Torsion on abelian varieties over large algebraic extensions of finitely generated extensions of Q

45 Pages·2017·0.5 MB·English

12 August 2017
Torsion on Abelian Varieties
over Large Algebraic Extensions
of Finitely Generated Extensions of Q
by
Moshe Jarden, Tel Aviv University, [email protected]
and
Sebastian Petersen, Kassel University, [email protected]
Abstract: Let K be a finitely generated extension of Q and A a non-zero abelian variety
˜ ˜
over K. Let K be the algebraic closure of K and Gal(K) = Gal(K/K) the absolute
˜
Galois group of K equipped with its Haar measure. For each σ ∈ Gal(K) let K(σ) be
˜
the fixed field of σ in K. We prove that for almost all σ ∈ Gal(K) there exist infinitely
˜
many prime numbers l such that A has a non-zero K(σ)-rational point of order l. This
completes the proof of a conjecture of Geyer-Jarden from 1978 in characteristic 0.
MR Classification: 12E30
Introduction
The goal of this work is to complete the proof of an old conjecture of Geyer-Jarden in
characteristic 0. The conjecture deals with a finitely generated field K of Q. We fix an
˜ ˜
algebraicclosureK ofK. Then,theabsolute Galois groupGal(K) = Gal(K/K)ofK
isaprofinitegroup. ItisequippedwithauniqueHaarmeasureµ withµ (Gal(K)) = 1
K K
[FrJ08, p. 378, Sec. 18.5]. For each positive integer e ≥ 1, the group Gal(K)e is equipped
with the product measure, which we also denote by µ . We say that a certain statement
K
holds for almost all σ ∈ Gal(K)e if the set of σ ∈ Gal(K)e for which that statement
holds has µ -measure 1. For each σ = (σ ,...,σ ) ∈ Gal(K)e, we consider the field
K 1 e
˜ ˜
K(σ) = {x ∈ K| σ x = x, i = 1,...,e}.
i
GivenanabelianvarietyAoverK andapositiveintegerm, wedenotethekernelof
(cid:83)∞
the multiplication of A by m with A . For a prime number l, we write A = A .
m l∞ i=1 li
Conjecture A ([GeJ78, p. 260, Conjecture]): Let K be a finitely generated field over
Q, let A be a non-zero Abelian variety over K, and let e be a positive integer. Then,
for almost all σ ∈ Gal(K)e the following holds:
˜
(a) If e = 1, then there exist infinitely many prime numbers l with A (K(σ)) (cid:54)= 0.
l
˜
(b) If e ≥ 2, then there exist only finitely many prime numbers l with A (K(σ)) (cid:54)= 0.
l
˜
(c) If e ≥ 1 and l is a prime number, then A (K(σ)) is finite.
l∞
B. Previous results. Conjecture A along with its analog to positive characteristics
has been proved in [GeJ78, p. 259, Thm. 1.1] when A is an elliptic curve. The analog of
the conjecture is true for an arbitrary abelian variety over a finite field [JaJ84, p. 114,
Prop. 4.2]. Note that the latter paper contains a proof of Part (a) of Conjecture A and
its analog to positive characteristic. Unfortunately, that proof is false as indicated in
[JaJ85].
Part (c) of Conjecture A along with its analog to positive characteristic and Part
(b) of the conjecture appears in [JaJ01, Main Theorem].
The main result of [GeJ05] considers a non-zero abelian variety A over a number
field K and says that there exists a finite Galois extension L of K such that for almost
˜
all σ ∈ Gal(L) there exist infinitely many primes l with A (K(σ)) (cid:54)= 0.
l
1
Finally, DavidZywina[Zyw16]improves[GeJ05]byprovingPart(a)ofConjecture
A for a number field K not only for almost all σ ∈ Gal(L) for some L as [GeJ05] does
but for almost all σ ∈ Gal(K).
