Open Homomorphisms of Global Solvably Closed Galois Groups

Proof. Since E(\mu_n)/E is abelian, E contains every root of unity. Adjoining an nth root of an element of E then gives a cyclic extension, so E is closed under radicals. Finite solvable Galois extensions can be built by towers of radical extensions after adjoining roots of unity; hence \Q^s\subset E.

The same tower argument shows that \Q^s contains all roots of unity and is closed under radicals. Any nontrivial finite abelian extension would have a cyclic subextension of prime degree, which Kummer theory excludes. Any nontrivial abelian algebraic extension would have a nontrivial finite abelian subextension. Thus \Q^s is solvably closed.

Finally, a finite solvable Galois extension of M lies in a finite solvable Galois extension of \Q: first take a solvable normal closure of M/\Q inside \Q^s, and then the compositum of the finitely many conjugate extensions. The group of this compositum over that normal closure embeds in a product of solvable groups. This proves the last assertion.□

Proof. An open subgroup is P_M for a number field M\subset\Q^s. An element centralizing P_M is a field automorphism of \Q^s implementing the identity automorphism of P_M. The identity field map also implements it, so uniqueness in Theorem 2.2 gives the assertion.□

Proof. Let p\ne\ell be the residue characteristic of v, and choose a topological generator \tau of a pro-\ell Sylow subgroup of I_v. Its image in source tame inertia generates the pro-\ell factor. The tame conjugation relation shows that

belongs to source wild inertia, a pro-p group. In the target tame quotient, the image of e also lies in the pro-\ell factor of tame inertia: both \varphi(\tau) and its conjugate do. Its image is therefore trivial, because it belongs to a pro-p subgroup and to a pro-\ell group. The nontrivial image of \varphi(\tau) in the torsion-free target pro-\ell tame factor then gives (2.1) by the target tame conjugation relation. The target residue characteristic is different from \ell by Theorem 2.4.□

Proof. The multiplicative hypothesis implies that the cyclic subgroup generated by every u^a with a\ne0 is Zariski dense in \mathbb G_m^n. Indeed, distinct Laurent monomials have distinct values at u^a. If a Laurent polynomial vanished at all nonnegative powers of u^a, a Vandermonde matrix applied to its finitely many monomials would force all its coefficients to vanish.

Suppose that the logarithms span a subspace of dimension r<n. The case r=0 contradicts the multiplicative hypothesis because the kernel of the logarithm on local units consists of roots of unity. Thus 1\leq r<n, and in particular n\geq2. We will obtain an algebraic determinant whose small \ell-adic absolute value is incompatible with its archimedean size.

First make all choices needed for the analytic estimate. Replacing all u_i by the same positive integer power preserves the hypotheses and the logarithmic rank. Such a power puts their coordinates in a neighborhood where exponential and logarithm are inverse. The \mathcal O_K-module generated by their logarithms is free of rank r; choose a basis b_1,\ldots,b_r\in K^n. After a further common \ell-power, these basis vectors have sufficiently small coordinates that

Here the basis is multiplied by the same power as the logarithms. Consequently there are z_i\in\mathcal O_K^r such that

From now on these powered points, the number field E, the local field K, the basis, and \eta are fixed. In particular, none depends on the degree parameter introduced below.

For an integer vector \alpha=(\alpha_1,\ldots,\alpha_n), the monomial X^\alpha on these points is represented by

It is analytic on a polydisc of radius greater than 1. If f_\alpha(z)=\sum_{\nu\in\Z_{\geq0}^r}c_{\alpha,\nu}z^\nu, the exponential series and |k!|_\ell^{-1}\leq\ell^{k/(\ell-1)} give the uniform bound

The bound is independent of \alpha, since every integer has \ell-adic absolute value at most 1. Moreover, for every a\in\Z_{\geq0}^m, the point z(a):=\sum_i a_i z_i lies in \mathcal O_K^r, and f_\alpha(z(a))=(u^a)^\alpha.

We next show that sufficiently many of these evaluations have full algebraic rank. For an integer D\geq1, put A=(D+1)^n and choose an integer S\geq1 such that

Consider the matrix with rows indexed by \alpha\in\{0,\ldots,D\}^n, columns indexed by a\in\{0,\ldots,nS\}^m, and entries (u^a)^\alpha. We claim that its rank is A.

