Some applications of Hasse principle for pseudoglobal fields
Some corollaries of the Hasse principle for Brauer group of a pseudoglobal field are obtained. In particular we prove Hasse-Minkowski theorem on quadratic forms over pseudoglobal field and the Hasse principle for quadratic forms of rank 2 or 3 over the field of fractions of an excellent two-dime...
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2004
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| Цитувати: | Some applications of Hasse principle for pseudoglobal fields / V. Andriychuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 1–8. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1564622019-06-19T01:26:29Z Some applications of Hasse principle for pseudoglobal fields Andriychuk, V. Some corollaries of the Hasse principle for Brauer group of a pseudoglobal field are obtained. In particular we prove Hasse-Minkowski theorem on quadratic forms over pseudoglobal field and the Hasse principle for quadratic forms of rank 2 or 3 over the field of fractions of an excellent two-dimensional henselian local domain with pseudofinite residue field. It is considered also the Galois group of maximal p-extensions of a pseudoglobal field. 2004 Article Some applications of Hasse principle for pseudoglobal fields / V. Andriychuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 1–8. — Бібліогр.: 13 назв. — англ. 1726-3255 1991 Mathematics Subject Classification: 11R58; 11EA12, 11R37. http://dspace.nbuv.gov.ua/handle/123456789/156462 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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English |
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Some corollaries of the Hasse principle for Brauer
group of a pseudoglobal field are obtained. In particular we prove
Hasse-Minkowski theorem on quadratic forms over pseudoglobal
field and the Hasse principle for quadratic forms of rank 2 or 3
over the field of fractions of an excellent two-dimensional henselian
local domain with pseudofinite residue field. It is considered also
the Galois group of maximal p-extensions of a pseudoglobal field. |
| format |
Article |
| author |
Andriychuk, V. |
| spellingShingle |
Andriychuk, V. Some applications of Hasse principle for pseudoglobal fields Algebra and Discrete Mathematics |
| author_facet |
Andriychuk, V. |
| author_sort |
Andriychuk, V. |
| title |
Some applications of Hasse principle for pseudoglobal fields |
| title_short |
Some applications of Hasse principle for pseudoglobal fields |
| title_full |
Some applications of Hasse principle for pseudoglobal fields |
| title_fullStr |
Some applications of Hasse principle for pseudoglobal fields |
| title_full_unstemmed |
Some applications of Hasse principle for pseudoglobal fields |
| title_sort |
some applications of hasse principle for pseudoglobal fields |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156462 |
| citation_txt |
Some applications of Hasse principle for pseudoglobal fields / V. Andriychuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 1–8. — Бібліогр.: 13 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT andriychukv someapplicationsofhasseprincipleforpseudoglobalfields |
| first_indexed |
2025-07-14T08:49:58Z |
| last_indexed |
2025-07-14T08:49:58Z |
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1837611613303078912 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2004). pp. 1 – 8
c© Journal “Algebra and Discrete Mathematics”
Some applications of Hasse principle for
pseudoglobal fields
V. Andriychuk
Communicated by M. Ya. Komarnytskyj
Abstract. Some corollaries of the Hasse principle for Brauer
group of a pseudoglobal field are obtained. In particular we prove
Hasse-Minkowski theorem on quadratic forms over pseudoglobal
field and the Hasse principle for quadratic forms of rank 2 or 3
over the field of fractions of an excellent two-dimensional henselian
local domain with pseudofinite residue field. It is considered also
the Galois group of maximal p-extensions of a pseudoglobal field.
Let K be an algebraic function field K in one variable with pseud-
ofinite [1] constant field k. We call such a field pseudoglobal. For pseu-
doglobal fields there is an analogue of global class field theory [2,3], in
particular, for such a field k we have the following exact sequence
0 −→ Br(K) −→
⊕
v∈V K
Br(Kv) −→ Q/Z −→ 0, (1)
where V K is the set of all valuations of K (trivial on the constant field
k), BrK (resp. BrKv) is the Brauer group of K (resp. of the completion
Kv of K at v ∈ V K).