We generalize Zywina’s result to an arbitrary finitely generated extension K of Q:
Theorem C: Let A be a non-zero abelian variety over a finitely generated extension
K of Q. Then, for almost all σ ∈ Gal(K) there exist infinitely many prime numbers l
˜
with A (K(σ)) (cid:54)= 0.
l
D. On the proof. Let g = dim(A). For each prime number l let ρ : Gal(K) →
A,l
GL (F ) be the l-ic representation (also called the mod-l representation) of Gal(K)
2g l
induced by the action of Gal(K) on the vector space A over F of dimension 2g.
l l
D1. Serre’s theorem. The proof of [GeJ05] uses the main result of [Ser86]. That result
deals with a number field K. Omong others, it gives a finite Galois extension L of K,
a positive integer n, and for each l a connected reductive subgroup H of GL such
l 2g,F
l
that (H (F ) : ρ (Gal(L))) divides n. In addition, the fields L(A ) with l ranging over
l l A,l l
all prime numbers are linearly disjoint over L. Another important feature of Serre’s
theorem is the existence of a set Λ of prime numbers of positive Dirichlet density, such
that H splits over F for each l ∈ Λ.
l l
D2. Borel-CantelliLemma. ForeachlletS = {σ ∈ Gal(L)| ρ (σ) has eigenvalue 1}.
l A,l
Then, [GeJ05]provestheexistenceofapositiveconstantcandasetΛofpositiveDirich-
let density such that µ (S ) > c for each l ∈ Λ. Thus, (cid:80) µ (S ) = ∞. In addition,
L l l l∈Λ L l
by D1, the sets S with l ranging over Λ are µ -independent. It follows from the Borel-
l L
Cantelli Lemma, that almost all σ ∈ Gal(L) lie in infinitely many S ’s with l ∈ Λ. Thus,
l
˜
for almost all σ ∈ Gal(L) there exist infinitely many l’s such that A (K(σ)) (cid:54)= 0, which
l
is the desired result over L.
D3. Zywina’s combinatorial approach. Zywina makes a more careful use of the Borel-
Cantelli Lemma. In [Zyw16] he chooses a set B of representatives of Gal(K) modulo
Gal(L). For each l and every β ∈ B he considers the set
U = {σ ∈ βGal(L)| ρ (σ) has eigenvalue 1}.
β,l A,l
2
Then, he constructs a positive constant c and a set Λ of prime numbers having positive
β
Dirichlet density such that
c
(1) µ (U ) ≥ for each l ∈ Λ .
K β,l β
l
Again, by the Borel-Cantelli Lemma, this leads to the conclusion that the µ -measure
K
of the set U of all σ ∈ Gal(K) that belong to infinitely many U is 1 . Since
β β,l [L:K]
the U ’s with β ∈ B are disjoint, it follows that for almost all σ ∈ Gal(K) there are
β
˜
infinitely many l’s such that A (K(σ)) (cid:54)= 0.
l
D4. Function fields. Now assume that K is a finitely generated extension of Q of
positive transcendence degree and choose a subfield E of K such that K/E is a regular
extension of transcendence degree 1. We wish to find a place of K/E with residue field
¯ ¯ ¯
K that induces a good reduction of A onto an abelian variety A over K such that
∼ ¯ ¯ ¯
(2) Gal(K(A )/K) = Gal(K(A )/K)
l l
for at least every l in a set of positive Dirichlet density.
D5. Hilbert irreducibility theorem. The first idea that comes into mind is to use
Hilbert Irreducibility Theorem. However, that theorem can take care of only finitely
many prime numbers, so it is of no use for our problem.
D6. Openness theorem. Instead, we choose a smooth curve S over E whose function
ˆ (cid:81)
field is K such that A has a good reduction along S and set K = K(A ), where
l∈L l
L is the set of al prime numbers. Using a combination of results of Anna Cadoret and
Akio Tamagawa that goes under the heading “openness theorem” (Proposition 1.6),
ˆ
we find a closed point s of S with an open decomposition group in Gal(K/K). Let
¯ ˆ (cid:81) ¯
K be the residue field of K at s and K = K (A ), where A is the reduction
s s l∈L s s,l s
of A at s. Then, there exists a finite extension K(cid:48) of K in Kˆ such that the reduction
modulo s induces an isomorphism Gal(Kˆ/K(cid:48)) ∼= Gal(Kˆ /K¯ ). This gives the desired
s s
isomorphism (2) for K(cid:48) rather than for K and for all prime numbers l.
D7. Serre’s theorem over K. Now we use a result of [GaP13] and find a finite Galois
extension L of K that contains K(cid:48) and satisfies the same reduction conditions that K(cid:48)
3
does and in addition the fields L(A ), with l ranging over all prime numbers, are linearly
l
disjoint over L.
