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OPEN HOMOMORPHISMS OF GLOBAL SOLVABLY CLOSED GALOIS GROUPS
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A complete, self-contained proof of Uchida's conjecture.
Every continuous open homomorphism between Galois groups of
solvably closed Galois extensions of number fields is induced
by a unique equivariant field embedding in the opposite
direction. No restriction on the kernel or extra compatibility
hypothesis is required.
1. CONVENTIONS, SOLVABLY CLOSED FIELDS, AND THE BLACK BOXES
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1.1 Standing notation.
Fix an algebraic closure Q-bar of Q. For a number field M inside
Q-bar, put G_M = Gal(Q-bar / M). Put
Gbig = G_Q = Gal(Q-bar / Q), P = Gal(Q^s / Q),
where Q^s is the maximal prosolvable extension of Q inside Q-bar.
For a number field M inside Q^s write P_M = Gal(Q^s / M).
All homomorphisms between profinite groups and all characters are
continuous. An OPEN homomorphism is one with open image; a continuous
surjection of profinite groups is a quotient map and is open.
We use finite primes unless infinity is named. For a finite prime v,
Nv denotes its absolute norm and [kappa(v):F_p] its absolute residue
degree, where p is the rational prime below v. Frobenius is
ARITHMETIC. Choosing a prolongation of v fixes a decomposition group
D_v and inertia group I_v; changing the prolongation conjugates both.
Whenever several such groups appear together, fixed compatible
prolongations are understood.
For M/Q finite Galois we let R = Gal(M/Q) and, for a subgroup J of
R acting on a characteristic-zero vector space carrying an R-action,
write
E_J = (1/|J|) sum_{g in J} g
for the averaging idempotent onto the J-invariants. We write res and
cor for restriction and corestriction on additive characters for a
finite extension, and Ad for conjugation. Ind denotes induction.
The l-adic logarithm on Z_l^x is the continuous homomorphism that
agrees with the power series near 1; it is normalized to vanish on
the finite torsion subgroup (the roots of unity).
1.2 Solvably closed fields. An algebraic extension E of a number
field is SOLVABLY CLOSED if it has no nontrivial abelian algebraic
extension. Q-bar and Q^s are the guiding examples.
LEMMA 1.1 (Q^s is solvably closed and minimal).
Every solvably closed algebraic extension E of Q contains Q^s. The
field Q^s is solvably closed. For every number field M inside Q^s,
the extension Q^s/M is the maximal prosolvable extension of M.
Consequently every additive l-adic character of G_M factors through
P_M.
Proof. Since E(mu_n)/E is abelian for every n, E contains all roots
of unity. Adjoining an n-th root of any element of E is then a cyclic
(hence abelian) extension, so E is closed under taking radicals.
Every finite solvable Galois extension can be built as a tower of
radical extensions after adjoining roots of unity; hence Q^s is
contained in E.
Conversely Q^s contains all roots of unity and is closed under
radicals by its definition as the union of finite solvable
extensions. Any nontrivial finite abelian extension of Q^s would have
a cyclic subextension of prime degree, which Kummer theory rules out
(the relevant roots of unity are already present). Any nontrivial
abelian algebraic extension contains a nontrivial finite abelian
subextension. So Q^s is solvably closed.
For the last claim, a finite solvable Galois extension of M lies in
a finite solvable Galois extension of Q: take a solvable normal
closure of M/Q inside Q^s, then the compositum of the finitely many
conjugate extensions; its group over the normal closure embeds in a
finite product of solvable groups. This shows Q^s/M is the maximal
prosolvable extension of M. QED
For a solvably closed Galois extension E/F, the action on roots of
unity defines the cyclotomic character
chi_{E/F} : Gal(E/F) -> Z-hat^x (Z-hat = profinite completion of Z).
We write chi_{M,l} for the usual l-adic character of G_M, and chi_l
for that of P.
1.3 The anabelian inputs. We record the five external facts. Four are
cited; the fifth (centralizer) is derived from the first.
THEOREM A (Uchida isomorphism theorem; Uchida 1979).
Let E_i/F_i be solvably closed Galois extensions of number fields. Every
isomorphism theta : Gal(E_1/F_1) -> Gal(E_2/F_2) is induced by a unique
field isomorphism j : E_2 -> E_1, with g(j) = j(theta(g)) for all g.
LEMMA B (centralizer of an open subgroup).
The centralizer in P of any open subgroup of P is trivial.
Proof. An open subgroup of P is P_M for a number field M inside Q^s.
An element of P 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 A gives the claim.
QED
DEFINITION (active prime). For a homomorphism phi : G_M -> P_N, an odd
prime l, and a prime v not above l of M, say v is ACTIVE AT l if phi
is nontrivial on a pro-l Sylow subgroup of I_v. This does not depend
on the choice of Sylow subgroup or prolongation, since such choices
are conjugate.
THEOREM C (Uchida's partial local correspondence; Uchida 1981).
Let M be a number field, N a number field inside Q^s, and phi : G_M ->
P_N open. Fix an odd prime l. Every prime v (not above l) active at l
determines a UNIQUE prime w (not above l) of N such that, with suitable
prolongations,
phi(D_v) is contained in D_w.
Moreover phi is injective on a pro-l Sylow subgroup of I_v and sends it
into I_w. All but finitely many primes of N arise from such active
primes of M.
We use the odd-prime case of Uchida's trichotomy together with his
Proposition 1. In that trichotomy nontrivial pro-l inertia is case
(ii); case (i) kills it; the torsion case (iii) needs l = 2. The
open-image hypothesis (not surjectivity onto P_N) suffices. We apply
the theorem after restricting to smaller open subgroups. The
cofinite target coverage is for a FIXED l; its exceptional set is not
asserted to be uniform in l, which is exactly the difficulty to be
overcome.
THEOREM D (Hoshi's cyclotomic criterion; Hoshi 2025).
Let E_i/F_i be solvably closed Galois extensions of number fields and
alpha : Gal(E_1/F_1) -> Gal(E_2/F_2) open. There is an equivariant field
embedding E_2 -> E_1 if and only if
chi_{E_2/F_2} o alpha = chi_{E_1/F_1} in Z-hat^x.
Uniqueness of the embedding, when it exists, is Uchida 1981,
Proposition 2.
The fifth input, the log-rank lemma, is proved in Section 2.3; it is a
special case of the l-adic Waldschmidt-Masser rank theorem but we give
a complete argument so the paper is self-contained.
1.4 Decomposition groups and local structure. We record that
decomposition groups in P_M really are full local absolute Galois
groups, so the usual language of tame inertia and residue exponents
applies.
