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Annihilating Ideals of Quadratic Forms over Local and Global Fields Klaas-Tido R uhl EPFL Galois Theory and Explicit Methods, First Annual Meeting in Leiden, 17 Sep - 21 Sep 2007 Klaas-Tido R uhl (EPFL) Annihilating Ideals over Global
Klaas-Tido R¨ uhl
EPFL
Galois Theory and Explicit Methods, First Annual Meeting in Leiden, 17 Sep - 21 Sep 2007
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 1 / 32
Introduction
It was already known by Witt that the Witt Ring of a field is integral. The same of course holds for the Witt-Grothendieck Ring of isometry classes of quadratic forms over a field. But only in 1987 did Lewis introduce specific annihilating polynomials. He showed that the polynomials Pn := (X − n)(X − n + 2) · · · (X + n) ∈ Z[X], n ∈ N0, annihilate all n-dimensional quadratic forms over an arbitrary field.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 2 / 32
Introduction
Since then a number of classes of quadratic forms have been identified for which there exist annihilating polynomials of lower degree. For example:
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 3 / 32
Introduction
Since then a number of classes of quadratic forms have been identified for which there exist annihilating polynomials of lower degree. For example: Positive quadratic forms, i.e. quadratic forms whose signatures are all positive (Lewis, 1992)
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 3 / 32
Introduction
Since then a number of classes of quadratic forms have been identified for which there exist annihilating polynomials of lower degree. For example: Positive quadratic forms, i.e. quadratic forms whose signatures are all positive (Lewis, 1992) Trace forms (Beaulieu & Palfrey, 1997; Lewis & McGarraghy, 2000)
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 3 / 32
Introduction
Since then a number of classes of quadratic forms have been identified for which there exist annihilating polynomials of lower degree. For example: Positive quadratic forms, i.e. quadratic forms whose signatures are all positive (Lewis, 1992) Trace forms (Beaulieu & Palfrey, 1997; Lewis & McGarraghy, 2000) Excellent forms (−, 2007)
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 3 / 32
Introduction
Some effort has also been put into calculating the annihilating ideal of Z[X] consisting of all polynomials which annihilate the whole Witt Ring, its torsion subgroup,
For example:
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 4 / 32
Introduction
Some effort has also been put into calculating the annihilating ideal of Z[X] consisting of all polynomials which annihilate the whole Witt Ring, its torsion subgroup,
For example: for the whole Witt Ring W (K), K a non-formally real field with level ≤ 16 (Ongenae & van Geel, 1997).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 4 / 32
Introduction
Some effort has also been put into calculating the annihilating ideal of Z[X] consisting of all polynomials which annihilate the whole Witt Ring, its torsion subgroup,
For example: for the whole Witt Ring W (K), K a non-formally real field with level ≤ 16 (Ongenae & van Geel, 1997). for the torsion subgroup Wt(K), K a field whose fundamental ideal I(K) fulfils a certain (strong) condition (de Wannemacker, 2006).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 4 / 32
Introduction
Some effort has also been put into calculating the annihilating ideal of Z[X] consisting of all polynomials which annihilate the whole Witt Ring, its torsion subgroup,
For example: for the whole Witt Ring W (K), K a non-formally real field with level ≤ 16 (Ongenae & van Geel, 1997). for the torsion subgroup Wt(K), K a field whose fundamental ideal I(K) fulfils a certain (strong) condition (de Wannemacker, 2006). for the fundamental ideal I(K), K a field with level ≤ 4 (de Wannemacker, 2006).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 4 / 32
Table of Contents
1
Preliminaries
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 5 / 32
