Witt equivalence of function fields of conics
Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely func...
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irk-123456789-1885532023-03-06T01:27:07Z Witt equivalence of function fields of conics Gladki, P. Marshall, M. Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works [5] and [6]. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields. 2020 Article Witt equivalence of function fields of conics / P. Gladki, M. Marshall // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 63–78. — Бібліогр.: 20 назв. — англ. 1726-3255 DOI:10.12958/adm1271 2000 MSC: Primary 11E81, 12J20; Secondary 11E04, 11E12 http://dspace.nbuv.gov.ua/handle/123456789/188553 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works [5] and [6]. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields. |
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Gladki, P. Marshall, M. |
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Gladki, P. Marshall, M. Witt equivalence of function fields of conics Algebra and Discrete Mathematics |
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Gladki, P. Marshall, M. |
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Gladki, P. |
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Witt equivalence of function fields of conics |
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Witt equivalence of function fields of conics |
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Witt equivalence of function fields of conics |
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Witt equivalence of function fields of conics |
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Witt equivalence of function fields of conics |
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witt equivalence of function fields of conics |
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Інститут прикладної математики і механіки НАН України |
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Witt equivalence of function fields of conics / P. Gladki, M. Marshall // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 63–78. — Бібліогр.: 20 назв. — англ. |
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Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT gladkip wittequivalenceoffunctionfieldsofconics AT marshallm wittequivalenceoffunctionfieldsofconics |
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2025-07-16T10:39:31Z |
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2025-07-16T10:39:31Z |
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1837799699905511424 |
| fulltext |
“adm-n3” — 2021/1/4 — 11:48 — page 63 — #69
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 30 (2020). Number 1, pp. 63–78
DOI:10.12958/adm1271
Witt equivalence of function fields of conics
P. Gładki and M. Marshall
Communicated by A. P. Petravchuk
Abstract. Two fields are Witt equivalent if, roughly speak-
ing, they have the same quadratic form theory. Formally, that is to
say that their Witt rings of symmetric bilinear forms are isomorphic.
This equivalence is well understood only in a few rather specific
classes of fields. Two such classes, namely function fields over global
fields and function fields of curves over local fields, were investigated
by the authors in their earlier works [5] and [6]. In the present work,
which can be viewed as a sequel to the earlier papers, we discuss
the previously obtained results in the specific case of function fields
of conic sections, and apply them to provide a few theorems of a
somewhat quantitive flavour shedding some light on the question of
numbers of Witt non-equivalent classes of such fields.
1. Introduction
One of the classical problems in bilinear algebra is to classify fields with
respect to Witt equivalence, that is equivalence defined by isomorphism
of their Witt rings of symmetric bilinear forms, which also includes fields
of characteristic two. This problem is, in fact, manageable only when
restricted to some specific classes of fields, which include trivial examples
of quadratically closed fields, real closed fields, and finite fields, the case
Murray Marshall passed away in May 2015. Our community lost a brilliant mathe-
matician and a wonderful man. We sorely miss him.
2000 MSC: Primary 11E81, 12J20; Secondary 11E04, 11E12.
Key words and phrases: symmetric bilinear forms, quadratic forms, Witt equiv-
alence of fields, function fields, conic sections, valuations, Abhyankar valuations.
https://doi.org/10.12958/adm1271
“adm-n3” — 2021/1/4 — 11:48 — page 64 — #70
64 Witt equivalence of conics
of local fields ([13]), global fields ([16], [17], [18]), function fields in one
variable over algebraically closed fields of characteristic 6= 2, and function
fields in one variable over real closed fields ([7], [12]).
The authors of the present paper attempted to add two more classes
of fields to this list, and investigated function fields over global fields [5]
and function fields of curves over local fields [6], and managed to show
that Witt equivalence of two function fields over global fields induces in a
canonical way a bijection v ↔ w between Abhyankar valuations v of K
having residue field not finite of characteristic 2 and Abhyankar valuations
w of L having residue field not finite of characteristic 2 ([5, Theorem 7.5]).
Subsequently, a variant of this theorem has been also established in the
local case ([6, Theorem 3.5]). Numerous corollaries providing some insight
into the question of how Witt equivalence in these cases is behaved have
been also drawn.
In the present paper we apply these results to take a closer look at
the question of the number of Witt non-equivalent classes of function
fields of conics, and provide some enumerative results in the case of conics
defined over certain number fields. Some of them generalize in a certain
way to the case of conics defined over arbitrary local fields. The main
new results of the paper are found in Section 4. Throughout the entire
exposition the authors use the language of hyperfields, which seem to
provide a natural and convenient language to study Witt equivalence. We
recall basic terminology and establish fundamental connections between
hyperfields, valuations and quadratic forms in Section 2. All of this is a
summary of Section 2 in [6], which, in turn, is a summary of Sections 2–6
in [5], and the reader more interested in all the technicalities is kindly
referred to consult author’s first paper [5] in the sequel. In Section 3 the
authors prove a few additional facts on function fields of conics, and cite
some old propositions that go back to Ernst Witt. The authors would
like to believe that their results can be thought of as extensions of these
beautiful, classical theorems by old masters.
