isAnisotropic -- determines whether a Grothendieck-Witt class is anisotropic

Synopsis

• Usage:

  isAnisotropic beta

• Inputs:
  • beta, a Grothendieck-Witt Class, denoted by $\beta\in\text{GW}(k)$, where $k$ is $\mathbb{Q}$, $ \mathbb{R}$, $\mathbb{C}$, or a finite field of characteristic not 2
• Outputs:
  • a Boolean value, whether $\beta$ is anisotropic

Description

Recall a symmetric bilinear form $\beta$ is said to be isotropic if there exists a nonzero vector $v$ for which $\beta(v,v) = 0$. Witt's decomposition theorem implies that a non-degenerate symmetric bilinear form decomposes uniquely into an isotropic and an anisotropic part. Certifying (an)isotropy is then an important computational problem when working with the Grothendieck-Witt ring.

Over $\mathbb{C}$, any form of rank two or higher contains a copy of the hyperbolic form, and hence is isotropic. Thus we can determine anisotropy simply by a consideration of rank.

i1 : isAnisotropic makeGWClass matrix(CC, {{3}})

o1 = true

i2 : isAnisotropic makeGWClass matrix(CC, {{2,0},{0,5}})

o2 = false

Forms over $\mathbb{R}$ are anisotropic if and only if all its diagonal entries are positive or are negative.

i3 : isAnisotropic makeGWClass matrix(RR, {{3,0,0},{0,5,0},{0,0,7}})

o3 = true

i4 : isAnisotropic makeGWClass matrix(RR, {{0,2},{2,0}})

o4 = false

Over finite fields, a form is anisotropic so long as it is nondegenerate, of rank $\le 2$ and not isomorphic to the hyperbolic form.

i5 : isAnisotropic makeGWClass matrix(GF(7), {{1,0,0},{0,1,0},{0,0,1}})

o5 = false

i6 : isAnisotropic makeGWClass matrix(GF(7), {{3,0},{0,3}})

o6 = true

Over $\mathbb{Q}$ things become a bit more complicated. We can exploit the local-to-global principle for isotropy (the Hasse-Minkowski principle), which states that a form is isotropic over $\mathbb{Q}$ if and only if it is isotropic over all its completions, meaning all the $p$-adic numbers and $\mathbb{R}$ [L05, VI.3.1]. We note, however, the classical result that all forms of rank $\ge 5$ in $\mathbb{Q}_p$ are isotropic [S73, IV Theorem 6]. Thus isotropy in this range of ranks is equivalent to checking it over the real numbers.

i7 : beta = makeGWClass matrix(QQ, {{1,0,2,0,3},{0,6,1,1,-1},{2,1,5,2,0},{0,1,2,4,-1},{3,-1,0,-1,1}});

i8 : isAnisotropic beta

o8 = false

i9 : getDiagonalClass beta

o9 = | 1 0 0  0  0   |
     | 0 6 0  0  0   |
     | 0 0 30 0  0   |
     | 0 0 0  -5 0   |
     | 0 0 0  0  671 |

o9 : GrothendieckWittClass

For forms of rank $\le 4$ the problem reduces to computing the maximum anisotropic dimension of the form over local fields. Ternary forms are isotropic away from primes dividing the coefficients of the form in a diagonal basis by e.g. [L05, VI.2.5(2)], so there are only finitely many places to check. Over these relevant primes, isotropy of a form $\beta \in \text{GW}(\mathbb{Q})$ over $\mathbb{Q}_p$ is equivalent to the statement that $(-1,-\text{disc}(\beta))_p = H(\beta)$ where $H(\beta)$ denotes the Hasse-Witt invariant attached to $\beta$ and $(-,-)_p$ is the Hilbert Symbol.

A binary form $q$ is isotropic if and only if it is isomorphic to the hyperbolic form, which implies in particular that the rank, signature, and discriminant of $q$ agree with that of $\mathbb{H}=\langle 1,-1\rangle$.

Citations:

• [S73] J.P. Serre, A course in arithmetic, Springer-Verlag, 1973.
• [L05] T.Y. Lam, Introduction to quadratic forms over fields, American Mathematical Society, 2005.

See also

• isIsotropic -- determines whether a Grothendieck-Witt class is isotropic
• getAnisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form
• getAnisotropicPart -- returns the anisotropic part of a Grothendieck-Witt class
• getWittIndex -- returns the Witt index of a symmetric bilinear form