Table of Contents
1
Preliminaries
2
Annihilating Polynomials
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 5 / 32
Table of Contents
1
Preliminaries
2
Annihilating Polynomials
3
Formally Real Fields
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 5 / 32
Table of Contents
1
Preliminaries
2
Annihilating Polynomials
3
Formally Real Fields
4
Local Fields
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 5 / 32
Table of Contents
1
Preliminaries
2
Annihilating Polynomials
3
Formally Real Fields
4
Local Fields
5
Global Fields
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 5 / 32
Preliminaries
Always
N does not contain 0. We use N0 := N ∪ {0}. We denote by K a field, char(K) = 2. All quadratic forms are regular (or non-degenerate).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 6 / 32
Preliminaries
Let ϕ, ψ be quadratic forms over K. We denote by a1, . . . , an the quadratic form over K associated to the diagonal matrix with entries a1, . . . , an ∈ K ∗.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 7 / 32
Preliminaries
Let ϕ, ψ be quadratic forms over K. We denote by a1, . . . , an the quadratic form over K associated to the diagonal matrix with entries a1, . . . , an ∈ K ∗. ϕ⊥ψ denotes the orthogonal sum of ϕ and ψ. ϕ ⊗ ψ denotes the tensor product of ϕ and ψ. For m ∈ N0 we define m × ϕ := ϕ⊥ . . . ⊥ϕ
.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 7 / 32
Preliminaries
Let ϕ, ψ be quadratic forms over K. We denote by a1, . . . , an the quadratic form over K associated to the diagonal matrix with entries a1, . . . , an ∈ K ∗. ϕ⊥ψ denotes the orthogonal sum of ϕ and ψ. ϕ ⊗ ψ denotes the tensor product of ϕ and ψ. For m ∈ N0 we define m × ϕ := ϕ⊥ . . . ⊥ϕ
. If ϕ and ψ are isometric we write ϕ ∼ = ψ. [ϕ] denotes the isometry class of ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 7 / 32
Preliminaries
The isometry classes of quadratic forms over K form a semi-ring W +(K) with addition [ϕ] + [ψ] := [ϕ⊥ψ] . and multiplication [ϕ] · [ψ] := [ϕ ⊗ ψ] .
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 8 / 32
Preliminaries
The isometry classes of quadratic forms over K form a semi-ring W +(K) with addition [ϕ] + [ψ] := [ϕ⊥ψ] . and multiplication [ϕ] · [ψ] := [ϕ ⊗ ψ] . By applying the Grothendieck Construction for semi-groups to W +(K) we
W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 8 / 32
Preliminaries
The elements of W (K) are formal differences [ϕ] − [ψ] and there exists an up to isometry unique anisotropic form χ over K and a unique m ∈ N such that [ϕ] − [ψ] = [χ] ± [m × H] , where H = 1, −1 is the hyperbolic plane.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 9 / 32
Preliminaries
The elements of W (K) are formal differences [ϕ] − [ψ] and there exists an up to isometry unique anisotropic form χ over K and a unique m ∈ N such that [ϕ] − [ψ] = [χ] ± [m × H] , where H = 1, −1 is the hyperbolic plane. We extend the notion of dimension to a ring homomorphism dim([ϕ] − [ψ]) := dim(ϕ) − dim(ψ) ∈ Z, where dim(ϕ) denotes the usual dimension of ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 9 / 32
Preliminaries
Denote by H the principal ideal of W (K) generated by [H]. The Witt Ring of K is defined as W (K) := W (K) /H.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 10 / 32
Preliminaries
Denote by H the principal ideal of W (K) generated by [H]. The Witt Ring of K is defined as W (K) := W (K) /H. If ϕ is a quadratic form over K, then {ϕ} will denote its equivalence class in W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 10 / 32
Preliminaries
Denote by H the principal ideal of W (K) generated by [H]. The Witt Ring of K is defined as W (K) := W (K) /H. If ϕ is a quadratic form over K, then {ϕ} will denote its equivalence class in W (K).
Note
The elements of W (K) classify anisotropic quadratic forms over K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 10 / 32
Annihilating Polynomials
Consider the canonical inclusion ι : Z − → W (K) defined for m ∈ N by m − → [m × 1] .