2. Hyperfields, valuations and Witt equivalence
Hyperfields seem to provide a convenient and very natural way to
describe Witt equivalence. In what follows we shall review the basic
concepts and definitions used later in the paper. By a hyperfield we shall
understand a system (H,+, ·,−, 0, 1), where H is a set, + is a multivalued
binary operation on H , i.e., a function from H×H to the set of all subsets
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P. Gładki, M. Marshall 65
of H , · is a binary operation on H , − : H → H is a function, and 0, 1 are
elements of H such that
(I) (H,+,−, 0) is a canonical hypergroup, i.e. for all a, b, c ∈ H,
(1) c ∈ a+ b ⇒ a ∈ c+ (−b),
(2) a ∈ b+ 0 iff a = b,
(3) (a+ b) + c = a+ (b+ c),
(4) a+ b = b+ a;
(II) (H, ·, 1) is a commutative monoid, i.e. for all a, b, c ∈ H,
(1) (ab)c = a(bc),
(2) ab = ba,
(3) a1 = a;
(III) a0 = 0 for all a ∈ H;
(IV) a(b+ c) ⊆ ab+ ac;
(V) 1 6= 0 and every non-zero element has a multiplicative inverse.
Hyperfields form a category with morphisms from H1 to H2, where
H1, H2 are hyperfields, defined to be functions α : H1 → H2 which satisfy
α(a + b) ⊆ α(a) + α(b), α(ab) = α(a)α(b), α(−a) = −α(a), α(0) = 0,
α(1) = 1. For a subgroup T of H∗ (throughout the paper, for a set A, we
shall always denote by A∗ the set A\{0}), where H is a hyperfield, denote
by H/mT the set of equivalence classes with respect to the equivalence
relation ∼ on H defined by
a ∼ b if and only if as = bt for some s, t ∈ T.
The operations on H/mT are the obvious ones induced by the correspond-
ing operations on H . Denote by a the equivalence class of a. Multiplication
is defined in a natural way, and addition is set as follows:
a ∈ b+ c if and only if as ∈ bt+ cu for some s, t, u ∈ T.
(H/mT,+, ·,−, 0, 1) is then a hyperfield that we shall refer to as quotient
hyperfield. For a hyperfield H = (H,+, ·,−, 0, 1) the prime addition on H
is defined by
a+′ b =
a+ b, if one of a, b is zero ,
a+ b ∪ {a, b}, if a 6= 0, b 6= 0, b 6= −a,
H, if a 6= 0, b 6= 0, b = −a.
For any hyperfield H := (H,+, ·,−, 0, 1), H ′ := (H,+′, ·,−, 0, 1) is also a
hyperfield [5, Proposition 2.1]. We shall call H ′ the prime of the hyperfield
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66 Witt equivalence of conics
H . The motivation for this definition comes from the following discussion:
let K be a field and define the quadratic hyperfield of K, denoted Q(K),
to be the prime of the hyperfield K/mK∗2. Now let W (K) be the Witt
ring of non-degenerate symmetric bilinear forms over K; see [13], [14]
or [20] for the definition in case char(K) 6= 2 and [9], [11] or [15] for the
definition in the general case. Recall that a (non-degenerate diagonal)
binary form over K is an ordered pair 〈a, b〉, a, b ∈ K∗/K∗2, and its
value set , denoted by DK〈a, b〉, is the set of non-zero elements of a+ b.
Now, a hyperfield isomorphism α between two quadratic hyperfields Q(K)
and Q(L), where K,L are fields, can be viewed as a group isomorphism
α : K∗/K∗2 → L∗/L∗2 such that α(−1) = −1 and
α(DK〈a, b〉) = DL〈α(a), α(b)〉 for all a, b ∈ K∗/K∗2.
Combining the results in [2], [8], and [14] one gets that two fields K
and L are Witt equivalent, denoted K ∼ L, iff Q(K) and Q(L) are
isomorphic as hyperfields. Moreover, a morphism ι : H1 → H2 between
two hyperfields H1 and H2 induces a morphism ι : H1/m∆ → H2 where
∆ := {x ∈ H∗
1 : ι(x) = 1}. The morphism ι is said to be a quotient
morphism if ι is an isomorphism, or, equivalently, if ι is surjective, and
ι(c) ∈ ι(a) + ι(b) if and only if cs ∈ at+ bu for some s, t, u ∈ ∆.
A morphism ι : H1 → H2 is said to be a group extension if ι is injective,
every x ∈ H∗
2\ι(H∗
1 ) is rigid , meaning 1 + x ⊆ {1, x}, and
ι(1 + y) = 1 + ι(y), for all y ∈ H1, y 6= −1.
