Explicit Construction of Self-Dual Integral Normal Bases for the Square-Root of the Inverse Different

TL;DR

This paper explicitly constructs self-dual integral normal bases for the square-root of the inverse different in certain abelian extensions of local fields, using Lubin-Tate formal groups and Dwork's exponential series.

Contribution

It provides an explicit construction of self-dual integral normal bases in new classes of abelian extensions of local fields, extending previous results to more general settings.

Findings

Explicit bases constructed for specific abelian extensions

Use of Dwork's exponential series in basis construction

Extension of known results to weakly ramified cases

Abstract

Let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the inverse different of $L/K$. For $p$ an odd prime and $L/\Q_p$ contained in certain cyclotomic extensions, Erez has described integral normal bases for $A_{L/\Q_p}$ that are self-dual with respect to the trace form. Assuming $K/\Q_p$ to be unramified we generate odd abelian weakly ramified extensions of $K$ using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.