MATH Seminar
| Title: Strong $u$-invariant and Period-Index bound |
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| Seminar: Algebra and Number Theory |
| Speaker: Shilpi Mandal of Emory University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a central simple algebra $A$ over $K$, there are two major invariants, viz., \textit{period} and \textit{index}.\\ \\ For a field $K$, define the \emph{Brauer $l$-dimension of $K$} for a prime number $l$, denoted by $\mathrm{Br}_l\mathrm{dim}(K)$, as the smallest $d \in \mathbb{N} \cup \{\infty\}$ 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 $\mathrm{ind}(A)$ divides $\mathrm{per}(A)^d$.\\ \\ If $K$ is a number field or a local field (a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$, for some prime number $p$), then classical results from class field theory tell us that $\mathrm{Br}_l\mathrm{dim}(K) = 1$. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Lieblich, 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 $K$, denoted by $u(K)$, is the maximal dimension of anisotropic quadratic forms over $K$. For example, $u(\mathbb{C}) = 1$; for $F$ a non-real global or local field, we have $u(F) = 1, 2, 4,$ or $8$, etc.\\ \\ In this talk, I will present similar bounds for the $\mathrm{Br}_l\mathrm{dim}$ and the strong $u$-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$. |
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