Justification. Fix a prime of Q-bar above a prime of M and form,
inside a local algebraic closure, the union of the completions of the
finite fields between M and Q^s. This union contains all roots of
unity and is closed under radicals (for a fixed exponent, a nonzero
local element is approximated by a global one in the defining field
and the ratio is made a power). So it has no cyclic extension. Every
finite Galois group over a nonarchimedean local field is solvable,
via wild inertia, tame inertia, and the unramified filtration. Every
closed subgroup of its prosolvable absolute Galois group is
prosolvable; a nontrivial such group has a nontrivial finite cyclic
quotient after passing through a finite solvable quotient. Hence the
union is the full local algebraic closure. The usual
decomposition-group identification proves the claim.
1.5 The local norm identity. For an active pair (v,w) from Theorem C
and g in D_v, let b(g) in Z-hat be the source residue exponent and
a(g) in Z-hat the target residue exponent of phi(g), measured using
Nv- and Nw-Frobenius respectively. (The target exponent is defined on
D_v by composition; we do NOT assume that phi sends all source
inertia into target inertia.)
LEMMA 1.2 (norm identity). For an active pair (v,w) at an odd prime l,
(Nv)^{b(g)} = (Nw)^{a(g)} in Z_l^x, for all g in D_v.
Powers by profinite integers are continuous powering in the compact
group Z_l^x.
Proof. Let p != l be the residue characteristic of v and choose a
topological generator tau of a pro-l Sylow subgroup of I_v. Its image
in source tame inertia generates the pro-l factor. The tame
conjugation relation gives
e = g tau g^{-1} tau^{-(Nv)^{b(g)}} lies in source wild inertia,
a pro-p group. In the target tame quotient the image of e also lies in
the pro-l factor of tame inertia (both phi(tau) and its conjugate do).
So its image is trivial, being in a pro-p subgroup and a pro-l group.
The nontrivial image of phi(tau) in the torsion-free target pro-l tame
factor then gives the identity by the target tame conjugation relation.
The target residue characteristic differs from l by Theorem C. QED
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2. ADDITIVE CHARACTERS AND FROBENIUS DETECTION
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We compare homomorphisms by evaluating additive l-adic characters at
Frobenius elements. The key fact: infinitely many primes of residue
degree one detect a character. We first prove the logarithm-rank
lemma, then the detection, averaging, and corestriction statements,
and finally describe a signed representation sitting inside the
character space.
2.1 Class-field description of characters.
For a number field M inside Q-bar, write
X_l(M) = Hom_cont(G_M, Q_l),
where Q_l is the additive group with trivial Galois action. Local
class field theory shows these characters are unramified at every
finite prime v not above l: the local unit group has a pro-p subgroup
of finite index for p != l and has no nonzero continuous homomorphism
to Q_l. Hence x(Fr_v) is defined for x in X_l(M) and v not above l.
If M/Q is Galois, R = Gal(M/Q) acts by
(g x)(h) = x(g-tilde^{-1} h g-tilde), g-tilde a lift of g in G_Q;
this is independent of the lift.
2.2 Logarithms and the rank lemma.
For a finite extension K of Q_l, normalize the absolute value by
|l|_l = l^{-1}. The l-adic logarithm on O_K^x is the continuous
homomorphism agreeing with the power series near 1; its kernel is the
finite group of roots of unity in K. Logarithms of torus points are
taken coordinatewise.
Class field theory expresses an additive character of G_M as a linear
form on M (x)_Q Q_l. A zero Frobenius value at a prime v not above l
forces this form to annihilate log b, where (b) = v^h for some h > 0.
For residue-degree-one primes over distinct rational primes unramified
in a Galois closure, the conjugate generators have disjoint ideal
supports; the next lemma turns their multiplicative independence into
full logarithmic rank.
The rank estimate is a special case of the l-adic Waldschmidt-Masser
rank theorem (Waldschmidt 1981, Masser 1981); we retain a proof via
Laurent's interpolation determinants (Laurent 1991; see also Dasgupta
2023, Theorems 5.1 and 5.8).
LEMMA 2.1 (log-rank). Let n,m >= 1, let E be a number field embedded
in a finite extension K of Q_l, and let
u_i = (u_{i1},...,u_{in}) in (E^x)^n, 1 <= i <= m,
have algebraic-integer coordinates that are units in K. Suppose that
for every a = (a_1,...,a_m) in Z^m minus {0}, the coordinates of
u^a := prod_i u_i^{a_i} are multiplicatively independent modulo roots
of unity. If m > n(n-1), then
span_K { log u_1, ..., log u_m } = K^n.
Proof. The multiplicative hypothesis makes the cyclic subgroup
generated by every u^a (a != 0) Zariski dense in (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 on its finitely many monomials would force all coefficients to
vanish.
Suppose 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 is the roots of unity. So 1 <= r < n and
in particular n >= 2. We produce an algebraic determinant whose small
l-adic absolute value clashes with its archimedean size.
Analytic preparation. Replacing all u_i by the same positive integer
power preserves the hypotheses and the log rank, and puts the
coordinates in a neighborhood where exponential and logarithm are
inverse. The O_K-module generated by the logarithms is free of rank r;
choose a basis b_1,...,b_r in K^n. After a further common l-power the
basis vectors have coordinates small enough that
rho := max_{j,t} |b_{t,j}|_l satisfies eta := rho l^{1/(l-1)} < 1.
(The basis is multiplied by the same power as the logarithms.) Then
there are z_i in O_K^r with
log u_i = sum_{t=1}^r z_{i,t} b_t.
From now on the powered points, the number field E, the local field K,
the basis, and eta are fixed; none depends on the degree parameter
introduced below.
The monomial functions. For an integer vector alpha =
(alpha_1,...,alpha_n), the monomial X^alpha on these points is
represented by
f_alpha(z) = exp( sum_{j=1}^n alpha_j sum_{t=1}^r b_{t,j} z_t ),
analytic on a polydisc of radius greater than 1. Writing
f_alpha(z) = sum_{nu in Z_{>=0}^r} c_{alpha,nu} z^nu, the exponential
series and |k!|_l^{-1} <= l^{k/(l-1)} give the uniform bound
|c_{alpha,nu}|_l <= eta^{|nu|}, |nu| = nu_1 + ... + nu_r. (2.1)
The bound is independent of alpha because every integer has l-adic
absolute value at most 1. Also, for every a in Z_{>=0}^m the point
z(a) := sum_i a_i z_i lies in O_K^r and f_alpha(z(a)) = (u^a)^alpha.
