Abstracts Mathematics

Hasse Principle for Hermitian Spaces

by Zhengyao Wu

This dissertation provides three results: (1) A Hasse principle for rational points of projective homogeneous spaces under unitary or special unitary groups associated to hermitian or skew hermitian spaces over function fields of p-adic curves; (2) A Springer-type theorem for isotropy of hermitian spaces over odd degree field extensions of function fields of p-adic curves; (3) Exact values of Hermitian u-invariants of quaternion or biquaternion algebras over function fields of p-adic curves. Chapter 1. Generalities  – 1  – 1.1. Central simple algebras and Brauer groups p.1  – 1.2. Hermitian spaces and Witt groups p.3  – 1.3. Algebraic groups and Rationality p.8  – 1.4. Galois cohomology and Principal homogeneous spaces p.15  – 1.5. Projective homogeneous spaces p.23  – 1.6. Morita invariance. p.28  – Chapter 2. Hasse principle of projective homogeneous spaces p.33  – 2.1. Maximal orders p.35  – 2.2. Complete regular local ring of dimension p.38  – 2.3. Patching and Hasse principle p.48  – Chapter 3. Springer's problem for odd degree extensions p.57  – 3.1. Reduction to the residue field58  – 3.2. Springer's theorem over local or global fields p.59  – 3.3. Springer's theorem over function fields of p-adic curves p.61  – Chapter 4. Hermitian u-invariants63  – 4.1. Hermitian u-invariants over complete discrete valued fields p.64  – 4.2. Division algebras over Ai(2)-fields p.72  – 4.3. Division algebras over semi-global fields p.75  – 4.4. Tensor product of quaternions over arbitrary fields p.77  – Bibliography p.81 Advisors/Committee Members: Raman, Parimala (Committee Member), Zureick-Brown, David M (Committee Member), Venapally, Suresh (Thesis Advisor).