For a field K we adopt the standard notation from valuation theory:
if v is a valuation on K, Γv denotes the value group, Av the valuation
ring, Mv the maximal ideal, Uv the unit group, Kv the residue field, and
π = πv : Av → Kv the canonical homomorphism, i.e., π(a) = a + Mv.
v is discrete rank one if Γv = Z. Denote T = (1 + Mv)K
∗2. Consider
the canonical group isomorphism α : UvK
∗2/(1 + Mv)K
∗2 → K∗
v/K
∗2
v
induced by
x ∈ Uv 7→ π(x) ∈ K∗
v .
and define ι : Q(Kv) → K/mT by ι(0) = 0 and ι(a) = α−1(a) for
a ∈ K∗
v/K
∗2
v . If v is non-trivial, then the map Q(K) → K/mT defined
by x 7→ xT is a quotient morphism and ι is a group extension. The
cokernel of the group embedding α−1 : K∗
v/K
∗2
v → K∗/T is equal to
K∗/UvK
∗2 ∼= Γv/2Γv. We reflect this by calling K/mT a group extension
“adm-n3” — 2021/1/4 — 11:48 — page 67 — #73
P. Gładki, M. Marshall 67
of Q(Kv) by the group Γv/2Γv. If v is non-trivial and char(Kv) 6= 2, then
K/mT can be naturally identified with Q(K̃v), where K̃v denotes the
henselization of (K, v). If v, v′ are valuations on K with v � v′, i.e. such
that v′ is a coarsening of v, meaning Av ⊆ Av′ , then Mv′ ⊆ Mv, and,
consequently, (1+Mv′)K
∗2 ⊆ (1+Mv)K
∗2. If we denote by v the valuation
on Kv′ induced by v, that is
v(πv′(a)) = v(a), for a ∈ Uv′ ,
then the valuations v and v have the same residue field. If v and v′ are
non-trivial and v′ is a proper coarsening of v, meaning Av ( Av′ , then
K/m(1+Mv)K
∗2 is a group extension of the hyperfield Kv′/m(1+Mv)K
∗2
v′
and the following diagram is commutative:
Q(K) // K/m(1 +Mv′)K
∗2 // K/m(1 +Mv)K
∗2
Q(Kv′)
OO
// Kv′/m(1 +Mv)K
∗2
v′
OO
Q(Kv)
OO
For a subgroup T of K∗ we say that x ∈ K∗ is T -rigid if T +Tx ⊆ T ∪Tx,
and denoting by
B(T ) := {x ∈ K∗ : either x or − x is not T -rigid}
we will refer to the elements of B(T ) as to the T -basic elements. If x ∈ K∗
is T -rigid and y = tx, for some t ∈ T , then y is T -rigid, so that B(T ) is a
union of cosets of T . If ±T = B(T ), and either −1 ∈ T or T is additively
closed, we shall say that the subgroup T is exceptional. If H ⊆ K∗ is a
subgroup containing B(T ), then there exists a subgroup Ĥ of K∗ such that
H ⊆ Ĥ and (Ĥ : H) 6 2, and a valuation v of K such that 1+Mv ⊆ T and
Uv ⊆ Ĥ . Moreover, Ĥ can be taken to be simply H , unless T is exceptional
[1, Theorem 2.16]. B(K∗2) is a subgroup of K∗, and in the case when
T = (1 +Mv)K
∗2, for some non-trivial valuation v of K, B(T ) ⊆ UvK
∗2
and
B(T ) = {x ∈ K∗ : x = ι(y) for some y ∈ B(K∗2
v )},
where ι : Q(Kv) →֒ K/mT is the morphism described above. B(T ) is
a group and the group isomorphism ι : K∗
v/K
∗2
v → UvK
∗2/T induces a
“adm-n3” — 2021/1/4 — 11:48 — page 68 — #74
68 Witt equivalence of conics
group isomorphism B(K∗2
v )/K∗2
v → B(T )/T . T is exceptional if and only
if K∗2
v is exceptional. The main result that explains how valuations in
Witt equivalent fields are matched is the following one:
Theorem 2.1 ([5, Theorem 5.3]). Suppose K, L are fields, α : Q(K) →
Q(L) is a hyperfield isomorphism and v is a valuation on K such that Γv
is finitely generated as an abelian group. Suppose either (i) the basic part
of (1 +Mv)K
∗2 is UvK
∗2 and (1 +Mv)K
∗2 is unexceptional, or (ii) the
basic part of (1 +Mv)K
∗2 is (1 +Mv)K
∗2 and (1 +Mv)K
∗2 has index 2
in UvK
∗2. Then there exists a valuation w on L such that the image of
(1 +Mv)K
∗2/K∗2 under α is (1 +Mw)L
∗2/L∗2 and (L∗ : UwL
∗2) > (K∗ :
UvK
∗2). If (i) holds, then the image of UvK
∗2/K∗2 under α is UwL
∗2/L∗2.
If K, L are fields, v and w are non-trivial, and α : Q(K) → Q(L)
is a hyperfield isomorphism such that the image of (1 + Mv)K
∗2/K∗2
under α is (1 +Mw)L
∗2/L∗2, then α induces a hyperfield isomorphism
K/m(1 +Mv)K
∗2 → L/m(1 +Mw)L
∗2 such that the diagram
Q(K) //
��
Q(L)
��
K/m(1 +Mv)K
∗2 // L/m(1 +Mw)L
∗2
(2.1)
commutes. If, in addition, the image of UvK
∗2/K∗2 under α is UwL
∗2/L∗2,
then α induces a hyperfield isomorphism Q(Kv) → Q(Lw) and a group
isomorphism Γv/2Γv → Γw/2Γw such that the diagrams
K/m(1 +Mv)K
∗2 // L/m(1 +Mw)L
∗2
Q(Kv)
OO
// Q(Lw)
OO
(2.2)
and
Q(K)∗ //
��
Q(L)∗
��
Γv/2Γv
// Γw/2Γw
(2.3)
both commute.
