Zero Cycles on Principal Homogeneous Spaces Jodi A. Black Emory - - PowerPoint PPT Presentation

Motivation Results

Zero Cycles on Principal Homogeneous Spaces

Jodi A. Black

Emory University

Ramification in Algebra and Geometry at Emory May 18, 2011

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

A Question of Serre

Serre’s Question (1962) Let k be a field and let {Li}m

i=1 be a set of finite field extensions

• f k of coprime degree. If G is a connected linear algebraic

group over k, does the canonical map H1(k, G) →

m

• i

H1(Li, G) have trivial kernel?

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Zero Cycles on Principal Homogeneous Spaces

There is a bijection between H1(k, G) and the set of isomorphism classes of principal homogeneous spaces under G over k. A positive answer to Serre’s question would give a positive answer to: Q(PHS): Let X be a principal homogeneous space under G over k. If X admits a zero cycle of degree one, does X have a k-rational point?

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Zero Cycles on Principal Homogeneous Spaces

There is a bijection between H1(k, G) and the set of isomorphism classes of principal homogeneous spaces under G over k. A positive answer to Serre’s question would give a positive answer to: Q(PHS): Let X be a principal homogeneous space under G over k. If X admits a zero cycle of degree one, does X have a k-rational point?

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

There are examples of other types of homogeneous varieties which may admit zero cycle of degree one but have no rational point. X a quasi-projective homogeneous variety (Florence, 2004) X a projective homogeneous variety (Parimala, 2005)

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Conjecture II

Serre’s Conjecture II (1962) Let k be a perfect field, cd(k) ≤ 2. Let G be a simply connected semisimple algebraic group over k. Then H1(k, G) = {1}.

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Serre’s Conjecture II is true if: G is a classical group (Merkurjev-Suslin,1990), (Bayer-Parimala, 1995) G is of type F4 or G2 (Serre, 1995) special cases of groups G of type 3,6D4, E6, E7 (Garibaldi, 2001) (Gille, 2001) (Chernousov, 2003) A positive answer to Serre’s question would imply a positive answer to Conjecture II for G split and of type E8.

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Serre’s Conjecture II is true if: G is a classical group (Merkurjev-Suslin,1990), (Bayer-Parimala, 1995) G is of type F4 or G2 (Serre, 1995) special cases of groups G of type 3,6D4, E6, E7 (Garibaldi, 2001) (Gille, 2001) (Chernousov, 2003) A positive answer to Serre’s question would imply a positive answer to Conjecture II for G split and of type E8.

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Known Results

The answer to Serre’s question is yes if: G is an abelian group (trivial case) G = O(q) (Springer’s theorem) G = PGLn (classical result) k is a number field (Sansuc, 1981) char(k) = 2 and G = Iso(A, σ) = {a ∈ A : σ(a)a = 1} (Bayer-Lenstra, 1990)

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Results

Theorem (B., 2011) The answer to Serre’s question is yes if:

1

char(k) = 2 and G is a simply connected or adjoint semisimple algebraic group of classical type

2

k is perfect, vcd(k) ≤ 2 and Gsc is of classical type, type F4 or type G2

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Key Ingredients Part I

Theorem (Gille, 1993) (Merkurjev, 1995) Let k be a perfect field. Let T be an algebraic k-torus and let G1 and G be a connected reductive k-groups such that the following sequence is exact: 1

G1 G

f

T 1

Then NL/k(f(RG(L)) ⊆ f(RG(k))

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Key Ingredients for Part II

Let k be a perfect field with vcd(k) ≤ 2 and let kv denote the real closure of k at an ordering v. Theorem (Bayer-Parimala, 1998) Let G be a simply connected semisimple group of classical type, type F4 or type G2 then H1(k, G) →

• v

H1(kv, G) is injective.

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

As a consequence of the results on similitudes used in the proof of the first result, we find that if (A, σ) and (A′, σ′) are central simple algebras with involution of the first kind over k which become isomorphic over an odd degree extension of k, then they are isomorphic over k

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces

Motivation Results

Thanks!

Jodi A. Black Zero Cycles on Principal Homogeneous Spaces