A complete description of the cohomological invariants of even genus hyperelliptic curves

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of ${\mathcal{H}}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification ${\bar{\mathcal{H}}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification ${\bar{\mathcal{H}}}_g$ and extend it to positive characteristic