Recall that the nominal transcendence degree of K is defined to be
ntd(K) :=
{
trdeg(K : Q) if char(K) = 0
trdeg(K : Fp)− 1 if char(K) = p 6= 0
.
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P. Gładki, M. Marshall 69
For an abelian group Γ, its rational rank of Γ, denoted rkQ(Γ), is defined
to be the dimension of the Q-vector space Γ⊗Z Q. If K is a function field
over k and v is a valuation on K, the Abhyankar inequality asserts that
trdeg(K : k) > rkQ(Γv/Γv|k) + trdeg(Kv : kv|k),
where v|k denotes the restriction of v to k. The valuation v is Abhyankar
(relative to k) if the actual equality holds in the Abhyankar inequality,
that is
trdeg(K : k) = rkQ(Γv/Γv|k) + trdeg(Kv : kv|k).
In this case it is known that Γv/Γv|k is finitely generated and Kv is a
function field over kv|k.
3. Function fields of conics
Let k be a field of characteristic 6= 2.
Proposition 3.1. Let a, b ∈ k∗ and assume that ax2 + by2 = 1 has no
k−rational points. Then k[x,y]
(ax2+by2−1)
is a principal ideal domain.
Proof. This is well known, see [4].
Set ka,b := qf k[x,y]
(ax2+by2−1)
, the quotient field of k[x,y]
(ax2+by2−1)
. We assume
always that a, b ∈ k∗.
Proposition 3.2. The field of constants of ka,b over k is equal to k.
Proof. Clearly ka,b = k(x)(
√
1−ax2
b
). Suppose f = f0+f1
√
1−ax2
b
, f0, f1 ∈
k(x), is algebraic over k. Then f = f0− f1
√
1−ax2
b
is also algebraic over k.
Consequently, f0 = (f + f)/2 and f2
0 − f2
1 (
1−ax2
b
) = ff are algebraic over
k. It follows that f2
1 (
1−ax2
b
) is algebraic over k, i.e., f1 = 0, and f0 ∈ k.
For a, b ∈ k∗, (a,b
k
) denotes the quaternion algebra over k, i.e., the
4-dimension central simple algebra over k generated by i, j subject to
i2 = a, j2 = b, ji = −ij. We identify quaternion algebras over k which
are isomorphic as k-algebras, equivalently, are equal as elements of the
Brauer group of k.
Proposition 3.3. The following are equivalent:
(1) (a,b
k
) = 1 (i.e., (a,b
k
) splits over k).
(2) 〈1,−a〉 ⊗ 〈1,−b〉 ∼ 0 over k.
“adm-n3” — 2021/1/4 — 11:48 — page 70 — #76
70 Witt equivalence of conics
(3) 1 ∈ Dk〈a, b〉.
(4) The conic ax2 + by2 = 1 has a rational point.
(5) ka,b is purely transcendental over k.
Proof. The equivalence of (1), (2), (3) and (4) is well-known from quadratic
form theory. If (p, q) is a rational point of ax2 + by2 = 1 then ka,b = k(z)
where z := y−q
x−p
. Conversely, if ka,b = k(z) then, choosing f(z), g(z), h(z) ∈
k[z] so that x = f(z)
h(z) , y = g(z)
h(z) , and choosing r ∈ k so that h(r) 6= 0,
one sees that (f(r)
h(r) ,
g(r)
h(r)) is a rational point of ax2 + by2 = 1. Note: This
argument fails if |k| < ∞, but the conclusion continues to hold even in
this case, since |k| < ∞ ⇒ the quadratic form 〈a, b〉 is k-universal.
From the definition of ka,b it is clear that 1 ∈ Dka,b〈a, b〉, so (a,b
k
)
splits over ka,b. Of course, 1 also splits over ka,b (since it splits over k).
Conversely one has the following:
Proposition 3.4 (E. Witt). The only quaternion algebras defined over
k which split over ka,b are (a,b
k
) and 1.
Proof. See [19, Satz, Page 465] or [10, Lemma 4.4].
We write K ∼=k L to indicate that the field extensions K,L of k are
k-isomorphic.
Proposition 3.5 (E. Witt). The following are equivalent:
(1) (a,b
k
) = ( c,d
k
).
(2) ka,b ∼=k kc,d.
Proof. Then implication (1) ⇒ (2) is [19, Satz, page 464]. The implication
(2) ⇒ (1) is immediate from the Proposition 3.4.
We will need to know which orderings of k extend to ka,b.
Lemma 3.6. An ordering < on k extends to an ordering on ka,b iff at
least one of a, b is positive at <.
Proof. One way is clear. If < extends to ka,b then, in ka,b, ax
2+by2 = 1 > 0
so at least one of a, b must be positive. Conversely, suppose at least one if
a, b is positive. Fix a real closure R of (k,<). Clearly ∃ x, y ∈ R satisfying
ax2 + by2 = 1. Then (a,b
k
) splits over k(x, y) and hence also over R, so
ka,b →֒ Ra,b ≡ R(t). Any one of the infinitely many orderings of R(t)
extends the ordering <.
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P. Gładki, M. Marshall 71
4. Application to function fields of conics
Let K be a function field in one variable (i.e. a function field K over k
satisfying trdeg(K : k) = 1) over a global field k (we do not assume that
k is the field of constants of K over k, i.e. the algebraic closure of k in K).
