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 RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . A form ϕ is isotropic if ϕ ( v ) = 0 for some v ∈ F n \ { 0 } , and is anisotropic otherwise. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . 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 ≃ ϕ an ⊥ i ( ϕ ) × � 1 , − 1 � where the anisotropic form ϕ an and the integer i ( ϕ ) are uniquely determined. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . 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 ≃ ϕ an ⊥ i ( ϕ ) × � 1 , − 1 � where the anisotropic form ϕ an and the integer i ( ϕ ) are uniquely determined. Let F ( ϕ ) be the function field of ϕ , the quotient field of F [ X 1 , . . . , X n ] / ( ϕ ( X 1 , . . . , X n )) , where ϕ �≃ � 1 , − 1 � and dim ϕ = n � 2. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . 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 ≃ ϕ an ⊥ i ( ϕ ) × � 1 , − 1 � where the anisotropic form ϕ an and the integer i ( ϕ ) are uniquely determined. Let F ( ϕ ) be the function field of ϕ , the quotient field of F [ X 1 , . . . , X n ] / ( ϕ ( X 1 , . . . , X n )) , where ϕ �≃ � 1 , − 1 � and dim ϕ = n � 2. i 1 ( ϕ ) := i ( ϕ F ( ϕ ) ) � i ( ϕ K ) for all K / F where ϕ K is isotropic. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 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 [ X 1 , . . . , X n ] . 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 ≃ ϕ an ⊥ i ( ϕ ) × � 1 , − 1 � where the anisotropic form ϕ an and the integer i ( ϕ ) are uniquely determined. Let F ( ϕ ) be the function field of ϕ , the quotient field of F [ X 1 , . . . , X n ] / ( ϕ ( X 1 , . . . , X n )) , where ϕ �≃ � 1 , − 1 � and dim ϕ = n � 2. i 1 ( ϕ ) := i ( ϕ F ( ϕ ) ) � i ( ϕ K ) for all K / F where ϕ K is isotropic. Question Given a form ϕ over F , which anisotropic forms over F are isotropic over F ( ϕ ) ? RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 2 / 11

Dimension criteria Theorem (Hoffmann) If ψ is anisotropic and dim ψ � 2 n < dim ϕ , then ψ F ( ϕ ) is anisotropic. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 3 / 11

Dimension criteria Theorem (Hoffmann) If ψ is anisotropic and dim ψ � 2 n < dim ϕ , then ψ F ( ϕ ) is anisotropic. Theorem (Karpenko, Merkurjev) Suppose that ψ is anisotropic and ψ F ( ϕ ) is isotropic. Then dim ψ − i 1 ( ψ ) � dim ϕ − i 1 ( ϕ ) . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 3 / 11

Dimension criteria Theorem (Hoffmann) If ψ is anisotropic and dim ψ � 2 n < dim ϕ , then ψ F ( ϕ ) is anisotropic. Theorem (Karpenko, Merkurjev) Suppose that ψ is anisotropic and ψ F ( ϕ ) is isotropic. Then dim ψ − i 1 ( ψ ) � dim ϕ − i 1 ( ϕ ) . Corollary (O’S) Let ϕ be anisotropic. The minimum dimension of the anisotropic forms over F that are isotropic over F ( ϕ ) is dim ϕ − i 1 ( ϕ ) + 1 . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 3 / 11

Dimension criteria Theorem (Hoffmann) If ψ is anisotropic and dim ψ � 2 n < dim ϕ , then ψ F ( ϕ ) is anisotropic. Theorem (Karpenko, Merkurjev) Suppose that ψ is anisotropic and ψ F ( ϕ ) is isotropic. Then dim ψ − i 1 ( ψ ) � dim ϕ − i 1 ( ϕ ) . Corollary (O’S) Let ϕ be anisotropic. The minimum dimension of the anisotropic forms over F that are isotropic over F ( ϕ ) is dim ϕ − i 1 ( ϕ ) + 1 . Given K / F , a form ψ over F is minimal K-isotropic if ψ K is isotropic and τ K is anisotropic for τ � ψ . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 3 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . Let P n F = { n -fold Pfister forms over F } . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . Let P n F = { n -fold Pfister forms over F } . Question For π ∈ P n F , which anisotropic forms over F are isotropic over F ( π ) ? RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . Let P n F = { n -fold Pfister forms over F } . Question For π ∈ P n F , which anisotropic forms over F are isotropic over F ( π ) ? τ is a neighbour of π ∈ P n F if τ ⊆ a π for a ∈ F × and dim τ > 1 2 dim π . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . Let P n F = { n -fold Pfister forms over F } . Question For π ∈ P n F , which anisotropic forms over F are isotropic over F ( π ) ? τ is a neighbour of π ∈ P n F if τ ⊆ a π for a ∈ F × and dim τ > 1 2 dim π . If ψ contains a neighbour of π ∈ P n F , then ψ F ( π ) is isotropic. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

The isotropy question for Pfister function fields π ≃ � 1 , a 1 � ⊗ . . . ⊗ � 1 , a n � with a 1 , . . . , a n ∈ F × is an n-fold Pfister form . Let P n F = { n -fold Pfister forms over F } . Question For π ∈ P n F , which anisotropic forms over F are isotropic over F ( π ) ? τ is a neighbour of π ∈ P n F if τ ⊆ a π for a ∈ F × and dim τ > 1 2 dim π . If ψ contains a neighbour of π ∈ P n F , then ψ F ( π ) is isotropic. Question If ψ is anisotropic and ψ F ( π ) is isotropic, where π ∈ P n F , when does ψ contain a neighbour of π ? RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 4 / 11

Excellence Given K / F , a form over F ϕ is K-excellent if ( ϕ K ) an ≃ γ K for some form γ over F . RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 5 / 11

Excellence Given K / F , a form over F ϕ is K-excellent if ( ϕ K ) an ≃ γ K for some form γ over F . K / F is excellent if every form over F is K -excellent. RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 5 / 11

Excellence Given K / F , a form over F ϕ is K-excellent if ( ϕ K ) an ≃ γ K 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? RAGE, Emory, 19 th May 2011 James O’Shea (Konstanz / Dublin) Isotropy over function fields of Pfister forms 5 / 11