Correction to: Canonical nonclassical Hopf-Galois module structure of nonabelian Galois extensions

We regret that we must issue a correction to the above article, which slightly reduces the scope of the main theorem.     

Galois structure and de Rham invariants of elliptic curves

Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois étale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive genus, we show that a given étale action of G on X extends to a numerically tame action on a regular model if and only if this is possible on the minimal model. Finally, we characterize the classes in Cl(OK[G]) which are realizable as the de Rham invariants for minimal models of elliptic curves when G has prime order.     

Hopf Galois扩张与Hopf模结构定理

设Ｈ是域ｋ上任意的Ｈｏｐｆ代数。本文首先讨论了右Ｈ＿扩张Ａ／Ａ￣（ｃｏＨ）与Ｈｏｐｆ模范畴，给出了Ａ／Ａ￣（ｃｏＨ）为右Ｈ－Ｇａｌｏｉｓ扩张的充分必要条件和Ｈｏｐｆ模范畴满足结构定理的若干等价条件．然后我们讨论了不可约作用与除环的Ｇａｌｏｉｓ扩张．     

Densities of quartic fields with even Galois groups

Let be the number of degree number fields with Galois group and whose discriminant satisfies . Under standard conjectures in diophantine geometry, we show that , and that there are monic, quartic polynomials with integral coefficients of height whose Galois groups are smaller than , confirming a question of Gallagher. Unconditionally we have , and that the -class groups of almost all Abelian cubic fields have size . The proofs depend on counting integral points on elliptic fibrations.

     

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

A module structure on cyclic cohomology of group graded algebras

Letk be a field of characteristic 0, and letB be an algebra overk which is graded by a discrete groupG. Let HC*(A) denote the cyclic cohomology of an algebraA overk. We prove that there is an HC*(kG)-module structure on HC*(B) which generalizes Connes' periodicity operator on HC*(B). This module structure also decomposes with respect to conjugacy classes and results in a natural generalization of the results of Burghelea and Nistor in the cases of group algebras and algebraic crossed product algebras, respectively. Moreover, the proofs given in this paper are purely analytic with explicit constructions which can be used in the calculation of the cyclic cohomology of topological twisted crossed product algebras.Research sponsored in part by NSF Grant DMS-9204005.