Note that I.Efrat [7] considers a more general situation where K is an
algebraic function field in one variable over a perfect pseudo-algebraically
closed constant field k and proves in that situation the exactness of the
sequence
0 −→ Br(K) −→
⊕
v∈V K
Br(Kh
v ) −→ G∨
k −→ 1, (2)
1991 Mathematics Subject Classification: 11R58; 11EA12, 11R37.
Key words and phrases: algebraic function field, Hasse principle, quadratic
form.
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.2 Some applications of hasse principle...
where G∨
k ' Homcont(Gk, Q/Z), Gk being the absolute Galois group of k,
and Kh
v is a fixed henselization of K at v ∈ V K .
The exact sequence (1) shows, in particular, that for a pseudoglobal
field K the map
Res : Br(K) −→
∏
v∈V K
Br(Kv) (3)
is injective, i.e. the Hasse principle for Brauer group holds over K.
Our first application of the Hasse principle for Brauer group of a
pseudoglobal field will be the analogue of the classical Hasse-Minkowski
theorem which asserts that a quadratic form defined over a global field
K is isotropic if and only if it is isotropic over all the completions of K.
This fact can be quickly proved by using the following proposition.
Proposition 1. Let K be a pseudoglobal field. Then:
(i) An element a ∈ K is a norm from a cyclic extension L/K if and
only if it is a norm everywhere locally.
(ii) Let S be a finite set of valuations of a global field K. Let m be a
positive integer, (p, char(K)) = 1, and a ∈ K∗. If a ∈ K∗m
v for all v /∈ S,
then a ∈ K∗m.
Proof. (i) For a cyclic extension L/K we get from (3) that there is an in-
jective map K∗/NL/KL∗ →
∏
v∈V K K∗
v/NLw/Kv
L∗
w, where for all v ∈ V K
w is a fixed extension of the valuation v to L, and Lw is the corresponding
completion.
(ii) We follow the argument used in [5, pp. 82–83, 275–276]. Let
L/K be an abelian extension, and G = Gal(L/K). First we show that if
Lw = Kv for almost all v ∈ V K then L = K. Suppose that K 6= L. Let
σ be a fixed generator of the absolute Galois group of the pseudofinite
constant field k. Let v ∈ V K , and let k(v) and k(w) be the residue
field of Kv and Lw respectively. Since almost all valuations of K are
unramified in L, we may assume v to be unramified in L. Denote by σw
the restriction of σ[k(v):k] to the field k(w). Then σw is a generator of
the cyclic group Gal(k(w)/k(v)) ' Gal(Lw/Kv) ⊂ G, note that σw does
not depend on the choice of extension w|v: if σ is fixed, then σw ∈ G is
uniquely determined by v, so we denote it by σv.
Let CK (resp. CL) be the idele class group of K (resp. L). By using
the isomorphism CK/NL/KCL ' G (cf. [3]) we see that for any finite set
of valuations S ⊂ V K the group G is generated by the elements σv, v /∈ S.
If there were exist only a finite set of valuations of K which does not split
completely in L, then by adding them to S we would obtain that all σv
are trivial for v /∈ S. This contradicts to the fact that σv, v /∈ S generate
the group G. Thus L = K.
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.V. Andriychuk 3
Let a ∈ K∗m
v for all v /∈ S. As in the classical case (cf. [5], p.82-83) it
is enough to consider the case where m is a power of a prime number and
the m-th roots of unity are in K. In that case the extension L = K( m
√
a)
is a Kummer extension, and we have Lw = Kv for all v /∈ S where w is
an extension of v to L. Then the above argument shows that L = K, i.e.
a ∈ K∗m.
Theorem 2. A nondegenerate quadratic form q over a pseudoglobal field
K, charK 6= 2, is isotropic if and only if it is isotropic over all the com-
pletions Kv of K.
Proof. Assume that the quadratic form q is isotropic over all the com-
pletions Kv of K. We shall argue by induction on n = rankq as in
([9], Appendix 3, and [10]). First, we consider the cases n = 1, 2, 3, 4.