Otherwise a nonzero polynomial P of degree at most D in each variable would vanish at every point u^a in this grid. Over an algebraic closure of E, define closed subsets of the torus by

These sets are nested and nonempty, since 1\in Z_n, whereas \dim Z_0\leq n-1. Hence some adjacent pair Z_j,Z_{j+1} has the same dimension d. Let V be a d-dimensional irreducible component of Z_{j+1}. For every b\in\{0,\ldots,S\}^m, the translate u^bV is contained in Z_j and has dimension d, so it is an irreducible component of Z_j. All these translates are distinct: an equality u^bV=u^{b'}V for b\ne b' would make V stable under every power of u^{b-b'}. The density proved above would then force V=\mathbb G_m^n, contrary to V\subseteq Z_0.

For completeness, a common zero set of polynomials of total degree at most \delta\geq1 in \mathbb A^n has at most \delta^n irreducible components. One can obtain this form of the affine Bézout bound by assigning a component Y the weight \delta^{\dim Y}\deg Y. Start with \mathbb A^n, of weight \delta^n, and impose the equations successively. A component contained in the next hypersurface retains its weight; a component not contained in it is cut into components of dimension one less, whose total degree is at most \delta\deg Y by Bézout. Thus the total weight does not increase, and removing redundant components can only decrease it. The same component bound holds after restriction to the torus. Applying it to Z_j, whose equations have total degree at most nD, contradicts (3.2). This proves the rank claim.

Choose a nonzero A\times A minor \Delta_D of the evaluation matrix. It is an algebraic integer in the fixed field E. Expand its rows using the convergent power series for f_\alpha. A nonzero term must involve A distinct monomials in the r variables z_1,\ldots,z_r, since repeated monomials give repeated rows in the corresponding evaluation determinant. The sum of the degrees of any A distinct such monomials is at least c_rA^{1+1/r} for all sufficiently large A: the number of monomials of degree at most t is \binom{t+r}{r}=O_r(t^r), so at least half of the A monomials have degree bounded below by a positive multiple of A^{1/r}. Their evaluations have absolute value at most 1. The nonarchimedean triangle inequality and (3.1), first for truncated series and then by convergence, therefore give a constant c>0 such that

At every archimedean embedding \tau:E\hookrightarrow\C, an entry of this minor has absolute value at most \exp(CDS), with C independent of D. Expanding the determinant thus gives

Let \lambda be the place of E induced by E\hookrightarrow K. Use absolute values at finite places extending the usual ones on \Q, and sum over all embeddings E\hookrightarrow\C, so that each complex place occurs twice. Writing p_v for the rational prime below a finite place v, the product formula reads

Every term in the second sum is nonpositive because \Delta_D is an algebraic integer. Retaining its \lambda-term and using (3.3)–(3.4) would imply

This is impossible as D\to\infty: the left side has order D^{n+n/r}, whereas the right side is O(D^{n+1+n/m}+D^n\log D), and

by m>n(n-1). This contradiction proves the lemma.□

Proof. We first recall the class-field-theoretic description of the character space. Let V_M=M\otimes_\Q\Q_\ell, and let U_M\subseteq V_M be the \Q_\ell-linear span of the logarithms of the global units, with logarithms taken at all places above \ell. Then

Indeed, pass to the inverse limit of the ray class sequences with moduli supported above \ell. Local logarithms identify the rationalized local unit groups with V_M; the global units impose exactly the subspace U_M. The ordinary class group and the factors at real places are finite, so contribute no additive \Q_\ell-characters. This description uses the actual logarithmic image of the units and makes no assertion about its dimension.

For v\nmid\ell, choose h_v>0 and b_v\in\mathcal O_M with (b_v)=v^{h_v}. Such a choice is possible because the ideal class group is finite. Principal reciprocity, with arithmetic Frobenius, gives for the linear form corresponding to x the identity

Here b_v is a unit at every place above \ell; all other local unit contributions are killed by x.

For (i), set n=[M:\Q] and choose m>n(n-1) of the zero primes, over distinct rational primes unramified in a Galois closure E of M. Let b_i be the elements just chosen for these primes, and let

All coordinates are algebraic integers and are units at every place of E above \ell. We verify the multiplicative hypothesis of Lemma 3.1 by valuations.

