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A complex CM period compares an algebraic invariant differential on a CM elliptic curve with its analytic lattice coordinate. At an ordinary prime, a $p$-adic CM period compares the same differential with the invariant differential of the multiplicative formal group: an isomorphism $\eta$ is normalized by ${\log }_{\hat{E}}\eta (T)={\Omega }_p\log (1+T)$. Matched periods make normalized critical Hecke L-function values algebraic and give precise p-adic L-function interpolation.

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.

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.

A Hecke character of a number field is a continuous character of its idèle class group. Its finite conductor records the local unit subgroups on which it is trivial. As an ideal character it is defined on fractional ideals prime to that conductor, with a prescribed rule on principal ideals coming from its archimedean character. A CM elliptic curve yields an algebraic Hecke character with one CM embedding in its infinity type of a Hecke character.

The L-function of a Hecke character is initially the displayed Euler product over prime ideals away from its conductor. The equivalent Dirichlet series sums $\psi (\mathfrak{a})N{\mathfrak{a}}^{-s}$ over integral ideals prime to the conductor. For a CM elliptic curve with its endomorphisms defined over the base, its Hasse-Weil L-function factors into the two conjugate Hecke L-functions.

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.

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.