We consider the set νK consisting of all non-trivial Abhyankar valuations
of K over k such that Kv is not finite of characteristic 2. Thus
νK = νK,0 ∪ νK,1 ∪ νK,2 ∪ νK,3 ∪ νK,4 (disjoint union)
where νK,0 is the set of valuations v of K such that Γv = Z and Kv is a
number field, νK,1 is the set of valuations v on K such that Γv = Z and
Kv is a global field of characteristic p 6= 0, 2, νK,2 is the set of valuations
v on K such that Γv = Z and Kv is a global field of characteristic 2, νK,3
is the set of valuations v on K such that Γv = Z× Z, Kv is a finite field
of characteristic 6= 2 and −1 /∈ K∗2
v and νK,4 is the set of valuations v
on K such that Γv = Z× Z, Kv is a finite field of characteristic 6= 2 and
−1 ∈ K∗2
v .
Of course, some of the sets νK,i may be empty. Specifically, if char(K) /∈
{0, 2} then νK,i = ∅ for i ∈ {0, 2} and if char(K) = 2 then νK,i = ∅ for
i ∈ {0, 1, 3, 4}.
Corollary 4.1 ([5, Corollary 8.1]). Suppose K, L are function fields in
one variable over global fields which are Witt equivalent via a hyperfield
isomorphism α : Q(K) → Q(L). Then for each i ∈ {0, 1, 2, 3, 4} there is a
uniquely defined bijection between νK,i and νL,i such that, if v ↔ w under
this bijection, then α maps (1 +Mv)K
∗2/K∗2 onto (1 +Mw)L
∗2/L∗2 and
UvK
∗2/K∗2 onto UwL
∗2/L∗2 for i ∈ {0, 1, 2, 3} and such that α maps
(1 +Mv)K
∗2/K∗2 onto (1 +Mw)L
∗2/L∗2 for i = 4.
Corollary 4.2 ([5, Corollary 8.2]). Let K ∼ L be function fields in one
variable over global fields k, ℓ respectively, with fields of constants k and
ℓ respectively. Then k ∼ ℓ except possibly in the case where k, ℓ are both
number fields. In the latter case assume there exists v ∈ νK,0 with Kv = k
and w ∈ νL,0 with Lw = ℓ. Then k ∼ ℓ.
Remark 4.3. The last assertion of Corollary 4.2 applies, in particular,
in the case where K = k(x), L = ℓ(x), where k, ℓ are number fields.
Let k be a number field. Every ordering of k is archimedean, so that
it corresponds to a real embedding k →֒ R. Denote by r1 the number of
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72 Witt equivalence of conics
real embeddings of k and by r2 the number of conjugate pairs of complex
embeddings of k. Then [k : Q] = r1 + 2r2. Furthermore, let
Vk := {r ∈ k∗ : (r) = a
2 for some fractional ideal a of k}.
Vk is a subgroup of k∗ and k∗2 ⊆ Vk. We will need the following result,
which is a version of [5, Theorem 8.6]:
Theorem 4.4. Suppose K and L are function fields of genus zero curves
over number fields with fields of constants k and ℓ respectively, and
α : Q(K) → Q(L) is a hyperfield isomorphism. Then
(1) r ∈ k∗/k∗2 iff α(r) ∈ ℓ∗/ℓ∗2.
(2) α induces a bijection between orderings P of k which extend to K
and orderings Q of ℓ which extend to L via P ↔ Q iff α maps P ∗/k∗2 to
Q∗/ℓ∗2.
(3) α maps Vk/k
∗2 to Vℓ/ℓ
∗2.
(4) [k : Q] = [ℓ : Q].
(5) K is purely transcendental over k iff L is purely transcendental
over ℓ. In this case, the map r 7→ α(r) defines a hyperfield isomorphism
between Q(k) and Q(ℓ), and the 2-ranks of the ideal class groups of k and
ℓ are equal.
First part the proof follows the same line of reasoning as the proof
of [5, Theorem 8.6], but we shall provide it here for the sake of the
completeness of the exposition. We will need two lemmas:
Lemma 4.5 ([5, Lemma 8.4]). The 2-rank of the group Vk/k
∗2 is r1+r2+t
where t is the 2-rank of the ideal class group of k.
Lemma 4.6 (see also [5, Lemma 8.5]). Suppose v is a discrete rank 1
valuation on k, and a, b ∈ k∗. There exists an Abhyankar extension of v
to ka,b such that v(ka,b
∗) = v(k∗).
Proof. Denote by k̂ = k̂v the completion of k at v. Suppose first that
(a,b
k
) splits over k̂. Then ka,b →֒ k̂a,b = k̂(x). Extend the valuation v to
k̂(x) by defining v(
∑n
i=0 aix
i) = min{v(ai) : 0 6 i 6 n} and v(f(x)
g(x) ) =
v(f(x))− v(g(x)).
Suppose now that (a,b
k
) does not split over k̂. Choose c ∈ k̂∗\k̂∗2 so
that k̂(
√
c) is the unique unramified quadratic extension of k̂. The group
D
k̂
〈1,−c〉 has index 2 in k̂∗, so ∃ d ∈ k̂∗ such that ( c,d
k̂
) 6= 1. Since there is
exactly one non-split quaternion algebra over k̂, we see that (a,b
k̂
) = ( c,d
k̂
).