Note that K¯ is again finitely generated over Q and the transcendence degree of
s
K¯ over Q is one less than that of K. Starting with Serre’s theorem for number fields
s
mentioned above and using induction on the transcendence degree over Q, we now prove
the theorem of Serre mentioned in D1 over our current field K.
D8. Strongly regular points. Having Serre’s theorem for our function field K at our
disposal, we now follow the proof of [Zyw16] to obtain the estimates (1) for our abelian
variety A/K. The proof contains a careful analysis of regular points of the reductive
groups H mentioned in Serre’s theorem for l ∈ Λ. It uses Zywina’s crucial observation
l
that if T is an F -split maximal torus of H and t ∈ T(F ), then tn ∈ ρ (Gal(L)).
l l l A,l
Moreover, if t is a regular element of H and T is the unique maximal torus of H that
l l
contains t, then the number of points t(cid:48) ∈ T(F ) with (t(cid:48))n = tn is at most nr, where
l
r = rank(H ) = dim(T). Finally, still following [Zyw16], we make use of the Lang-
l
Weil estimates (or rather the more accurate version of these estimates that [Zyw16]
provides) to prove that “most of the points” of ρ (Gal(K)) are regular points of H
A,l l
whose characteristic polynomials have “maximal numbers of roots in F ” (We may refer
l
to these points as “strongly regular”).
D9. Serre’s density theorem. At some point of the proof, [Zyw16] uses the Chebotarev
density theorem for number fields to choose a prime of K whose Artin class is equal
to a previously chosen conjugacy class in Gal(L(A )/K) (where L is the number field
l
mentioned in Serre’s theorem for number field). Instead, we use Serre’s generalization
of the Chebotarev density theorem (Proposition 3.5) to our function field K in order to
find a prime p of K with the same properties as above.
Acknowledgement: Part of this wark was done during research visits of the first
author at the University of Kassel. We thank Wulf-Dieter Geyer and Aharon Razon for
careful reading of this work and the referee for helpful comments to an earlier version
of it.
4
1. Adelic Openness
Let K be a finitely generated transcendental extension of Q and A an abelian variety
over K. We consider K as a function field of one variable over a field E. Using results
of Cadoret and Tamagawa, we prove that there exists a finite extension K(cid:48) of K in
ˆ (cid:81)
K = K(A ), with l ranging over all prime numbers, such that the reduction modulo
l l
“almost every valuation v of K(cid:48) over E” maps the group Gal(K(cid:48)(A )/K(cid:48)), for each l,
l
isomorphically onto the corresponding group with respect to the reduced objects.
To be more specific, let E be a finitely generated extension of Q, S an absolutely
integral smooth curve over E, K = E(S) the function field of S, and A an abelian
variety over K of dimension g > 0 with good reduction along S [Shi98, p. 95, Prop. 25].
Let A(K˜) be the abelian group of all K˜-rational points of A. For each m ∈ N let A
m
˜
be the kernel of multiplication of A by m. By [Mil85, p. 116, Remark 8.4], A (K) is a
m
free Z/mZ-module of rank 2g. Moreover, since A is defined over K, each σ ∈ Gal(K)
˜ ˜
gives rise to an automorphism of A(K) that leaves A (K) invariant.
m
We denote the set of all prime numbers by L. For each l ∈ L let T (A) =
l
lim A (K˜) be the Tate module of A associated with l. Then, A (K˜) ∼= F2g and
←− li l l
T (A) ∼= Z2g, so Aut(A ) ∼= GL (F ) and Aut(T (A)) ∼= GL (Z ). Thus, the action of
l l l 2g l l 2g l
˜
Gal(K) on A(K) mentioned in the preceding paragraph gives rise to homomorphisms
(1) ρ : Gal(K) → GL (F ), ρ : Gal(K) → GL (Z ).