Put G=\Gal(E/\Q) and H=\Gal(E/M). For a selected degree-one prime v_i and a prime \mathfrak p_i of E above it, the decomposition group D_i is contained in H. In fact, the rational prime is unramified in E, and the degree-one condition says that [D_i:D_i\cap H]=1. The support of v_i\mathcal O_E is indexed by H/D_i. Its translates corresponding to the embeddings M\hookrightarrow E, indexed by G/H, have supports indexed by the disjoint sets gH/D_i. Distinct indices i involve distinct rational primes. It follows that the mn numbers \sigma(b_i) have pairwise disjoint nonempty supports of their principal ideals. They are therefore multiplicatively independent modulo roots of unity. In particular, for any nonzero a\in\Z^m, the n coordinates of u^a are multiplicatively independent modulo roots of unity.

Embed E in a finite extension K/\Q_\ell. Under the splitting

the vector \log b_i becomes \log u_i. Lemma 3.1 says that these vectors span the right side. Equations (3.5) and (3.6) therefore force \lambda=0, and hence x=0.

For (ii), put C=\langle\delta\rangle, F=M^C, and y=E_Cx. Since \delta fixes every indicated prime v, averaging preserves its Frobenius value, so y(\Fr_v)=0. The restriction–corestriction identity gives

Thus y is the restriction of z=|C|^{-1}\cor_{M/F}y\in\X{\ell}{F}. At an indicated prime, the decomposition group of M/\Q is exactly C. The prime u=v\cap F consequently has absolute residue degree one, and

Infinitely many distinct u occur because M/F is finite. Part (i) gives z=0, and therefore y=0.

For (iii), suppose that infinitely many zero primes exist. Let E/\Q be a finite Galois closure of M, put R=\Gal(E/\Q), and let y=\res_{E/M}x. Discarding finitely many primes, choose primes of E above these zero primes whose rational primes are unramified in E. Restriction multiplies a Frobenius value by the relative residue degree, so they are zero primes for y. An infinite subset has one fixed Frobenius element \delta\in R. By (ii), E_{\langle\delta\rangle}y=0, and hence E_Ry=0. It follows that

Restriction on additive characters across a finite extension is injective: a character that vanishes on a subgroup of finite index has finite image in the torsion-free group \Q_\ell, and so is zero. On the other hand, transitivity and the corestriction–restriction identity give

a contradiction.□

Proof. By the normal basis theorem, V_M=M\otimes_\Q\Q_\ell is the regular representation of R, as is its dual. The naturality of (3.5) gives the second embedding.

The equivariant Dirichlet unit theorem [1, Section 3] identifies the representation \mathcal O_M^\times\otimes_\Z\Q_\ell with the permutation representation on the archimedean places minus its trivial summand. Since these places are indexed by R/\langle c\rangle, writing U^{\mathrm{alg}}=\mathcal O_M^\times\otimes_\Z\Q_\ell gives

The character identity follows first over the real numbers from the usual logarithmic unit embedding, and hence over \Q_\ell by semisimplicity in characteristic zero. In particular, U^{\mathrm{alg}} is self-dual. Its \ell-adic logarithmic image U_M is a quotient of U^{\mathrm{alg}}; no injectivity of the logarithm on this tensor product is needed.

The decomposition of the regular representation induced from the two characters of \langle c\rangle is

Since U_M^\vee is a subrepresentation of (U^{\mathrm{alg}})^\vee\simeq U^{\mathrm{alg}}, semisimplicity and \X{\ell}{M}\simeq(V_M/U_M)^\vee show that \X{\ell}{M} contains \Q_\ell\oplus W_M. This proves the first embedding in (3.7).

Finally, realize W_M as the direct sum of the signed lines indexed by the left cosets of \langle c\rangle. An element acting trivially must fix the line of the identity coset, and therefore belongs to \langle c\rangle. The nonidentity element c acts by -1 on that line. Thus only the identity acts trivially on W_M, proving faithfulness.□

Proof. Conjugation by t\in T induces a continuous automorphism of U/N=A. By Theorem 2.2, applied to \Q^s/(\Q^s)^A, this automorphism is conjugation by a unique element \beta(t) of P. Indeed, the implementing field automorphism of \Q^s fixes \Q. Uniqueness shows both that \beta is a homomorphism and that \beta(u)=\alpha_0(u) for u\in U. It also shows that any extension of \alpha_0|_U to T must equal \beta: its values must induce these same conjugations on A.

