotomic units identifies the purely local quotientIts mechanism is visible in Q2. The Coleman unit-measure exact sequence, on its even part, has no roots-of-unity kernel or first-moment cokernel, since complex conjugation acts by $-1$ on those terms. It identifies $\mathcal{U}$ with the measure algebra $\Omega$. The measure comes from the Frobenius-corrected logarithmwhose kth moment is $(1-p^{k-1}){\delta }_k(f)$. This follows by applying $D^k$ and the chain rule to the two logarithms. For the units in Q2 and even $k$, use $\zeta (1-k)=-B_k/k$ to obtainUniqueness of moments therefore identifies their measures with $(p-1)[2](1-[a]){\zeta }_p$. The factors $(p-1)[2]$ are units in $\Omega$, and the differences $1-[a]$ generate $I$. Thus the closed cyclotomic-unit module maps exactly to $I{\zeta }_p$. This local interpolation calculation explains Iwasawa's quotient theorem, including the augmentation correction at the trivial character.

What remains globally is made explicit by class field theory. The class-field unit sequence givesBreak this into two short exact sequences. Multiplicativity of characteristic ideals yieldsConsequently, using the local theorem, the main conjecture is equivalent toThis is the characteristic-ideal bridge from local units to the main conjecture. It shows why the local quotient theorem alone is insufficient: it does not measure how many local cyclotomic-unit classes are supplied by global units. The Euler-system divisibility for real cyclotomic class modules supplies one containment, while the cyclotomic unit-index and norm-defect comparisons upgrade it to equality. Neither vanishing of class groups nor the Vandiver conjecture is assumed.

There is also the Iwasawa class-number growth theorem, often called Iwasawa's theorem in discussions of these towers. For sufficiently large $n$,The Iwasawa invariants $\mu ,\lambda$ come from the characteristic series of $Y$; Ferrero-Washington theorem gives $\mu =0$ in this abelian cyclotomic setting. This explains eventual linear growth of the p-adic class-number exponent. It is weaker than the main conjecture: the latter determines the complete characteristic ideal, including distinguished polynomial factors, rather than only their total degree and p-power content. The local unit theorem and the class-number growth theorem are thus important inputs and consequences in the framework, but neither should be mistaken for the global ideal equality itself.

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.

A one-variable main conjecture for CM elliptic curves compares a torsion arithmetic Iwasawa module with a measure interpolating special Hecke L-function values. The classical bounded-measure formulation uses a prime of good reduction which splits in the CM field, hence gives ordinary reduction. We first describe the one-prime division-tower formulation, including the analytic construction, and then explain its cyclotomic Selmer counterpart and the change at supersingular primes.

For a precise basic setting, take $E/K$ with complex multiplication by ${\mathcal{O}}_K$, $K$ of class number one, and a split prime $p>3$ of good reduction, prime to the Hecke conductor $\mathfrak{f}$. Write $p{\mathcal{O}}_K=\mathfrak{p}\bar{\mathfrak{p}}$, choose the $p$-adic embedding corresponding to $\mathfrak{p}$, and let $\pi ={\psi }_E(\mathfrak{p})$. Its formal group of an elliptic curve has height one, with $[\pi ](t)\equiv t^p(\mathrm{mod}\mathfrak{p})$. The one-prime division towerhas $\Delta$ of order prime to $p$ and $\Gamma \simeq {\mathbb{Z}}_p$. Its CM character $\kappa :G\to {\mathbb{Z}}_p^{\times }$ describes the action on the prime-primary Tate module. Let $F=F_{\infty }^{\Gamma }$ and $\Lambda =I[[\Gamma ]]\simeq I[[T]]$. We extend coefficients to the complete unramified ring $I$ to include character values and a CM period.

