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} ∼ = H2(Gk, k

×)

The Brauer group of a variety X is Br X := H2

et(X, Gm).

Example Let X be a smooth proper model of y2 + z2 = (3 − x2)(x2 − 2). The element (−1, x2 − 2)2 = (−1, 3 − x2)2 = (−1, 1 − 2/x2)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: Br0 X := im(Br k → Br X) algebraic elements: Br1 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 = Br1 X.
• 2. Br X is purely transcendental, i.e. Br1 X = Br0 X.

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems

Problem Let X be a variety such that

• Br X

Br0 X

• [m] is finite.

Determine generators for

• Br X

Br0 X

• [m].

Problem Let A ∈ Br X

Br0 X . Determine whether or not A is trivial in Br X Br0 X .

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems

Problem Let X be a variety such that

• Br X

Br0 X

• [m] is finite.

Determine generators for

• Br X

Br0 X

• [m].

Problem Let A ∈ Br X

Br0 X . Determine whether or not A is trivial in Br X Br0 X .

If Br X = Br1 X and Pic(X) ∼ →

• Pic X

Gk, then one could try to make Br1 X Br0 X

∼

→ H1(Gk, Pic X) explicit.

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Motivating problems

Problem Let X be a variety such that

• Br X

Br0 X

• [m] is finite.

Determine generators for

• Br X

Br0 X

• [m].

Problem Let A ∈ Br X

Br0 X . Determine whether or not A is trivial in Br X Br0 X .

Now consider the case where Br1 X = Br0 X. For Problem 2, if k is a global field, then one can compute X(Ak)A. If X(Ak)A X(Ak), then A is nontrivial in

Br X Br0 X .

What if X(Ak)A = X(Ak)?

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, Br X =

• V ⊆X

vertical prime divisor

ker

• Br C

∂V

→ H1(κ(V ), Q/Z)

• .

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 = Br1 C and Br0 C = 0. Thus, we have an isomorphism Br C

∼

→ H1(GK, Pic CK). Using the inclusion of J := Jac(CK) into Pic CK, and the multiplication by m map on J, we obtain

J(K) mJ(K)

H1(GK, J[m])

g

• H1(GK, J)[m]
• (Br C) [m]

H1(GK, Pic(CK))[m]

∼

• Possible approach:

Step 1: Find explicit generators for H1(Gk, 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 H1(Gk, J[m]). Step 2: Make g explicit Step 3: Compute residue maps ∂V . Olivier Wittenberg ’04: W = P1, C(K) = ∅, m = 2, J[2] ⊂ J(K). Evis Ieronymou ’09: W = P1, C(K) = ∅, m = 2, J[2] ⊂ J(K), partial progress when m = 4.

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Previous work

Possible approach: Step 1: Find explicit generators for H1(Gk, J[m]). Step 2: Make g explicit Step 3: Compute residue maps ∂V . Olivier Wittenberg ’04: W = P1, C(K) = ∅, m = 2, J[2] ⊂ J(K). Evis Ieronymou ’09: W = P1, C(K) = ∅, m = 2, J[2] ⊂ J(K), 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 H1(Gk, J[m]). Step 2: Make g explicit Step 3: Compute residue maps ∂V .

J(K) 2J(K)

H1(GK, J[2])

g

• H1(GK, J)[2]
• (Br C)[2]

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Connection with descent

Possible approach: Step 1: Find explicit generators for H1(Gk, J[m]). Step 2: Make g explicit Step 3: Compute residue maps ∂V .

J(K) 2J(K)

H1(GK, J[2])

g

• H1(GK, J)[2]
• (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 H1(Gk, J[2])/Q. Step 2’: Make h explicit Step 3 : Compute residue maps ∂V .

