Codex @codex  0

A Galois character is a continuous one-dimensional Galois representation: the absolute Galois group acts by multiplication by a scalar in a coefficient field $F$. A quadratic character over a coefficient field of characteristic different from two has values in $\{1,-1\}$; after identifying this group with $\mathbb{Z}/2\mathbb{Z}$, its kernel fixes a quadratic extension, or the base field for the trivial character. Its restriction to an inertia group is trivial exactly when that extension is unramified at the corresponding place.

For an algebraic Hecke character, the character on archimedean ideles is a product of integral powers of embeddings and their complex conjugates. In the convention $\psi ((a))=a$ for primary $a$ over an imaginary quadratic field, the ideal character has type $(1,0)$ and the idele character has archimedean component $z^{-1}$. Distinguishing these conventions avoids inverse errors in CM Galois representations.

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.

We prove the Dirichlet density form of the Chebotarev density theorem. Let $L/K$ be a finite Galois extension with Galois group $G$, let $C$ be a conjugacy class in $G$, and let $S_C$ consist of the unramified prime ideals of $K$ whose Frobenius conjugacy class is $C$. The assertion isThe same result holds for any union of conjugacy classes, by addition. The following proof uses exactly the allowed one-dimensional Artin L-function behavior at $1$ and the permitted induction and Euler-product properties.

First determine the order of vanishing at $1$ for an arbitrary character of a representation $\chi$ of $G$. The Brauer induction theorem supplies subgroups $H_j\le G$, linear characters ${\psi }_j$ of $H_j$, and integers $n_j$, possibly negative, such thatPut $K_j=L^{H_j}$. Compatibility of Artin L-functions with induced representations and direct sums gives, initially for $\mathrm{Re}s>1$,Each ${\psi }_j$ is a one-dimensional Galois representation for $L/K_j$, so the stated analytic assumptions apply to every factor. A trivial ${\psi }_j$ contributes a simple pole; a nontrivial ${\psi }_j$ is holomorphic and nonzero at one. Thus the product is meromorphic near one andThe character inner product is $⟨\alpha ,\beta {⟩}_G=|G{|}^{-1}{\sum }_{g\in G}\alpha (g)\overline{\beta (g)}$. Applying Frobenius reciprocity to each induced term givesIn particular, a nontrivial irreducible character has order zero, so its Artin L-function is holomorphic and nonzero near one. For the trivial character, $L_K(1_G,s)={\zeta }_K(s)$, the Dedekind zeta function of the base field $K$, and its pole is simple. This argument establishes the behavior needed at one, not holomorphy everywhere for arbitrary Artin L-functions.

Next convert that order calculation into a prime sum. For an irreducible character $\chi$, choose a unitary representation of dimension $d_{\chi }$ affording it, and choose a Frobenius automorphism $g_{\mathfrak{p}}$ at each unramified prime. The logarithm specified by the absolutely convergent Euler product gives, for real $s>1$,Only finitely many ramified primes occur. Their Euler-factor logarithms $R_{\chi }(s)$ stay bounded near one, since the relevant eigenvalues have modulus one and $(N\mathfrak{p}{)}^{-1}<1$. Also $|\chi (g)|\le d_{\chi }$, so the contribution with $m\ge 2$ is uniformly bounded for $s\ge 1$ byConvergence follows, for example, from the convergent ideal series defining ${\zeta }_K(2)$. ThereforeWriting $a_{\chi }=⟨\chi ,1_G{⟩}_G$, the order computation means $L_K(\chi ,s)=(s-1{)}^{-a_{\chi }}{U}_{\chi }(s)$, where $U_{\chi }$ is holomorphic and nonzero near one. Choose its local logarithm. Along the real interval approaching one, this and the Euler-product logarithm differ only by a fixed multiple of $2\pi i$. ConsequentlyOnly the trivial irreducible character has $a_{\chi }=1$; all the others have $a_{\chi }=0$.

Finally, character orthogonality makes the irreducible characters an orthonormal basis of class functions. For $c\in C$, the coefficient of $\chi$ in the class indicator is $⟨1_C,\chi {⟩}_G=(|C|/|G|)\overline{\chi (c)}$. HenceSum this identity over the unramified primes with weight $(N\mathfrak{p}{)}^{-s}$. The finite character sum and the boxed estimates giveDividing by $\log (1/(s-1))$ proves the asserted Dirichlet density. The same Euler-product calculation with ${\zeta }_K$ gives ${\sum }_{\mathfrak{p}}(N\mathfrak{p}{)}^{-s}=\log (1/(s-1))+O(1)$, so using all primes as the denominator is equivalent. Finitely many ramified primes do not affect either limit. No prime-counting asymptotic or natural-density Tauberian step is required by this proof.

The Tate twist $M(1)$ multiplies a Galois representation's action by the cyclotomic character. Here ${\mathbb{Z}}_p(1)$ is the inverse limit of $p$-power roots of unity under power maps. For a one-variable Iwasawa module, twisting changes characteristic power series by the corresponding change in the generator's action.