Open problems on central simple algebras

Abstract

Abstract. We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners. Motivated by these and other developments, we now present an updated list of open problems. Some of these problems are-or have become-special cases of much more general problems for algebraic groups. To keep our task manageable, we (mostly) restrict our attention to central simple algebras and PGL n . The following is an idiosyncratic list that originated during the Brauer group conference held at Kibbutz Ketura in January 2010. We thank the participants in that problem session for their contributions to this text. We especially thank Darrel Haile,

On Brauer p-dimensions and index-exponent relations over finitely-generated field extensions

“…As in [2], Sect. 4, we say that a field E is of finite Brauer p-dimension Brd p (E) = n, for a fixed p ∈ P, if n is the least integer ≥ 0, for which ind(D) ≤ exp(D) n whenever D ∈ d(E) and [D] ∈ Br(E) p .…”

$E_8$, the most exceptional group

“…I]. Specifically, the Platonic solids correspond to the embeddings of the alternating group on four or five letters, PSL(2, 3) or PSL (2,5), and the symmetric group on five letters, PGL(2, 3), in SO(3), and these embeddings are part of a series of embeddings of such subgroups in simple Lie groups, including the case of E 8 (k); see [142] and [144].…”

“…The theta-series for this lattice, θ(q) = v∈Q q v 2 /2 , is the fourth Eisenstein series, which provides a connection with the j-invariant via the formula 24 , where η denotes Dedekind's eta-function. 5 For more on this lattice and how it fits into the rest of mathematics, see [38] or [52].…”

“…(See [5] for context and discussion.) Philippe Gille showed that this question, for the case p = 5, can be rephrased as a statement about groups of type E 8 ; see [80].…”

Signatures, sums of hermitian squares and positive cones on algebras with involution

“…The study of central simple algebras with involution was initiated by Albert in the 1930s [1] and is still a topic of current research as testified by The Book of Involutions [19]; see also [10] and the copious references therein for a list of open problems in this area. A large part of present day research in algebras with involution is driven by the deep connections with linear algebraic groups, first observed by Weil [35]; see also Tignol's 2 ECM exposition [34].…”