J(K) 2J(K)

H1(GK, J[2])

g

• H1(GK, J)[2]
• H1(GK, J[2])/Q

h

(Br C)[2]

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 y2 = f (x) where deg(f ) = 4. Let L be the degree 4 ´ etale algebra K[α]/(f (α)). The following diagram commutes,

J(K) 2J(K)

• x−α
• H1(GK, J[2])
• H1(GK, J)[2]
• ker
• N :

L× L×2K × → K × K ×2

• h

(Br C)[2],

where h: ℓ → Cork(CL)/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 y2 = f (x) where deg(f ) = 4. Let L be the degree 4 ´ etale algebra K[α]/(f (α)). The following diagram commutes,

J(K) 2J(K)

• x−α
• H1(GK, J[2])
• H1(GK, J)[2]
• ker
• N :

L× L×2K × → K × K ×2

• h

(Br C)[2],

where h: ℓ → Cork(CL)/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 y2 = f (x) where deg(f ) = 4. Let L be the degree 4 ´ etale algebra K[α]/(f (α)). The following diagram commutes,

J(K) 2J(K)

• x−α
• H1(GK, J[2])
• H1(GK, J)[2]
• ker
• N :

L× L×2K × → K × K ×2

• h

(Br C)[2],

where h: ℓ → Cork(CL)/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] → µ2(L) µ2(K)

N

→ µ2(K) → 0. 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 y2 = f (x) where deg(f ) = 4. Let L be the degree 4 ´ etale algebra K[α]/(f (α)). The following diagram commutes,

J(K) 2J(K)

• x−α
• H1(GK, J[2])
• H1(GK, J)[2]
• ker
• N :

L× L×2K × → K × K ×2

• h

(Br C)[2],

where h: ℓ → Cork(CL)/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 y2 = f (x) where deg(f ) = 4. Let L be the degree 4 ´ etale algebra K[α]/(f (α)). The following diagram commutes,

J(K) 2J(K)

• x−α
• H1(GK, J[2])
• H1(GK, J)[2]
• ker
• N :

L× L×2K × → K × K ×2

• h

(Br C)[2],

where h: ℓ → Cork(CL)/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

Definition of (ker N)S−unr

Let S be a finite set of points of W containing all places of bad reduction of y2 = f (x). Definition An element ℓ ∈ L× is S-unramified if for all t ∈ W \ S and for all P, Q ∈ Zt vP(ℓ) ≡ vQ(ℓ) (mod 2) An element is ramified if it is not S-unramified. Proposition h−1(Br X) ⊆ (ker N)S-unr.

Bianca Viray (Brown University) Transcendental Brauer elements and descent

h−1(Br X) ⊆ (ker N)S-unr

Proof Let ℓ be a ramified element of (ker N). Then ∃t ∈ W such that vP1(ℓ) ≡ vP2(ℓ) ≡ 0 (mod 2), vP3(ℓ) ≡ vP4(ℓ) ≡ 1 (mod 2), where Zt = {P1, P2, P3, P4}, for some ordering of the Pi. We claim that h(ℓ) = Cork(CL)/k(C) ((ℓ, x − α)2) is ramified at Xt. Recall that Xt is a smooth genus 1 curve with model y2 = ft(x) and that ft(α(Pi)) = 0 for i = 1, . . . , 4. A computation shows that ∂Xt(h(ℓ)) = x − α(P3) x − α(P4) ∈ κ(Xt)× κ(Xt)×2 . If ∂Xt(h(ℓ)) is trivial, then [(α(P3), 0)] − [(α(P4), 0)] is trivial in Pic(Xt). This is not the case, so h(ℓ) / ∈ Br X.

Bianca Viray (Brown University) Transcendental Brauer elements and descent

Finiteness of (ker N)S−unr

Proof of finiteness of (ker N)S−unr Let t0 ∈ W be a fixed point on good reduction. Let ℓ0 ∈ L be such that vP(ℓ) ≡ 1 (mod 2) if and only if P ∈ Zt0. Let Leven := {ℓ : div(ℓ) = 2D, and [D] ∈ Jac(Z)}; note that Leven is finite of order 22g(Z). Now take any element ℓ ∈ (ker N)S−unr. By multiplying ℓ by elements in L×2 · K × · ℓZ

0 · Leven, we may assume div(ℓ) is

supported only on the fibers of bad reduction of C. Since there are only finitely many possibilities for these divisors (modulo div(L×2K ×)), (ker N)S−unr is finite.

Bianca Viray (Brown University) Transcendental Brauer elements and descent