An Alternating Property for Higher Brauer Groups

对称积与范数簇的弱逼近

Weak approximation of symmetric products and norm varieties

本文研究数域上簇的对称积及其光滑射影模型是否保持带Brauer-Manin障碍的弱逼近性质，并将同一方法推广到范数簇。

Vanishing of Massey products and Brauer groups

Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Minac, and Tan, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

关于线性代数群的近强逼近性质

On almost strong approximation for linear algebraic groups

本文研究了线性代数群的近强逼近性质，通过Brauer群条件提出充要条件，并利用该条件将ASA控制在地方密度和分裂域上。

Manin obstruction to strong approximation for homogeneous spaces

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group Br(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélè…

Etale Steenrod operations and the Artin-Tate pairing

We prove a 1966 conjecture of Tate concerning the Artin-Tate pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen-Stoll on the Cassels-Tate pairing. Our method is based on studying a connection between the Artin-Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate t…