Fundamental Groups of Curves in Characteristic p

David Harbater
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David Harbater: University of Pennsylvania, Mathematics Department

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 656-666 from Springer

Abstract: Abstract Consider the following general problem: Given a smooth affine curve U over an algebraically closed field k, find the fundamental group πA(U). When k = C, U is a Riemann surface, and π1 can be computed using loops. If U is obtained by deleting S = {ξ0, ⋯, ξ r } from a compact Riemann surface X of genus g, we thus obtain classically that π1 has generators a1, ⋯, a g , b1, ⋯, bg, ⋯, c0, ⋯, c r subject to the single relation $$\Pi _{j = 1}^g[{a_j},{b_j}]\Pi _{i = 0}^r{c_i} = 1$$ Π j = 1 g [ a j , b j ] Π i = 0 r c i = 1 . This is isomorphic to the free group on 2g + r generators, so πA (U) is the set of finite groups with 2g + r generators. Thus, these are the Galois groups of finite unramified Galois covers of U, or equivalently of finite branched covers of X with branch locus disjoint from U.

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_59

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