Isotropy over function fields of Pfister forms James OShea - - PowerPoint PPT Presentation

Isotropy over function fields of Pfister forms

James O’Shea

Universität Konstanz / University College Dublin

Ramification in Algebra and Geometry at Emory

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 1 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn].

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn]. A form ϕ is isotropic if ϕ(v) = 0 for some v ∈ F n \ {0}, and is anisotropic otherwise.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn]. A form ϕ is isotropic if ϕ(v) = 0 for some v ∈ F n \ {0}, and is anisotropic otherwise. Every form ϕ has a decomposition ϕ ≃ ϕan ⊥ i(ϕ) × 1, −1 where the anisotropic form ϕan and the integer i(ϕ) are uniquely determined.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn]. A form ϕ is isotropic if ϕ(v) = 0 for some v ∈ F n \ {0}, and is anisotropic otherwise. Every form ϕ has a decomposition ϕ ≃ ϕan ⊥ i(ϕ) × 1, −1 where the anisotropic form ϕan and the integer i(ϕ) are uniquely determined. Let F(ϕ) be the function field of ϕ, the quotient field of F[X1, . . . , Xn]/(ϕ(X1, . . . , Xn)), where ϕ ≃ 1, −1 and dim ϕ = n 2.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn]. A form ϕ is isotropic if ϕ(v) = 0 for some v ∈ F n \ {0}, and is anisotropic otherwise. Every form ϕ has a decomposition ϕ ≃ ϕan ⊥ i(ϕ) × 1, −1 where the anisotropic form ϕan and the integer i(ϕ) are uniquely determined. Let F(ϕ) be the function field of ϕ, the quotient field of F[X1, . . . , Xn]/(ϕ(X1, . . . , Xn)), where ϕ ≃ 1, −1 and dim ϕ = n 2. i1(ϕ) := i(ϕF(ϕ)) i(ϕK) for all K/F where ϕK is isotropic.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

The isotropy question for function fields

F field, char(F) = 2. A (regular quadratic) form over F is a homogeneous quadratic polynomial ϕ ∈ F[X1, . . . , Xn]. A form ϕ is isotropic if ϕ(v) = 0 for some v ∈ F n \ {0}, and is anisotropic otherwise. Every form ϕ has a decomposition ϕ ≃ ϕan ⊥ i(ϕ) × 1, −1 where the anisotropic form ϕan and the integer i(ϕ) are uniquely determined. Let F(ϕ) be the function field of ϕ, the quotient field of F[X1, . . . , Xn]/(ϕ(X1, . . . , Xn)), where ϕ ≃ 1, −1 and dim ϕ = n 2. i1(ϕ) := i(ϕF(ϕ)) i(ϕK) for all K/F where ϕK is isotropic.

Question

Given a form ϕ over F, which anisotropic forms over F are isotropic

• ver F(ϕ)?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 2 / 11

Dimension criteria

Theorem (Hoffmann)

If ψ is anisotropic and dim ψ 2n < dim ϕ, then ψF(ϕ) is anisotropic.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 3 / 11

Dimension criteria

Theorem (Hoffmann)

If ψ is anisotropic and dim ψ 2n < dim ϕ, then ψF(ϕ) is anisotropic.

Theorem (Karpenko, Merkurjev)

Suppose that ψ is anisotropic and ψF(ϕ) is isotropic. Then dim ψ − i1(ψ) dim ϕ − i1(ϕ).

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 3 / 11

Dimension criteria

Theorem (Hoffmann)

If ψ is anisotropic and dim ψ 2n < dim ϕ, then ψF(ϕ) is anisotropic.

Theorem (Karpenko, Merkurjev)

Suppose that ψ is anisotropic and ψF(ϕ) is isotropic. Then dim ψ − i1(ψ) dim ϕ − i1(ϕ).

Corollary (O’S)

Let ϕ be anisotropic. The minimum dimension of the anisotropic forms

• ver F that are isotropic over F(ϕ) is dim ϕ − i1(ϕ) + 1.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 3 / 11

Dimension criteria

Theorem (Hoffmann)

If ψ is anisotropic and dim ψ 2n < dim ϕ, then ψF(ϕ) is anisotropic.

Theorem (Karpenko, Merkurjev)

Suppose that ψ is anisotropic and ψF(ϕ) is isotropic. Then dim ψ − i1(ψ) dim ϕ − i1(ϕ).

Corollary (O’S)

Let ϕ be anisotropic. The minimum dimension of the anisotropic forms

• ver F that are isotropic over F(ϕ) is dim ϕ − i1(ϕ) + 1.

Given K/F, a form ψ over F is minimal K-isotropic if ψK is isotropic and τK is anisotropic for τ ψ.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 3 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form. Let PnF = {n-fold Pfister forms over F}.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form. Let PnF = {n-fold Pfister forms over F}.