Full algebraic rank of enough evaluations. For an integer D >= 1 put
A = (D+1)^n and choose S >= 1 with
(S+1)^m > (nD)^n, S = O(D^{n/m}). (2.2)
Consider the matrix with rows indexed by alpha in {0,...,D}^n, columns
indexed by a in {0,...,nS}^m, and entries (u^a)^alpha. We claim its
rank is A.
Otherwise a nonzero polynomial P of degree at most D in each variable
vanishes at every point u^a in this grid. Over an algebraic closure of
E define closed subsets of the torus by
Z_j = { x in (G_m)^n : P(u^a x) = 0 for all a in {0,...,jS}^m },
0 <= j <= n.
These are nested and nonempty since 1 lies in Z_n, while dim Z_0 <=
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,...,S}^m the translate u^b V lies in Z_j and has dimension d,
so it is an irreducible component of Z_j. The translates are distinct:
an equality u^b V = u^{b'} V with b != b' would make V stable under
every power of u^{b-b'}, and the density proved above would force V =
(G_m)^n, contradicting V contained in Z_0.
Degree bound for components. A common zero set of polynomials of total
degree at most delta >= 1 in affine n-space has at most delta^n
irreducible components. Sketch: assign a component Y the weight
delta^{dim Y} deg Y. Start with affine n-space, of weight delta^n, and
impose equations successively. A component lying in the next
hypersurface keeps its weight; a component not lying in it is cut into
components of dimension one less, whose total degree is at most
delta deg Y by Bezout. Thus the total weight never increases, and
removing redundant components only decreases it. The same bound holds
after restricting to the torus. Applying it to Z_j, whose equations
have total degree at most nD, contradicts (2.2). This proves the rank
claim.
The minor. Choose a nonzero A x A minor Delta_D of the evaluation
matrix. It is an algebraic integer in the fixed field E. Expand its
rows using the power series for f_alpha. A nonzero term must involve A
DISTINCT monomials in z_1,...,z_r, since repeated monomials give
repeated rows in the corresponding evaluation determinant. The degree
sum of any A distinct such monomials is at least c_r A^{1+1/r} for
large A: the number of monomials of degree at most t is
binomial(t+r,r) = O_r(t^r), so at least half of the A monomials have
degree at least a positive multiple of A^{1/r}, and their evaluations
have absolute value at most 1. The nonarchimedean triangle inequality
and (2.1), first for truncated series then by convergence, therefore
give a constant c > 0 with
log |Delta_D|_l <= -c A^{1+1/r}. (2.3)
Archimedean size. At every archimedean embedding tau : E -> C an entry
of this minor has absolute value at most exp(CDS), with C independent
of D. Expanding the determinant,
log |tau(Delta_D)| <= C' A D S + A log A. (2.4)
Contradiction via the product formula. Let lambda be the place of E
induced by E -> K. Use absolute values at finite places extending the
usual ones on Q and sum over all embeddings E -> C, so each complex
place occurs twice. The product formula reads
0 = sum_{tau} log |tau(Delta_D)|
+ sum_{v finite} [E_v : Q_{p_v}] log |Delta_D|_v,
where p_v is the rational prime below v. Every term in the second sum
is nonpositive because Delta_D is an algebraic integer. Retaining its
lambda-term and using (2.3), (2.4) would give
c [E_lambda : Q_l] A^{1+1/r} <= [E:Q] ( C' A D S + A log A ).
This is impossible as D -> infinity: the left side is of order
D^{n+n/r}, while the right side is O(D^{n+1+n/m} + D^n log D), and
n/r >= n/(n-1) > 1 + n/m
by m > n(n-1). The contradiction proves the lemma. QED
2.3 Frobenius detection, averaging, and corestriction.
The class-field description identifies the character space as follows.
Let V_M = M (x)_Q Q_l and let U_M be the Q_l-span of the logarithms of
the global units (at all places above l). Then
X_l(M) = { lambda in Hom_{Q_l}(V_M, Q_l) : lambda(U_M) = 0 }. (2.5)
Indeed, pass to the inverse limit of the ray-class sequences with
moduli supported above l. Local logarithms identify the rationalized
local unit groups with V_M, and the global units impose exactly U_M.
The ordinary class group and the factors at real places are finite and
contribute no additive Q_l-characters. This uses the actual
logarithmic image of the units; it asserts nothing about its dimension.
For v not above l choose h_v > 0 and b_v in O_M with (b_v) = v^{h_v}
(possible since the ideal class group is finite). Principal
reciprocity, with arithmetic Frobenius, gives for the linear form
lambda corresponding to x the identity
lambda(log b_v) = -h_v x(Fr_v). (2.6)
Here b_v is a unit at every place above l, and all other local unit
contributions are killed by x.
PROPOSITION 2.2 (Frobenius detection). Let M be a number field, l a
rational prime, and x in X_l(M).
(i) If x(Fr_v) = 0 at infinitely many primes v not above l of
absolute residue degree one, then x = 0.
(ii) Suppose M/Q is Galois and fix delta in Gal(M/Q). If x(Fr_v) = 0
at infinitely many primes v not above l lying over rational
primes unramified in M whose Frobenius in Gal(M/Q) is delta,
then E_{<delta>} x = 0.
(iii) If cor_{M/Q} x != 0, then x(Fr_v) = 0 at only finitely many
primes v not above l, with no condition on residue degree.
Proof. (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 from (2.6) for these primes and
u_i = ( sigma(b_i) )_{sigma : M -> E} in (E^x)^n.
All coordinates are algebraic integers and units at every place of E
above l. We verify the multiplicative hypothesis of Lemma 2.1 by
valuations.
Put G = Gal(E/Q), H = Gal(E/M). For a selected degree-one prime v_i
and a prime p_i of E above it, the decomposition group D_i lies in H:
the rational prime is unramified in E and the degree-one condition
says [D_i : D_i cap H] = 1. The support of v_i O_E is indexed by H/D_i.
Its translates corresponding to the embeddings M -> E (indexed by G/H)
have supports indexed by the disjoint sets gH/D_i. Distinct indices i
involve distinct rational primes. So the mn numbers sigma(b_i) have
pairwise disjoint nonempty supports of their principal ideals, hence
are multiplicatively independent modulo roots of unity. In particular,
for every 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 of Q_l. Under the splitting
(M (x)_Q Q_l) (x)_{Q_l} K = prod_{sigma : M -> E} K,
the vector log b_i becomes log u_i. Lemma 2.1 says these vectors span
the right side. Equations (2.5) and (2.6) then force lambda = 0, hence
x = 0.
(ii) Put C = <delta>, F = M^C, and y = E_C x. Since delta fixes every
indicated prime v, averaging preserves its Frobenius value, so
y(Fr_v) = 0. The restriction-corestriction identity gives
res_{M/F} cor_{M/F} y = sum_{g in C} g y = |C| y.