When n = 1, there is nothing to prove. When n = 2, we may suppose
that q = X2 − aY 2, and use Proposition 1 (ii) for m = 2. If n = 3,
after multiplying q by nonzero element from K, we may assume that
q = X2 − aY 2 − bZ2. The latter form represents zero in K if and only if
b is a norm from the field K(
√
a), so for n = 3 Theorem 1 follows from
Proposition 1 (i). Finally, let n = 4. In this case we may suppose that
q = X2 − bY 2 − cZ2 + acT 2. (4)
Form (4) represents 0 if and only if c as an element of K(
√
ab) is a
norm from K(
√
a,
√
b) ([10], 193-194). Thus Theorem 1 is established for
1 ≤ n ≤ 4.
Now let n ≥ 5. Write the form q as follows
q(X1, . . . , Xn) = a1X
2
1 + a2X
2
2 − r(X3, . . . , Xn). (5)
The form r has rank n−2 ≥ 3. Similarly to the classical case of quadratic
forms over global fields, the form r represents 0 for almost all v ∈ V K .
It suffices to show this for quadratic forms of rank 3. Let r = b1Y
2
1 +
b2Y
2
2 + b3Y
2
3 ; let S = {v ∈ V K | ∃i ∈ {1, 2, 3} v(bi) 6= 0}. S is a finite
set, and for all v /∈ S we can reduce r modulo v to obtain a quadratic
form r = b1Y
2
1 + b2Y
2
2 + b3Y
2
3 of rank 3 over a pseudofinite field k which
represents 0 over k (such statement is true over any finite field, thus it is
true over a pseudofinite field k, because the pseudofinite fields are infinite
models of finite fields). Henceforth, for all v /∈ S Hensel’s lemma implies
that the form r represents 0 in Kv for all v /∈ S.
Since the subgroup K∗2
v is open in K∗
v with respect to v-adic topology,
and r represents every element in the coset c ·k∗2 if it represents cv ∈ K∗
v ,
then it follows that r represents the elements in a nonempty open subset
of K∗
v .
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.4 Some applications of hasse principle...
Consider any v ∈ S. Since the form (5) represents 0 in Kv, there
exists cv ∈ K∗
v such that both forms r and a1X
2
1 +a2X
2
2 represent it. So,
there exist x1(v), . . . , xn(v) ∈ K∗
v such that
a1x1(v)2 + a2x2(v)2 = r(x3(v), . . . , xn(v)) = cv.
According to weak approximation theorem, we can find elements
x1, x2 ∈ K∗ which are close enough to x1(v), x2(v) for all v ∈ S, so
that c = a1x
2
1 + a2x
2
2 is close enough to cv to be represented by the form
r.
Thus the form cY 2−r represents 0 in Kv for v ∈ S. Since r represents
0 in Kv for v /∈ S, it represents all elements in Kv for v /∈ S. So, cY 2 − r
represents 0 in Kv for all v ∈ V K . By induction, cY 2 − r represents 0 in
K. It follows that q represents 0 in K.
Recall that two quadratic forms are said to be equivalent if one can
be obtained from the other by an invertible change of variables.
Corollary 3. Two nondegenerate quadratic forms q and q′ over a pseu-
doglobal field K are equivalent over K if and only if q and q′ are equivalent
over all the completions Kv, v ∈ V K .
Proof. Use induction on n = rankq = rankq′ exactly as in the case of
global field (cf. [9], p.150 or [10], p.209).
Corollary 4. Any nondegenerate 5-dimensional quadratic form over a
pseudoglobal field K is isotropic.
Proof. Let q = r(X1, . . . , X4) − aX2
5 . Using the local class field theory
for general local field [13] it is easy to prove that a nondegenerate 4-
dimensional quadratic form over a general local field F (i.e. complete
discrete valued field with quasifinite residue field represents every nonzero
element of F . It follows that a nondegenerate 5-dimensional quadratic
form over a pseudoglobal field K represents 0 over all the completions
Kv, v ∈ V K .
