Transcendental Brauer elements and descent on elliptic surfaces - PowerPoint PPT Presentation

Transcendental Brauer elements and descent on elliptic surfaces Bianca Viray Brown University Ramification in Algebra and Geometry Emory University May 18, 2011 Bianca Viray (Brown University) Transcendental Brauer elements and descent

The Brauer group Let k be a field of characteristic 0, and let X be a smooth, projective, geometrically integral variety over k . Definition The Brauer group of a field k is Br k := { central simple algebras over k } × ) = H 2 ( G k , k ∼ The Brauer group of a variety X is Br X := H 2 et ( X , G m ) . Example Let X be a smooth proper model of y 2 + z 2 = (3 − x 2 )( x 2 − 2) . The element ( − 1 , x 2 − 2) 2 = ( − 1 , 3 − x 2 ) 2 = ( − 1 , 1 − 2 / x 2 ) 2 ∈ Br k ( X ) is contained in Br X . Bianca Viray (Brown University) Transcendental Brauer elements and descent

Elements in the Brauer group Elements in Br X can be divided into 3 categories. constant elements: Br 0 X := im(Br k → Br X ) algebraic elements: Br 1 X := ker(Br X → Br X ) transcendental elements: all other elements We have two extreme cases: 1. Br X is purely algebraic, i.e. Br X = Br 1 X . 2. Br X is purely transcendental, i.e. Br 1 X = Br 0 X . Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems Problem � � Br X Let X be a variety such that [ m ] is finite. Br 0 X � � Br X [ m ] . Determine generators for Br 0 X Problem Let A ∈ Br X Br X Br 0 X . Determine whether or not A is trivial in Br 0 X . Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems Problem � � Br X Let X be a variety such that [ m ] is finite. Br 0 X � � Br X [ m ] . Determine generators for Br 0 X Problem Let A ∈ Br X Br X Br 0 X . Determine whether or not A is trivial in Br 0 X . If Br X = Br 1 X and Pic( X ) ∼ � G k , then one could try to � → Pic X make Br 1 X ∼ → H 1 ( G k , Pic X ) Br 0 X explicit. Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems Problem � � Br X [ m ] is finite. Let X be a variety such that Br 0 X � � Br X Determine generators for [ m ] . Br 0 X Problem Let A ∈ Br X Br X Br 0 X . Determine whether or not A is trivial in Br 0 X . Now consider the case where Br 1 X = Br 0 X . For Problem 2, if k is a global field, then one can compute X ( A k ) A . If X ( A k ) A � X ( A k ), then A is nontrivial in Br X Br 0 X . What if X ( A k ) A = X ( A k )? Bianca Viray (Brown University) Transcendental Brauer elements and descent

Previous work Most of the progress has been made when X is an elliptic surface. General setup From now on, assume that k = k . Assume that there exists a curve W , and map π : X → W such that the generic fiber is a smooth curve C / K := k ( W ). By the purity theorem, � ∂ V � � → H 1 ( κ ( V ) , Q / Z ) Br X = ker Br C . V ⊆ X vertical prime divisor So if we can solve Problems 1 and 2 for C , we can compute the residue maps ∂ V to solve Problems 1 and 2 for X . Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � � Previous work General setup (cont.) By Tsen’s theorem, Br C = Br 1 C and Br 0 C = 0. Thus, we have ∼ → H 1 ( G K , Pic C K ). an isomorphism Br C Using the inclusion of J := Jac( C K ) into Pic C K , and the multiplication by m map on J , we obtain J ( K ) � H 1 ( G K , J [ m ]) H 1 ( G K , J )[ m ] � � � � mJ ( K ) g ∼ H 1 ( G K , Pic( C K ))[ m ] (Br C ) [ m ] Possible approach: Step 1: Find explicit generators for H 1 ( G k , J [ m ]). Step 2: Make g explicit Step 3: Compute residue maps ∂ V . Bianca Viray (Brown University) Transcendental Brauer elements and descent

Previous work Possible approach: Step 1: Find explicit generators for H 1 ( G k , J [ m ]). Step 2: Make g explicit Step 3: Compute residue maps ∂ V . Olivier Wittenberg ’04: W = P 1 , C ( K ) � = ∅ , m = 2, J [2] ⊂ J ( K ). ’09: W = P 1 , C ( K ) = ∅ , m = 2 , J [2] ⊂ J ( K ), Evis Ieronymou partial progress when m = 4. Bianca Viray (Brown University) Transcendental Brauer elements and descent