Thus (a,b
k
) splits over k̂(
√
c) and we can proceed as before, but with k̂
replaced by k̂(
√
c).
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P. Gładki, M. Marshall 73
We now proceed to the proof of Theorem 4.4.
Proof. By our hypothesis, K = ka,b for some a, b ∈ k∗ and L = ℓc,d for
some c, d ∈ ℓ∗. If r ∈ k∗/k∗2, then r ∈ UvK
∗2/K∗2 for all v ∈ νK,0. This
implies α(r) ∈ UwL
∗2/L∗2 for all w ∈ νL,0. Since ℓ[x,y]
(cx2+dy2−1)
is a principal
ideal domain with unit group ℓ∗ this implies, in turn, that α(r) ∈ ℓ∗/ℓ∗2.
Thus α induces a group isomorphism between k∗/k∗2 and ℓ∗/ℓ∗2. This
proves (1). Suppose P is an ordering of k which extends to an ordering P1
of K. Since α is a hyperfield isomorphism, there exists a unique ordering
Q1 of L such that α(P ∗
1 /K
∗2) = Q∗
1/L
∗2. Denote by Q the restriction of
Q1 to ℓ. Clearly α(P ∗/k∗2) = Q∗/ℓ∗2. This proves (2). Lemma 4.6 implies
that
Vk/k
∗2 = {r ∈ k∗/k∗2 : r ∈ UvK
∗2/K∗2 ∀v ∈ νK,1 ∪ νK,2},
so (3) is clear. Observe that if v ↔ w, v ∈ νK,0, w ∈ νL,0, the diagram
(∗) K∗
v/K
∗2
v
// L∗
w/L
∗2
w
k∗/k∗2
OO
// ℓ∗/ℓ∗2
OO
is commutative. The vertical arrows are the maps induced by the field
embeddings k →֒ Kv, ℓ →֒ Lw. Since the top arrow in (∗) defines a
hyperfield isomorphism between Q(Kv) and Q(Lw) we know that [Kv :
Q] = [Lw : Q]. Choose v, w with [Kv : Q] = [Lw : Q] minimal. If (a,b
k
)
splits over k then Kv = k and [Kv : Q] = [k : Q]. If (a,b
k
) is non-split
over k then (a,b
k
) splits over a quadratic extension, so [Kv : k] = 2,
i.e., [Kv : Q] = 2[k : Q]. Thus we see that either [k : Q] = [ℓ : Q],
2[k : Q] = [ℓ : Q], or [k : Q] = 2[ℓ : Q]. Suppose 2[k : Q] = [ℓ : Q],
i.e., [Kv : Q] = 2, and Lw = ℓ. Then the left vertical arrow in (∗) has a
non-trivial kernel, but the right vertical arrow in (∗) is an isomorphism, a
contradiction. Thus 2[k : Q] = [ℓ : Q] is impossible. A similar argument
shows that [k : Q] = 2[ℓ : Q] is impossible. This proves (4). If (a,b
k
) is non-
split and ( c,d
ℓ
) is split, then [Kv : k] = 2 and Lw = ℓ so 2[k : Q] = [ℓ : Q],
contradicting what was proved in (4). This proves the first statement of (5).
If (a,b
k
) and ( c,d
ℓ
) are both split then k = Kv, ℓ = Lw. In this case it follows
from the commutativity of (∗) and what was already proved that the map
r 7→ α(r) defines a hyperfield isomorphism between Q(k) and Q(ℓ). Since
it is well-known that r1 and r2 are invariant under Witt equivalence, the
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74 Witt equivalence of conics
last assertion of (5) is immediate now, from (3) and Lemma 4.5. This
completes the proof.
Question 1: Is it true that every hyperfield isomorphism α : Q(ka,b) →
Q(ℓc,d) induces a hyperfield isomorphism between Q(k) and Q(ℓ)? I.e., is
the hypothesis that (a,b
k
) and ( c,d
ℓ
) are split really necessary?
Question 2: For a given number field k, are there infinitely many Witt
inequivalent fields of the form ka,b, a, b ∈ k∗? All we are able to prove in
this regard is the following:
Theorem 4.7. Let k be a number field, r = the number of orderings of
k, w = the number of Witt inequivalent fields of the form ka,b, a, b ∈ k∗.
Then
w >
2 if − 1 ∈ Dk〈1, 1〉,
3 if − 1 /∈ Dk〈1, 1〉, k is not formally real,
r + 3 if k is formally real.
Proof. For a prime p of k (finite or infinite), denote by k̂p the completion
of k at p.
Case 1: −1 ∈ Dk〈1, 1〉. Fix a, b ∈ k∗ so that (a,b
k
) does not split over
k. E.g., fix some finite prime p of k and choose a, b ∈ k∗ so that a /∈ k̂∗2p
and b /∈ D
k̂p
〈1,−a〉. Theorem 4.4 (5) implies that ka,b 6∼ k1,1.
