Local Global Principles for Torsors over Arithmetic Curves Julia - - PowerPoint PPT Presentation

Local Global Principles for Torsors over Arithmetic Curves

Julia Hartmann

RWTH Aachen University

• jt. work with D. Harbater, University of Pennsylvania

and D. Krashen, University of Georgia

Patching

Patching in algebraic terms: F ≤ F1, F2 ≤ F0 fields. Base change functor Θ : Vect(F) − → Vect(F1) ×Vect(F0) Vect(F2).

RAGE 2011 Julia Hartmann

Patching

Patching in algebraic terms: F ≤ F1, F2 ≤ F0 fields. Base change functor Θ : Vect(F) − → Vect(F1) ×Vect(F0) Vect(F2). patching problems

RAGE 2011 Julia Hartmann

Patching

Patching in algebraic terms: F ≤ F1, F2 ≤ F0 fields. Base change functor Θ : Vect(F) − → Vect(F1) ×Vect(F0) Vect(F2). solutions patching problems

RAGE 2011 Julia Hartmann

Patching

Patching in algebraic terms: F ≤ F1, F2 ≤ F0 fields. Base change functor Θ : Vect(F) − → Vect(F1) ×Vect(F0) Vect(F2). Question: When do patching problems have solutions?

RAGE 2011 Julia Hartmann

Existence of Solutions

Let F ≤ F1, F2 ≤ F0, and consider a patching problem (V1, V2, φ)

RAGE 2011 Julia Hartmann

Existence of Solutions

Let F ≤ F1, F2 ≤ F0, and consider a patching problem (V1, V2, φ), i.e., Vi is an Fi-vector space and φ : V1 ⊗F1 F0 → V2 ⊗F2 F0 is an isomorphism

RAGE 2011 Julia Hartmann

Existence of Solutions

Let F ≤ F1, F2 ≤ F0, and consider a patching problem (V1, V2, φ), i.e., Vi is an Fi-vector space and φ : V1 ⊗F1 F0 → V2 ⊗F2 F0 is an isomorphism Proposition: Suppose that F1 ∩ F2 = F. Then the following are equivalent:

• The patching problem has a solution.

RAGE 2011 Julia Hartmann

Existence of Solutions

Let F ≤ F1, F2 ≤ F0, and consider a patching problem (V1, V2, φ), i.e., Vi is an Fi-vector space and φ : V1 ⊗F1 F0 → V2 ⊗F2 F0 is an isomorphism Proposition: Suppose that F1 ∩ F2 = F. Then the following are equivalent:

• The patching problem has a solution.
• V1 ∩ V2 has the maximum dimension dimFi(Vi).

RAGE 2011 Julia Hartmann

Existence of Solutions

Let F ≤ F1, F2 ≤ F0, and consider a patching problem (V1, V2, φ), i.e., Vi is an Fi-vector space and φ : V1 ⊗F1 F0 → V2 ⊗F2 F0 is an isomorphism Proposition: Suppose that F1 ∩ F2 = F. Then the following are equivalent:

• The patching problem has a solution.
• V1 ∩ V2 has the maximum dimension dimFi(Vi).

Notice: A solution (if it exists) is automatically unique.

RAGE 2011 Julia Hartmann

A Criterion for Patching

Let F ≤ F1, F2 ≤ F0. Proposition: The base change Θ is an equivalence of categories if and only if

RAGE 2011 Julia Hartmann

A Criterion for Patching

Let F ≤ F1, F2 ≤ F0. Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F1 ∩ F2 ≤ F0 (Intersection)

RAGE 2011 Julia Hartmann

A Criterion for Patching

Let F ≤ F1, F2 ≤ F0. Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F1 ∩ F2 ≤ F0 (Intersection) (2) n ∈ N, g ∈ GLn(F0) ⇒ gi ∈ GLn(Fi) s.t. g = g1g2 (Factorization)

RAGE 2011 Julia Hartmann

A Criterion for Patching

Let F ≤ F1, F2 ≤ F0. Proposition: The base change Θ is an equivalence of categories if and only if (1) F = F1 ∩ F2 ≤ F0 (Intersection) (2) n ∈ N, g ∈ GLn(F0) ⇒ gi ∈ GLn(Fi) s.t. g = g1g2 (Factorization) From now on, we will assume that our fields satisfy these two conditions.

