The ramification group filtrations of certain function field extensions

Jeffrey A. Castañeda and Qingquan Wu

Vol. 276 (2015), No. 2, 309–320

DOI: 10.2140/pjm.2015.276.309

Abstract

We investigate the ramification group filtration of a Galois extension of function fields, if the Galois group satisfies a certain intersection property. For finite groups, this property is implied by having only elementary abelian Sylow $p$-subgroups. Note that such groups could be nonabelian. We show how the problem can be reduced to the totally wild ramified case on a $p$-extension. Our methodology is based on an intimate relationship between the ramification groups of the field extension and those of all degree-$p$ subextensions. Not only do we confirm that the Hasse–Arf property holds in this setting, but we also prove that the Hasse–Arf divisibility result is the best possible by explicit calculations of the quotients, which are expressed in terms of the different exponents of all those degree-$p$ subextensions.

Keywords

function fields, ramification groups, filtrations

Mathematical Subject Classification 2010

Primary: 11R58

Secondary: 11R32, 11S15

Milestones

Received: 22 May 2014

Revised: 10 December 2014

Accepted: 10 February 2015

Published: 15 July 2015