A,l 2g l A,l∞ 2g l
(cid:83)∞
Since Ker(ρ ) = Gal(K(A )) and Ker(ρ ) = Gal(K(A )) = Gal( K(A )),
A,l l A,l∞ l∞ i=1 li
the homomorphism ρ (resp. ρ ) (also called the l-ic and the l-adic representations
A,l A,l∞
of Gal(K)) induces (under an abuse of notation) a homomorphism ρ : Gal(N/K) →
A,l
GL (F ) (resp. ρ : Gal(N/K) → GL (Z )) for each Galois extension N of K that
2g l A,l∞ 2g l
contains K(A ) (resp. K(A )).
l l∞
We denote the set of closed points of S by S . By Hilbert Nullstellensatz,
closed
S is an infinite set.
closed
Since S is a smooth curve, each s ∈ S induces a discrete valuation v of K
closed s
¯ ˜
with residue field K which is a finite extension of E in E [Lan58, p. 151, Thm. 1] and
s
˜ ˜
where E is the algebraic closure of E in K.
5
Let K = K be the maximal Galois extension of K which is unramified along
ur ur,S
˜
S and observe that E ⊆ K , because char(E) = 0. Thus, Gal(K /K) is the ´etale
ur ur
¯
fundamental group of S. Since char(K ) = 0 for each s ∈ S , [SeT68, Thm. 1]
s closed
implies that
(2) K(A ) ⊆ K for each m ∈ N.
m ur
By what we said above, ρ and ρ give rise to homomorphisms
A,l A,l∞
ρ : Gal(K /K) → Aut(A ), ρ : Gal(K /K) → Aut(T (A)).
l ur l l∞ ur l
Writing π : Aut(T (A)) → Aut(A ) for the epimorphism defined by the reduction
l l l
GL (Z ) → GL (F ) modulo l, we have ρ = π ◦ ρ . Further, the products of
2g l 2g l l l l∞
the ρ ’s, the ρ ’s, and the π ’s, with l ranging over L, give rise to homomorphisms that
l l∞ l
fit into the following commutative diagram:
(3) Gal(K /K)
(cid:118)(cid:118)(cid:109)(cid:109)(cid:109)(cid:109)(cid:109)ρ(cid:109)∞(cid:109)(cid:109)(cid:109)(cid:109)(cid:109)(cid:109)(cid:109) ur (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)ρ(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:40)(cid:40)
(cid:81) π (cid:47)(cid:47) (cid:81)
Aut(T (A)) Aut(A )
l∈L l l∈L l
Next we consider a point s ∈ S and choose an extension v of v to K .
closed s,ur s ur
˜ ˜
Since E ⊆ K , the residue field of v is E. For each Galois extension L of K in K
ur s,ur ur
we consider the decomposition group of v | over K,
s,ur L
D = {σ ∈ Gal(L/K)| for all x ∈ L: v (σx) ≥ 0 ⇐⇒ v (x) ≥ 0}.
s,L/K s,ur s,ur
Since v /v is unramified, reduction modulo the prime ideal of the valuation ring of
s,ur s
¯
v gives rise to an isomorphism ϕ : D → Gal(K ) [EnP10, second paragraph
s,ur s s,K /K s
ur
of page 123 and the “first exact sequence” on page 124].
(4) Let ψ : Gal(K¯ ) → D be the inverse of ϕ . For each l ∈ L we con-
s s s,K /K s
ur
¯
sider the homomorphism ρ = ρ ◦ ψ : Gal(K ) → Aut(T (A)). It satisfies
l∞,s l∞ s s l
¯
ρ (Gal(K )) = ρ (D ). We also consider the homomorphisms
l∞,s s l∞ s,K /K
ur
(cid:89) (cid:89)
¯ ¯
ρ = ρ◦ψ : Gal(K ) → Aut(A ) and ρ = ρ ◦ψ : Gal(K ) → Aut(T (A)).
s s s l ∞,s ∞ s s l
l∈L l∈L
6
¯ ¯
They satisfy ρ (Gal(K )) = ρ(D ) and ρ (Gal(K )) = ρ (D ).
s s s,K /K ∞,s s ∞ s,K /K
ur ur
The following result of Anna Cadoret is the main theorem of [Cad15], rewritten
in our notation:
Proposition 1.1: We consider a point s ∈ S . If there exists l ∈ L such that
closed
¯ ¯
the group ρ (Gal(K )) is open in ρ (Gal(K /K)), then ρ (Gal(K )) is open in
l∞,s s l∞ ur ∞,s s
ρ (Gal(K /K)).
