Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Abstract

Let K be a finite extension of Q p , let L / K be a finite abelian Galois extension of odd degree and let O L be the valuation ring of L. We define A L / K to be the unique fractional O 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.