In this article we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application we give a purely algebraic proof of some 2-adic theta identities which describe the set of theta null points...

We construct examples of K3 surfaces of geometric Picard rank 1. Our method is a refinement of that of R. van Luijk [vL]. It is based on an analysis of the Galois module structure on étale cohomology. This allows to abandon the original limitation to cases of Picard rank 2 after reduction modulo p. Furthermore, the use of Galois data enables us to construct examples which require significantly ...

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F . We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.

This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over Fq (in defining characteristic) and derives explicit additive polynomials for the extensions. Our unified approach is based on results of Matzat which give bounds for Galois groups of Frobenius modules and uses the structure and representation theory of th...

The topology of an elementary abelian extension of local fields with one ramification break is, since there is only one break, rather symmetric with respect to Galois action. In this paper, we consider a particularly symmetric sub-class, which we call one-dimensional and in characteristic p is linked to the Artin-Schreier equation xp f −x = β. The utility of this additional symmetry is illustra...

Abstract. We define a variant of normal basis, called a Galois scaffolding, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic p, called one-dimensional, that, in a particular sense, are as simple as cyclic degree p extensions, and prove the sta...

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodu...

We introduce a new class of noncommutative rings-Galois orders, realized as certain subrings of invariants in skew semigroup rings, and develop their structure theory. The class of Galois orders generalizes classical orders in noncommutative rings and contains many classical objects, such as the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra, assoc...