relevantPrimes -- outputs a list of primes at which the Hasse-Witt invariants of a symmetric bilinear form may be non-trivial

Synopsis

• Usage:

  relevantPrimes(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, denoted by $\beta\in\text{GW}(\mathbb{Q})$
• Outputs:
  • a list, a finite list of primes $(p_1,\ldots,p_r)$ for which the Hasse-Witt invariants $\phi_p(\beta)$ may be nontrivial

Description

It is a classical result that the Hasse-Witt invariants of a quadratic form are equal to 1 for all but finitely many primes (see e.g. [S73, IV Section 3.3]. As the Hasse-Witt invariants are computed as a product of Hilbert symbols of the pairwise entries appearing on a diagonalization of the symbol, it suffices to consider primes dividing diagonal entries.

i1 : beta = diagonalForm(QQ,(6,7,22));

i2 : relevantPrimes(beta)

o2 = {2, 3, 7, 11}

o2 : List

Citations:

• [S73] J.P. Serre, A course in arithmetic, Springer-Verlag, 1973.