Continuity on the open subgroup U implies continuity on T. Moreover, \beta(T) contains A and is therefore open in P. A continuous homomorphism of profinite groups is open onto its image, so \beta is open.□

Proof. Choose a quotient R_1=\Gal(L_1/\Q) dominating R_0 such that every H_i has nontrivial image in R_1, the kernel of P\to R_1 is contained in A, and L_1 contains \Q(i). These are finitely many compatible conditions on an open normal subgroup of P. Put n=|R_1|.

We will enlarge R_1 while keeping element orders bounded by 2n. At the same time, the images of the H_i and the conjugacy class of complex conjugation will grow. Growth of the H_i makes the sum of their invariant spaces small. Growth of the conjugacy class makes the signed representation supply enough D-invariant vectors killed by E_C to escape that sum.

We use the regular-module auxiliary extension construction appearing in Uchida’s proof [14, pp. 360–362], with quantitative bounds on its invariant spaces. More precisely, we construct a further quotient with kernel consisting of m copies of the regular \mathbb F_2[R_1]-module, where m will be chosen below. Choose m distinct rational primes that split completely in L_1. For each of them, select one prime of L_1 above it. Approximation gives elements a_1,\ldots,a_m\in L_1^\times such that a_j has valuation one at its selected prime and valuation zero at every other prime in these m sets. The mn elements

have independent square classes: their valuations at the selected sets of primes form a permutation of the identity matrix modulo two. Adjoin their square roots and call the resulting field L. Kummer theory gives

as R_1-modules. The dual module supplied by Kummer theory is again the regular permutation module. The field L is Galois over \Q because all conjugates of the a_j were included; its group is solvable, so L\subset\Q^s. No splitting of (4.2) is required.

Every element of R has order at most 2n, since its nth power lies in the exponent-two group V. Write A_R and H_{i,R} for the images of A and H_i in R. The choice of R_1 gives V\subset A_R. Choose h_i\in H_{i,R} with nonidentity image g_i\in R_1, of order d_i>1. As H_{i,R} is normal in A_R,

On m regular modules the rank of g_i-1 is mn(1-1/d_i)\geq mn/2. Consequently

If c\in R is a complex conjugation, its image in R_1 is a nontrivial involution because L_1 is totally imaginary. Conjugation by V alone therefore gives at least 2^{mn/2} distinct conjugates of c; thus

Set \eta=2^{-mn/2} and write X=\X{\ell}{L}. By Proposition 3.3, X embeds in the regular \Q_\ell[R]-representation and contains

The regular-representation bound and (4.3) give

The character of W takes the value |R|/2 at the identity, the value -|R|/(2|c^R|) on c^R, and zero elsewhere. Hence, for every subgroup J\subset R,

For D<C=\langle\gamma\rangle, the map E_C:W^D\to W^C is surjective. Since |D|\leq |C|/2 and |C|\leq2n, (4.6) implies

Choose the integer m\geq1 so large that

This single choice works for all \gamma and all proper D<C. The last dimension is then greater than (4.5), so its kernel contains a vector outside the indicated sum. This vector gives (4.1). If C is trivial, there is no proper subgroup to consider.□

Proof. Choose R by Lemma 4.2 and choose any lift \gamma of \gamma_0. Put C=\langle\gamma\rangle. Take an open normal subgroup U'\trianglelefteq\mathcal G contained in U\cap\beta^{-1}(P_L); for example, take the core of this open subgroup in \mathcal G. Let F'/\Q be the finite Galois extension with G_{F'}=U'. The map

is open.

Chebotarev’s theorem supplies infinitely many primes w' of L, over distinct rational primes q\ne\ell unramified in L, with Frobenius exactly \gamma. Theorem 2.4 applied to \beta|_{U'} pairs all but finitely many of them with \ell-active primes v' of F', away from \ell. Uniqueness of the target prime for an active source prime implies that infinitely many v' occur. Since only finitely many primes of F' lie over a given rational prime, we may pass to a subset with distinct source rational primes p, all unramified in F'. Passing to another infinite subset, we may also assume that their Frobenius elements in \Gal(F'/\Q) are a fixed element \delta.

