HasseWittInvariant -- outputs the Hasse-Witt invariant for a prime p for the quadratic form of the Grothendieck-Witt class

Synopsis

• Usage:

  HasseWittInvariant(beta, p)

• Inputs:
  • beta, a Grothendieck Witt Class, denoted by $\beta\in\text{GW}(\mathbb{Q})$
  • p, an integer, an integral prime number
• Outputs:
  • an integer, the Hasse-Witt invariant for $\beta$ for the prime $p$

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 $\prod_{i<j} \left((a_i,a_j)_p \right)$ 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 $\beta \cong \left\langle a_1,\ldots,a_n\right\rangle$ then 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 invariant at primes dividing diagonal entries.

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

i2 : HasseWittInvariant(beta, 7)

o2 = 1