Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.

The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri–Iwaniec–Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves.

This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri–Iwaniec–Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

We study Rees algebras of modules within a fairly general framework. We introduce an approach through the notion of Bourbaki ideals that allows the use of deformation theory. One can talk about the (essentially unique) generic Bourbaki ideal I(E) of a module E which, in many situations, allows one to reduce the nature of the Rees algebra of E to that of its Bourbaki ideal I(E). Properties such as Cohen–Macaulayness, normality and being of linear type are viewed from this perspective. The known numerical invariants, such as the analytic spread, the reduction number and the analytic deviation, of an ideal and its associated algebras are considered in the case of modules. Corresponding notions of complete intersection, almost complete intersection and equimultiple modules are examined in some detail. Special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way. For these modules one is able to estimate the reduction number and other invariants in terms of the Buchsbaum–Rim multiplicity.

For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

Using variational measures, we characterize additive interval functions which are indefinite Henstock–Kurzweil integrals in the Euclidean space, answering a problem posed by W. B. Jurkat and R. W. Knizia and independently by Lee Peng-Yee and Chew Tuan-Seng. This also extends a result of Bongiorno, Di Piazza and Skvortsov.

We establish the correspondence between Euclidean differential geometry of submanifolds in $\mathbb{R}^n$ and projective differential geometry of submanifolds in $\mathbb{R}^{n +1 }$ under stereographic projection to quadrics of revolution (and to a more general class of quadrics called quasi-revolution quadrics). V. D. Sedykh found a relation between Lagrangian and Legendrian singularities by stereographic projection to a sphere in Euclidean space. We generalise Sedykh's results in several directions, for instance:

(1) such a relation holds for any stereographic projection of a hyperplane to a quasi-revolution quadric in $\mathbb{R}^n \times \mathbb{R}$, where here $\mathbb{R}^n$ denotes Euclidean space;

(2) when the quadric is a paraboloid the relation between Lagrangian and Legendrian singularities is the most natural one, and the calculations, formulas and proofs are simpler;

(3) using the classical theory of poles, polars and polar duality, we construct the natural isomorphism between the front of the Lagrange submanifold of the normal map (considered as a subvariety in $J^0(\mathbb{R}^n) = \mathbb{R}^n \times \mathbb{R}$) and the front of the Legendre submanifold of the tangential map (considered as a subvariety in the space $(\mathbb{R}^{n + 1})^\vee$ of affine n-dimensional subspaces in $\mathbb{R}^{n + 1}$).

As a consequence of our results we obtain a formula to calculate the vertices of smooth curves in $\mathbb{R}^n$. Our results may be applied to calculate and study umbilic points of surfaces in $\mathbb{R}^3$, or more generally to study the contact of submanifolds in Euclidean space $\mathbb{R}^n$ with k-spheres, for $k = 1,\ldots,n - 1$, in terms of the contact of submanifolds in $\mathbb{R}^{n + 1}$ with $(k + 1)$-planes.

We remark that it is possible to obtain additional geometric information when stereographic projection is replaced by an inversion.

A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside in 1900.

We study conjugacy invariants for completely positive maps that are inspired by the concept of curvature introduced for commuting d-tuples of contractions by Arveson.

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m > 2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m > 2$, on three real parameters along with an arbitrary Kähler–Einstein metric $h$ in complex dimension $m - 1$. We provide an explicit description of all these local-isometry types, for any given $h$. This result is derived from a more general local classification theorem for metrics admitting functions that we call special Kähler–Ricci potentials.