Let $F$ be arbitrary field and $A$ a central simple (finite-dimensional) $F$-algebra. Knowing index of $A$ (i.e. $\sqrt{Dim_F D}$, where $A \cong M_r (D)$ and $D$ is underlying division algebra) how it changes under scalar extension is often an important source of structural imformation about $A$; but the index is often very difficult to compute, and $ind(D\otimes_F L)$ for fields $F\subseteq L$ even more so. Since $ind(D\otimes_F L) | ind(D)$, a formula for $ind(D\otimes_F L)$ is often called an index reduction formula.
A lot of index reduction formulas were found in case where $L$ are of special type for example in case they are function fields of some special projective varieties (Blanchet, Schofield, van den Bergh, Saltman, Tao, Wadsworth, Panin, Merkurjev). The significance of these results could be ullustrated for example by the fact they were used to solve the well known problem of the existence of fields with arbitrary even $u$-invariant.
Another problem closely related to searching of index reduction formulas is the problem of computation of indices of smooth projective curves. It is very complicated open problem even for special curves and their fields of definition. The case I mean to consider in my talk is the case of hyperelliptic curves defined over local fields. In this case (it is known by Lichtenbaum) one has that for a hyperelliptic curve $X$ defined over a local field $F$ the index of $X$ is the order of the kernel of the natural homomorphism from $BrF$ to $BrF(X)$, where $F(X)$ is the function field of $X$. (Of course if the last homomorphism is injective, so the corresponding index reduction formula is trivial, otherwise it is more complicated). In my talk I plan to give an information on my joint with J. van Geel results concerning indices of $X$ (to appear in Manuscripta Mathematica) and my own ones on index reduction formulas for such curves.