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials
Consider the canonical inclusion ι : Z − → W (K) defined for m ∈ N by m − → [m × 1] . Usually we will simply write m for its image via ι in W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials
Consider the canonical inclusion ι : Z − → W (K) defined for m ∈ N by m − → [m × 1] . Usually we will simply write m for its image via ι in W (K).
Definition
A polynomial P = znX n + · · · + z0 ∈ Z[X] is called annihilating polynomial
P([ϕ]) := zn [ϕ]n + · · · + z1 [ϕ] + zn = 0 ∈ W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials
Consider the canonical inclusion ι : Z − → W (K) defined for m ∈ N by m − → [m × 1] . Usually we will simply write m for its image via ι in W (K).
Definition
A polynomial P = znX n + · · · + z0 ∈ Z[X] is called annihilating polynomial
P([ϕ]) := zn [ϕ]n + · · · + z1 [ϕ] + zn = 0 ∈ W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 11 / 32
Annihilating Polynomials
In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N0 he defined Pn := (X − n)(X − n + 2) · · · (X + n − 2)(X + n) ∈ Z[X]. He furthermore proved that Pn annihilates all quadratic forms of dimension n over an arbitrary field K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials
In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N0 he defined Pn := (X − n)(X − n + 2) · · · (X + n − 2)(X + n) ∈ Z[X]. He furthermore proved that Pn annihilates all quadratic forms of dimension n over an arbitrary field K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials
In 1987 David Lewis introduced the following annihilating polynomials: For n ∈ N0 he defined Pn := (X − n)(X − n + 2) · · · (X + n − 2)(X + n) ∈ Z[X]. He furthermore proved that Pn annihilates all quadratic forms of dimension n over an arbitrary field K. These polynomials constitute the base for all our following observations.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 12 / 32
Annihilating Polynomials
Proposition
For every quadratic form ϕ over K there exists a unique polynomial Qϕ ∈ Z[X] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials
Proposition
For every quadratic form ϕ over K there exists a unique polynomial Qϕ ∈ Z[X] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Qϕ is monic.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials
Proposition
For every quadratic form ϕ over K there exists a unique polynomial Qϕ ∈ Z[X] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Qϕ is monic. Furthermore there exists some m ∈ N such that mQϕ is an annihilating polynomial of ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials
Proposition
For every quadratic form ϕ over K there exists a unique polynomial Qϕ ∈ Z[X] which divides all annihilating polynomials of ϕ and has maximal degree among all polynomials with this properties. This polynomial Qϕ is monic. Furthermore there exists some m ∈ N such that mQϕ is an annihilating polynomial of ϕ. Idea of Proof. Use the greatest common divisor and B´ ezout’s identity.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 13 / 32
Annihilating Polynomials
Definition
The Qϕ from the proposition is called embracing polynomial of ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials
Definition
The Qϕ from the proposition is called embracing polynomial of ϕ. The ideal Annϕ ⊂ Z[X] of all annihilating polynomials of ϕ is called annihilating ideal of ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials
Definition
The Qϕ from the proposition is called embracing polynomial of ϕ. The ideal Annϕ ⊂ Z[X] of all annihilating polynomials of ϕ is called annihilating ideal of ϕ. Qϕ is called embracing polynomial since Annϕ ⊂ (Qϕ).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 14 / 32
Annihilating Polynomials
Proposition
For every n-dimensional quadratic form ϕ over K, the factor X − n ∈ Z[X] divides Qϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 15 / 32
Annihilating Polynomials
Proposition
For every n-dimensional quadratic form ϕ over K, the factor X − n ∈ Z[X] divides Qϕ. Sketch of Proof. Recall that Qϕ is a product of linear factors. The claim follows since dim : W (K) → Z is a ring homomorphism and Z is an integral domain.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 15 / 32
