unQVNTS (Vermont) - Fall 2024

Shilpi Mandal (Emory University)

October 24th, 2024, 3:15PM - 4:10PM

Title: Strong u-invariant and Period-Index Bounds

For a central simple algebra A over a field K, there are two major invariants, viz., period and index. For a field K, the Brauer-l-dimension of K for a prime number l, is the smallest natural number d such that for every finite field extension L/K and every central simple L-algebra A (of period a power of l), we have that index(A) divides period(A)^d.

If K is a number field or a local field, then classical results from class field theory tell us that the Brauer-l-dimension of K is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for K a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields K, Parimala-Suresh have given some bounds.

Also, the u-invariant of K is the maximal dimension of anisotropic quadratic forms over K. For example, the u-invariant of $\mathbb{C}$ is 1, for F a non-real global or local field the u-invariant of F is 1, 2, 4, or 8, etc.

In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field K with residue field $\kappa$.