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Let $R=K\cap {\mathrm{End}}_k(E)$. If $\alpha \in R$ maps the Tate module into ${\ell }^n{T}_{\ell }(E)$, it kills $E[{\ell }^n]$ and factors through the quotient isogeny $[{\ell }^n]$. The resulting endomorphism is $\alpha /{\ell }^n$, belongs to $K$ and is defined over $k$, hence belongs to $R$. Approximation in $R\otimes {\mathbb{Z}}_{\ell }$ now identifies the displayed stabilizer. This does not assume that the integral Tate module is free over a possibly nonmaximal CM order.

An abelian variety of dimension $g$ has complex multiplication when its rational endomorphism ring contains a commutative semisimple $\mathbb{Q}$-algebra of dimension $2g$. For an elliptic curve in characteristic zero, this means its geometric endomorphism ring of an elliptic curve is an order in an imaginary quadratic field. The resulting rank-one Tate module representations connect torsion points to class field theory.

For an elliptic curve over a number field, its Hasse-Weil L-function is the product of the local factors determined by its Tate module. At a good finite place, the factor is $(1-a_{v}N{v}^{-s}+N{v}^{1-2s}{)}^{-1}$, with $a_v=Nv+1-\#E(\kappa (v))$. At a bad place use the inertia-invariant subspace and an auxiliary prime different from the residue characteristic. For a CM elliptic curve with all endomorphisms defined over the base, this factors as the product of the two conjugate Hecke L-functions.

Fix a separable closure $k^s$. The Tate module and rational Tate module arewhere the transition maps are multiplication by $\ell$. Since $\ell$ is different from the characteristic, $E[{\ell }^n](k^s)\simeq (\mathbb{Z}/{\ell }^n{)}^2$ and these transition maps are surjective. Choosing compatible bases gives $T_{\ell }(E)\simeq {\mathbb{Z}}_{\ell }^2$, and $T_{\ell }(E)/{\ell }^n{T}_{\ell }(E)\simeq E[{\ell }^n]$. Both spaces carry the natural continuous action of the absolute Galois group of $k$ and of the base-defined endomorphism ring of an elliptic curve.

For (i), choose a nonrational $a\in R$. Such an element exists because the intersection of the given two-dimensional rational subalgebra with the integral endomorphism ring contains a full lattice. The characteristic equation of an elliptic endomorphism isSince $a\in K$ is nonrational, its minimal polynomial over $\mathbb{Q}$ has degree two. Comparing the two monic equations gives $\mathrm{tr}(a)=a+\bar{a}$ and $\mathrm{deg}(a)=a\bar{a}$. The same is the characteristic polynomial on the rational Tate module: the Weil pairing gives determinant $\mathrm{deg}(a)$, and applying it to $1-a$ gives trace $1+\mathrm{deg}(a)-\mathrm{deg}(1-a)$. Thus on $V_{\ell }(E)$ the characteristic polynomial isIf $K_{\ell }$ is a quadratic field over ${\mathbb{Q}}_{\ell }$, a unital $K_{\ell }$-action on this two-dimensional ${\mathbb{Q}}_{\ell }$-space makes it a one-dimensional $K_{\ell }$-space. If $K_{\ell }\simeq {\mathbb{Q}}_{\ell }\times {\mathbb{Q}}_{\ell }$, the two roots above are distinct in ${\mathbb{Q}}_{\ell }$ and their eigenspaces each have dimension one. The two idempotent factors therefore have equal rank one, so the module is again free of rank one. This also proves faithfulness in the split case. Hence

For (ii), first prove an integral divisibility statement. If $a\in R$ maps $T_{\ell }(E)$ into ${\ell }^n{T}_{\ell }(E)$, then its induced action on $E[{\ell }^n]$ is zero. The kernel of an isogeny of $[{\ell }^n]$ is therefore contained in the kernel of $a$. The quotient property of this isogeny of elliptic curves gives a unique endomorphism $b$ withThis factorization is defined over $k$: applying any descent automorphism preserves it, and uniqueness fixes $b$. Inside the rational endomorphism ring, $b=a/{\ell }^n$ belongs to $K$, so $b\in R$. ConsequentlyHere the prime-to-characteristic torsion is étale, so vanishing on its geometric points is vanishing on the entire kernel scheme.

Every element of $R_{\ell }$ preserves the Tate module by continuity. Conversely suppose $\varphi \in K_{\ell }$ preserves it. Choose $n\ge 0$ with ${\ell }^n\varphi \in R_{\ell }$, and approximate it by $a\in R$ modulo ${\ell }^n{R}_{\ell }$. Write ${\ell }^n\varphi =a+{\ell }^{n}c$ with $c\in R_{\ell }$. Then $a{T}_{\ell }(E)\subseteq {\ell }^n{T}_{\ell }(E)$, so the divisibility statement gives $a/{\ell }^n\in R$. It follows that $\varphi =a/{\ell }^n+c\in R_{\ell }$. Thus the CM Tate module stabilizer isThis does not presume freeness of the integral Tate module over a nonmaximal order.