Corollary 5. Let A be a central simple algebra of exponent a power of
2 over a pseudoglobal field K. Then over any finite extension of K the
exponent of A is equal to the index of A.
Proof. This follows from [6, Prop. 7].
Remark 6. Any pseudoglobal field is a C2-field (cf. [4]), and this im-
plies Corollaries 4 and 5. Moreover, the exponent of every central simple
algebra over a pseudoglobal field is equal to its index.
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.V. Andriychuk 5
Let k be a field, and let X be a curve defined over k. The Brauer
group Br(X) of X is the kernel of homomorphism BrK −→ ⊕v∈VK
BrKv,
where K is the function field of X (cf. [11], Appendix A).
Proposition 7. Let K be a pseudoglobal field over constant field k, then
the following equivalent properties hold:
i) the reciprocity law holds for K/k;
ii) for any finite cyclic extension L/K the sequence
Br(L/K) −→ ⊕v∈V K Br(Lw/Kv) → [L : K]−1Z/Z −→ 0
is exact;
iii) for any finite cyclic extension L/K, H1(Gal(L/K), Br(Y )) = 0,
where Br(Y ) is the Brauer group of a smooth projective curve Y with
function field L;
iv) for any finite cyclic extension L/K the map
K∗/NL/KL∗ −→
⊕
v∈V K
K∗
v/NLw/Kv
L∗
w
is injective;
v) H1(G(k), JacC(ks)) = 0, where G(k) is the absolute Galois group
of k, and JacC(ks) is the Jacobian of any complete smooth curve C over
k;
vi) Br(C) = 0 for any complete smooth curve C over k.
Proof. For a pseudoglobal field K/k property i) was proved in [3] as well
as the equivalence of i) and iv), property iv) was also stated in Proposi-
tion 1 (i). The equivalence of i), ii), and iii) was proved in Proposition
А.12 [11, p.167], and the equivalence of iv),v),vi) in Proposition А.13 [11,
p.168].
Condition vi) of Proposition 7 has important applications to the
quadratic forms and to the period-index problem of algebras on curves
over discretely valued fields. Namely, using the results from [12] we have.
Proposition 8. Let C be a curve defined over a general local field K
with pseudofinite residue field k. Let K(C) be its function field, and let
(n, chark) = 1. Let α ∈ Br(K(C)) be an element of order n in the Brauer
group of K(C). Then the index of α divides n2.
Proof. i) By Proposition 7 vi) Br(C) = 0 for any smooth projective curve
C defined over k. Then by [12], Theorem 3.5 the index of α divides n2.
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.6 Some applications of hasse principle...
On the other hand, Theorem 3.1 from [6] on quadratic forms over
fields of fractions of excellent two-dimensional henselian local domains
with either separably closed or finite residue field k holds also in the case
of pseudofinite residue field.
Proposition 9. Let A be an excellent two-dimensional henselian local
domains with pseudofinite residue field k in which 2 is invertible. Let K
be the field of fraction of A, and let q be a quadratic form of rank 2 or 3
over K. Then q is isotropic over K if and only if it is isotropic over all
completions of K with respect to rank 1 discrete valuations.
Proof. The only step in the proof of the corresponding result in [6] (The-
orem 3.1) which uses the specific of the field K is the assertion that if
certain element of exponent 2 in Br(K) is unramified, then it is trivial.
Denoting the unramified Brauer group by Brnr(K) we have the natural
inclusions Brnr(K) ⊂ Br(X) ⊂ Br(K), where X is a regular model of A
with special fiber X0 → Spec(k). By Theorem 1.3 of [6] the restriction
map Br(X) → Br(X0) induces an isomorphism on l-primary subgroups
for any prime l different from p = chark. Further, Proposition 7 vi)
implies that Br(X0) = 0, so Brnr(K) is a p-primary group.
