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''invariant of Arf''
 
''invariant of Arf''
  
An invariant of a quadratic form modulo 2, given on an integral lattice endowed with a bilinear skew-symmetric form. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132301.png" /> be an integral lattice of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132302.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132303.png" /> be a form for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132304.png" />. There exists bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132305.png" />, called symplectic bases, in which the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132306.png" /> reduces to block-diagonal form: The diagonal contains the blocks
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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
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132307.png" /></td> </tr></table>
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\left({\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}}\right)
 
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$$
 
i.e.
 
i.e.
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132308.png" /></td> </tr></table>
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\psi(e_i,f_i) = -\psi(f_i,e_i)  = 1
 
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$$
 
while the other entries are zero.
 
while the other entries are zero.
  
 
Suppose that a mapping
 
Suppose that a mapping
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a0132309.png" /></td> </tr></table>
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\psi_0 : L \rightarrow \mathbf{Z}/2\mathbf{Z}
 
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$$
is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323010.png" /> such that
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is given on $L$ such that
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323011.png" /></td> </tr></table>
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\psi_0(x+y) = \psi_0(x) + \psi_0(y) + \psi(x,y)\ \pmod2
 
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$$
 
(a  "quadratic form modulo 2" ). The expression
 
(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
 +
$$
 +
\psi_0(e_1) = \psi_0(f_1) = 1 \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323012.png" /></td> </tr></table>
 
 
is then called an Arf-invariant [[#References|[1]]]. If this expression equals zero, then there is a symplectic basis on all elements of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323013.png" /> vanishes; if this expression equals one, then there is a symplectic basis on all elements of which, except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323015.png" />, the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323016.png" /> vanishes, while
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323017.png" /></td> </tr></table>
 
  
 
====References====
 
====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>
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<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====
 
====Comments====
See [[#References|[a1]]], Appendix 1, for material concerning the Arf-invariant for inner product spaces over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323018.png" /> of characteristic 2 in relation to the Witt algebra of quadratic inner product spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013230/a01323019.png" />.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Milnor,  D. Husemoller,  "Symmetric bilinear forms" , Springer  (1973)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 07:17, 24 December 2017

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)
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