Question

For π ∈ PnF, which anisotropic forms over F are isotropic over F(π)?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form. Let PnF = {n-fold Pfister forms over F}.

Question

For π ∈ PnF, which anisotropic forms over F are isotropic over F(π)? τ is a neighbour of π ∈ PnF if τ ⊆ aπ for a ∈ F × and dim τ > 1

2 dim π.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form. Let PnF = {n-fold Pfister forms over F}.

Question

For π ∈ PnF, which anisotropic forms over F are isotropic over F(π)? τ is a neighbour of π ∈ PnF if τ ⊆ aπ for a ∈ F × and dim τ > 1

2 dim π.

If ψ contains a neighbour of π ∈ PnF, then ψF(π) is isotropic.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

The isotropy question for Pfister function fields

π ≃ 1, a1 ⊗ . . . ⊗ 1, an with a1, . . . , an ∈ F × is an n-fold Pfister form. Let PnF = {n-fold Pfister forms over F}.

Question

For π ∈ PnF, which anisotropic forms over F are isotropic over F(π)? τ is a neighbour of π ∈ PnF if τ ⊆ aπ for a ∈ F × and dim τ > 1

2 dim π.

If ψ contains a neighbour of π ∈ PnF, then ψF(π) is isotropic.

Question

If ψ is anisotropic and ψF(π) is isotropic, where π ∈ PnF, when does ψ contain a neighbour of π?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 4 / 11

Excellence

Given K/F, a form over F ϕ is K-excellent if (ϕK)an ≃ γK for some form γ over F.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 5 / 11

Excellence

Given K/F, a form over F ϕ is K-excellent if (ϕK)an ≃ γK for some form γ over F. K/F is excellent if every form over F is K-excellent.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 5 / 11

Excellence

Given K/F, a form over F ϕ is K-excellent if (ϕK)an ≃ γK for some form γ over F. K/F is excellent if every form over F is K-excellent.

Question

Which forms ϕ are such that F(ϕ)/F is excellent?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 5 / 11

Excellence

Given K/F, a form over F ϕ is K-excellent if (ϕK)an ≃ γK for some form γ over F. K/F is excellent if every form over F is K-excellent.

Question

Which forms ϕ are such that F(ϕ)/F is excellent?

Theorem (Hoffmann, Knebusch)

Let ϕ be an anisotropic form over F. Then ϕ is F(ϕ)-excellent iff ϕ is a Pfister neighbour.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 5 / 11

Excellence

Given K/F, a form over F ϕ is K-excellent if (ϕK)an ≃ γK for some form γ over F. K/F is excellent if every form over F is K-excellent.

Question

Which forms ϕ are such that F(ϕ)/F is excellent?

Theorem (Hoffmann, Knebusch)

Let ϕ be an anisotropic form over F. Then ϕ is F(ϕ)-excellent iff ϕ is a Pfister neighbour.

Theorem (Arason for n = 2)

Let π ∈ PnF for n 2. Then F(π)/F is excellent.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 5 / 11

Function fields of conics

Let π ∈ P2F be anisotropic.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 6 / 11

Function fields of conics

Let π ∈ P2F be anisotropic.

Theorem (Hoffmann, Van Geel)

Every minimal F(π)-isotropic form has odd dimension, and every odd number n 3 is the dimension of some minimal F(π)-isotropic form, for certain F and π.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 6 / 11

Function fields of conics

Let π ∈ P2F be anisotropic.

Theorem (Hoffmann, Van Geel)

Every minimal F(π)-isotropic form has odd dimension, and every odd number n 3 is the dimension of some minimal F(π)-isotropic form, for certain F and π.

Theorem (Hoffmann, Lewis, Van Geel)

ϕ anisotropic is a minimal F(π)-isotropic iff sπ(ϕ) = dim ϕ−1

2

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 6 / 11

Function fields of conics

Let π ∈ P2F be anisotropic.

Theorem (Hoffmann, Van Geel)

Every minimal F(π)-isotropic form has odd dimension, and every odd number n 3 is the dimension of some minimal F(π)-isotropic form, for certain F and π.