Now let us turn to the Galois group of maximal p-extensions of a
pseudoglobal field. The cohomological approach for describing the Galois
groups for p-extensions of local and global fields was elaborated by Koch
in [8]. It is known that any group can be described in terms of generators
and relations. We recall some definitions from [7] and [8].
Let p be a prime number, G be a pro-p-group, Hn(G, Z/pZ) :=
Hn(G). The number of generators of G is dimZ/pZH1(G). The num-
ber of relations of G is dimZ/pZH2(G). Let GK be the absolute Ga-
lois group of the field K, and GK(p) be its p-component.We denote
Hn(K) = Hn(GK(p), Z/pZ).
In particular, a pro-p-group G is free if and only if H2(G) = 0.
Recall that a field k is called pseudo-algebraically closed (PAC) if
each nonempty variety over k has a k-rational point (pseudofinite field
is a perfect PAC field whose absolute Galois group is isomorphic to Ẑ).
I.Efrat [7] considers the Hasse principle for Brauer group of arbitrary
extension of a perfect PAC field of relative trancendence degree 1 and
proves the following result.
Proposition 10. ([7], Corollary 3.6) Let k be a PAC field and let K
be an extension of k of relative trancendence degree 1. Then the restriction
homomorphism
Res : H2(K) −→
∏
v∈V K
H2(K̂v)
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.V. Andriychuk 7
is injective, where K̂v = K(p) ∩ Kh
v , K(p) is the composite of all finite
Galois extensions of p-power degree, and Kh
v is a henselization of K at v.
As an immediate corollary, we have the following theorem.
Theorem 11. Suppose that K is a pseudoglobal field. Let p be a prime
number. Let G be the Galois group of the maximal p-extension of K, and
for v ∈ V K let Gv be the corresponding decomposition group. Then the
restriction homomorphism defines an injective map
ϕ∗ : H2(G) −→
∑
v∈V K
H2(Gv).
Proof. It suffices to note that by Lemma 3.3 [7] the image of the restric-
tion map H2(G) → ∏
v∈V K H2(Gv) actually lies in
∑
v∈V K H2(Gv).
Corollary 12. Let w be any valuation of pseudoglobal field K, and let
ϕ∗
w : H2(G) →
∑
v 6=w
H2(Gv)
be the map induced by ϕ∗, where the item H2(Gw) is omitted in the direct
sum. Suppose that K contains the p-th roots of 1. Then the map ϕ∗
w is
injective.
Proof. It suffices to note that by the Hasse principle the map
H2(G, K̂∗)p →
∑
v 6=w
H2(Gv, K̂
∗
p)p
remains injective.
Finally, consider the maximal p-extensions of a pseudoglobal field with
given ramification.
Let K be an algebraic function field in one variable over constant
field k, S be any set of valuations of the field K. Let GS be the Galois
group of the maximal p-extension KS of K, unramified outside S. The
field KS is the composite of all finite p-extensions of K with ramification
only in the set S. To the map ϕ∗ from Theorem 11 there corresponds
the map ϕ∗
S : H2(GS) → ∑
v∈S H2(Gv), induced by the morphisms
ϕ∗
v : Gv → G → GS . Denote the kernel of ϕ∗
S by ШS . The group ШS
can be nontrivial, but in the case of a global field it is finite. Moreover,
it is a subgroup of the finite group BS := Char(VS/K∗p), where VS =
{α ∈ K∗ | (α) = a
p, α ∈ kp
v ∀v ∈ S}, and (α) is a principal divisor
corresponding to α.
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.8 Some applications of hasse principle...
It is known [8] that in the case of a global field there is a natural em-
bedding of the group ШS into the group BS . Here two natural questions
arise. Is there such an embedding in the case of a pseudoglobal field? Is
the group ШS finite for a pseudoglobal field?
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Contact information
V. Andriychuk Department of Mathematics and Mechanic,
Lviv Ivan Franko University, Lviv, Ukraine
E-Mail: v_andriychuk@mail.ru,
topos@franko.lviv.ua
Received by the editors: 24.02.2004
and final form in 01.06.2004.
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