We establish a connection between the theory of cyclotomic ideal class groups and the theory of “geometric” Galois modules and obtain results on the Galois module structure of coherent cohomology groups of Galois covers of varieties over Z. In particular, we show that an invariant that measures the obstruction to the existence of a virtual normal integral basis for the coherent cohomology of su...

Mazur’s principle gives a criterion under which an irreducible mod l Galois representation arising from a classical modular form of level Np (with p prime to N) also arises from a classical modular form of level N . We consider the analogous question for Galois representations arising from certain unitary Shimura varieties. In particular, we prove an analogue of Mazur’s principle for U(2, 1) Sh...

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the p-adic group ring Zp[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h(Spec(L))(r),Z[G]). The same methods can also be applied to other conjectures concerning the vanishing of certa...

We prove a capitulation result for locally free class groups of orders of abelian group algebras over number fields. As a corollary, we obtain an “abelian arithmetically disjoint capitulation result” for the Galois module structure of rings of integers.

We determine the Hopf Galois structures on a field extension of degree twice an odd prime square and classify corresponding skew left braces. Besides we separable extensions allowing cyclic structure number these structures.

One of the fundamental questions in current field theory, related to Grothendieck’s conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro...

Example 1. Let us fix now an example which we will use throughout the paper; it is an abelian extension, so it does not have quite all of the structure of the general case, but it should still illustrate our results. Consider the extension Q(ζ15)/Q, which is a Galois extension of degree 8. We will just write ζ for ζ15 from now on. The Galois group Gal(Q(ζ)/Q) can be identified with (Z/15Z)∗, wh...

In this work, we are interested in the tame version of the Fontaine–Mazur conjecture. By viewing the pro-p-proup GS as a quotient of a Galois extension ramified at p and S, we obtain a connection between the conjecture studied here and a question of Galois structure. Moreover, following a recent work of A. Schmidt, we give some evidence of links between this conjecture, the étale cohomology and...

This article is concerned with proving a refined function field analogue of the Coates-Sinnott conjecture, formulated in the number field context in 1974. Our main theorem calculates the Fitting ideal of a certain even Quillen K-group in terms of special values of L-functions. The techniques employed are directly inspired by recent work of Greither and Popescu in the equivariant Iwasawa theory ...

Recall the premise of the previous lecture. We considered (A, i,Φ) an abelian variety with CM structure, defined over Q, which (with all the relevant data) descended to A0/K where K is a large enough finite Galois extension of E. For an element σ ∈ Gal(Q/E) the Galois twist (A0 , i,Φ) is another CM abelian variety of the same type, and Jeremy and Brandon constructed an explicit isogeny ξσ,P : A...