Generalized period-index problem with an application to quadratic forms

Let $F$ be the function field of a curve over a complete discretely valued field. Let $\mathrm{\ell }$ be a prime not equal to the characteristic of the residue field. Given a finite subgroup $B$ in the $\mathrm{\ell }$ torsion part of the Brauer group ${}_{\mathrm{\ell }}Br(F)$, we define the index of $B$ as the minimum of the degrees of field extensions which split all elements in $B$. In this manuscript, we give an upper bound for the index of any finite subgroup $B$ in terms of arithmetic invariants of $F$. As a simple application of our result, given a quadratic form $q\mathrm{/}F$, where $F$ is the function field of a curve over an $n$-local field, we provide an upper bound to the minimum of degrees of field extensions $L\mathrm{/}F$ so that the Witt index of $q\otimes L$ becomes the largest possible.