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# 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, , 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

## Ways to use HasseWittInvariant :

• HasseWittInvariant(GrothendieckWittClass,ZZ)
• HasseWittInvariant(List,ZZ)

## For the programmer

The object HasseWittInvariant is .