For (iii), every element of $R$ is defined over $k$, so the Galois representation commutes with $R$ and therefore with $K_{\ell }$. By (i), its action on $V_{\ell }(E)$ is multiplication by an element of $K_{\ell }^{\times }$. The action and its inverse preserve $T_{\ell }(E)$; by (ii), this scalar and its inverse belong to $R_{\ell }$. Hence

The construction must first specify the field of definition of the complex multiplication. Let $E/F$ be a CM elliptic curve and let $K$ be its geometric CM field, embedded by its action on an invariant differential. In characteristic zero this action is injective. Consequently every geometric endomorphism is defined over $F^{\prime }=FK$: a Galois automorphism fixing $FK$ fixes its differential scalar and therefore fixes the endomorphism itself. Initially assume $F$ already contains $K$; we return to the other case below.

Choose a complex embedding and an analytic complex uniformization of an elliptic curvewhere $\mathfrak{a}$ is a lattice in $K$, stable under the CM order $R$. It need not be the maximal order or a principal ideal. Its torsion is described by the finite adelic quotients of this lattice.

The input from the CM reciprocity theorem is precise: if $s$ is an idele of $F$, $n=N_{F/K}s$, and $\sigma ={\mathrm{Art}}_F(s)$, then $\sigma$ takes the CM torus to the torus with lattice $n_f^{-1}L$; on torsion the comparison is induced by multiplication by $n_f^{-1}$. This reciprocity statement comes from the ideal action on CM lattices: an ideal $\mathfrak{b}$ gives the isogeny to ${\mathfrak{b}}^{-1}L$, composition multiplies ideals, and adding torsion level structure removes automorphism ambiguity. The resulting action on algebraic CM points is the ray-class Artin action. This is the deep CM input, not an assertion that an arbitrary abelian Galois representation automatically comes from an algebraic Hecke character.

Because $\sigma$ fixes $F$, the algebraic curve $E^{\sigma }$ is $E$. Comparing the reciprocity uniformization with the original one therefore gives multiplication by a scalarThe scalar belongs to $K$: a complex scalar carrying one lattice contained in $K$ to another is the quotient of two nonzero lattice elements. Its torsion action isThe uniqueness of the torsion-compatible comparison is important: two possible scalars would induce the same action on all torsion, hence their difference would annihilate torsion of arbitrarily large order and be zero. Multiplying the torsion-action formulas for $s_f$ and $t_f$ now proves $a(s_f{t}_f)=a(s_f)a(t_f)$.

For $b\in F^{\times }$, Artin reciprocity sends its principal idele to the identity. The archimedean factors are complex and connected, so its finite part also has trivial Artin image. The preceding formula therefore givesIt follows thatis trivial on principal ideles and defines a character of the idèle class group of $F$. This establishes the multiplicativity and principal-idele condition, rather than merely prescribing Frobenius values and hoping that they extend.

We must still prove continuity and a finite conductor. For a finite place $v$ and a prime $\ell$ different from its residue characteristic, an idele supported on the units at $v$ has trivial $\ell$-component of its norm. Thus its action on the Tate module is exactly multiplication by $a(u)$. At a place of good reduction, inertia acts trivially, so $a(u)=1$. At a bad place, a CM curve has potentially good reduction, and inertia on the prime-to-residue-characteristic Tate module has finite image. Hence $a$ is trivial on an open subgroup of the local units there. There are only finitely many bad places. Together these facts give an open finite-congruence kernel on the unit ideles and prove continuity. The archimedean norm factor is already continuous. The potentially-good-reduction and inertia facts are the reduction-theoretic inputs in Serre and Tate's treatment of CM abelian varieties.

Consequently ${\psi }_E$ is a Hecke character. In ideal notation, for $b\equiv 1(\mathrm{mod}\mathfrak{f})$,so its infinity type of a Hecke character is the type determined by the chosen CM embedding, with the inverse archimedean norm in the idele convention. Nonmaximal orders merely contribute additional finite-level conditions; the proof retains their lattice and does not replace it by the maximal-order lattice without justification.

At a good prime $v$, choose a uniformizer idele and a prime $\ell$ different from the residue characteristic. Its norm has trivial $\ell$-component, so arithmetic Frobenius acts on the Tate module by ${\psi }_E(v)$. Specialization of prime-to-residue-characteristic torsion identifies this with the reduced Frobenius isogeny. ThusThe corresponding Hasse-Weil L-function factors as $L(E/F,s)=L({\psi }_E,s)L({\bar{\psi }}_E,s)$, including the local factors determined by inertia. This explains why the Hecke character encodes both the torsion representations and the Euler factors.

If $F$ does not contain $K$, apply the construction over $FK$. The two conjugate CM characters are exchanged by the nontrivial automorphism of $FK/F$, and the two-dimensional Galois representation over $F$ is their induced representation. In that case the natural description is $L(E/F,s)=L({\psi }_E/FK,s)$ by induction, rather than a pair of one-dimensional characters over $F$. Question 3 supplies a concrete example: $\mathbb{Q}$ does not define $[i]$, whereas $\mathbb{Q}(i)$ does. This completes the existence argument for an arbitrary CM curve over a number field, with its field of definition correctly accounted for.

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.

Tensoring a Tate module with the P-adic numbers gives its rational Tate module. For a CM elliptic curve with base-defined multiplication by $K$, it is free of rank one over $K\otimes {\mathbb{Q}}_{\ell }$, including when this algebra is a product of two fields. In the split case the two distinct roots of a nonrational CM endomorphism each have a one-dimensional eigenspace.