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 [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

[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 $F$ of characteristic 2 in relation to the Witt algebra of quadratic inner product spaces over $F$.

References

[a1]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)