Let $X$ be the Galois group over $F_{\infty }$ of the maximal abelian pro-$p$ extension unramified outside primes above $\mathfrak{p}$. Let $A$ be the analogous everywhere-unramified group, equivalently the norm inverse limit of the $p$-parts of the ideal class groups. Write $U$ for the norm limit of principal local units at $\mathfrak{p}$, $\mathcal{E}$ for the closures of global units in $U$, and $\mathcal{C}$ for the closures of elliptic units. Roots of unity and finite initial-layer errors are retained when formulating integral statements; after passage to a finite-character component they do not change a characteristic ideal.

The class-field unit sequence, obtained by taking norm limits in global Artin reciprocity, givesIn this ordinary one-prime tower, the unit and global-unit terms have the same Iwasawa rank, and the displayed quotients and $X,A$ are torsion on their character components. The equalityis the global-units/class-groups form of the CM main theorem. Multiplicativity of characteristic ideals in the sequence then givesThe analytic construction must identify the latter ideal with a genuinely interpolating p-adic L-function; it is not enough to rename an arbitrary characteristic power series as an L-function.

Here is that construction. For an auxiliary ideal $\mathfrak{a}$ prime to $6p\mathfrak{f}$, a suitably normalized rational elliptic function has divisor a multiple ofTranslate by a generator of the conductor torsion and take the product over its ray-class orbit. This produces a rational function $f_{\mathfrak{a}}(t)$ defined over $K$ which is a unit series at the identity. Evaluating it on compatible ${\mathfrak{p}}^n$-division points gives elliptic units. The reason for the norm relations is geometric: multiplication by $\pi$ groups the translated zeros and poles into fibers, so the product over $E[\mathfrak{p}]$ has the divisor of $f_{\mathfrak{a}}([\pi ]t)$. The normalization fixes the constant, giving the local identityThis is also the norm relation of the Coleman power series for a CM formal group interpolating the resulting local units.

An integral logarithmic correction makes the passage from those units to a bounded measure explicit. In the chosen split completion $K_{\mathfrak{p}}={\mathbb{Q}}_p$, putSince $[\pi ]t\equiv t^p(\mathrm{mod}p)$ and the coefficients of $f$ reduce to ${\mathbb{F}}_p$, the ratio is in $1+p{\mathbb{Z}}_p[[t]]$. For $p>2$, its logarithm divided by $p$ is integral. Constants are treated using the usual $p$-adic logarithm on units. This proves the integrality needed for the Frobenius-corrected logarithm; applying an uncorrected logarithm alone would not prove boundedness.

Choose an isomorphism $\eta :{\hat{\mathbb{G}}}_m\to \hat{E}$ over $I$, and normalize the CM period byThen $A_f(T)={\mathcal{L}}_f(\eta (T))\in I[[T]]$ is the Amice transform of a unique $I$-valued p-adic measure on ${\mathbb{Z}}_p$. The norm relation above givesthe first logarithmic sum is $\log f([\pi ]t)$ and the second is the same sum, because $[\pi ]Q=0$ and there are $p$ points. On the measure side the sum selects $p{\mathbb{Z}}_p$ and multiplies by $p$, so this identity says that the measure is supported on ${\mathbb{Z}}_p^{\times }$. Transport it to $G$ by the CM character and project to the desired component.

The moments can now be calculated rather than postulated. With $D=(1+T)d/dT$,The factor ${\pi }^k$ appears because $[\pi ]$ is multiplication by $\pi$ in the additive formal-logarithm coordinate. The remaining derivative is a Coates–Wiles homomorphism. The complex product for the elliptic function makes it a sum of shifted inverse powers of lattice elements. Grouping these lattice elements by ray-class ideals identifies that sum with $(k-1)!L_{\mathfrak{f}}({\bar{\psi }}_E^k,k)/{\Omega }_{\infty }^k$, multiplied by the auxiliary factor $N\mathfrak{a}-{\psi }_E(\mathfrak{a}{)}^k$. Here $L_{\mathfrak{f}}$ omits primes dividing the original conductor, including when the conductor of the power character becomes smaller.