So y is the restriction of z = |C|^{-1} cor_{M/F} y in X_l(F). At an
indicated prime the decomposition group of M/Q is exactly C, so the
prime u = v cap F has absolute residue degree one, and
0 = y(Fr_v) = [kappa(v):kappa(u)] z(Fr_u).
Infinitely many distinct u occur because M/F is finite. Part (i) gives
z = 0, hence y = 0.
(iii) Suppose infinitely many zero primes exist. Let E/Q be a finite
Galois closure of M, R = Gal(E/Q), 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_{<delta>} y = 0, hence E_R y = 0. Therefore
res_{E/Q} cor_{E/Q} y = |R| E_R y = 0.
Restriction of additive characters across a finite extension is
injective: a character vanishing on a finite-index subgroup has finite
image in the torsion-free group Q_l, hence is zero. On the other hand,
transitivity and the corestriction-restriction identity give
cor_{E/Q} y = cor_{M/Q} cor_{E/M} res_{E/M} x = [E:M] cor_{M/Q} x != 0,
a contradiction. QED
2.4 A signed representation in the character space.
The description (2.5) also controls the supply of characters as the
number field varies. The regular representation bounds dimensions of
subgroup invariants from above; the signed representation below bounds
them from below and makes the action faithful.
PROPOSITION 2.3 (signed representation). Let M/Q be a finite totally
imaginary Galois extension, R = Gal(M/Q), and c in R a complex
conjugation. For every rational prime l there are R-equivariant
embeddings
W_M := Ind_{<c>}^R (sign) >--> X_l(M) >--> Q_l[R], (2.7)
where sign(c) = -1. In particular R acts faithfully on X_l(M).
Proof. By the normal basis theorem V_M = M (x)_Q Q_l is the regular
representation of R, as is its dual; the naturality of (2.5) gives the
second embedding.
The equivariant Dirichlet unit theorem (Bass 1966, Section 3)
identifies O_M^x (x)_Z Q_l with the permutation representation on the
archimedean places minus its trivial summand. Since these places are
indexed by R/<c>, writing U^{alg} = O_M^x (x)_Z Q_l gives
Q_l (+) U^{alg} = Ind_{<c>}^R 1.
The character identity follows first over R from the usual
logarithmic unit embedding, then over Q_l by semisimplicity in
characteristic zero. In particular U^{alg} is self-dual. Its l-adic
logarithmic image U_M is a QUOTIENT of U^{alg}; no injectivity of the
logarithm on this tensor product is needed.
The decomposition of the regular representation induced from the two
characters of <c> is
Q_l[R] = Ind_{<c>}^R 1 (+) W_M = Q_l (+) U^{alg} (+) W_M.
Since U_M^dual is a subrepresentation of (U^{alg})^dual = U^{alg},
semisimplicity and X_l(M) = (V_M / U_M)^dual show X_l(M) contains
Q_l (+) W_M. This is the first embedding in (2.7).
Finally realize W_M as the direct sum of the signed lines indexed by
the left cosets of <c>. An element acting trivially fixes the line of
the identity coset, hence lies in <c>; the nonidentity element c acts
by -1 on that line. So only the identity acts trivially, proving
faithfulness. QED
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3. EXTENSION TO THE NORMALIZER, AND GENERATION BY FROBENIUS
=====================================================================
Let F be a number field and let alpha_0 : G_F -> P be a continuous
open homomorphism. We first enlarge an open part of its domain to the
normalizer of the kernel. For the enlarged map we prove surjectivity
and equality of logarithmic cyclotomic characters for an infinite
congruence class of l.
3.1 Extension to the normalizer.
Choose an open normal subgroup U of Gbig contained in G_F and put
A = alpha_0(U), N = ker(alpha_0 |_U), T = N_{Gbig}(N).
A is open in P. Since U is contained in T, the normalizer T is open
and closed in Gbig; write T = G_k.
LEMMA 3.1 (extension to the normalizer). There is a unique continuous
open homomorphism beta : T -> P whose restriction to U is alpha_0|_U.
Proof. Conjugation by t in T induces a continuous automorphism of
U/N = A. By Theorem A applied to Q^s/(Q^s)^A, this automorphism is
conjugation by a unique beta(t) in P (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
any extension of alpha_0|_U to T equals beta, since 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 open in P; a continuous homomorphism of
profinite groups is open onto its image, so beta is open. QED
We keep U, A, N, T, k, beta fixed. List the conjugates of N other than
N itself as
N_i = t_i N t_i^{-1} (1 <= i <= r), t_i not in T, H_i = beta(N_i).
There are finitely many since U normalizes N. Normality of U in Gbig
makes each N_i lie in U and be normal in U, so H_i is normal in A. Each
H_i is nontrivial: otherwise t_i N t_i^{-1} is contained in N, and
iterating gives a descending chain returning to N (some power of t_i
lies in U and normalizes N), forcing all inclusions to be equalities,
contrary to t_i not in T. We allow r = 0 (then the indexed sums below
are zero).
3.2 A finite quotient with enough characters.
The next lemma constructs characters that distinguish a proper
subgroup of a cyclic group while avoiding the invariant spaces of the
H_i; the latter prevents cancellation by conjugates of N when we
average a pulled-back character.
LEMMA 3.2 (quotient amplification). Fix an odd prime l and a finite
quotient R_0 of P. There is a finite Galois extension L/Q inside Q^s,
totally imaginary, with R = Gal(L/Q) dominating R_0, satisfying: for
every gamma in R, with C = <gamma>, and every proper subgroup D < C,
there is x in X_l(L) with
x in X_l(L)^D, E_C x = 0, x not in sum_{i=1}^r X_l(L)^{H_i}. (3.1)
Here each H_i acts through its image in R.
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 -> R_1 lies 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 enlarge R_1 keeping element orders at most 2n. The images of the
H_i and the conjugacy class of complex conjugation grow: larger H_i
make the sum of their invariant spaces small, larger conjugacy class
makes the signed representation supply enough D-invariant vectors
killed by E_C to escape that sum.
This is the regular-module auxiliary extension of Uchida 1979,
pp. 360-362, with quantitative bounds. Take a further quotient with
kernel m copies of the regular F_2[R_1]-module, m chosen below. Choose m
distinct rational primes completely split in L_1; for each, select one
prime of L_1 above it. Approximation gives a_1,...,a_m in L_1^x 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 g(a_j) (g in R_1,
1 <= j <= m) have independent square classes: their valuations at the
selected sets of primes form a permutation of the identity matrix mod 2.
