## Outline of Professor Hanke's Workshop Lectures- Lecture 1: Quadratic Forms and Equivalence
- Definitions of Quadratic Forms over a ring, and expression in terms of matrices
- Quadratic forms over a field
- Orthogonal decomposition
- Isotropic/anisotropic subspaces and hyperbolic planes
- Witt's theorem
- Uniqueness up to isomorphism of the anisotropic subspace
- Lecture 2: Quadratic Forms over a local field
- Definitions of a local field
- Invariants of a p-adic quadratic space (up to isomorphism)
- Dimension and discriminant squareclass (with a warning about the varying definitions of discriminant)
- Definition of the Hilbert symbol and Hasse invariant
- Local invariants over
**R**and**C**
- Lecture 3: Quadratic forms over
**Q**and**Z**- Statement of the Hasse
*local-global*principle - Hasse principle holds over
**Q**(and over a number field or function field) - Failure of the Hasse principle over
**Z**and**Z**_{p} - Local normal forms over
**Z**_{p}and the Jordan decomposition - Class numbers and finiteness by reduction theory
- Statement of the Hasse
- Lecture 4: Miscellaneous Topics
- Generation of the orthogonal group by reflections
- Spinor norms and spinor genera
- Mass Formulas
- Theta functions and representations of forms by forms
- Binary quadratic forms, quadratic extensions and composition laws
The url of this page is fgarvan@math.ufl.edu |