RAGE 2011 Julia Hartmann

More equivalences of categories

Suppose Θ is an equivalence of categories. Then so are the base change functors for the following categories:

• associative finite dimensional F-algebras
• commutative finite dimensional F-algebras
• separable commutative F-algebras
• G-Galois F-algebras
• central simple F-algebras
• differential modules, Frobenius modules
• . . .

RAGE 2011 Julia Hartmann

Patching of Division Algebras

Corollary: If Θ is an equivalence of categories, then the natural map Br(F) → Br(F1) ×Br(F0) Br(F2) is a group isomorphism.

RAGE 2011 Julia Hartmann

Patching of Division Algebras

Corollary: If Θ is an equivalence of categories, then the natural map Br(F) → Br(F1) ×Br(F0) Br(F2) is a group isomorphism. New aspect: An object is globally trivial if and only if it is locally trivial. So patching leads to local-global principles.

RAGE 2011 Julia Hartmann

Patching Torsors

Let F ≤ F1, F2 ≤ F0 be as above. Let G be a linear algebraic group defined over F.

RAGE 2011 Julia Hartmann

Patching Torsors

Let F ≤ F1, F2 ≤ F0 be as above. Let G be a linear algebraic group defined over F. A patching problem of G-torsors is a triple (T1, T2, φ) where each Ti is a GFi-torsor and φ : T1 ×F1 F0 → T2 ×F2 F0 is an isomorphism of GF0-torsors.

RAGE 2011 Julia Hartmann

Patching Torsors

Let F ≤ F1, F2 ≤ F0 be as above. Let G be a linear algebraic group defined over F. A patching problem of G-torsors is a triple (T1, T2, φ) where each Ti is a GFi-torsor and φ : T1 ×F1 F0 → T2 ×F2 F0 is an isomorphism of GF0-torsors. Notice: The coordinate ring F[G] is not a finite dimensional F-algebra (unless G is finite), so this is different from the situation considered before.

RAGE 2011 Julia Hartmann

Patching Torsors

Let F ≤ F1, F2 ≤ F0 be as above. Let G be a linear algebraic group defined over F. A patching problem of G-torsors is a triple (T1, T2, φ) where each Ti is a GFi-torsor and φ : T1 ×F1 F0 → T2 ×F2 F0 is an isomorphism of GF0-torsors. Theorem: Every G-torsor patching problem has a unique solution.

RAGE 2011 Julia Hartmann

Galois Cohomology

Let F be any field, and let G be a linear algebraic group over F.

RAGE 2011 Julia Hartmann

Galois Cohomology

Let F be any field, and let G be a linear algebraic group over F. Recall that there is a 1 − 1 correspondence {G − torsors overF} / ∼ = ↔ H1(F, G) := H1(Gal(F sep/F), G(F sep))

RAGE 2011 Julia Hartmann

Galois Cohomology

Let F be any field, and let G be a linear algebraic group over F. Recall that there is a 1 − 1 correspondence {G − torsors overF} / ∼ = ↔ H1(F, G) := H1(Gal(F sep/F), G(F sep)) Notice that both are sets with a distinguished point.

RAGE 2011 Julia Hartmann

An exact sequence

Let F, Fi be as before. We have seen: There are exact sequences of pointed sets 1

G(F) G(F1) × G(F2)

id ·()−1

G(F0)

and H1(F, G)

H1(F1, G) × H1(F2, G)

• H1(F0, G)

RAGE 2011 Julia Hartmann

An exact sequence

Let F, Fi be as before. Theorem: There is an exact sequence of pointed sets 1

G(F) G(F1) × G(F2) G(F0)

• H1(F, G)

H1(F1, G) × H1(F2, G)

• H1(F0, G)

RAGE 2011 Julia Hartmann

An exact sequence

Let F, Fi be as before. Theorem: There is an exact sequence of pointed sets 1

H0(F, G) H0(F1, G) × H0(F2, G) H0(F0, G)

• H1(F, G)

H1(F1, G) × H1(F2, G)

• H1(F0, G)

RAGE 2011 Julia Hartmann

Local-Global-Principles and Factorization

Corollary: There is a local-global principle for G-torsors over F if and only if factori- zation holds in G.

RAGE 2011 Julia Hartmann

Local-Global-Principles and Factorization

Corollary: There is a local-global principle for G-torsors over F if and only if factori- zation holds in G. So far, this is all abstract.

RAGE 2011 Julia Hartmann

Our Setup - Simplest Case

T = k[[t]] a complete discrete valuation ring with residue field k X = P1

T with closed fibre X = P1 k

F = k((t))(x), F1 = frac(k[x−1][[t]]), F2 = k((x, t)), F0 = k((x))((t)) Then F ≤ F1, F2 ≤ F0 satisfy the intersection and factorization condition.