Fix a source prolongation above each v', and use it also for the prime v=v'|_k. The original correspondence at v' can be made with this prolongation: changing the source prolongation conjugates the target group by an element of \beta(U')\subset P_L, which preserves the finite prime w'. Activity passes from v' to v, since the relevant inertia subgroup for U' is contained in that for T. Apply Theorem 2.4 at v using this same source prolongation. If w_1 is the restriction to L of its target prolongation, then

Uniqueness of the target prime for v' at the level of L gives w_1=w'. Consequently the corresponding target rational prime is q, and the image of D_v in R lies in C. These changes of prolongations preserve v' and w', so they preserve the previously selected finite Frobenius elements \delta and \gamma.

Put H=T/U'\subset\mathcal G/U' and let f be the residue degree of v over p. Figure 1 records the primes involved. Here \delta=\Fr(v'/p) and \gamma=\Fr(w'/q) belong to the finite Galois groups over \Q. The symbols \Fr_{v'} and \Fr_{w'} used below instead denote local arithmetic Frobenius over F' and L, respectively.

In the decomposition group \langle\delta\rangle\subset\Gal(F'/\Q) we have

Indeed, the residue degree is the orbit size of the distinguished coset in the action of \langle\delta\rangle on \Gal(F'/\Q)/H. This description does not require H to be normal. In particular f is constant on the chosen infinite set. The homomorphism \beta induces H\to R, since \beta(U')\subset P_L; define

This notation is independent of the lift of \delta^f to T. A local lift of rational Frobenius to the fth power is a Frobenius lift at v, so the preceding containment gives d\in C.

We claim that D=\langle d\rangle equals C. Suppose that D<C, and choose x as in (4.1). Pullback along \beta|_{U'} gives a linear map

The map j is injective: its defining homomorphism has open image in P_L, and an additive \Q_\ell-valued character vanishing on an open subgroup vanishes everywhere. Frobenius evaluation at w' is invariant under \gamma, because \gamma fixes w' and acts trivially by conjugation on its unramified local quotient. Therefore

The character x is unramified at w', so this equality says that it vanishes on the entire decomposition group there. It follows that y(\Fr_{v'})=0 for every prime in our infinite set. Proposition 3.2(ii) now gives

To see why this contradicts the choice of x, write a=|\langle\delta\rangle| and Y=j(\X{\ell}{L}). Pullback is T-equivariant, where the action on the target character space is through \beta. In particular y is fixed by N. In the sum defining (4.8), the powers \delta^j belonging to H are exactly those for which f\mid j, by (4.7). Each such power acts as the identity on y, because x is D-invariant. If f\nmid j, choose a lift t of \delta^j to \mathcal G. Then t\notin T, and \delta^jy is fixed by tNt^{-1}. Multiplying (4.8) by a thus yields

It remains to return this membership to the target character space. The finite group U/U' acts on \X{\ell}{F'}, and its subspace Y is stable. Semisimplicity supplies a U-equivariant projection

Every N_i lies in U, so \pi carries N_i-invariants into Y^{N_i}. Equivariance and injectivity of j give

Applying \pi to the displayed membership and then applying j^{-1} contradicts the last condition in (4.1). Only U-equivariance of the projection is used; no normality of T in \mathcal G is needed. Thus D=C, proving the proposition.□

Proof. Let R_0 be any finite quotient of P and let \gamma_0\in R_0. Proposition 4.3 produces an element d in the image of \beta in a further quotient R such that \langle d\rangle=\langle\gamma\rangle, where \gamma lifts \gamma_0. Thus \gamma, and hence \gamma_0, belongs to the image. Since \beta(T) is compact, surjectivity onto every finite quotient implies \beta(T)=P.

Dirichlet’s theorem on primes in arithmetic progressions shows that \mathcal L is infinite. Fix \ell\in\mathcal L, take

and choose \gamma_0 to generate this cyclic group. The proposition gives infinitely many active primes v of k over distinct rational primes p\ne\ell, for which the image of a Frobenius lift \sigma_v acts on \mu_\ell by a generator. Let q be the corresponding target rational prime and let f=[\kappa(v):\mathbb F_p], so Nv=p^f. If a(\sigma_v)\in\widehat{\Z} is the target residue exponent, Lemma 2.6 gives

Its reduction modulo \ell says that p^f generates \mathbb F_\ell^\times. In particular \gcd(f,\ell-1)=1. But 1\leq f\leq[k:\Q], and every prime divisor of any f>1 in this range divides m_0, hence divides \ell-1. Consequently f=1.

