Twists of symmetric bundles

In 1984 J.-P. Serre proved a formula relating the Hasse-Witt invariant of the quadratic form $x \to Tr_{E/K}(x^2)$ associated to a finite separable extension of E  of a field K of characteristic $\neq$ 2 with the second Stiefel-Whitney class of the permutation representation of the Galois group of E. The similarity between this formula and the one he obtained when considering a covering of Riemann surfaces with odd ramification led him to pose the question of the existence of a general result containing the previous ones as particular cases. After recalling the for mulas of Fr\"hlich  and Serre for quadratic forms on fields,  we will  present results obtained in a joint work with B. Erez and M.J. Taylor. Working with symmetric bundles attached to certain etale or tame coverings of schemes, we will present formulas for the invariants attached to these forms which provide a positive answer to Serre"s question. These results generalize some previous work of Esnault, Kahn and Viehweg.