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
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
|||C. Arf, "Untersuchungen über quadratischen Formen in Körpern der Charakteristik 2, I" J. Reine Angew. Math. , 183 (1941) pp. 148–167|
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 .
|[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