Case 2: −1 /∈ Dk〈1, 1〉, k not formally real. By hypothesis, (−1,−1
k
) is
not split over k. By the Hasse norm theorem there exists a finite prime p
such that (−1,−1
k
) is not split over k̂p. By Hilbert reciprocity there exists
a finite prime q 6= p such that (−1,−1
k
) is not split over k̂q. Fix b ∈ k∗ so
that b is sufficiently close to −1 at p and b is sufficiently close to 1 at
q. Then b is minus a square in k̂p and b is a square in k̂q. If there exists
a hyperfield isomorphism α : Q(k−1,−1) → Q(kb,−1) then, since (−1,−1
k
)
splits over k−1,−1 and α(−1) = −1, it follows that (−1,−1
k
) splits over kb,−1.
According to Proposition 3.4, this implies (−1,−1
k
) = 1 or (−1,−1
k
) = ( b,−1
k
),
i.e., (−1,−1
k
) = 1 or (−b,−1
k
) = 1. Since (−1,−1
k
) does not split over k̂p and
(−b,−1
k
) does not split over k̂q this is a contradiction. Thus k−1,−1 6∼ kb,−1.
Since (−1,−1
k
) 6= 1 and ( b,−1
k
) 6= 1 one also has that k−1,−1 6∼ k1,1 and
kb,−1 6∼ k1,1.
Case 3: Suppose k is formally real. For each integer 0 6 i 6 r choose
ai ∈ k∗ so that ai > 0 for exactly i of the orderings of k. Without loss of
generality, a0 = −1, ar = 1. By Lemma 3.6 exactly i of the orderings of
“adm-n3” — 2021/1/4 — 11:48 — page 75 — #81
P. Gładki, M. Marshall 75
k extend to kai,−1, so kai,−1 6∼ kaj ,−1 if i 6= j by Theorem 4.4 (2). This
proves w > r+1. Fix 0 6 i 6 r so that r−i is odd (e.g., take i = r−1). By
Hilbert reciprocity there exists a finite prime p of k such that (ai,−1
k
) does
not split over k̂p. Pick ai ∈ k∗ sufficiently close to 1 at p and sufficiently
close to ai at each ordering of k. Then ai is a square in k̂p and has the
same sign as ai at each ordering. By Hilbert reciprocity there exists a
finite prime q 6= p so that (ai,−1
k
) does not split over k̂q. Pick br ∈ k∗ so
that br is close to ar(= 1) at each ordering of k and such that br is close to
ai at p. Then kbr,−1 6∼ kar,−1, by Theorem 4.4 (5) (because (ar,−1
k
) splits
over k but ( br,−1
k
) does not split over k̂p). Also, all orderings of k extend
to kbr,−1, so kbr,−1 6∼ kaj ,−1 for 0 6 j < r, by Theorem 4.4 (2). Finally,
pick b0 ∈ k∗ so that b0 is close to a0(= −1) at each ordering of k, b0 is
close to ai at p and b0 is close to −ai at q. A similar argument to that
used in Case 2 shows that kb0,−1 6∼ k−1,−1. Theorem 4.4 (2) shows that
kb0,−1 6∼ kaj ,−1 for 0 < j 6 r and kb0,−1 6∼ kbr,−1.
Question 3: For a fixed integer n > 1, are there infinitely many Witt
inequivalent fields ka,b, k a number field, [k : Q] = n, a, b ∈ k∗?
Following the argument of [5, Corollary 8.8] we are able to partially
answer this question in case n = 2: let d be a square free integer and
denote by N the number of prime integers that ramify in Q(
√
d). This is
equal to the number of prime divisors of the discriminant of Q(
√
d). Recall
that the discriminant of Q(
√
d) is d if d ≡ 1 mod 4 and 4d otherwise.
Then the 2-rank of the class group of Q(
√
d) is
{
N − 2 if d > 0 and d /∈ DQ〈1, 1〉,
N − 1 otherwise.
See [3, Corollary 18.3] for the proof. In particular, there are infinitely
many possible values for the 2-rank of the class number for fields of the
sort Q(
√
d), d ∈ Q∗\Q∗2. Combining this with Theorem 4.4 yields:
Corollary 4.8. There are infinitely many Witt inequivalent fields of the
form k(x), k a quadratic extension of Q.
Finally, we consider an application of Corollary 4.1 and Theorem 4.4
to fields of the form Qa,b.
Proposition 4.9. Suppose α : Q(Qa,b) → Q(Qc,d) is a hyperfield isomor-
phism. Then, for each prime integer p, α(p) = ±q for some prime integer
q, and p = 2 ⇒ q = 2.
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76 Witt equivalence of conics
Proof. Let K = Qa,b, L = Qc,d. Theorem 4.4 shows that r 7→ α(r) defines
a group automorphism of Q∗/Q∗2. For r ∈ Q∗/Q∗2, define
SK(r) := {v ∈ νK,1 ∪ νK,2 : r /∈ UvK
∗2}.
Note that SK(±1) is the empty set. Let p be a prime. The set SK(±p) is
non-empty (by Lemma 4.6) and is minimal among all non-empty SK(r).
It follows that SL(α(p)) is non-empty and minimal among all non-empty
SL(s), s ∈ Q∗/Q∗2, so α(p) = ±q, for some prime q. Note also that if
p = 2, then SK(p) is a subset of νK,2. Then SL(±q) is a subset of νL,2, so
q = 2.
