Serre’s Conjecture II

Serre's Conjecture II predicts that if $k$ is a perfect field whose Galois cohomological dimension is at most $2$, and $G$ is a semisimple simply connected linear algebraic group over $k$, then the Galois cohomology set $H^1(k,G)$ is trivial; equivalently, every $G$-torsor over $\operatorname{Spec}k$ has a $k$-rational point. Serre formulated the conjecture in his 1962 course on Galois cohomology [Serre1962Cohomologie].

The conjecture has been established for local fields and for totally imaginary global fields, and for several classes of geometric fields as well as for many types of groups; Gille's survey gives a detailed account of the known cases and of the case-by-case methods, following the classification of semisimple groups, that produced them [Gille2010SerreII]. More recently, transfer principles in Galois cohomology have been developed to propagate known cases to new fields [IzquierdoLucchiniArteche2025].

The conjecture remains open in general. The outstanding cases include groups of exceptional type over arbitrary fields of cohomological dimension $2$, where no uniform argument for the vanishing of $H^1$ is available.