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

Erik Jarl Pickett

Published 2010-07-02Version 1

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.