Adjoin their square roots; call the result L. Kummer theory gives
1 -> V -> R = Gal(L/Q) -> R_1 -> 1, V = F_2[R_1]^m (3.2)
as R_1-modules; the Kummer dual is again the regular permutation
module. L is Galois over Q because all conjugates of the a_j were
included; its group is solvable, so L lies in Q^s. No splitting of
(3.2) is required.
Every element of R has order at most 2n (its n-th 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 contained in A_R. Pick 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,
(g_i - 1) V = [h_i, V] is contained in H_{i,R}.
On m regular modules the rank of g_i - 1 is mn(1 - 1/d_i) >= mn/2, so
|H_{i,R}| >= 2^{mn/2}. (3.3)
A complex conjugation c in R has nontrivial image in R_1 (as L_1 is
totally imaginary), so conjugation by V alone gives at least 2^{mn/2}
distinct conjugates of c:
|c^R| >= 2^{mn/2}. (3.4)
Set eta = 2^{-mn/2} and X = X_l(L). By Proposition 2.3, X embeds in the
regular Q_l[R]-module and contains W = Ind_{<c>}^R(sign). The
regular-representation bound and (3.3) give
dim( sum_{i=1}^r X^{H_i} ) <= sum_i |R|/|H_{i,R}| <= r eta |R|. (3.5)
The character of W is |R|/2 at the identity, -|R|/(2|c^R|) on c^R, and
0 elsewhere. Hence for every subgroup J of R,
| dim W^J - |R|/(2|J|) | <= eta |R| / 2. (3.6)
For D < C = <gamma> the map E_C : W^D -> W^C is surjective. Since
|D| <= |C|/2 and |C| <= 2n, (3.6) gives
dim ker(E_C |_{W^D}) = dim W^D - dim W^C
>= (|R|/2)(1/|D| - 1/|C|) - eta|R|
>= |R|( 1/(4n) - eta ).
Choose m so large that (r+1) 2^{-mn/2} < 1/(4n). One such m works for
all gamma and all proper D < C. The last dimension exceeds (3.5), so the
kernel contains a vector outside the sum. This vector gives (3.1). If
C is trivial there is no proper subgroup to consider. QED
3.3 Generation by local Frobenius images.
Containment of a source decomposition group in a target one places its
Frobenius image in a cyclic subgroup of a finite quotient. We show that,
after enlarging the quotient, infinitely many such images generate the
whole cyclic subgroup.
PROPOSITION 3.3 (Frobenius generation). Fix an odd prime l, a finite
quotient R_0 of P, and gamma_0 in R_0. There are a quotient
P ->> R = Gal(L/Q) ->> R_0 and a lift gamma of gamma_0 in R such that:
there are infinitely many l-active primes v of k, over distinct rational
primes p != l, for which compatible prolongations can be chosen so that
(1) the corresponding target rational prime q != l is unramified in L,
and the target prolongation restricts to a prime w' of L whose
Frobenius in R is gamma;
(2) the image in R of an arithmetic Frobenius lift at v generates
<gamma>.
Proof. Choose R by Lemma 3.2 and any lift gamma of gamma_0; put
C = <gamma>. Take an open normal subgroup U' of Gbig contained in
U cap beta^{-1}(P_L) (e.g. the core of this open subgroup in Gbig). Let
F'/Q be the finite Galois extension with G_{F'} = U'; the map
beta|_{U'} : G_{F'} -> P_L is open.
By Chebotarev infinitely many primes w' of L, over distinct rational
primes q != l unramified in L, have Frobenius exactly gamma. Theorem C
applied to beta|_{U'} pairs all but finitely many of them with l-active
primes v' of F' (away from l). Uniqueness of the target prime for an
active source prime makes infinitely many v' occur. Since only finitely
many primes of F' lie over a given rational prime, pass to a subset
with distinct source rational primes p, all unramified in F'. Pass to a
further infinite subset on which the Frobenius elements in Gal(F'/Q)
are a fixed element delta.
Fix a source prolongation above each v' and use it for 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') contained in P_L, which preserves the finite prime
w'. Activity passes from v' to v because the relevant inertia subgroup
for U' is contained in that for T. Apply Theorem C at v with this same
source prolongation. If w_1 is the restriction to L of its target
prolongation, then
beta(D_{v'}) is contained in beta(D_v) cap P_L is contained in D_{w_1}.
Uniqueness of the target prime for v' at the level of L gives w_1 = w'.
So the corresponding target rational prime is q, and the image of D_v
in R lies in C. These changes preserve v' and w', hence preserve the
previously selected Frobenius elements delta and gamma.
Put H = T/U' inside Gbig/U' and let f be the residue degree of v over
p. 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 denote
local arithmetic Frobenius over F' and L respectively.
In the decomposition group <delta> of Gal(F'/Q),
<delta> cap H = <delta^f>, f = min{ j >= 1 : delta^j in H }. (3.7)
Indeed, the residue degree is the orbit size of the distinguished coset
in the action of <delta> on Gal(F'/Q)/H; this does not require H normal.
In particular f is constant on the chosen infinite set. The homomorphism
beta induces H -> R (since beta(U') is contained in P_L); define
d = beta(delta^f) in R,
independent of the lift of delta^f to T. A local lift of rational
Frobenius to the f-th power is a Frobenius lift at v, so the containment
gives d in C.
Claim D = <d> equals C. Suppose D < C and choose x as in (3.1).
Pullback along beta|_{U'} gives a linear map
j : X_l(L) -> X_l(F'), y = j(x).
It is injective: the defining homomorphism has open image in P_L, and an
additive Q_l-valued character vanishing on an open subgroup vanishes
everywhere. Frobenius evaluation at w' is invariant under gamma, since
gamma fixes w' and acts trivially by conjugation on its unramified local
quotient. Hence
x(Fr_{w'}) = (E_C x)(Fr_{w'}) = 0.
x is unramified at w', so this says x vanishes on the whole decomposition
group there; thus y(Fr_{v'}) = 0 for every prime in our infinite set.
Proposition 2.2(ii) gives
E_{<delta>} y = 0. (3.8)
To get a contradiction, write a = |<delta>| and Y = j(X_l(L)). Pullback
is T-equivariant with the target action through beta, so y is fixed by
N. In the sum defining (3.8), the powers delta^j in H are exactly those
with f | j, by (3.7). Each such power acts as the identity on y because
x is D-invariant. If f does not divide j, choose a lift t of delta^j to
Gbig; then t is not in T and delta^j y is fixed by t N t^{-1}. Multiplying
(3.8) by a yields
0 = (a/f) y + sum_{0 <= j < a, f not dividing j} delta^j y,
with y in sum_{i=1}^r X_l(F')^{N_i}.