Normalize the rational functions and matching complex and $p$-adic periods consistently, and remove the auxiliary factors by the distribution relations. They are evaluations of $N\mathfrak{a}-[\mathfrak{a}]$ in the Iwasawa algebra. On each finite-character component an auxiliary ideal can be chosen making this element a unit: its residues at $\mathfrak{p}$ and $\bar{\mathfrak{p}}$ can be prescribed independently by the Chinese remainder theorem. The resulting integral measure ${\mu }_E$ satisfies, on its branch and with $k$ in the corresponding residue class,The exact normalization of periods matters for this equality; multiplication by an integral unit does not affect the ideal in the main conjecture. Finite-order twists are obtained by inserting a character in the ray-class sums, with its local Gauss sum. The lattice calculation proves algebraicity of the normalized complex values, while the integral series construction proves their $p$-adic interpolation. These are different steps.

The Coleman measure map identifies $U_{\chi }$ with a rank-one Iwasawa algebra lattice, up to finite errors, and takes ${\mathcal{C}}_{\chi }$ to the ideal generated by the corresponding power series ${\mathcal{L}}_{\chi }$. The conjecture therefore has the concrete formIts proof uses the Euler system of elliptic units. Adjoining auxiliary ray moduli supplies norm relations with factors $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$. Derivative operators in the auxiliary cyclic groups produce Kolyvagin classes; their controlled valuations relate a new auxiliary prime to its ideal class. Chebotarev density theorem permits the required Frobenius choices. Successively choosing these primes bounds the class-group elementary divisors by the elliptic-unit index, giving the difficult characteristic-ideal divisibility. The opposite comparison comes from the class-number/unit-index formula and passage through the tower. Their equality, combined with the exact sequence above and the logarithmic computation, is the proof mechanism. The ordinary split-prime setting of this formulation is specified in Rubin's one-variable main-conjecture paper.

For completeness, the familiar cyclotomic formulation concerns a different tower. If $E/\mathbb{Q}$ has complex multiplication and good ordinary reduction at $p$, put ${\Gamma }_{\mathrm{cyc}}=\mathrm{Gal}({\mathbb{Q}}_{\infty }/\mathbb{Q})$ andFor a finite layer $L$, its Selmer group of an elliptic curve iswhere $S$ contains the bad primes and primes above $p$; at infinite level take the direct limit. Thus it imposes the local Kummer conditions and sits between rational points tensored with ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ and the $p$-primary Tate–Shafarevich group. Its dual ${\mathcal{X}}_E$ is a torsion Iwasawa module, and the cyclotomic main conjecture isWith compatible reciprocity conventions, the ordinary function has the finite-character interpolationwhere $\alpha$ is the $p$-adic unit root of $X^2-a_{p}X+p$ and $\tau (\chi )={\sum }_{a\mathrm{mod}{p}^n}\chi (a)e^{2\pi ia/p^n}$; characters of this cyclotomic ${\mathbb{Z}}_p$-extension are even. The two-variable CM measure specializes to this line after the CM-character twist. Kummer theory, class-field descriptions of the prime-primary local conditions and control of finite-layer errors identify its algebraic counterpart with the displayed Selmer dual. The one-prime division tower is not itself the cyclotomic tower, so this requires the twist and specialization argument, not just changing its name.

Finally, at a good inert prime the curve has supersingular reduction. The ordinary unit root is absent, the unmodified cyclotomic Selmer dual is not torsion, and the ordinary bounded-function argument does not apply. For CM curves over $\mathbb{Q}$ and odd supersingular $p$, signed local trace conditions and signed functions give instead $\mathrm{char}{\mathcal{X}}_E^{\pm }=(L_p^{\pm }(E))$, as proved in Pollack and Rubin's supersingular main-conjecture theorem. Thus the ordinary hypotheses and the analytic local correction are essential parts of the essay's main-conjecture statement.