Katherine F. Stevenson

Abhyankar’s Local Conjecture on Fundamental Groups

This paper proves the remaining open case of Abhyankars higher dimensional conjecture on local fundamental groups in characteristic p ((Ab2), (Ab3)). This conjecture, which is analogous to Abhyankars conjectures on global fundamental groups, proposed that a finite group G is a Galois group over k((x1 ,...,x n))((x1 ··· xr) −1 ) if and only if its maximal prime-to-p quotient is, pro- vided n ≥ 2a nd 1≤ r ≤ n .F or r> 1, this conjecture was disproven in (HP). Here we prove that the conjecture is true in the case r = 1. So the Galois groups over k((x1 ,...,x n))(x −1 1 ) are precisely the cyclic-by-quasi-p groups.