∞ ur
Our goal is to prove the assumption of Proposition 1.1, hence to make the con-
sequence of that theorem valid. To this end we combine two theorems of Cadoret and
Akio Tamagawa:
Proposition 1.2 (Cadoret-Tamagawa): Given l ∈ L and d ∈ N, we set
S(d) = {s ∈ S | [K¯ : E] ≤ d} and consider the set
closed s
¯
S = {s ∈ S | ρ (Gal(K )) is not open in ρ (Gal(K /K))}.
l closed l∞,s s l∞ ur
Then, S ∩S(d) is finite.
l
Proof: By [CaT12, Thm. 5.1], ρ is a GSRP-representation. In other words, the
l∞
˜
maximal abelian quotient of each open subgroup of ρ (Gal(K /EK)) is finite. It
l∞ ur
follows from [CaT13, Thm. 1.1] that S ∩S(d) is finite, as claimed.
l
Corollary 1.3: There exists s ∈ S such that the group ρ (Gal(K¯ )) is open
closed ∞,s s
in ρ (Gal(K /K)).
∞ ur
Proof: Since K is the function field of the curve S over E, there exists t ∈ K which is
transcendental over E such that d = [K : E(t)] < ∞. For all but finitely many elements
t¯∈ E, the map t → t¯gives rise to a point s ∈ S such that [K¯ : E] ≤ d. Hence,
closed s
S(d) is infinite.
Nowwechoosel ∈ L. ByProposition1.2andtheprecedingparagraph, S(d)(cid:114)S is
l
¯
infinite. Thus,thereexistss ∈ S suchthatρ (Gal(K ))isopeninρ (Gal(K /K)).
closed l∞,s s l∞ ur
¯
It follows from Proposition 1.1 that ρ (Gal(K )) is open in ρ (Gal(K /K)), as
∞,s s ∞ ur
claimed.
7
Corollary 1.4: There exists s ∈ S such that the group ρ (Gal(K¯ )) is open in
closed s s
ρ(Gal(K /K)).
ur
Proof: Let s be a point in S that satisfies the conclusion of Corollary 1.3. Then,
closed
by (4), the commutative diagram (3) extends to a commutative diagram
¯
(5) Gal(K )
(cid:65)s
(cid:123) (cid:65)
(cid:123) (cid:65)
(cid:123) (cid:65)
(cid:123) (cid:65)
(cid:123)(cid:123)(cid:123)(cid:123) (cid:15)(cid:15) ψs (cid:65)(cid:65)(cid:65)(cid:65)
(cid:123) (cid:65)
(cid:125)(cid:125)(cid:123)(cid:123)(cid:123)(cid:118)(cid:118)(cid:109)(cid:123)(cid:109)(cid:123)(cid:109)(cid:123)ρ(cid:109)(cid:123)∞(cid:123)(cid:109)ρ(cid:123),(cid:109)∞s(cid:123)(cid:109)(cid:123)(cid:109)(cid:109)(cid:109)(cid:109)(cid:109)G(cid:109)al(Kur/K(cid:80)(cid:80))(cid:80)(cid:80)(cid:80)(cid:80)(cid:65)(cid:80)ρρ(cid:65)(cid:80)s(cid:65)(cid:80)(cid:65)(cid:80)(cid:65)(cid:65)(cid:80)(cid:65)(cid:80)(cid:65)(cid:40)(cid:40)(cid:65)(cid:65)(cid:32)(cid:32)
(cid:81) π (cid:47)(cid:47) (cid:81)
Aut(T (A)) Aut(A ) .
l∈L l l∈L l
In particular,
¯ ¯
π(ρ (Gal(K ))) = ρ (Gal(K )) and π(ρ (Gal(K /K))) = ρ(Gal(K /K)).
∞,s s s s ∞ ur ur
¯
By Corollary 1.3, ρ (Gal(K )) is open in ρ (Gal(K /K)). By [FrJ08, p. 5], π is an
∞,s s ∞ ur
¯
open map. Therefore, ρ (Gal(K )) is open in ρ(Gal(K /K)).
s s ur
Setup 1.5: We interpret Corollary 1.4 in terms of Galois groups. To this end we fix
¯
a point s ∈ S such that ρ (Gal(K )) is open in ρ(Gal(K /K)). Since A has
closed s s ur
good reduction at s, its reduction A with respect to v is an abelian variety over
s s
¯
K , in particular, it is non-empty and absolutely integral [Shi98, p. 83, Section 11.1].