Annihilating Polynomials
Let ϕ be a quadratic form over K. K Pythagorean. Then W (K) is torsion free. = ⇒ We have Qϕ([ϕ]) = 0 and therefore Annϕ = (Qϕ).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials
Let ϕ be a quadratic form over K. K Pythagorean. Then W (K) is torsion free. = ⇒ We have Qϕ([ϕ]) = 0 and therefore Annϕ = (Qϕ).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials
Let ϕ be a quadratic form over K. K Pythagorean. Then W (K) is torsion free. = ⇒ We have Qϕ([ϕ]) = 0 and therefore Annϕ = (Qϕ). K not formally real. Then an element of W (K) is torsion if and only if its dimension is 0. = ⇒ Qϕ = X − n ∈ Z[X]
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Annihilating Polynomials
Let ϕ be a quadratic form over K. K Pythagorean. Then W (K) is torsion free. = ⇒ We have Qϕ([ϕ]) = 0 and therefore Annϕ = (Qϕ). K not formally real. Then an element of W (K) is torsion if and only if its dimension is 0. = ⇒ Qϕ = X − n ∈ Z[X]
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 16 / 32
Formally Real Fields
In this section K is a formally real field. XK is the space of orderings of K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 17 / 32
Formally Real Fields
In this section K is a formally real field. XK is the space of orderings of K. For each A ∈ XK there exists a signature homomorphism W (K) − → Z defined by {a} − →
if a >A 0, −1
This homomorphism will be denoted by signA.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 17 / 32
Formally Real Fields
Let ϕ be a quadratic form of dimension n over K. Set Ssign
ϕ
:= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields
Let ϕ be a quadratic form of dimension n over K. Set Ssign
ϕ
:= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields
Let ϕ be a quadratic form of dimension n over K. Set Ssign
ϕ
:= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z. We have −n ≤ signA({ϕ}) ≤ n for all A ∈ XK. = ⇒ Ssign
ϕ
is finite.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields
Let ϕ be a quadratic form of dimension n over K. Set Ssign
ϕ
:= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z. We have −n ≤ signA({ϕ}) ≤ n for all A ∈ XK. = ⇒ Ssign
ϕ
is finite. = ⇒ We can define Psign
ϕ
:=
ϕ
(X − s) ∈ Z[X].
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields
Let ϕ be a quadratic form of dimension n over K. Set Ssign
ϕ
:= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z. We have −n ≤ signA({ϕ}) ≤ n for all A ∈ XK. = ⇒ Ssign
ϕ
is finite. = ⇒ We can define Psign
ϕ
:=
ϕ
(X − s) ∈ Z[X].
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 18 / 32
Formally Real Fields
Theorem
If ϕ is an n-dimensional quadratic form over a formally real field K, then Qϕ = Psign
ϕ
.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields
Theorem
If ϕ is an n-dimensional quadratic form over a formally real field K, then Qϕ = Psign
ϕ
. Proof. We have dim(Psign
ϕ
([ϕ])) = 0 and signA(Psign
ϕ
([ϕ])) = 0 for all a ∈ XK. = ⇒ Psign
ϕ
([ϕ]) is torsion.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields
Theorem
If ϕ is an n-dimensional quadratic form over a formally real field K, then Qϕ = Psign
ϕ
. Proof. We have dim(Psign
ϕ
([ϕ])) = 0 and signA(Psign
ϕ
([ϕ])) = 0 for all a ∈ XK. = ⇒ Psign
ϕ
([ϕ]) is torsion. Since Qϕ([ϕ]) is torsion and Qϕ is a product of linear factors = ⇒ Psign
ϕ
divides Qϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields
Theorem
If ϕ is an n-dimensional quadratic form over a formally real field K, then Qϕ = Psign
ϕ
. Proof. We have dim(Psign
ϕ
([ϕ])) = 0 and signA(Psign
ϕ
([ϕ])) = 0 for all a ∈ XK. = ⇒ Psign
ϕ
([ϕ]) is torsion. Since Qϕ([ϕ]) is torsion and Qϕ is a product of linear factors = ⇒ Psign
ϕ
divides Qϕ. = ⇒ Psign
ϕ
= Qϕ
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 19 / 32
Formally Real Fields
Let ϕ be a quadratic form over R with n = dim(ϕ). Set s := sign({ϕ}) (there exists only one ordering of R). Since R is Pythagorean, we have already seen that Annϕ = (Qϕ).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields
Let ϕ be a quadratic form over R with n = dim(ϕ). Set s := sign({ϕ}) (there exists only one ordering of R). Since R is Pythagorean, we have already seen that Annϕ = (Qϕ). Hence by the theorem Annϕ =
if s = n, ((X − s)(X − n))
