On wild ramification in quaternion extensions

Abstract

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain -1 (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare and can occur only when the refined ramification filtration is, in two different ways, extreme.