Let k be an algebraically closed field of characteristic p>0. Let X be a smooth projective curve of genus g over k, let B be a non-empty set of r points of X, and let U=X-B. For a finite group G, let p(G) denote the subgroup generated by the p-subgroups of G. Abhyankar's conjecture for affine curves. The group G is the Galois group of an unramified cover of U if and only if G/p(G) has a generating set of size at most 2g+r-1. This conjecture is presented in the paper as proved by Serre, Raynaud, and Harbater, so its status is solved.

Abhyankar's conjecture for affine curves

Abhyankar posed the conjecture in 1957. It characterizes exactly which finite groups occur as Galois groups of unramified covers of an affine curve, using the genus, the number of deleted points, and the group's contribution from characteristic- $p$ subgroups.

Known results

Serre, 1990: solved the soluble-group case for the affine line.
Raynaud, 1994: proved the affine-line case in full.
Harbater, 1994: extended the result to arbitrary smooth affine curves using formal patching.
The necessary direction follows from Grothendieck's tame fundamental-group result.

1994 resolution

The cited literature reports that Raynaud and Harbater proved the stated criterion, with stronger versions controlling ramification at a selected boundary point. Later expositions from 2017--2020 continue to describe the conjecture as affirmatively solved; the earlier 2014 source's contrary “open” label is superseded.

Current status (as of September 2026): the conjecture is reported as proved for all affine curves, with no remaining open case identified in the supplied sources.

Abhyankar's conjecture for affine curves

Primary source

Progress summary

Solutions 0