Taking logarithms in (4.10) gives

The difference is an element of \X{\ell}{k} vanishing at infinitely many primes of absolute residue degree one. Proposition 3.2(i) proves (4.9).□

Proof. Fix r\in\mathcal L. Proposition 3.3 supplies a nonzero character \psi\in\X{r}{B} on which \Gal(B/\Q) acts by its nontrivial character. By Proposition 3.2(i), \psi(\Fr_w) is nonzero at all but finitely many degree-one primes w\nmid r. Fix such a w above a split rational prime q\ne r, and put

The denominator is nonzero: the kernel of the logarithm on \Z_r^\times consists of roots of unity, and the positive rational integer q is not a root of unity. Thus z\ne0, and \psi-zx_0 vanishes on the decomposition group at w, since it is unramified there and vanishes on Frobenius.

Consider its pullback

The pullback of \psi transforms by sign under G_k/G_K. Its corestriction to k is therefore zero: restricting that corestriction back to K gives the sum of the two translates, and restriction is injective for additive characters with characteristic-zero coefficients. Theorem 4.4 identifies \beta_B^*x_0 with \log\chi_{K,r}. Consequently

For every v\in\mathcal V_w away from r, decomposition containment gives y(\Fr_v)=0, regardless of which auxiliary prime makes the pair active. Proposition 3.2(iii) permits only finitely many such v. There are also only finitely many primes of K above r, proving the assertion.□

Proof. Integral Kummer theory gives

The inverse-limit assertion follows from the finite-level Kummer isomorphisms; the degree-zero groups are finite and satisfy the Mittag–Leffler condition. Every continuous rational cocycle has bounded image, since G_F is compact, and hence becomes integral after multiplication by one power of \ell. The same observation for coboundaries gives rationalization of this integral cohomology. At a prime outside S, inertia acts trivially on \Z_\ell(1), and its Kummer evaluation is the valuation times the tame \ell-adic character. These integral inertia homomorphisms are torsion-free. Thus unramifiedness after rationalization is exactly the vanishing of all integral valuation coordinates outside S.

For completeness, put U_S=\mathcal O_{F,S}^\times and let

Writing \Cl_S(F) for the S-ideal class group, there is an exact sequence

The last map sends [a] to the ideal class of \operatorname{div}_S(a)/\ell^n, where \operatorname{div}_S records the valuations outside S. The transition from level n+1 to level n induces multiplication by \ell on these obstruction classes. Since \Cl_S(F) is finite, their inverse limit is zero. It follows that the kernel of all outside-S valuations in the completed multiplicative group is precisely \varprojlim_n U_S/U_S^{\ell^n}. The argument takes place in the full completed multiplicative group; it does not assume finite valuation support for its elements. Finally U_S is finitely generated, so rationalizing its completion gives U_S\otimes\Q_\ell. This proves (5.1). All the constructions commute with field automorphisms preserving S, giving the asserted equivariance.□

Proof. Fix \ell\in\mathcal L and any finite set J\subset\mathcal M_\ell of primes away from \ell. For each w\in J, choose a_w\in B^\times and h_w>0 with

Its compatible Kummer cocycles define a class

They factor through P_B because \Q^s contains the roots of unity and all the required radicals. Valuations at the distinct primes w show that these classes are linearly independent. Each is unramified at every finite prime other than w and the primes above \ell.

Write

By Theorem 4.4, \epsilon takes values in \mu_{\ell-1}\subset\Q_\ell^\times. Let K'/K be the cyclic extension fixed by its kernel, and let \Delta=\Gal(K'/K). Pullback along the surjection \beta_B is injective on H^1, with pulled-back coefficient module \Q_\ell(\rho). Restriction to G_{K'} remains injective: its kernel is a first cohomology group of the finite group \Delta, which vanishes in characteristic zero. We therefore obtain independent classes

Under the natural \Delta-action on this cohomology, they all lie in the \epsilon^{-1} eigenspace. Indeed the original coefficient module is \Q_\ell(1)\otimes\epsilon, so invariance for its twisted action becomes the eigenvalue \epsilon^{-1} after restriction.