Theorem (Hoffmann, Lewis, Van Geel)

ϕ anisotropic is a minimal F(π)-isotropic iff sπ(ϕ) = dim ϕ−1

2

Theorem (Hoffmann, Lewis, Van Geel)

Let π ≃ 1, −a ⊗ 1, −b and ϕ be anisotropic with dim ϕ = 5. Then ϕ is minimal F(π)-isotropic iff ϕ is a neighbour of π ⊗ 1, −x for some x ∈ F × and indF(c(ϕ) ⊗F (a, b)) = 4.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 6 / 11

Excellence and minimal F(π)-isotropy

Theorem (Izhboldin)

For n 3, there exist F and π ∈ PnF such that F(π)/F is not excellent.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 7 / 11

Excellence and minimal F(π)-isotropy

Theorem (Izhboldin)

For n 3, there exist F and π ∈ PnF such that F(π)/F is not excellent. Izhboldin’s non F(π)-excellent forms are minimal F(π)-isotropic forms

• f dimension 2n.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 7 / 11

Excellence and minimal F(π)-isotropy

Theorem (Izhboldin)

For n 3, there exist F and π ∈ PnF such that F(π)/F is not excellent. Izhboldin’s non F(π)-excellent forms are minimal F(π)-isotropic forms

• f dimension 2n.They are twisted Pfister forms.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 7 / 11

Excellence and minimal F(π)-isotropy

Theorem (Izhboldin)

For n 3, there exist F and π ∈ PnF such that F(π)/F is not excellent. Izhboldin’s non F(π)-excellent forms are minimal F(π)-isotropic forms

• f dimension 2n.They are twisted Pfister forms.

Theorem (Hoffmann, O’S)

Let π ∈ PnF and ψ be a minimal F(π)-isotropic form. Then ψ is F(π)-excellent iff ψ ⊆ π ⊗ γ for some form γ and dim ψ > 1

2dim(π ⊗ γ).

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 7 / 11

Excellence and minimal F(π)-isotropy

Theorem (Izhboldin)

For n 3, there exist F and π ∈ PnF such that F(π)/F is not excellent. Izhboldin’s non F(π)-excellent forms are minimal F(π)-isotropic forms

• f dimension 2n.They are twisted Pfister forms.

Theorem (Hoffmann, O’S)

Let π ∈ PnF and ψ be a minimal F(π)-isotropic form. Then ψ is F(π)-excellent iff ψ ⊆ π ⊗ γ for some form γ and dim ψ > 1

2dim(π ⊗ γ).

Φ(n) = sup

• m ∈ N
• Every form of dimension m is F(π)-excellent

for every F and π ∈ PnF

• James O’Shea (Konstanz / Dublin)

Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 7 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

Φ(n) = ∞ for n 2 (Arason)

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

Φ(n) = ∞ for n 2 (Arason)

Proposition (O’S)

Let n 3, π ∈ PnF and ϕ be anisotropic with dim ϕ Φ(n). Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

Φ(n) = ∞ for n 2 (Arason)

Proposition (O’S)

Let n 3, π ∈ PnF and ϕ be anisotropic with dim ϕ Φ(n). Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

Proposition (Hoffmann, Izhboldin)

6 Φ(3) 7.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

Φ(n) = ∞ for n 2 (Arason)

Proposition (O’S)

Let n 3, π ∈ PnF and ϕ be anisotropic with dim ϕ Φ(n). Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

Proposition (Hoffmann, Izhboldin)

6 Φ(3) 7.

Theorem (O’S)

2n−1 + 1 Φ(n) 2n − 2n−3 for every n 4.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Excellence and minimal F(π)-isotropy

Φ(n) = sup{m ∈ N| Every form of dim m is F(π)-excellent ∀ F and π ∈ PnF}

Φ(n) = ∞ for n 2 (Arason)

Proposition (O’S)

Let n 3, π ∈ PnF and ϕ be anisotropic with dim ϕ Φ(n). Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

Proposition (Hoffmann, Izhboldin)

6 Φ(3) 7.

Theorem (O’S)

2n−1 + 1 Φ(n) 2n − 2n−3 for every n 4.

Proposition (O’S)

Let π ∈ PnF and ϕ be such that ϕF(π) is isotropic and dim ϕ 2n. If ϕ is F(π)-excellent, then ϕ contains a neighbour of π.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 8 / 11

Real fields of finite Hasse number

˜ u(F) = sup{dim ϕ | ϕ anisotropic and totally indefinite over F}.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 9 / 11

Real fields of finite Hasse number

˜ u(F) = sup{dim ϕ | ϕ anisotropic and totally indefinite over F}.

Question

Let F be such that ˜ u(F) < ∞ and let π ∈ PnF. Which anisotropic forms over F are isotropic over F(π)?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 9 / 11

Real fields of finite Hasse number

˜ u(F) = sup{dim ϕ | ϕ anisotropic and totally indefinite over F}.

Question

Let F be such that ˜ u(F) < ∞ and let π ∈ PnF. Which anisotropic forms over F are isotropic over F(π)?

Theorem (O’S)

Let ϕ anisotropic be such that dim ϕ > ˜ u(F) and let π ∈ PnF. Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 9 / 11

Real fields of finite Hasse number

˜ u(F) = sup{dim ϕ | ϕ anisotropic and totally indefinite over F}.

