In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hasse invariant (or Hasse–Witt invariant) of a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
''Q'' over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' takes values in the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:-
* Alfred Brauer (1894–1985), German-American mathematician, brother of Richard
* Andreas Brauer (born 1973), German film producer
* Arik ...
Br(''K''). The name "Hasse–Witt" comes from
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
and
Ernst Witt.
The quadratic form ''Q'' may be taken as a
diagonal form
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is
:\sum_^n a_i ^m\
for some given degree ''m''.
Such forms ''F'', and the hypersurfaces ''F'' ...
:Σ ''a''
''i''''x''
''i''2.
Its invariant is then defined as the product of the classes in the Brauer group of all the
quaternion algebras
:(''a''
''i'', ''a''
''j'') for ''i'' < ''j''.
This is independent of the diagonal form chosen to compute it.
[Lam (2005) p.118]
It may also be viewed as the second
Stiefel–Whitney class of ''Q''.
Symbols
The invariant may be computed for a specific
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
φ taking values in the group C
2 = .
[Milnor & Husemoller (1973) p.79]
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 C
2.
[Serre (1973) p.36] The invariants of a quadratic forms over a local field are precisely the dimension,
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
and Hasse invariant.
[Serre (1973) p.39]
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
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
s coming from real embeddings.
[Conner & Perlis (1984) p.16]
See also
* Hasse–Minkowski theorem
References
*
*
*
*
* {{cite book , first=Jean-Pierre , last=Serre , authorlink=Jean-Pierre Serre , title=A Course in Arithmetic , series=
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard s ...
, volume=7 , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, year=1973 , isbn=0-387-90040-3 , zbl=0256.12001 , url-access=registration , url=https://archive.org/details/courseinarithmet00serr
Quadratic forms