View source for 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 $L$ be an integral lattice of dimension $k=2m$ and let $\psi$ be a form for which $\psi(x,y) = -\psi(y,x)$. There exists bases $(e_1,f_1,\ldots,e_m,f_m)$, called symplectic bases, in which the matrix of $\psi$ reduces to block-diagonal form: The diagonal contains the blocks
$$
\left({\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}}\right)
$$
i.e.
$$
\psi(e_i,f_i) = -\psi(f_i,e_i)  = 1
$$
while the other entries are zero.

Suppose that a mapping
$$
\psi_0 : L \rightarrow \mathbf{Z}/2\mathbf{Z}
$$
is given on $L$ such that
$$
\psi_0(x+y) = \psi_0(x) + \psi_0(y) + \psi(x,y)\ \pmod2
$$
(a  "quadratic form modulo 2" ). The expression
$$
\sum_{i=1}^m \psi_0(e_i)\psi_0(f_i)
$$
is then called an Arf-invariant [[#References|[1]]]. If this expression equals zero, then there is a symplectic basis on all elements of which $\psi_0$ vanishes; if this expression equals one, then there is a symplectic basis on all elements of which, except $e_1$ and $f_1$, the form $\psi_0$ vanishes, while
$$
\psi_0(e_1) = \psi_0(f_1) = 1 \ .
$$


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  C. Arf,   "Untersuchungen über quadratischen Formen in Körpern der Charakteristik 2, I"  ''J. Reine Angew. Math.'' , '''183'''  (1941)  pp. 148–167</TD></TR>
</table>



====Comments====
See [[#References|[a1]]], Appendix 1, for material concerning the Arf-invariant for inner product spaces over a field $F$ of characteristic 2 in relation to the Witt algebra of quadratic inner product spaces over $F$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,   D. Husemoller,   "Symmetric bilinear forms" , Springer  (1973)</TD></TR>
</table>

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