Question

Let F be such that ˜ u(F) < ∞ and let π ∈ PnF. Which anisotropic forms over F are isotropic over F(π)?

Theorem (O’S)

Let ϕ anisotropic be such that dim ϕ > ˜ u(F) and let π ∈ PnF. Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

Proposition (O’S)

Let π ∈ PnF. Then F(π)/F is excellent iff every form over F of dimension at most ˜ u(F) is F(π)-excellent.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 9 / 11

Real fields of finite Hasse number

˜ u(F) = sup{dim ϕ | ϕ anisotropic and totally indefinite over F}.

Question

Let F be such that ˜ u(F) < ∞ and let π ∈ PnF. Which anisotropic forms over F are isotropic over F(π)?

Theorem (O’S)

Let ϕ anisotropic be such that dim ϕ > ˜ u(F) and let π ∈ PnF. Then ϕF(π) is isotropic iff ϕ contains a neighbour of π.

Proposition (O’S)

Let π ∈ PnF. Then F(π)/F is excellent iff every form over F of dimension at most ˜ u(F) is F(π)-excellent.

Proposition (Hoffmann)

Let F be such that ˜ u(F) 6. Then F(π)/F is excellent for all π ∈ PnF.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 9 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power. Pfister neighbours have maximal splitting.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power. Pfister neighbours have maximal splitting.

Question

Are all forms with maximal splitting (of a given dimension) necessarily Pfister neighbours?

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power. Pfister neighbours have maximal splitting.

Question

Are all forms with maximal splitting (of a given dimension) necessarily Pfister neighbours?

Proposition (Hoffmann)

For each n 3, there exists a form of dimension 2n−1 + 2n−3 which has maximal splitting but is not a Pfister neighbour.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power. Pfister neighbours have maximal splitting.

Question

Are all forms with maximal splitting (of a given dimension) necessarily Pfister neighbours?

Proposition (Hoffmann)

For each n 3, there exists a form of dimension 2n−1 + 2n−3 which has maximal splitting but is not a Pfister neighbour.

Theorem (Izhboldin, Vishik)

Let n 4 and ϕ be such that 2n − 7 dim ϕ 2n. Then ϕ has maximal splitting iff ϕ is a Pfister neighbour.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

Pfister neighbours and maximal splitting

A form ϕ has maximal splitting if dim ϕ − i1(ϕ) is a 2-power. Pfister neighbours have maximal splitting.

Question

Are all forms with maximal splitting (of a given dimension) necessarily Pfister neighbours?

Proposition (Hoffmann)

For each n 3, there exists a form of dimension 2n−1 + 2n−3 which has maximal splitting but is not a Pfister neighbour.

Theorem (Izhboldin, Vishik)

Let n 4 and ϕ be such that 2n − 7 dim ϕ 2n. Then ϕ has maximal splitting iff ϕ is a Pfister neighbour. γ is a neighbour of π ∈ PnF iff γ has maximal splitting, dim γ 2n and γF(π) is isotropic.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 10 / 11

A related result

Theorem (O’S)

Let π ∈ PnF, ψ and ϕ ≃ π ⊥ −c1, −d be anisotropic, for c, d ∈ F ×.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 11 / 11

A related result

Theorem (O’S)

Let π ∈ PnF, ψ and ϕ ≃ π ⊥ −c1, −d be anisotropic, for c, d ∈ F ×. Suppose that dim ψ max{2n+1 − 7, dim ϕ + 1}.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 11 / 11

A related result

Theorem (O’S)

Let π ∈ PnF, ψ and ϕ ≃ π ⊥ −c1, −d be anisotropic, for c, d ∈ F ×. Suppose that dim ψ max{2n+1 − 7, dim ϕ + 1}. Then ϕF(ψ) is isotropic iff c1, −d represents x ∈ F × such that ψ is a neighbour of π ⊗ 1, −x.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 11 / 11

A related result

Theorem (O’S)

Let π ∈ PnF, ψ and ϕ ≃ π ⊥ −c1, −d be anisotropic, for c, d ∈ F ×. Suppose that dim ψ max{2n+1 − 7, dim ϕ + 1}. Then ϕF(ψ) is isotropic iff c1, −d represents x ∈ F × such that ψ is a neighbour of π ⊗ 1, −x.

Conjecture (Izhboldin)

Let π ∈ PnF, ψ and ϕ ≃ π ⊥ −c1, −d be anisotropic, for c, d ∈ F ×. Suppose that dim ψ dim ϕ + 1. Then ϕF(ψ) is isotropic iff c1, −d represents x ∈ F × such that ψ is a neighbour of π ⊗ 1, −x.

James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms RAGE, Emory, 19th May 2011 11 / 11