Local Brauer invariant of a quaternion algebra

For a quaternion algebra $B$ over a local field $F$ of characteristic different from two, its local Brauer invariant is

\operatorname{inv}_F(B)= \begin{cases} 0 & B\cong M_2(F),\\ \tfrac12\pmod{\mathbb Z} & B\text{ is a division algebra}, \end{cases} \qquad\text{in }\mathbb Q/\mathbb Z.

It is the restriction of the local invariant map on the Brauer group. For $F=\mathbb C$ , only the value zero occurs.

Sign versus additive invariant

When $\operatorname{char}F\ne2$ , the Hilbert symbol of a presentation takes the value $+1$ for invariant zero and $-1$ for invariant $\tfrac12$ . The additive invariant is not the sign itself.

Global consistency

For a quaternion algebra over a number field $K$ , the invariants of $B\otimes_KK_v$ sum to zero in $\mathbb Q/\mathbb Z$ . This expresses the even parity of the ramification set.

The rational Hamilton algebra has local invariant $\tfrac12$ at $2$ and at $\infty$ , and zero elsewhere. Their sum is $1$ , which is zero modulo $\mathbb Z$ .

References

John Voight, Quaternion Algebras, author's text, Remark 13.4.3 and Remark 14.6.10, especially the quaternion specialization following (14.6.11).