Hasse invariant of a quadratic form

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"Hasse–Witt invariant" redirects here. For the invariant of algebraic curves, see Hasse–Witt matrix.

In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.

The quadratic form Q may be taken as a diagonal form

Σ aixi2.

Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras

(ai, aj) for i < j.

This is independent of the diagonal form chosen to compute it.[1]

It may also be viewed as the second Stiefel–Whitney class of Q.

Symbols[edit]

The invariant may be computed for a specific symbol φ taking values ±1 in the group C2.[2]

In the context of quadratic forms over a local field, the Hasse invariant may be defined using the Hilbert symbol, the unique symbol taking values in C2.[3] The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.[4]

For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.[5]

References[edit]

  1. ^ Lam (2005) p.118
  2. ^ Milnor & Husemoller (1973) p.79
  3. ^ Serre (1973) p.36
  4. ^ Serre (1973) p.39
  5. ^ Conner & Perlis (1984) p.16