Hasse invariant of a quadratic form

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.

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

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

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
  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
  • Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017.
  • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
  • Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
  • O'Meara, O.T. (1973). Introduction to quadratic forms. Die Grundlehren der mathematischen Wissenschaften. 117. Springer-Verlag. ISBN 3-540-66564-1. Zbl 0259.10018.
  • Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.
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