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields
Let ϕ be a quadratic form over R with n = dim(ϕ). Set s := sign({ϕ}) (there exists only one ordering of R). Since R is Pythagorean, we have already seen that Annϕ = (Qϕ). Hence by the theorem Annϕ =
if s = n, ((X − s)(X − n))
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Formally Real Fields
Let ϕ be a quadratic form over R with n = dim(ϕ). Set s := sign({ϕ}) (there exists only one ordering of R). Since R is Pythagorean, we have already seen that Annϕ = (Qϕ). Hence by the theorem Annϕ =
if s = n, ((X − s)(X − n))
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 20 / 32
Local Fields
In this section K is a local field with finite residue field.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
In this section K is a local field with finite residue field.
Proposition
Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Annϕ ⊂ Z[X] is given as follows:
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
In this section K is a local field with finite residue field.
Proposition
Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Annϕ ⊂ Z[X] is given as follows: Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
In this section K is a local field with finite residue field.
Proposition
Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Annϕ ⊂ Z[X] is given as follows: Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
In this section K is a local field with finite residue field.
Proposition
Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Annϕ ⊂ Z[X] is given as follows: Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
In this section K is a local field with finite residue field.
Proposition
Let ϕ be a quadratic form over K, n := dim(ϕ), ϕ ∼ = n × 1. Then a complete and minimal set of generators for Annϕ ⊂ Z[X] is given as follows: Annϕ =
if det(ϕ) is a sum of two squares in K,
If ϕ ∼ = n × 1, then Annϕ = (X − n).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 21 / 32
Local Fields
Let ϕ ∼ = a1, . . . , an be a quadratic form over K. The determinant of ϕ is defined as det(ϕ) := a1 · · · an := a1 · · · an(K ∗)2 ∈ K ∗/(K ∗)2.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields
Let ϕ ∼ = a1, . . . , an be a quadratic form over K. The determinant of ϕ is defined as det(ϕ) := a1 · · · an := a1 · · · an(K ∗)2 ∈ K ∗/(K ∗)2. The Hasse invariant of ϕ is defined as s(ϕ) :=
ai, aj K
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields
Let ϕ ∼ = a1, . . . , an be a quadratic form over K. The determinant of ϕ is defined as det(ϕ) := a1 · · · an := a1 · · · an(K ∗)2 ∈ K ∗/(K ∗)2. The Hasse invariant of ϕ is defined as s(ϕ) :=
ai, aj K
It is well-known that quadratic forms over a local field are classified by dimension, determinant and discriminant.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields
Let ϕ ∼ = a1, . . . , an be a quadratic form over K. The determinant of ϕ is defined as det(ϕ) := a1 · · · an := a1 · · · an(K ∗)2 ∈ K ∗/(K ∗)2. The Hasse invariant of ϕ is defined as s(ϕ) :=
ai, aj K
It is well-known that quadratic forms over a local field are classified by dimension, determinant and discriminant. The ideal I(K) ⊂ W (K) consisting of equivalence classes of even dimensional quadratic forms is called fundamental ideal of W (K).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 22 / 32
Local Fields
Idea of Proof. Calculate the Hasse invariants and determinants of 2(X − n), (X − n)2, 4(X − n) resp. (X − n + 2)(X − n) applied to ϕ.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 23 / 32
Local Fields
Idea of Proof. Calculate the Hasse invariants and determinants of 2(X − n), (X − n)2, 4(X − n) resp. (X − n + 2)(X − n) applied to ϕ. Then compare these Hasse invariants (resp. determinants) with the Hasse invariants (resp. determinants) of the hyperbolic forms 2n × H, 2n2 × H, 4n × H resp. (2n2 + 2n) × H.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 23 / 32
Local Fields
In the case that the residue field K has characteristic = 2, we can reformulate the proposition using the valuation vK of K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields
In the case that the residue field K has characteristic = 2, we can reformulate the proposition using the valuation vK of K.
