Renee Bell, University of Pennsylvania
Authors: Renee Bell, Jeremy Booher, Will Chen, Yuan Liu
2022 AWM Research Symposium
WiAG: Women in Algebraic Geometry

The étale fundamental group $\pi_1^{et}$ 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 $\pi_1^{et}$ 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.

Back to Search Research Symposium Abstracts