Abstract

The étale fundamental group  in algebraic geometry formalizes an analogy between Galois theory and topology, extending our intuition to spaces in which loops, as defined traditionally, do not yield meaningful information. For a curve X over an algebraically closed field of characteristic 0, finite quotients of  can be described solely in topological terms, but in characteristic p, dramatic differences and new phenomena have inspired many conjectures, including Abhyankar's conjectures. Let k be an algebraically closed field of characteristic p and let X be the projective line over k with three points removed. In joint work with Booher, Chen, and Liu, we show that for each prime p ≥ 5, there are families of tamely ramified covers with monodromy the symmetric group S_n or alternating group A_n for infinitely many n, producing these covers from moduli spaces of elliptic curves, and relating the fiber of these covers to the Markoff surface.

Link information

https://us02web.zoom.us/j/81197758280?pwd=MnFRRVN4N1pNNUdSeDU1Wm9YTzMzUT09

Meeting ID: 811 9775 8280
Passcode: ag****ncts     (**** = Euler number of a smooth quintic threefold)