Date:
Mon, 18/11/202414:30-15:30
Location:
Ross 70
Title: On the Hahn-Witt series and their generalizations
Abstract: There is a canonical construction of an algebraic closure of $\Q_p,$ namely, the algebraic closure inside the field $HW(\bar{F}_p)$ of Hahn-Witt series of $\bar{F}_p.$ Informally, the Hahn-Witt series are expressions of the form $\sum_{i\in\Q} [a_i] p^i,$ where the set of $i$ such that $a_i
e 0$ is well-ordered, and $[a_i]$ are Teichmuller representatives. This field is equipped with a natural automorphism $\varphi,$ coming from the Frobenius automorphism of $\bar{F}_p.$ I will explain that the action of $\varphi$ on the $p$-power roots of unity is given by $\varphi(\zeta)=\zeta^{-1},$ answering a question of Kontsevich.
More generally, I will explain a similar result for an arbitrary non-archimedean local field $K$ with a chosen uniformizer $\pi.$ The proof uses only the standard facts from local class field theory, Lubin-Tate theory and wild ramification theory. The results are contained in the preprint https://arxiv.org/abs/2406.19163.
Zoom link
https://huji.zoom.us/j/88037279712?pwd=N3MwWW5RYzRTZHg4K0U2bS80Rmxjdz09