Number Theory Seminar

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

Date, time & place Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Naoki Imai, Shane Kelly

2015/02/18

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Piotr Achinger (University of California, Berkeley)
Wild ramification and $K(\pi, 1)$ spaces (English)
[ Abstract ]
A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.