The Brauer-Manin Obstruction of del Pezzo surfaces of degree 4 - PowerPoint PPT Presentation

The Brauer-Manin Obstruction of del Pezzo surfaces of degree 4 Manar Riman General Exam University of Washington August 9, 2016 Manar Riman Brauer-Manin Obstruction August 9, 2016

Overview Motivation 1 The Brauer-Manin Obstruction 2 The Brauer Group The Brauer-Manin Obstruction BSD Example The Main Theorem 3 Manar Riman Brauer-Manin Obstruction August 9, 2016

Motivation Let X be a nice variety, i.e, a smooth, projective, geometrically integral variety over a global field k with set of places Ω k . We are interested in existence of k -rational points on X . X p k q : “ { k -rational points on X } X p A k q : “ { adelic points on X } . If X is nice then X p A k q “ ś v P Ω X p k v q . If X p k q ‰ H then X p k v q ‰ H for every v P Ω k . The converse when it holds for a class of varieties is called the Hasse principle . The Hasse principle is interesting because finding k v points is an easier question. Question Does the Hasse principle always hold? Manar Riman Brauer-Manin Obstruction August 9, 2016

Motivation Let X be a nice variety, i.e, a smooth, projective, geometrically integral variety over a global field k with set of places Ω k . We are interested in existence of k -rational points on X . X p k q : “ { k -rational points on X } X p A k q : “ { adelic points on X } . If X is nice then X p A k q “ ś v P Ω X p k v q . If X p k q ‰ H then X p k v q ‰ H for every v P Ω k . The converse when it holds for a class of varieties is called the Hasse principle . The Hasse principle is interesting because finding k v points is an easier question. Question Does the Hasse principle always hold? Answer It holds for some classes of varieties, but not in general. Manar Riman Brauer-Manin Obstruction August 9, 2016

Example of the failure of the Hasse principle Let X be the intersection of two quadrics in P 4 given by the equations s 2 “ xy ` 5 z 2 X : s 2 ´ 5 t 2 “ x 2 ` 3 xy ` 2 y 2 . X is an example of the failure of the Hasse principle, i.e, X p A Q q ‰ H and X p Q q “ H . Manar Riman Brauer-Manin Obstruction August 9, 2016

Manin constructed a smaller set X p A k q Br depending on the Brauer group such that: X p k q Ă X p A k q Br Ă X p A k q . We say that there is a Brauer-Manin obstruction to the Hasse principle for X if X p A k q ‰ H and X p A k q Br “ H . Question Is the Brauer-Manin obstruction the only obstruction to the Hasse principle? Manar Riman Brauer-Manin Obstruction August 9, 2016

Manin constructed a smaller set X p A k q Br depending on the Brauer group such that: X p k q Ă X p A k q Br Ă X p A k q . We say that there is a Brauer-Manin obstruction to the Hasse principle for X if X p A k q ‰ H and X p A k q Br “ H . Question Is the Brauer-Manin obstruction the only obstruction to the Hasse principle? Answer No, the first counter example was constructed by Skorobogatov in 1999 [Sko99]. Manar Riman Brauer-Manin Obstruction August 9, 2016

However Colliot-Th´ el` ene and Sansuc conjectured that: Colliot-Th´ el` ene and Sansuc [CTS81] For a geometrically rational variety X over a number field, the Brauer-Manin obstruction is the only obstruction to the Hasse principle. Assuming Schinzel’s hypothesis and the finiteness of the Tate-Shafarevich groups for elliptic curves, Wittenberg proved this conjecture for some cases of X which assumed Br X “ Br k [Wit07]. We will study a consequence of the conjecture for a nice intersection of two quadrics in P 4 also referred to as a del Pezzo surface of degree 4 . Manar Riman Brauer-Manin Obstruction August 9, 2016

Springer’s Theorem [Lam05] Let Q be a quadric over k and L { k an odd degree extension. Let X : “ V p Q q . Then X p L q ‰ H ñ X p k q ‰ H . Amer-Brumer Theorem [Lam05] Let Q 1 and Q 2 be two quadrics over k . Let X “ V p Q 1 , Q 2 q be the intersection of Q 1 and Q 2 over k , and X λ “ V p Q 1 ` λ Q 2 q a quadric over k p λ q where λ is an indeterminant. Then X p k q ‰ H ð ñ X λ p k p λ qq ‰ H . If X is the intersection of two quadrics, and L { k is an odd degree extension then we deduce that: Spr AB AB X p L q ‰ H ù ñ X λ p L p λ qq ‰ H ù ñ X λ p k p λ qq ‰ H ù ñ X p k q ‰ H . Manar Riman Brauer-Manin Obstruction August 9, 2016

X p L q ‰ H ñ X p k q ‰ H Theorem Assume CTS conjecture. Let X be an intersection of two quadrics over k , and L { k an odd degree extension. Then X p A k q Br “ H ù ñ X p A L q Br “ H . Reason: X p A k q Br “ H ñ X p k q “ H ñ X p L q “ H CTS ñ X p A L q Br “ H ù ù Our goal is to prove the theorem unconditionaly. Proving the theorem serves as evidence for Colliot-Th´ el` ene and Sansuc’s conjecture. Manar Riman Brauer-Manin Obstruction August 9, 2016

