This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
In broad terms the principle investigator proposes to study fields which support division algebras in their Brauer groups that are seen as exotic. As an example, in one part of the project the PI proposes to study the existence of non-crossed product division algebras as well as indecomposable division algebras over the function field of a p-adic curve. The general technique that the PI proposes is to study the division algebras that exist over completions of the function field and use a splitting map from the Brauer group of the completed function field to the Brauer group of the function field to lift the division algebras. In another part of the project the PI proposes to continue to study the connection between degeneracy of a matrix defining an abelian crossed product and decomposability of the abelian crossed product. The PI will follow up on an observation which she made which shows a not yet fully understood connection between degeneracy of the matrix, decomposability of the abelian crossed product and torsion in the 2nd Chow group of the associated Severi-Brauer variety. Lastly, the PI proposes to study fields for which division algebras of small degree over that field can be distinguished by their splitting fields. In particular, the PI will consider this question over function fields of K3 surfaces.
The origins of division algebras can be traced back to Hamilton's discovery of the quaternions in 1843. Hamilton constructed his quaternion algebra to generalize the complex numbers and apply it to mechanics in three dimensional space. Since this discovery, quaternion algebras have been generalized to finite dimensional division algebras over a field, Azumaya algebras over a ring, and even sheaves of Azumaya algebras over a scheme. In each case the isomorphism classes of the objects are in 1-1 correspondence with a group, the Brauer group. Along the way the study of these algebras has involved many mathematical tools including Galois cohomology, valuation theory, number theory and algebraic geometry, just to name a few. The types of division algebras that exist over a given field can be seen as a measure of complexity or robustness of the field. In this project the PI proposes to study this complexity by considering the types of division algebras that exist over particular fields.
