Fields with bounded Brauer 2-torsion index
September 28, 2022
Online seminar Quadratic Forms, Linear Algebraic Groups and Beyond, Wedneys 8:30-9:30am,
Alexander Merkurjev’s groundbreaking results from the 1980ies showed that the u-invariant (maximal dimension of an anisotropic quadratic form) of a field is related to the degrees of division algebras of exponent 2. This led Bruno Kahn to conjecture that the u-invariant is bounded in terms of the 2-symbol length (the number of quaternion algebras necessary to represent an arbitrary element in the 2-torsion of the Brauer group). He showed that if every central simple algebra of exponent 2 over a field F of characteristic not 2 is equivalent to a tensor product of n quaternion algebras, then every (2n+2)-fold Pfister form over F is split after adjoining the square-root of -1.
In my talk I want to present a variation and refinement of this observation. Assuming that every central division algebra of exponent 2 over F has degree at most 2^n, I show that every (2n+2)-fold Pfister form is hyperbolic if -1 is a sum of squares in F and that it is in any case equal to 8 times a (2n-1)-fold Pfister form. The proof of this result is based on computations with trace forms of central simple algebras. Using this fact, one can now remove in a result of Daniel Krashen from 2016, which characterises fields for which all symbol lengths in the Milnor K-groups modulo 2 of a field are bounded, the condition that -1 be a square. In fact, these fields are the same as those with finite u-invariant (in the sense of Elman-Lam) and finite stability index. This latter result is joint work in progress with Saurabh Gosavi.