Arf-invariant
invariant of Arf
An invariant of a quadratic form modulo 2, given on an integral lattice endowed with a bilinear skew-symmetric form. Let be an integral lattice of dimension
and let
be a form for which
. There exists bases
, called symplectic bases, in which the matrix of
reduces to block-diagonal form: The diagonal contains the blocks
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i.e.
![]() |
while the other entries are zero.
Suppose that a mapping
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is given on such that
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(a "quadratic form modulo 2" ). The expression
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is then called an Arf-invariant [1]. If this expression equals zero, then there is a symplectic basis on all elements of which vanishes; if this expression equals one, then there is a symplectic basis on all elements of which, except
and
, the form
vanishes, while
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References
[1] | C. Arf, "Untersuchungen über quadratischen Formen in Körpern der Charakteristik 2, I" J. Reine Angew. Math. , 183 (1941) pp. 148–167 |
Comments
See [a1], Appendix 1, for material concerning the Arf-invariant for inner product spaces over a field of characteristic 2 in relation to the Witt algebra of quadratic inner product spaces over
.
References
[a1] | J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) |
Arf-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arf-invariant&oldid=12338