Return this membership to the target character space. The finite group
U/U' acts on X_l(F') and its subspace Y is stable. Semisimplicity gives
a U-equivariant projection pi : X_l(F') -> Y. Every N_i lies in U, so pi
carries N_i-invariants into Y^{N_i}; equivariance and injectivity of j
give Y^{N_i} = j(X_l(L)^{H_i}). Applying pi to the membership and then
j^{-1} contradicts the last condition in (3.1). Only U-equivariance of
pi is used; no normality of T in Gbig is needed. So D = C, proving the
proposition. QED
3.4 Surjectivity and cyclotomic logarithms.
The generation statement gives the two properties needed downstream.
First, it removes any restriction on the target image. Second, a
congruence condition on l forces the source primes to have residue
degree one, so the detection theorem applies.
THEOREM 3.4 (normalized map). The extension beta : G_k -> P is
surjective. Put
m_0 = [k:Q]!, Lset = { l odd prime : l = 1 (mod m_0) }.
Lset is infinite, and for every l in Lset,
log chi_l o beta = log chi_{k,l} as additive characters on G_k. (3.9)
Here chi_l is the l-adic cyclotomic character on P and chi_{k,l} the
ordinary one on G_k.
Proof. Let R_0 be any finite quotient of P and gamma_0 in R_0.
Proposition 3.3 produces d in the image of beta in a further quotient R
with <d> = <gamma>, where gamma lifts gamma_0. Thus gamma, hence
gamma_0, lies in the image. Since beta(T) is compact, surjectivity onto
every finite quotient gives beta(T) = P.
Dirichlet's theorem shows Lset is infinite. Fix l in Lset, take
R_0 = Gal(Q(mu_l)/Q) = F_l^x, and choose gamma_0 generating this cyclic
group. The proposition gives infinitely many active primes v of k over
distinct rational primes p != l for which the image of a Frobenius lift
sigma_v acts on mu_l by a generator. Let q be the corresponding target
rational prime and f = [kappa(v):F_p], so Nv = p^f. If a(sigma_v) in
Z-hat is the target residue exponent, Lemma 1.2 gives
p^f = q^{a(sigma_v)} in Z_l^x. (3.10)
Its reduction mod l says p^f generates F_l^x, so gcd(f, l-1) = 1. But
1 <= f <= [k:Q], and every prime divisor of any f > 1 in this range
divides m_0, hence divides l-1. Thus f = 1.
Taking logarithms in (3.10),
(log chi_l o beta)(sigma_v) = log p = log chi_{k,l}(sigma_v).
The difference is an element of X_l(k) vanishing at infinitely many
primes of absolute residue degree one. Proposition 2.2(i) proves (3.9).
QED
=====================================================================
4. MATCHING RESIDUE CHARACTERISTICS
=====================================================================
We now show that almost every prime of Q(i) above a split rational
prime has an active partner of the same residue characteristic, the
auxiliary prime being allowed to vary. Two finiteness statements make
this possible: almost every such prime has only finitely many possible
partners, and the number of primes missed at a fixed l is uniformly
bounded.
Keep the surjection beta : G_k -> P and Lset from Theorem 3.4, and put
B = Q(i), G_K = beta^{-1}(P_B), beta_B = beta|_{G_K} : G_K ->> P_B.
Then K/k is quadratic. An ACTIVE PAIR AT l in this section means a
source prime v of K active at l for beta_B together with its
corresponding prime w of B from Theorem C (so both primes are away
from l). For a prime w of B define
V_w = { v : (v,w) is an active pair at some l in Lset }.
4.1 Finite fibers and a uniform bound on missed primes.
LEMMA 4.1 (finite fibers). For all but finitely many primes w of B
above rational primes split in B, the set V_w is finite.
Proof. Fix r in Lset. Proposition 2.3 supplies a nonzero character psi
in X_r(B) on which Gal(B/Q) acts by its nontrivial character. By
Proposition 2.2(i), psi(Fr_w) is nonzero at all but finitely many
degree-one primes w not above r. Fix such a w above a split rational
prime q != r and put
x_0 = (log chi_r)|_{P_B}, z = psi(Fr_w) / log_r q.
The denominator is nonzero: the kernel of the logarithm on Z_r^x is the
roots of unity and the positive integer q is not one. So z != 0 and
psi - z x_0 vanishes on the decomposition group at w (unramified there,
vanishes on Frobenius).
Consider its pullback y = beta_B^x(psi - z x_0) in X_r(K). The pullback
of psi transforms by sign under G_k/G_K; its corestriction to k is zero,
because restricting that corestriction back to K gives the sum of the
two translates and restriction is injective. Theorem 3.4 identifies
beta_B^x x_0 with log chi_{K,r}. Consequently
cor_{K/Q}(y) = -z [K:Q] log chi_{Q,r} != 0.
For every v in V_w away from r, decomposition containment gives
y(Fr_v) = 0, regardless of which auxiliary prime makes the pair active.
Proposition 2.2(iii) permits only finitely many such v; there are also
only finitely many primes of K above r. QED
For l in Lset let M_l be the set of primes of B occurring in no active
pair at l (this includes the primes above l). We bound #M_l depending
only on [K:Q]. The proof places independent classes attached to missed
primes in one character space of an S-unit group.
LEMMA 4.2 (unramified Kummer classes). Let F be a number field, l a
prime, S a finite set of places of F containing all places above l and
all archimedean places. Then Kummer theory gives a natural isomorphism
H^1_cont(G_F, Q_l(1))_{unramified outside S}
= O_{F,S}^x (x)_Z Q_l. (4.1)
It is equivariant for every group of field automorphisms of F
preserving S, with the natural action on cohomology.
Proof. Integral Kummer theory gives H^1_cont(G_F, Z_l(1)) =
lim_n F^x/(F^x)^{l^n}. The limit assertion follows from the finite-level
Kummer isomorphisms; the degree-zero groups are finite and
Mittag-Leffler. Every continuous rational cocycle has bounded image
(G_F compact) and becomes integral after multiplying by one power of l;
the same for coboundaries rationalizes the integral cohomology. At a
prime outside S, inertia acts trivially on Z_l(1) and its Kummer
evaluation is the valuation times the tame l-adic character; these
integral inertia maps are torsion-free. So unramifiedness after
rationalization is exactly vanishing of all integral valuation
coordinates outside S.
For completeness put U_S = O_{F,S}^x and
H_n = { [a] in F^x/(F^x)^{l^n} : ord_v(a) = 0 (mod l^n) for every v not in S }.
