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getHasseWittInvariant -- computes the Hasse-Witt invariant at a prime p for the quadratic form of the Grothendieck-Witt class

Description

The Hasse-Witt invariant of a diagonal form $\langle a_1,\ldots,a_n\rangle$ over a field $K$ is defined to be the product $\prod_{i<j} \left(a_i,a_j\right)_p $ where $(-,-)_p$ is the Hilbert symbol.

The Hasse-Witt invariant of a form will be equal to 1 for almost all primes. In particular, after diagonalizing a form $\beta \cong \left\langle a_1,\ldots,a_n\right\rangle$ the Hasse-Witt invariant at a prime $p$ will be 1 automatically if $p\nmid a_i$ for all $i$. Thus we only have to compute the Hasse-Witt invariant at primes dividing diagonal entries.

i1 : beta = makeGWClass matrix(QQ, {{1,4,7},{4,3,-1},{7,-1,5}});
i2 : getHasseWittInvariant(beta, 7)

o2 = 1

See also

Ways to use getHasseWittInvariant:

  • getHasseWittInvariant(GrothendieckWittClass,ZZ)
  • getHasseWittInvariant(List,ZZ)

For the programmer

The object getHasseWittInvariant is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/GWInvariantsDoc.m2:59:0.