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). Assume char(K) = 2. (a) If ϕ ∼ = n × 1, then Annϕ = (X − n).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields
In the case that the residue field K has characteristic = 2, we can reformulate the proposition using the valuation vK of K.
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). Assume char(K) = 2. (a) If ϕ ∼ = n × 1, then Annϕ = (X − n). (b) If ϕ ∼ = n × 1, and
(i) if −1 ∈ (K ∗)2, then Annϕ =
.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Local Fields
In the case that the residue field K has characteristic = 2, we can reformulate the proposition using the valuation vK of K.
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). Assume char(K) = 2. (a) If ϕ ∼ = n × 1, then Annϕ = (X − n). (b) If ϕ ∼ = n × 1, and
(i) if −1 ∈ (K ∗)2, then Annϕ =
. (ii) if −1 ∈ (K ∗)2, then Annϕ =
if vK(det(ϕ)) is even,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 24 / 32
Global Fields
In this section K is a global field.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields
In this section K is a global field. Let V be the set of equivalence classes of absolute values of K. For every ν ∈ V choose a representative | · |ν of the class ν.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields
In this section K is a global field. Let V be the set of equivalence classes of absolute values of K. For every ν ∈ V choose a representative | · |ν of the class ν. Denote by Kν the completion of K with respect to | · |ν.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields
In this section K is a global field. Let V be the set of equivalence classes of absolute values of K. For every ν ∈ V choose a representative | · |ν of the class ν. Denote by Kν the completion of K with respect to | · |ν. We can write V as the disjoint union V = VR ∪ VC ∪ Vfin such that Kν = R for ν ∈ VR, C for ν ∈ VC, local for ν ∈ Vfin.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 25 / 32
Global Fields
Let ϕ be an n-dimensional quadratic form over K, and let f ∈ Z[X]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕKν for all ν ∈ V .
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields
Let ϕ be an n-dimensional quadratic form over K, and let f ∈ Z[X]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕKν for all ν ∈ V . Since AnnϕKν = (X − n) for all ν ∈ VC, and since X − n divides every annihilating polynomial of ϕ, we do not have to take into account the completions Kν for ν ∈ VC.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields
Let ϕ be an n-dimensional quadratic form over K, and let f ∈ Z[X]. By the Hasse-Minkowski Theorem f is an annihilating polynomial of ϕ ⇐ ⇒ f is an annihilating polynomial of ϕKν for all ν ∈ V . Since AnnϕKν = (X − n) for all ν ∈ VC, and since X − n divides every annihilating polynomial of ϕ, we do not have to take into account the completions Kν for ν ∈ VC.
Proposition
Annϕ =
AnnϕKν
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 26 / 32
Global Fields
Assume that VR = ∅. = ⇒ K is formally real.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields
Assume that VR = ∅. = ⇒ K is formally real. There exists a one-to-one correspondence VR ← → XK between the real completions of K and the space of orderings of K.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields
Assume that VR = ∅. = ⇒ K is formally real. There exists a one-to-one correspondence VR ← → XK between the real completions of K and the space of orderings of K. More specifically: Every signature homomorphism W (K) → Z factors uniquely as
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields
Assume that VR = ∅. = ⇒ K is formally real. There exists a one-to-one correspondence VR ← → XK between the real completions of K and the space of orderings of K. More specifically: Every signature homomorphism W (K) → Z factors uniquely as W (K)
εν
− − − → W (R)
sign
− − − → Z, where εν is induced by the completion K ֒ → Kν, and sign is the usual signature homomorphism over R.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 27 / 32
Global Fields
Recall that Ssign
ϕ
= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z and Psign
ϕ
=
ϕ
(X − s) ∈ Z[X].