With Cl_S(F) the S-ideal class group, there is an exact sequence
0 -> U_S/U_S^{l^n} -> H_n -> Cl_S(F)[l^n] -> 0; (4.2)
the last map sends [a] to the ideal class of div_S(a)/l^n (div_S records
valuations outside S). The transition from level n+1 to n multiplies
these obstruction classes by l; since Cl_S(F) is finite their inverse
limit is zero. So the kernel of all outside-S valuations in the
completed multiplicative group equals lim_n U_S/U_S^{l^n}. (This takes
place in the full completed group; finite valuation support is not
assumed for its elements.) Finally U_S is finitely generated, so
rationalizing its completion gives U_S (x) Q_l. This proves (4.1); all
constructions commute with field automorphisms preserving S. QED
Uchida 1981 (Proposition 1) bounded missed primes by a unit rank. We
refine this by keeping the classes in one character space of a cyclic
extension, whose multiplicity is bounded independently of l even when
the degree grows.
PROPOSITION 4.3 (uniform bound). For every l in Lset,
#{ w in M_l : w not above l } <= 2[K:Q], and #M_l <= 2[K:Q] + 2.
Proof. Fix l in Lset and a finite set J of primes of M_l away from l.
For each w in J choose a_w in B^x and h_w > 0 with (a_w) = w^{h_w}. Its
Kummer cocycles define a class
kappa_w in H^1_cont(P_B, Q_l(1)).
They factor through P_B because Q^s contains the roots of unity and the
required radicals. Valuations at the distinct w show linear
independence. Each is unramified at every finite prime other than w and
the primes above l.
Write rho = chi_l o beta_B and eps = rho chi_{K,l}^{-1}. By Theorem 3.4,
eps takes values in mu_{l-1} inside Q_l^x. Let K'/K be the cyclic
extension fixed by its kernel, Delta = Gal(K'/K). Pullback along the
surjection beta_B is injective on H^1 with coefficients Q_l(rho).
Restriction to G_{K'} stays injective: its kernel is an H^1 of the
finite group Delta, which vanishes in characteristic zero. So we obtain
independent classes
c_w in H^1_cont(G_{K'}, Q_l(1)), w in J.
Under the Delta-action they all lie in the eps^{-1} eigenspace: the
coefficient module is Q_l(1) (x) eps, so invariance becomes the
eigenvalue eps^{-1} after restriction.
Claim each c_w is unramified away from l. If c_w is ramified at a prime
u not above l of K', with v below it in K, the cyclotomic module is
trivial on I_u, so the restricted cocycle is a nonzero continuous map
from I_u to Q_l(1). Choose a pro-l Sylow J_u of I_u on which it is
nonzero and a pro-l Sylow J_v of I_v containing J_u. Then beta_B(J_v) is
nontrivial, so v is active at l; Theorem C sends J_v into inertia at a
corresponding prime t not above l of B. Since t occurs in an active pair
t is not in J, whereas kappa_w is unramified at t. Inertia at t acts
trivially on the coefficient module, so the Kummer cocycle vanishes on
that inertia group, hence on J_u, a contradiction. This proves the
claim.
Let S be the archimedean places of K' plus its primes above l. By Lemma
4.2 the classes c_w give independent elements of the eps^{-1} eigenspace
of O_{K',S}^x (x) Q_l. The equivariant Dirichlet S-unit theorem (Bass
1966) gives the character of the full S-unit module as [Q_l[S]] - [Q_l].
Each Delta-orbit in S contributes at most one copy of the
one-dimensional eps^{-1} to the permutation representation (a stabilizer
H contributes one if eps^{-1} is trivial on H, else zero). There are
#(S/Delta) = r_1(K) + r_2(K) + #{ v of K : v | l } <= 2[K:Q]
orbits; subtracting the trivial representation cannot increase the
multiplicity. So the independent c_w lie in a space of dimension at most
2[K:Q], giving #J <= 2[K:Q]. Since J was arbitrary this bounds all missed
primes away from l; at most two primes of B lie above l, giving the
second bound. QED
The bound depends on K but neither on l nor on the degree of K'/K. We
can then pick one auxiliary prime excluding every possible partner of
several specified target primes, contradicting the bound. The needed
Kummer construction is next.
4.2 Simultaneous separation by power residues.
LEMMA 4.4 (power separation). Let m,d >= 1, let q_1,...,q_n be distinct
rational primes, and for each j let P_j be a finite set of rational
primes not containing q_j. Given any finite set of rational primes to
avoid, there are infinitely many odd primes l = 1 (mod m), outside that
set and all displayed prime sets, such that
p^f mod l is not in < q_j mod l >
(1 <= j <= n, p in P_j, 1 <= f <= d).
The sets P_j may overlap and may contain q_i for i != j.
Proof. Choose distinct auxiliary primes s_j larger than m, d, and every
prime in the lists, and put C = Q(zeta_{m prod_j s_j}). For each j the
classes of the distinct rational primes in {q_j} union P_j are
independent in C^x/(C^x)^{s_j}: at any prime in these lists the
ramification index in C/Q divides [Q(zeta_m):Q] (the other cyclotomic
conductors are prime to that prime), and this index is prime to s_j;
valuations at the separate primes give the independence.
Kummer theory identifies the group over C of
E_j = C( q_j^{1/s_j}, p^{1/s_j} : p in P_j ) with an elementary abelian
s_j-group having one coordinate per listed prime. Choose sigma_j over C
fixing q_j^{1/s_j} and acting nontrivially on each p^{1/s_j}. The E_j/C
are linearly disjoint (degrees are powers of distinct primes), so the
sigma_j combine to sigma in Gal(E/C), E their compositum.
E/Q is Galois, and conjugation in Gal(E/Q) preserves whether each
specified radical coordinate of sigma is trivial: conjugation only
raises its root-of-unity multiplier to a power prime to s_j. Chebotarev
supplies infinitely many rational primes l with Frobenius in the
conjugacy class of sigma, avoiding the prescribed finite set, all listed
primes, and 2. They split completely in C, so l = 1 (mod m) and
l = 1 (mod s_j). At these primes q_j is an s_j-th power mod l and every
p in P_j is not.
The quotient of F_l^x by its s_j-th powers has prime order s_j, so a
nontrivial class stays nontrivial under raising to any 1 <= f <= d < s_j,
while every power of q_j is an s_j-th power. This gives the separation.
Overlaps between lists impose no conflict: their conditions live in
linearly disjoint Kummer extensions with distinct prime exponents. QED
PROPOSITION 4.5 (same characteristic). For all but finitely many primes
w of B above rational primes q split in B, there are a prime v of K
above the same q and an auxiliary prime l in Lset with (v,w) an active
pair for beta_B.
