Hasse Principle for Hermitian Spaces Open Access

Wu, Zhengyao (2016)

Permanent URL: https://etd.library.emory.edu/concern/etds/sn009z38c?locale=en


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.

Table of Contents

Chapter 1. Generalities. 1

1.1. Central simple algebras and Brauer groups. 1

1.2. Hermitian spaces and Witt groups. 3

1.3. Algebraic groups and Rationality. 8

1.4. Galois cohomology and Principal homogeneous spaces. 15

1.5. Projective homogeneous spaces. 23

1.6. Morita invariance. 28

Chapter 2. Hasse principle of projective homogeneous spaces. 33

2.1. Maximal orders. 35

2.2. Complete regular local ring of dimension. 38

2.3. Patching and Hasse principle. 48

Chapter 3. Springer's problem for odd degree extensions. 57

3.1. Reduction to the residue field. 58

3.2. Springer's theorem over local or global fields. 59

3.3. Springer's theorem over function fields of p-adic curves. 61

Chapter 4. Hermitian u-invariants. 63

4.1. Hermitian u-invariants over complete discrete valued fields. 64

4.2. Division algebras over Ai(2)-fields. 72

4.3. Division algebras over semi-global fields. 75

4.4. Tensor product of quaternions over arbitrary fields. 77

Bibliography. 81

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