Function fields of conics defined over local fields have been investigated
by the authors in their earlier work [6]. There, the following versions of
Corollaries 4.1 and 4.2 are established for local fields:
Theorem 4.10 ([6, Theorem 3.5]). Suppose K, L are function fields
in one variable over local fields of characteristic 6= 2 which are Witt
equivalent via a hyperfield isomorphism α : Q(K) → Q(L). Then for each
i ∈ {0, 1, 2, 3} there is a uniquely defined bijection between µK,i and µL,i
such that, if v ↔ w under this bijection, then α maps (1 +Mv)K
∗2/K∗2
onto (1 +Mw)L
∗2/L∗2 and UvK
∗2/K∗2 onto UwL
∗2/L∗2 for i ∈ {0, 1, 2}
and such that α maps (1 +Mv)K
∗2/K∗2 onto (1 +Mw)L
∗2/L∗2 for i = 3.
Theorem 4.11 ([6, Theorem 3.6]). Let K ∼ L be function fields in one
variable over local fields k and ℓ respectively, with fields of constants k
and ℓ respectively. Then k ∼ ℓ except possibly when k, ℓ are both dyadic
local fields. In the latter case if there exists v ∈ µK,0 with Kv = k and
w ∈ µL,0 with Lw = ℓ then k ∼ ℓ.
In view of the abovementioned results, we are able to slightly extend
the results of Proposition 4.9 to the local case:
Theorem 4.12. Suppose k, ℓ are local fields of characteristic 6= 2, a, b ∈ k∗,
c, d ∈ ℓ∗. Then ka,b ∼ ℓc,d ⇒ k ∼ ℓ.
Proof. By Theorem 4.11 it suffices to deal with the case where k, ℓ are both
dyadic. Let α : Q(ka,b) → Q(ℓc,d) be some hyperfield isomorphism. Making
use of the bijection between µK,0 and µL,0 induced by α, and arguing as in
the proof of Theorem 4.4 we see that α induces an isomorphism between
k∗/k∗2 and ℓ∗/ℓ∗2. This implies [k : Q̂2] = [ℓ : Q̂2]. Thus k ∼ ℓ iff k, ℓ have
the same level i.e. the minimal number of summands in a representation
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P. Gładki, M. Marshall 77
of −1 as a sum of squares. The level of a dyadic local field is 1, 2 or 4. If
k has level 4 then [k : Q̂2] = [ℓ : Q̂2] is odd so ℓ has level 4. If k has level
1 then ka,b and consequently also ℓc,d has level 1. Since ℓ is algebraically
closed in ℓc,d this implies ℓ has level 1.
Theorem 4.13. Suppose k is a local field of characteristic 6= 2, a, b, c, d ∈
k∗. Then ka,b ∼ kc,d ⇒ (a,b
k
) = ( c,d
k
) except possibly in the case when k is
p-adic of level 1, for some odd prime p.
Proof. If k = C there is only one quaternion algebra and the result
is obvious. Otherwise, there are two quaternion algebras, one split and
one non-split. Suppose K = ka,b, L = kc,d, (
a,b
k
) split, ( c,d
k
) non-split.
Suppose k = R. Then K is formally real and L is non-real (of level
2), so K 6∼ L. Suppose now that k is dyadic. Suppose K ∼ L. Fix a
hyperfield isomorphism α : Q(K) → Q(L). Fix v ∈ µK,0 with Kv = k
and let w be the corresponding element of µL,0. Then k = Kv ∼ Lv so
[k : Q̂2] = [Kv : Q̂2] = [Lw : Q̂2]. Since k ⊆ Lw, this forces Lw = k,
i.e., ( c,d
k
) splits, a contradiction. Suppose now that k is p-adic, p 6= 2,
and k has level 2. Since k has level 2, we may assume c = π, where
v0(π) = 1, and d = −1. Thus ∃ f, g ∈ L∗ such that π = f2 + g2. For
each w ∈ µL,1, w(π) = v0(π) = 1 is odd, so w(f2) = w(g2) < w(f2 + g2),
i.e., −1 is a square in Lw, for all w ∈ µL,1. Suppose K ∼ L. Fix a
hyperfield isomorphism α : Q(K) → Q(L). Then the induced one-to-one
correspondence v ↔ w between µK,1 and µL,1 and the induced hyperfield
isomorphisms Q(Kv) → Q(Lw) imply −1 is a square in Kv for all v ∈ µK,1.
Define one particular such v as follows: Since (a,b
k
) splits,K = k(x). Extend
v0 to K by defining v(
∑n
i=0 aix
i) = min{v0(ai) : i ∈ {0, . . . , n}}. Clearly
v ∈ µK,1 and Kv = kv0(x). But then Kv and kv0 both have the same level,
contradicting the fact that Kv has level 1 and kv0 has level 2.
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Contact information
Paweł Gładki Institute of Mathematics, University of Silesia,
ul. Bankowa 14, 40-007 Katowice, Poland, and
Department of Computer Science, AGH
University of Science and Technology, al.
Mickiewicza 30, 30-059 Kraków, Poland
E-Mail(s): pawel.gladki@us.edu.pl
Received by the editors: 25.10.2018.
mailto:pawel.gladki@us.edu.pl
P. Gładki, M. Marshall
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