Quaternion rings and ternary quadratic forms John Voight University - PowerPoint PPT Presentation

Quaternion rings and ternary quadratic forms John Voight University of Vermont RAGE Emory University 19 May 2011

Quaternion rings?

Quaternion rings? How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring?

Quaternion rings? How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring? i 2 = j 2 = k 2 = ijk = − 1

Quaternion rings? How should one define a quaternion ring if the coefficients can come from an arbitrary commutative ring? i 2 = j 2 = k 2 = ijk = − 1 Sir William Rowan Hamilton (1805-1865)

Quaternion algebras over fields

Quaternion algebras over fields Let F be a field with char F � = 2.

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection   Similarity classes of   regular ternary quadratic forms q over F  

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F q ( x , y , z ) = ax 2 + by 2 + cz 2

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F q ( x , y , z ) = ax 2 + by 2 + cz 2 �→ C 0 ( q )

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F � − bc , − ac � q ( x , y , z ) = ax 2 + by 2 + cz 2 �→ C 0 ( q ) = F

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F � − bc , − ac � q ( x , y , z ) = ax 2 + by 2 + cz 2 �→ C 0 ( q ) = F � a , b � nrd : B 0 → F ← � B = F

Quaternion algebras over fields Let F be a field with char F � = 2. Then there is a functorial bijection � Isomorphism classes of   Similarity classes of �   ∼ regular ternary quadratic − → quaternion algebras forms q over F   B over F � − bc , − ac � q ( x , y , z ) = ax 2 + by 2 + cz 2 �→ C 0 ( q ) = F � a , b � nrd : B 0 → F ← � B = F nrd( xi + yj + zij ) = − ax 2 − by 2 + abz 2

Quaternion rings?

Quaternion rings? Let R be a commutative ring.

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )).

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module.

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like?

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like? 1. Azumaya (central R -simple) algebra of rank 4 over R .

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like? 1. Azumaya (central R -simple) algebra of rank 4 over R . 2. Crossed products,

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like? 1. Azumaya (central R -simple) algebra of rank 4 over R . � a , b � 2. Crossed products, e.g. B = . R

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like? 1. Azumaya (central R -simple) algebra of rank 4 over R . � a , b � 2. Crossed products, e.g. B = . R 3. Quaternion orders (if R is a domain),

Quaternion rings? Let R be a commutative ring. Let B be an R -algebra (an associative ring with 1 equipped with R ֒ → Z ( B )). Suppose that B is a finitely generated, locally free R -module. So, what does a quaternion ring over R look like? 1. Azumaya (central R -simple) algebra of rank 4 over R . � a , b � 2. Crossed products, e.g. B = . R 3. Quaternion orders (if R is a domain), subrings B ⊆ B ⊗ R Frac( R ).

Standard involutions and exceptional rings

Standard involutions and exceptional rings A standard involution : B → B

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B .

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution.

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings .

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n .

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M with multiplication law x · y =

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M with multiplication law x · y = t ( x ) y for all x , y ∈ M

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M with multiplication law x · y = t ( x ) y for all x , y ∈ M is an R -algebra with standard involution x = t ( x ) − x for all x ∈ M .

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M with multiplication law x · y = t ( x ) y for all x , y ∈ M is an R -algebra with standard involution x = t ( x ) − x for all x ∈ M . In particular, xx = x ( t ( x ) − x ) = 0 for all x ∈ M .

Standard involutions and exceptional rings A standard involution : B → B is an involution such that xx ∈ R for all x ∈ B . From now on, let B have a standard involution. First, some decidedly non-quaternion rings: exceptional rings . Let t : M → R be R -linear with rk( M ) = n . Then B = R ⊕ M with multiplication law x · y = t ( x ) y for all x , y ∈ M is an R -algebra with standard involution x = t ( x ) − x for all x ∈ M . In particular, xx = x ( t ( x ) − x ) = 0 for all x ∈ M . For an exceptional ring, we have charpoly( x ; T ) = T ( T − t ( x )) n in B