Outline for section 2 Motivation 1 The Brauer-Manin Obstruction 2 The Brauer Group The Brauer-Manin Obstruction BSD Example The Main Theorem 3 Manar Riman Brauer-Manin Obstruction August 9, 2016

The Brauer Group of a field Definition 1 (algebraic) We define the Brauer group of a field Br k to be: t CSA { k u Br k : “ Brauer Equivalence . 1 iff A b k M n p k q » A 1 b k M m p k q for some n , m . We say A „ A Definition 2 (cohomological) Let k s be the seperable closure of k . Then Br k “ H 2 p Gal p k s { k q , k ˆ s q “ H 2 et p k , G m q . ´ Manar Riman Brauer-Manin Obstruction August 9, 2016

Example Let L { k be a cyclic extension of degree n. Fix a generator σ of Gal p L { k q and a P k ˆ . We define the cyclic algebra p σ, a q to be p σ, a q : “ L r x s σ x n ´ a where L r x s σ is the twisted polynomial ring with multiplication defined as x ℓ “ ℓ σ x . Example Let a , b P k ˚ . The quaternion algebra p a , b q 2 is generated by i , j as a k -algebra such that i 2 “ a , j 2 “ b and ij “ ´ ji . It is a cyclic algebra corresponding to k p? a q{ k . Theorem [GS06] The algebra p σ, a q is trivial in Br k if and only if a P Norm L { k p L ˆ q . Manar Riman Brauer-Manin Obstruction August 9, 2016

Brauer group of a local field The Brauer group of a nonarchimedean local field k v is completely determined by the invariant isomorphism from class field theory: „ inv v : Br k v Ý Ñ Q { Z . If L w is a finite extension of k v then the following diagram commutes inv kv Br k v Ý Ý Ý Ý Ñ Q { Z § § § § đ r L w : k v s đ inv Lw Br L w Ý Ý Ý Ý Ñ Q { Z . Example [Mil] Let L w { k v be an unramified cyclic extension and σ P Gal p L w { k v q . Then v p a q inv k p σ, a q “ r L w : k v s P Q { Z . Manar Riman Brauer-Manin Obstruction August 9, 2016

Brauer group of a global field For k a global field, the fundamental exact sequence of global class field theory completely characterizes Br k : ř v inv v 0 Ñ Br k Ñ ‘ v P Ω k Br k v Ý Ý Ý Ý Ñ Q { Z . Manar Riman Brauer-Manin Obstruction August 9, 2016

Brauer group of a scheme Let X be a scheme. As a generalization of the Brauer group of a field we define the cohomological Brauer group of X as follows. Definition Br X : “ H 2 et p X , G m q . ´ In practice we often use the following exact sequence for a regular integral noetherian scheme X to test whether an algebra A P Br K p X q is in Br X . 0 Ñ Br X Ñ Br k p X q ‘B x Ñ ‘ x P X p 1 q H 1 p k p x q , Q { Z q Ý Ý where X p 1 q is the set of codimension 1 points of X and k p x q is the residue field corresponding to x . Manar Riman Brauer-Manin Obstruction August 9, 2016

Brauer group of a scheme Let X be a scheme. As a generalization of the Brauer group of a field we define the cohomological Brauer group of X as follows. Definition Br X : “ H 2 et p X , G m q . ´ In practice we often use the following exact sequence for a regular integral noetherian scheme X to test whether an algebra A P Br K p X q is in Br X . 0 Ñ Br X Ñ Br k p X q ‘B x Ñ ‘ x P X p 1 q H 1 p k p x q , Q { Z q Ý Ý where X p 1 q is the set of codimension 1 points of X and k p x q is the residue field corresponding to x . Fact: Let L { K p X q be cyclic and of prime degree p that is unframified at x . Then p L { K p X q , a q P ker B x iff x splits completely in L or v x p a q ” 0 p p q . Manar Riman Brauer-Manin Obstruction August 9, 2016

Let A P Br X . We define X p A k q A : “ tp P v q P X p A k q : ÿ inv v p A p P v qq “ 0 u . v P Ω k Definition The Brauer-Manin set is X p A k q Br : “ X p A k q A . č A P Br X Manar Riman Brauer-Manin Obstruction August 9, 2016

The Brauer-Manin Obstruction X p A k q A : “ tp P v q P X p A k q : ÿ inv v p A p P v qq “ 0 u v P Ω k It follows from the fundamental sequence of class field theory X p k q X p A k q ev A ev A ř v inv v 0 Br k ‘ v Br k v Q { Z that X p k q Ă X p A k q Br Ă X p A k q . Manar Riman Brauer-Manin Obstruction August 9, 2016

The Brauer-Manin Obstruction X p A k q A : “ tp P v q P X p A k q : ÿ inv v p A p P v qq “ 0 u v P Ω k It follows from the fundamental sequence of class field theory X p k q X p A k q ev A ev A ř v inv v 0 Br k ‘ v Br k v Q { Z that X p k q Ă X p A k q Br Ă X p A k q . Definition We say that there is a Brauer-Manin obstruction to the Hasse principle for X if X p A k q ‰ H and X p A k q Br “ H . Manar Riman Brauer-Manin Obstruction August 9, 2016