Previous work Possible approach: Step 1: Find explicit generators for H 1 ( G k , J [ m ]). Step 2: Make g explicit Step 3: Compute residue maps ∂ V . Olivier Wittenberg ’04: W = P 1 , C ( K ) � = ∅ , m = 2, J [2] ⊂ J ( K ). ’09: W = P 1 , C ( K ) = ∅ , m = 2 , J [2] ⊂ J ( K ), Evis Ieronymou partial progress when m = 4. Remark The assumption that J [2] ⊂ J ( K ) simplifies Step 1. Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � Connection with descent Possible approach: Step 1: Find explicit generators for H 1 ( G k , J [ m ]). Step 2: Make g explicit Step 3: Compute residue maps ∂ V . J ( K ) � H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � g � � � � � � � � � � (Br C )[2] Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � Connection with descent Possible approach: Step 1: Find explicit generators for H 1 ( G k , J [ m ]). Step 2: Make g explicit Step 3: Compute residue maps ∂ V . J ( K ) � H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � g � � � � � � � � � � (Br C )[2] Idea Use ideas from descent to tackle Steps 1 and 2. Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � �� Connection with descent Possible approach: Step 1’: Find explicit generators for H 1 ( G k , J [2]) / Q . Step 2’: Make h explicit Step 3 : Compute residue maps ∂ V . J ( K ) � H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � g � � � � � � � � � � � h � (Br C )[2] H 1 ( G K , J [2]) / Q Idea Use ideas from descent to tackle Steps 1’ and 2’. Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � �� � Main theorem Theorem (V.) Assume that C has a model y 2 = f ( x ) where deg( f ) = 4 . Let L be the degree 4 ´ etale algebra K [ α ] / ( f ( α )) . The following diagram commutes, J ( K ) H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � � � x − α � � � � � � � � � � � h L × 2 K × → K × L × � (Br C )[2] , ker N : K × 2 where h : ℓ �→ Cor k ( C L ) / k ( C ) (( ℓ, x − α ) 2 ) . Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � �� � Main theorem Theorem (V.) Assume that C has a model y 2 = f ( x ) where deg( f ) = 4 . Let L be the degree 4 ´ etale algebra K [ α ] / ( f ( α )) . The following diagram commutes, J ( K ) H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � � � x − α � � � � � � � � � � � h L × 2 K × → K × L × � (Br C )[2] , ker N : K × 2 where h : ℓ �→ Cor k ( C L ) / k ( C ) (( ℓ, x − α ) 2 ) . Moreover, there is a finite group (ker N ) S-unr such that h − 1 (Br X ) ⊆ (ker N ) S-unr . Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � �� � Main theorem Theorem (V.) Assume that C has a model y 2 = f ( x ) where deg( f ) = 4 . Let L be the degree 4 ´ etale algebra K [ α ] / ( f ( α )) . The following diagram commutes, J ( K ) H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � � � x − α � � � � � � � � � � � h L × 2 K × → K × L × � (Br C )[2] , ker N : K × 2 where h : ℓ �→ Cor k ( C L ) / k ( C ) (( ℓ, x − α ) 2 ) . Moreover, there is a finite group (ker N ) S-unr such that h − 1 (Br X ) ⊆ (ker N ) S-unr . Bianca Viray (Brown University) Transcendental Brauer elements and descent

Sketch of proof Sketch of proof: commutative diagram The idea of this proof comes from [Poonen-Schaefer ’97]. Let L := L ⊗ K K . There is an exact sequence 0 → J [2] → µ 2 ( L ) N → µ 2 ( K ) → 0 . µ 2 ( K ) After computing long exact sequences from cohomology and applying Tsen’s theorem, we obtain the commutative diagram in the theorem Bianca Viray (Brown University) Transcendental Brauer elements and descent

� � �� � Main theorem Theorem (V.) Assume that C has a model y 2 = f ( x ) where deg( f ) = 4 . Let L be the degree 4 ´ etale algebra K [ α ] / ( f ( α )) . The following diagram commutes, J ( K ) H 1 ( G K , J [2]) H 1 ( G K , J )[2] � � � � 2 J ( K ) � � � � x − α � � � � � � � � � � � h L × 2 K × → K × L × � (Br C )[2] , ker N : K × 2 where h : ℓ �→ Cor k ( C L ) / k ( C ) (( ℓ, x − α ) 2 ) . Moreover, there is a finite group (ker N ) S-unr such that h − 1 (Br X ) ⊆ (ker N ) S-unr . Bianca Viray (Brown University) Transcendental Brauer elements and descent