Capitulation for locally free class groups of orders of group algebras over number fields

We prove a capitulation result for locally free class groups of orders of group algebras over number fields. As a corollary, we obtain an "arithmetically disjoint capitulation result" for the Galois module structure of rings of integers.Comment: 9 pages, 7 figures, uses xy-pic. v2 covers the non-abelian case. v3 includes numerous minor corrections following referee's report. v4 corrects error in Remark 1.

An Equivariant Main Conjecture in Iwasawa Theory and Applications

We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in the 1970s and studied extensively from a Galois module structure point of view in our recent work. We prove that the new Iwasawa modules are of projective dimension 1 over the appropriate profinite group rings. In the abelian case, we prove an Equivariant Main Conjecture, identifying the first Fitting ideal of the Iwasawa module in question over the appropriate profinite group ring with the principal ideal generated by a certain equivariant p-adic L-function. This is an integral, equivariant refinement of the classical Main Conjecture over totally real number fields proved by Wiles in 1990. Finally, we use these results and Iwasawa co-descent to prove refinements of the (imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away from their 2-primary components, in the most general number field setting. All of the above is achieved under the assumption that the relevant prime p is odd and that the appropriate classical Iwasawa mu-invariants vanish (as conjectured by Iwasawa.)Comment: 52 page

Minimal Hopf-Galois Structures on Separable Field Extensions

In Hopf-Galois theory, every $H$-Hopf-Galois structure on a field extension $K/k$ gives rise to an injective map $\mathcal{F}$ from the set of $k$-sub-Hopf algebras of $H$ into the intermediate fields of $K/k$. Recent papers on the failure of the surjectivity of $\mathcal{F}$ reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. This paper surveys and illustrates group-theoretical methods to determine $H$-Hopf-Galois structures on finite separable extensions in the extreme situation when $H$ has only two sub-Hopf algebras.Comment: 10 page

On totally real Hilbert-Speiser Fields of type C_p

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote the cyclic group of prime order p. We show that if p >= 7 (or p=5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C_p.Comment: 8 pages, latex, minor revisions following referee's repor