We study the equivariant local epsilon constant conjecture, denoted by C na EP (N/K, V ), as formulated in various forms by Kato, Benois and Berger, Fukaya and Kato and others, for certain 1-dimensional twists T = Z p (χ nr )(1) of Z p (1). Following ideas of recent work of Izychev and Venjakob we prove that for T = Z p (1) a conjecture of Breuning is equivalent to C na EP (N/K, V ). As our main result we show the validity of C na EP (N/K, V ) for certain wildly and weakly ramified abelian extensions N/K. A crucial step in the proof is the construction of an explicit representative of RΓ(N, T ).

The Steinitz class of a number field extension K /k is an ideal class in the ring of integers O k of k, which, together with the degree [K : k] of the extension determines the O k -module structure of O K . We call R t (k, G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A -groups inductively, starting with abelian groups and then considering semidirect products of A -groups with abelian groups of relatively prime order and direct products of two A -groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A -groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine R t (k, D n ) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

We study the local epsilon constant conjecture as formulated by Breuning in [3]. This conjecture fits into the general framework of the equivariant Tamagawa number conjecture (ETNC) and should be interpreted as a consequence of the expected compatibility of the ETNC with the functional equation of Artin-L-functions.Let K/Q p be unramified. Under some mild technical assumption we prove Breuning's conjecture for weakly ramified abelian extensions N/K with cyclic ramification group. As a consequence of Breuning's local-global principle we obtain the validity of the global epsilon constant conjecture as formulated in [1] and of Chinburg's Ω(2)-conjecture as stated in [9] for certain infinite families F/E of weakly and wildly ramified extensions of number fields.

Given a finite group G and a number field k, a well-known conjecture asserts that the set Rt(k, G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper, we investigate an explicit candidate for Rt(k, G) when G is of odd order. More precisely, we define a subgroup W(k, G) of the class group of k and we prove Rt(k, G) ⊆ W(k, G). We show that equality holds for all groups of odd order for which a description of Rt(k, G) is known so far. Furthermore, by refining techniques introduced by Cobbe [Steinitz classes of tamely ramified galois extensions of algebraic number fields, J. Number Theory 130 (2010) 1129-1154], we use the Shafarevich-Weil Theorem in cohomological class field theory, to construct some tame Galois extensions with a given Steinitz class. In particular, this allows us to prove the equality Rt(k, G) = W(k, G) when G is a group of order dividing 4 , where is an odd prime.