Abstract. The famous theorem of Amitsur characterizes finite-dimensional central division algebras over a given field that have the same splitting fields, including infinite-dimensional ones. The situation changes dramatically if one allows only finite-dimensional splitting fields or just the maximal subfields of the division algebras at hand. To quantify these issues, one defines the genus gen(D) of a central division algebra D of degree n over a field K as the set of classes [D'] in the Brauer group Br(K) represented by a central division K-algebra D' of degree n having the same maximal subfields as D. I will review the results on gen(D) obtained in the last several years, including the finiteness theorem for gen(D) when K is a finitely generated field. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups and report on some recent progress in this direction.
On the notion of genus for division algebras and algebraic groups
Date
Location
Amitsur Memorial Symposium 2020, Open University of Israel