s
Moreover, by the last paragraph of [Shi98, p. 70], dim(A ) = dim(A) = g. We write
s
ˆ (cid:81) ˆ (cid:81) ¯ ˆ
K = K(A ) and K = K (A ). By (2), K ⊆ K . Moreover, by [SeT68,
l∈L l s l∈L s s,l ur
p. 495, Lemma 2], for each l ∈ L, reduction modulo s induces an isomorphism A (K˜) →
l
¯
A (K(cid:102)).
s,l s
ˆ ˆ ˆ
We denote the restriction of v to K by vˆ . Then, K is the residue field of K
s,ur s s
with respect to vˆ . Also, D is the image of D under the restriction map
s s,Kˆ/K s,Kur/K
res: Gal(K /K) → Gal(Kˆ/K). We write K(cid:48) for the fixed field of D in Kˆ (and
ur s,Kˆ/K
note that K(cid:48) depends on s). Then, ψ induces a monomorphism ψˆ : Gal(Kˆ /K¯ ) →
s s s s
Gal(Kˆ/K) whose image is Gal(Kˆ/K(cid:48)). Let ϕˆ : Gal(Kˆ/K(cid:48)) → Gal(Kˆ /K¯ ) be the
s s s
ˆ ˜ ¯
inverse of ψ . Again, by [SeT68, p. 495, Lemma 2], the isomorphism A (K) → A (K(cid:102))
s l s,l s
8
induces an isomorphism α : Aut(A ) → Aut(A ) that commutes with the action of
l l s,l
Gal(Kˆ/K(cid:48)). Thus, α ◦ ρ | = ρ ◦ ϕˆ | for each l ∈ L. The
l A,l Gal(Kˆ/K(cid:48)) As,l s Gal(Kˆs/K¯s)
(cid:81) (cid:81)
product of the α ’s gives rise to an isomorphism α: Aut(A ) → Aut(A ).
l l∈L l l∈L s,l
Proposition 1.6: In the notation of Setup 1.5 and in particular with the choice of the
closed point s of S made in the Setup, K(cid:48) is a finite extension of K in Kˆ.
(cid:81)
Proof: Observe that ρ: Gal(K /K) → Aut(A ) naturally decomposes as ρ =
ur l∈L l
ˆ (cid:81) ˆ
ρˆ◦res , where ρˆ: Gal(K/K) → Aut(A ) is defined by the action of Gal(K/K)
K /Kˆ l∈L l
ur
ˆ (cid:81)
on the A ’s. Since K = K(A ), the homomorphism ρˆ is injective.
l l∈L l
Similarly,wewriteρ(cid:48): Gal(Kˆ /K¯ ) → (cid:81) Aut(A )forthecorrespondingmono-
s s s l∈L s,l
¯
morphism associated with K and A . It fits into the following commutative diagram:
s s
ρ
(cid:43)(cid:43)
res (cid:47)(cid:47) ˆ ρˆ (cid:47)(cid:47) (cid:81)
(6) Gal(K /K) Gal(K/K) Aut(A )
(cid:79)(cid:79)ur (cid:79)(cid:79) l∈L l
ψs ψˆs α
(cid:15)(cid:15)
Gal(K¯ ) res (cid:47)(cid:47) Gal(Kˆ /K¯ ) ρ(cid:48)s (cid:47)(cid:47) (cid:81) Aut(A ).
s s s (cid:51)(cid:51) l∈L s,l
ρ
s
Note that in the notation of Corollary 1.4, ρ(cid:48) ◦res = ρ . We use Corollary 1.4 in
s K(cid:102)¯ /Kˆ s
s s
˜ ¯
order to choose s ∈ S(E) such that the group ρ (Gal(K )) is open in ρ(Gal(K /K)).
s s ur
Since both restrictions maps in (6) are surjective, α−1(ρ(cid:48)(Gal(Kˆ /K¯ ))) is open in
s s s
ρˆ(Gal(Kˆ/K)). Since ψˆ is injective, since α is bijective, and since both ρ(cid:48) and ρˆ are
s s
ˆ ˆ ¯ ˆ
injective, the group D = ψ (Gal(K /K )) is open in Gal(K/K). It follows that
s,Kˆ/K s s s
K(cid:48), which is the fixed field of D in Kˆ, is a finite extension of K in Kˆ, as claimed.
s,Kˆ/K
9