RAGE 2011 Julia Hartmann

Our Setup

T a complete discrete valuation ring X a regular connected projective T-curve with closed fibre X and function field F F1: related to the set P of closed points of X at which distinct irreducible components meet F2: related to the set of components of the complement of P in X F0: related to the set of branches at points of P.

RAGE 2011 Julia Hartmann

Our Setup

T a complete discrete valuation ring X a regular connected projective T-curve with closed fibre X and function field F F1: related to the set P of closed points of X at which distinct irreducible components meet F2: related to the set of components of the complement of P in X F0: related to the set of branches at points of P. These are in general not fields, but finite direct products of fields. For ex- ample, F1 is the product over the fraction fields FP of the complete local rings at points P ∈ P.

RAGE 2011 Julia Hartmann

Our Setup

T a complete discrete valuation ring X a regular connected projective T-curve with closed fibre X and function field F F1: related to the set P of closed points of X at which distinct irreducible components meet F2: related to the set of components of the complement of P in X F0: related to the set of branches at points of P. Then F ≤ F1, F2 ≤ F0 satisfy the intersection and factorization condition.

RAGE 2011 Julia Hartmann

The kernel of the local-global principle

Consider F, Fi as above, and let G be a linear algebraic group defined over

• F. Recall:

Factorization in G is equivalent to the a local-global principle for H1(F, G)

RAGE 2011 Julia Hartmann

The kernel of the local-global principle

Consider F, Fi as above, and let G be a linear algebraic group defined over

• F. Recall:

Factorization in G is equivalent to the a local-global principle for H1(F, G) So in general, define XP(X, G) = ker

• H1(F, G) → H1(F1, G) × H1(F2, G)
• RAGE 2011

Julia Hartmann

What we can say about XP(X, G)

Let G be a rational linear algebraic group over F, i.e., every component of G is an F-rational variety. Then:

• XP(X, G) is finite and has an explicit description in terms of the re-

duction graph Γ of X.

RAGE 2011 Julia Hartmann

What we can say about XP(X, G)

Let G be a rational linear algebraic group over F, i.e., every component of G is an F-rational variety. Then:

• XP(X, G) is finite and has an explicit description in terms of the re-

duction graph Γ of X.

• Using this description it can be shown that it is trivial if and only if

either G is connected or Γ is a tree.

RAGE 2011 Julia Hartmann

What we can say about XP(X, G)

Let G be a rational linear algebraic group over F, i.e., every component of G is an F-rational variety. Then:

• XP(X, G) is finite and has an explicit description in terms of the re-

duction graph Γ of X.

• Using this description it can be shown that it is trivial if and only if

either G is connected or Γ is a tree.

• XP(X, G) agrees with the a priori bigger kernel X0(X, G) of

H1(F, G) →

• P∈X

H1(FP, G)

RAGE 2011 Julia Hartmann

• Both kernels are independent of the choice of a regular model X for F;

so may write e.g. X0(F, G)

RAGE 2011 Julia Hartmann

• Both kernels are independent of the choice of a regular model X for F;

so may write e.g. X0(F, G)

• The kernel X(F, G) of the local global map w.r.t. all discrete valuations

contains X0(F, G), and the cokernel of this inclusion can be described in terms of „local kernels“.

RAGE 2011 Julia Hartmann

• Both kernels are independent of the choice of a regular model X for F;

so may write e.g. X0(F, G)

• The kernel X(F, G) of the local global map w.r.t. all discrete valuations

contains X0(F, G), and the cokernel of this inclusion can be described in terms of „local kernels“.

• This cokernel vanishes e.g. if

1) The residue field of T is algebraically closed of characteristic zero. 2) G0 is defined and reductive over a regular model X of F. In particular, X(F, G) is finite in these cases and we know exactly when it is trivial.

RAGE 2011 Julia Hartmann

Applications

Let F be the function field of a regular projective curve over a complete discrete valuation ring. We obtain:

• Local Global Principles for isotropy of quadratic forms of dimension at

least three in terms of patches, points of the closed fibre, or valuations (Colliot-Thélène, Parimala, Suresh)

• Results on the u-invariant
• Description of X0(F, O(q)) and X(F, O(q)) for quadratic forms q
• Results on the period-index problem

RAGE 2011 Julia Hartmann

Thank you for your attention!

RAGE 2011 Julia Hartmann