Introduction
xvii
The case of groups of type E 7 is of special interest. If the Tits algebra of a
semisimple simply connected k-group of type E 7 is of index ≤ 4, we prove the
conjecture in Theorem 8.3.2. The only remaining case is that of index 8, and we
show how this case is related to a problem about hermitian forms over biquaternion
algebras (Sect. 8.3.3). Finally, the case of groups of type E 8 , is discussed and related
to other open questions.
***
The contents of the monograph are as follows: The first chapter reviews some
basic material and concepts, such as Weil restriction and algebraic tori, etc.
Chapter 2 is devoted to Borel-Tits theory of reductive groups (parabolic subgroups,
root systems, etc.). The third chapter deals with the various embeddings of μ n in a
given semisimple algebraic group and splitting fields for algebraic groups. Chapter 4
focuses on fields of small separable cohomological dimension and norm groups of
certain varieties.
The study of Serre’s conjecture II starts in Chap. 5 and concludes in Chap. 8.
Finally, in Chap. 9, we deal with applications of the results and link conjecture II to
other open questions.