# Arf-invariant

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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 i.e. while the other entries are zero.

Suppose that a mapping is given on such that (a "quadratic form modulo 2" ). The expression is then called an Arf-invariant . 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 How to Cite This Entry:
Arf-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arf-invariant&oldid=12338
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article