Indices of inseparability and refined ramification breaks

Abstract

Let K be a finite extension of Q p which contains a primitive pth root of unity ζ p . Let L/K be a totally ramified (Z/pZ) 2 -extension which has a single ramification break b. In [2] Byott and Elder defined a "refined ramification break" b * for L/K. In this paper we prove that if p > 2 and the index of inseparability i 1 of L/K is not equal to p 2 b − pb then b * = i 1 − p 2 b + pb + b.In [1, 2], Byott and Elder described an alternative method for supplying missing ramification data by defining refined lower ramification breaks for extensions with fewer than n ordinary breaks. Suppose L/K is a totally ramified (Z/pZ) 2 -extension with a single (ordinary) ramification break b. Then L/K has one refined break b * , which is computed in [2] under the assumption that K contains a primitive pth root of unity. Byott and Elder also showed that the Galois module structure of O L determines b * in certain cases.In this paper we study the relationship between the index of inseparability i 1 of L/K and the refined ramification break b * . In particular, when p > 2 and i 1 = p 2 b − pb we give a formula which expresses b * in terms of i 1 . Our approach is based on the methods given in [8] for computing i 1 in terms of the norm group N L/K (L × ). We relate these methods to the Byott-Elder formula for b * using Vostokov's formula [9] for computing the Kummer pairing , p : K × × K × → µ p . The calculations are simplified somewhat through the use of the Artin-Hasse exponential series E p (X).The author would like to thank the referee for writing a very careful and thorough review of this paper.