Outline of Professor Hanke's Workshop Lectures

  • Lecture 1: Quadratic Forms and Equivalence
    1. Definitions of Quadratic Forms over a ring, and expression in terms of matrices
    2. Quadratic forms over a field
    3. Orthogonal decomposition
    4. Isotropic/anisotropic subspaces and hyperbolic planes
    5. Witt's theorem
    6. Uniqueness up to isomorphism of the anisotropic subspace

  • Lecture 2: Quadratic Forms over a local field
    1. Definitions of a local field
    2. Invariants of a p-adic quadratic space (up to isomorphism)
    3. Dimension and discriminant squareclass (with a warning about the varying definitions of discriminant)
    4. Definition of the Hilbert symbol and Hasse invariant
    5. Local invariants over R and C

  • Lecture 3: Quadratic forms over Q and Z
    1. Statement of the Hasse local-global principle
    2. Hasse principle holds over Q (and over a number field or function field)
    3. Failure of the Hasse principle over Z and Zp
    4. Local normal forms over Zp and the Jordan decomposition
    5. Class numbers and finiteness by reduction theory

  • Lecture 4: Miscellaneous Topics
    1. Generation of the orthogonal group by reflections
    2. Spinor norms and spinor genera
    3. Mass Formulas
    4. Theta functions and representations of forms by forms
    5. Binary quadratic forms, quadratic extensions and composition laws

The url of this page is http://qseries.org/fgarvan/quadformsconf/workshop-program/hanke-lectures.html.
Created by F.G. Garvan (fgarvan@math.ufl.edu) on Wednesday, February 04, 2009.
Last update made Wed Feb 4 14:55:39 EST 2009.

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