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 28 / 32
Global Fields
Recall that Ssign
ϕ
= {signA({ϕ}) | A ∈ XK } ∪ {n} ⊂ Z and Psign
ϕ
=
ϕ
(X − s) ∈ Z[X]. By our previous observations we obtain:
Lemma
The signature polynomial Psign
ϕ
annihilates ϕKν for all ν ∈ VR.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 28 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (a) If ϕ ∼ = n × 1, then Annϕ = (X − n).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (a) If ϕ ∼ = n × 1, then Annϕ = (X − n). (b) If ϕ ∼ = n × 1, and
(i) if |Ssign
ϕ
| = 1, then Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (a) If ϕ ∼ = n × 1, then Annϕ = (X − n). (b) If ϕ ∼ = n × 1, and
(i) if |Ssign
ϕ
| = 1, then Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (a) If ϕ ∼ = n × 1, then Annϕ = (X − n). (b) If ϕ ∼ = n × 1, and
(i) if |Ssign
ϕ
| = 1, then Annϕ =
if det(ϕ) is a sum of two squares in K,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 29 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (b) If ϕ ∼ = n × 1, and
(ii) if |Ssign
ϕ
| = 2 with Ssign
ϕ
= {s, n}, then Annϕ =
(X − s)(X − n)2 if s ≡ n (mod 4) and det(ϕKν) is not a sum of two squares in Kν for some ν ∈ Vfin,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (b) If ϕ ∼ = n × 1, and
(ii) if |Ssign
ϕ
| = 2 with Ssign
ϕ
= {s, n}, then Annϕ =
(X − s)(X − n)2 if s ≡ n (mod 4) and det(ϕKν) is not a sum of two squares in Kν for some ν ∈ Vfin,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (b) If ϕ ∼ = n × 1, and
(ii) if |Ssign
ϕ
| = 2 with Ssign
ϕ
= {s, n}, then Annϕ =
(X − s)(X − n)2 if s ≡ n (mod 4) and det(ϕKν) is not a sum of two squares in Kν for some ν ∈ Vfin,
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields
Theorem
Let ϕ be a quadratic form over K, n = dim(ϕ). (b) If ϕ ∼ = n × 1, and
(ii) if |Ssign
ϕ
| = 2 with Ssign
ϕ
= {s, n}, then Annϕ =
(X − s)(X − n)2 if s ≡ n (mod 4) and det(ϕKν) is not a sum of two squares in Kν for some ν ∈ Vfin,
(iii) if |Ssign
ϕ
| ≥ 3, then Annϕ = (Psign
ϕ
).
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 30 / 32
Global Fields
Sketch of Proof. The claims (a) and (b).(iii) follow directly from our previous observations.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 31 / 32
Global Fields
Sketch of Proof. The claims (a) and (b).(iii) follow directly from our previous observations. For the proof of (b).(i) one shows that one does not have to take into account the real completions Kν for ν ∈ VR.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 31 / 32
Global Fields
Sketch of Proof. The claims (a) and (b).(iii) follow directly from our previous observations. For the proof of (b).(i) one shows that one does not have to take into account the real completions Kν for ν ∈ VR. The proof of (b).(ii) involves some Hasse invariant calculations and arguments analogous to those used to prove the classification of annihilating ideals over local fields.
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 31 / 32
Klaas-Tido R¨ uhl (EPFL) Annihilating Ideals over Global Fields GTEM - Annual Meeting 2007 32 / 32