We claim that every c_w is unramified away from \ell. Suppose that it is ramified at a prime u\nmid\ell of K', and let v be the prime below it in K. The cyclotomic module is trivial on I_u, so the restricted cocycle is a nonzero continuous homomorphism from I_u to the additive group \Q_\ell(1). Choose a pro-\ell Sylow subgroup J_u\subset I_u on which it is nonzero, and a pro-\ell Sylow subgroup J_v\subset I_v containing J_u. Then \beta_B(J_v)\ne1, so v is active at \ell. Theorem 2.4 sends J_v into inertia at a corresponding prime t\nmid\ell of B. Since t occurs in an active pair, t\notin J, whereas \kappa_w is unramified at t. Inertia at t acts trivially on the coefficient module, so the Kummer cocycle itself vanishes on that inertia group. Its pullback therefore vanishes on J_u, a contradiction. This proves the claim.

Let S consist of the archimedean places of K' and its primes above \ell. By Lemma 5.2, the classes c_w give independent elements of the \epsilon^{-1} eigenspace in \mathcal O_{K',S}^{\times}\otimes\Q_\ell. It remains to bound this eigenspace. The equivariant Dirichlet S-unit theorem [1, Section 3] gives the character of the full S-unit module as

Each \Delta-orbit in S contributes at most one copy of the one-dimensional character \epsilon^{-1} to the permutation representation: for a stabilizer H\subset\Delta, its multiplicity in \Q_\ell[\Delta/H] is one if it is trivial on H, and zero otherwise. There are

such orbits. Subtracting the trivial representation cannot increase this multiplicity. Hence the independent classes c_w lie in a space of dimension at most 2[K:\Q], and \#J\le2[K:\Q]. Since J was arbitrary, this bounds all missed primes away from \ell. There are at most two primes of B above \ell, giving the second bound.□

Proof. Choose distinct auxiliary primes s_j larger than m, d, and every prime in the lists, and put

For each j, the classes of the distinct rational primes in \{q_j\}\cup\mathcal P_j are independent in C^\times/(C^\times)^{s_j}. Indeed, at any prime in those lists, the ramification index in C/\Q divides [\Q(\zeta_m):\Q]: the other cyclotomic conductors are prime to that prime. This index is prime to s_j, and valuations at the separate rational primes give the asserted independence.

Kummer theory therefore identifies the Galois group over C of

with an elementary abelian s_j-group having one independent coordinate for each listed prime. Choose \sigma_j\in\Gal(E_j/C) which fixes q_j^{1/s_j} and acts nontrivially on every p^{1/s_j} with p\in\mathcal P_j. The extensions E_j/C are linearly disjoint, since their degrees are powers of distinct primes. Thus these automorphisms combine to an element \sigma\in\Gal(E/C), where E is their compositum.

The extension E/\Q is Galois. Moreover, conjugation in \Gal(E/\Q) preserves whether each of the specified radical coordinates of \sigma is trivial: a conjugation only raises its root-of-unity multiplier to a power prime to the corresponding s_j. Chebotarev’s theorem supplies infinitely many rational primes \ell with Frobenius in the conjugacy class of \sigma, avoiding the prescribed finite set, all listed primes, and 2. They split completely in C, so \ell\equiv1\pmod m and \ell\equiv1\pmod{s_j}. At these primes, q_j is an s_jth power modulo \ell and every p\in\mathcal P_j is not.

The quotient of \F_\ell^\times by its s_jth powers has prime order s_j. A nontrivial class stays nontrivial on raising it to any exponent 1\le f\le d<s_j. Every power of q_j, however, is an s_jth power. This proves the displayed separation. In particular, overlaps between different lists impose no conflict: their conditions belong to linearly disjoint Kummer extensions with different prime exponents.□

Proof. Suppose that there are infinitely many exceptions. Lemma 5.1 allows us to choose

among them, above distinct split rational primes q_1,\ldots,q_n, with every \mathcal V_{w_j} finite. Let \mathcal P_j be the set of rational residue characteristics of the primes in \mathcal V_{w_j}. The failure of the asserted conclusion means q_j\notin\mathcal P_j.

Apply Lemma 5.4 with m=m_0 and d=[K:\Q]. It gives \ell\in\mathcal L, away from all the primes involved, such that

because Nv=p^f with p\in\mathcal P_j and 1\le f\le[K:\Q]. If w_j occurred in an active pair at this \ell, its source prime would lie in \mathcal V_{w_j}. Applying Lemma 2.6 to a lift of source Frobenius and reducing modulo \ell would give

contrary to (5.3). All n primes w_j therefore belong to \mathcal M_\ell, contradicting Proposition 5.3.□

Proof. Fix a finite Galois extension L/\Q inside \Q^s containing B, and put R=\Gal(L/\Q). Choose a finite Galois extension F'/\Q containing K and L such that \beta(G_{F'})\subset P_L. This is possible by taking the core in \mathcal G of the appropriate open subgroup of T. Fix r\in\mathcal L.

