HilbertSymbolReal -- computes the Hilbert symbol of two rational numbers over the reals

Synopsis

• Usage:

  HilbertSymbolReal(a,b,p)

• Inputs:
  • a, a rational number, any non-zero integer or rational number, considered as an element of $\mathbb{Q}_p$
  • b, a rational number, any non-zero integer or rational number, considered as an element of $\mathbb{Q}_p$
• Outputs:
  • an integer, the Hilbert symbol $(a,b)_{\mathbb{R}}$

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 $\prod_{i<j} \phi(a_i,a_j)$ where $\phi \colon K \times K \to \left\{\pm 1\right\}$ is any symbol (see e.g. [MH73, III.5.4] for a definition). It is a classical result of Hilbert that over a local field of characteristic not equal to two, there is one and only symbol, $(-,-)_p$ called the Hilbert symbol ([S73, Chapter III]) computed as follows:

$(a,b)_{\mathbb{R}} = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } {\mathbb{R}}^3 \\ -1 & \text{otherwise.} \end{cases}$

$(a,b)_{\mathbb{R}}$ will equal 1 unless both $a,\,b$ are negative.

Consider the example, that $z^2=-3x^2-2y^2/3$ does not admit a non-zero solution. Thus:

i1 : HilbertSymbolReal(-3,-2/3) == -1

o1 = true

Computing Hasse-Witt invariants is a key step in classifying symmetric bilinear forms over the rational numbers, and in particular certifying their (an)isotropy.

Citations:

• [S73] J.P. Serre, A course in arithmetic, Springer-Verlag, 1973.
• [MH73] Milnor and Husemoller, Symmetric bilinear forms, Springer-Verlag, 1973.