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/Qp 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.

Sufficient conditions for large Galois scaffolds

“…It was shown by B. Erez [Ere91] that this ideal is locally free over the group ring O K [Gal(L/K)]. This led several subsequent authors (see for example [Vin03], [Pic09]) to investigate the square root of the inverse different in weakly ramified extensions, both of number fields and of local fields. The valuation ring, and its maximal ideal, in a weakly ramified (but not necessarily totally ramified) extension of local fields are studied as Galois modules in [Joh15].…”

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

“…From Lemma 4.2 we know that x un x tot will be a self-dual integral normal basis for A L ′ /K . From Lemma 4.3 we know that T r L ′ /L (x un x tot ) will be a self-dual element of L, and so using [16] Lemma 8 we just need to show that T r L ′ /L (x un x tot ) ∈ A L/K . It is therefore sufficient to show that T r L ′ /L (A L ′ /K ) ⊆ A L/K .…”

“…We use these methods to give explicit constructions for self-dual integral normal bases for A L/K whenever L/K is abelian and p = 2. We remark that some of our constructions still work for p = 2 and the only case needed for completeness is L/K unramified with [L : K] = r i where r is an odd prime and i ∈ N. We should also remark that the constructions in [5] and [16] are probably a lot more useful in terms of calculations of invariants such as resolvents and Galois Gauss sums.…”

“…The first is due to Erez, with L/Q p and L contained in some cyclotomic extension of Q p , [5]. The second is due to the author and is a generalisation of Erez's results using Dwork's power series in extensions of local fields generated by Lubin-Tate formal groups, [16]. Specifically, the author studies cyclic degree p extensions, M/K, of local fields where K/Q p is unramified.…”

Formal Group Exponentials and Galois Modules in Lubin–Tate Extensions

“…The main motivation for this paper came from new progress in the theory of Galois module structure. Indeed, explicit descriptions of local integral Galois module generators due to Erez [5] and Pickett [16] have recently been used to make progress with open questions on integral Galois module structure in wildly ramified extensions of number fields (see [18] and [23]).…”