The maps \beta and \sigma, restricted to G_{F'}, give two pullbacks

Both are injective: their underlying homomorphisms have open image, and an additive \Q_r-valued character vanishing on a finite-index subgroup is zero. For a\in P let \sigma_a=\Ad(a)\circ\sigma and let i_a be its pullback. Since L/\Q is Galois, \sigma_a(G_{F'})\subset P_L. The map i_a depends only on a modulo P_L, because inner automorphisms of P_L act trivially on additive characters.

Frobenius evaluations. Let q\ne r range over rational primes split completely in F'. By Proposition 5.5, for all but finitely many such q there is a prime v of K above q whose decomposition group maps into a decomposition group at a prime w of B above the same q. Choose a prime v' of F' above v and compatible prolongations. Then

for a prime u of L above w. Every prime here has absolute residue degree one, because q splits completely in F'.

Choose a lift h of arithmetic Frobenius in D_{v'}. Let a_q\in\widehat{\Z} be the residue exponent of \beta(h) at u. Equation (6.1) gives

The number \log_r q is nonzero: q\in\Z_r^\times is a positive rational integer larger than one and is not a root of unity. Consequently (a_q)_r=1. Every x\in\X rL is unramified at u, so

This calculation uses only decomposition containment from the active correspondence. It does not require that its auxiliary prime be r.

Under ordinary restriction, the same Frobenius maps to Frobenius at another prime of L above q. An element of R carries this prime to u. Choose a lift a\in P of that element. Then

The choice of a modulo P_L may initially depend on q. Since R is finite, an infinite set of distinct q uses one fixed choice. For each x, the difference i_\beta(x)-i_a(x) therefore vanishes at infinitely many degree-one primes of F'. Proposition 3.2 gives

From characters to the finite group. The field F'/\Q is Galois, so T acts on \X r{F'} by conjugation. The two pullbacks intertwine this action with the respective actions on \X rL induced by \beta and \sigma_a. If i denotes their common injective value in (6.4), then for t\in T and x\in\X rL,

The field L is totally imaginary because it contains B. By Proposition 3.3, R acts faithfully on \X rL. Hence

One conjugation for all finite quotients. For each such L, let

We have proved that C_L is nonempty. It is closed, and indeed clopen, since the defining condition depends only on a modulo P_L. If L contains finitely many fields L_1,\ldots,L_s, then C_L\subset\bigcap_j C_{L_j}. Composita thus give the finite-intersection property. Compactness of P supplies an element in every C_L. The extensions L containing B are cofinal among finite Galois subextensions of \Q^s/\Q, so this element gives the asserted equality in P.□

Proof. As in Lemma 4.1, choose an open normal subgroup U of \mathcal G contained in G_F, and extend \varphi|_U to the normalizer of its kernel. Proposition 6.1 shows that, after one conjugation of \varphi, it agrees on U with ordinary restriction \sigma:G_F\to P.

For g\in G_F and u\in U, normality of U gives

Thus \sigma(g)^{-1}\varphi(g) centralizes \sigma(U)=\varphi(U), an open subgroup of P. Lemma 2.3 makes this element trivial. The equality holds for every g\in G_F.□

Proof of Theorem 1.1. Choose copies of the given fields inside \overline{\Q}. By Lemma 2.1, each E_i contains \Q^s. Form the composite

where the outer arrows are restrictions. The first is surjective. The last has open image, since its composite with G_{F_2}\twoheadrightarrow G_2 is ordinary restriction from an open subgroup of \mathcal G to P. Continuous surjective homomorphisms between profinite groups are open, so (6.5) is open.

By Theorem 6.2, this composite is conjugate to ordinary restriction. Cyclotomic characters are unchanged by conjugation. All roots of unity lie in \Q^s, so the full cyclotomic characters factor through the groups in (6.5). Surjectivity of the first arrow therefore yields

Theorem 2.5 gives an equivariant embedding E_2\hookrightarrow E_1. Its uniqueness is Uchida’s theorem cited after Theorem 2.5.□