Proof. Suppose there are infinitely many exceptions. Lemma 4.1 lets us
choose w_1,...,w_n, n > 2[K:Q] + 2, among them, above distinct split
rational primes q_1,...,q_n, with every V_{w_j} finite. Let P_j be the
set of rational residue characteristics of primes in V_{w_j}. Failure of
the conclusion means q_j not in P_j.
Apply Lemma 4.4 with m = m_0 and d = [K:Q]. It gives l in Lset, away
from all primes involved, with
Nv mod l not in < q_j mod l > (v in V_{w_j}, 1 <= j <= n), (4.3)
because Nv = p^f with p in P_j and 1 <= f <= [K:Q]. If w_j occurred in
an active pair at this l, its source prime would lie in V_{w_j}; applying
Lemma 1.2 to a lift of source Frobenius and reducing mod l would give
Nv mod l in < Nw_j mod l > = < q_j mod l >,
contrary to (4.3). So all n primes w_j lie in M_l, contradicting
Proposition 4.3. QED
=====================================================================
5. IDENTIFYING THE HOMOMORPHISM, AND THE MAIN THEOREM
=====================================================================
We convert the prime correspondence into equality of group
homomorphisms, using one fixed coefficient prime r while the auxiliary
prime witnessing activity may vary.
Retain beta : T = G_k -> P, the set Lset, and the fields B = Q(i) and K
from Sections 3-4; thus G_K = beta^{-1}(P_B). Theorem 3.4 gives
log chi_r o beta = log chi_{k,r} (r in Lset). (5.1)
Write sigma : T -> P for ordinary restriction.
PROPOSITION 5.1 (identify normalized). There is a in P with
beta = Ad(a) o sigma on T.
Proof. Fix a finite Galois extension L/Q inside Q^s containing B, and
R = Gal(L/Q). Choose a finite Galois F'/Q containing K and L with
beta(G_{F'}) contained in P_L (take the core in Gbig of the appropriate
open subgroup of T). Fix r in Lset.
The maps beta and sigma on G_{F'} give two pullbacks
i_beta, i_sigma : X_r(L) -> X_r(F'). (5.2)
Both are injective: the 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) o sigma and i_a its pullback. Since
L/Q is Galois, sigma_a(G_{F'}) is contained in P_L. i_a depends only on a
mod P_L, because inner automorphisms of P_L act trivially on additive
characters.
Frobenius evaluations. Let q != r range over rational primes split
completely in F'. By Proposition 4.5, 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 v' of F'
above v and compatible prolongations. Then
beta(D_{v'}) is contained in D_u contained in P_L,
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'}, and let a_q in Z-hat
be the residue exponent of beta(h) at u. Equation (5.1) gives
(a_q)_r log_r q = log_r q. The number log_r q is nonzero (q in Z_r^x is
a positive integer > 1, not a root of unity), so (a_q)_r = 1. Every x in
X_r(L) is unramified at u, so
i_beta(x)(Fr_{v'}) = x(Fr_u). (5.3)
This uses only decomposition containment; it does not require the
auxiliary prime to be r.
Under ordinary restriction the same Frobenius maps to Frobenius at
another prime of L above q; some element of R carries this prime to u.
Choose a lift a in P of that element. Then
i_beta(x)(Fr_{v'}) = i_a(x)(Fr_{v'}) for every x in X_r(L).
The choice of a mod P_L may 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) vanishes at infinitely many degree-one primes of F'.
Proposition 2.2 gives
i_beta = i_a. (5.4)
From characters to the finite group. F'/Q is Galois, so T acts on
X_r(F') by conjugation. The two pullbacks intertwine this action with
the actions on X_r(L) induced by beta and sigma_a. If i is their common
injective value (5.4), then for t in T and x in X_r(L),
i(beta(t) . x) = t . i(x) = i(sigma_a(t) . x).
L is totally imaginary (it contains B), so by Proposition 2.3 R acts
faithfully on X_r(L). Hence
beta(t) mod P_L = sigma_a(t) mod P_L (t in T).
One conjugation for all finite quotients. For each such L let
C_L = { a in P : beta(t) = Ad(a)(sigma(t)) (mod P_L) for every t in T }.
C_L is nonempty, closed and clopen (it depends only on a mod P_L). If L
contains L_1,...,L_s then C_L is contained in the intersection of the
C_{L_j}; composita 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
beta = Ad(a) o sigma on T. QED
THEOREM 5.2 (prosolvable rigidity). Let F be a number field inside
Q-bar. Every continuous open homomorphism phi : G_F -> P is conjugate in
P to ordinary restriction.
Proof. As in Lemma 3.1, choose an open normal subgroup U of Gbig
contained in G_F and extend phi|_U to the normalizer of its kernel.
Proposition 5.1 shows that, after one conjugation, phi agrees on U with
ordinary restriction sigma : G_F -> P.
For g in G_F and u in U, normality of U gives
phi(g) sigma(u) phi(g)^{-1} = phi(g u g^{-1})
= sigma(g u g^{-1}) = sigma(g) sigma(u) sigma(g)^{-1}.
Thus sigma(g)^{-1} phi(g) centralizes sigma(U) = phi(U), an open subgroup
of P. Lemma B makes this element trivial, so phi(g) = sigma(g) for every
g in G_F. QED
5.1 The main theorem.
THEOREM 5.3 (main). Let F_1,F_2 be number fields and E_i/F_i be
(possibly infinite) solvably closed Galois extensions. Every continuous
open homomorphism
alpha : Gal(E_1/F_1) -> Gal(E_2/F_2)
is induced by a unique field embedding j : E_2 -> E_1.
(A field embedding j : E_2 -> E_1 induces alpha if g o j = j o alpha(g)
for all g in Gal(E_1/F_1). In particular such j sends F_2 into F_1.)
Proof. Choose copies of the given fields inside Q-bar. By Lemma 1.1 each
E_i contains Q^s. Form the composite
G_{F_1} ->> G_1 --alpha--> G_2 ->> P, (5.5)
the outer arrows being restrictions. The first is surjective. The last
has open image because its composite with the surjection G_{F_2} ->> G_2
is ordinary restriction from an open subgroup of Gbig to P. Continuous
surjective maps of profinite groups are open, so the composite is open.
By Theorem 5.2 the composite is conjugate to ordinary restriction.
Cyclotomic characters are unchanged by conjugation, and all roots of
unity lie in Q^s, so the full cyclotomic characters factor through the
groups in (5.5). Surjectivity of the first arrow therefore yields
chi_{E_2/F_2} o alpha = chi_{E_1/F_1} in Z-hat^x.
Theorem D (Hoshi) gives an equivariant embedding E_2 -> E_1. Its
uniqueness is Uchida 1981, Proposition 2. QED
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END OF PAPER
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