Hasse invariant for the tame Brauer group of a higher local field

CYCLIC LENGTH IN THE TAME BRAUER GROUP OF THE FUNCTION FIELD OF A p-ADIC CURVE

Let F be the function field of a smooth curve over the p-adic number field Qp. We show that for each prime-to-p number n the n-torsion subgroup H(F, μn) = nBr(F ) is generated by Z/n-cyclic classes; in fact the Z/n-length is equal to two. It follows that the Brauer dimension of F is two (first proved in [20]), and any F -division algebra of period n and index n is decomposable.

The Hasse Principle and the Brauer-manin Obstruction for Curves

We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the c...

The Brauer-manin Obstruction to the Hasse Principle

Acknowledgements My greatest gratitude goes out to my supervisor Dr Daniel Chan. He helped me pick the topic for this thesis which turned out to be extremely interesting, helped me with the writing of this thesis and most importantly never refused to spend time with me to help me learn mathematics. I thank him deeply for this. I would to also like thank my high school maths teacher Ian Denton f...

The Brauer-Hasse-Noether theorem in historical perspective

The Equivariant Brauer Group of a Group

We consider the Brauer group BM(k, G) of a group G (finite or infinite) over a commutative ring k with identity. A split exact sequence 1 −→ Br′(k) −→ BM′(k, G) −→ Gal(k, G) −→ 1 is obtained. This generalizes the Fröhlich-Wall exact sequence ([7, 8])from the case of a field to the case of a commutative ring, and generalizes the PiccoPlatzeck exact sequence ([13]) from the finite case of G to th...