Cohomology classes on moduli of curves from Theta Characteristics

Using Galois-Stiefel-Whitney classes of theta characteristics we show that over a totally real base field the moduli stack of smooth genus $g$ curves and the moduli stack of principally polarized abelian varieties of dimension $g$ have nontrivial cohomological invariants and \'etale cohomology classes in degree respectively $2^{g-2}, 2^{g-1}$ and $2^{g-1}$. We also compute the pullback from the Brauer group of $\mathcal{M}_3$ to that of $\mathcal{H}_3$ over a general field of characteristic different from $2$.