Ambiguous divisor classes in function fields

Let $\text{L}\text{K}$ be a Galois extension of algegraic function fields in one variable with Galois group G. Let J_K and J_L be the divisor classes of degree zero in K and L, respectively. A study is made of the kernel and cokernel of the natural map from J_K to $\text{J}{}_{\mathrm{L}}{}^{\mathrm{G}}$.     

Localizability of the embedding problem with symplectic kernel

In this paper we consider the question of how much information is supplied by local solutions to a global embedding problem for the special case in which the normal subgroup belonging to the given group extension is the projective symplectic group PSp(2m, q). It is proved that for suitable Galois extensions K of a given number field k (which constitute part of the data of the embedding problem), the local solutions in a sense determine whether or not an extension K ? K, Galois over k, with $\text{G(}\text{L}\text{K}\text{) ≈ PSp(2m, q)}$, represents a global solution to the embedding problem.     

Zonal measure algebras on isotropy irreducible homogeneous spaces

This paper analyzes the convolution algebra $\text{M(K}\text{G}\text{K}\text{)}$ of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when $\text{M(K}\text{G}\text{K}\text{)}$ is commutative and $\text{G}\text{K}$ is isotropy irreducible. A basic result for such (G, K) is that the convolution of dim $\text{G}\text{K}$ continuous (on $\text{G}\text{K}$) zonal measures is absolutely continuous. Using this, the spectrum (maximal ideal space) of $\text{M(K}\text{G}\text{K}\text{)}$ is determined and shown to be in 1-1 correspondence with the bounded Borel spherical functions. Also, certain asymptotic results for the continuous spherical functions are derived. For the special case when G is compact, all the idempotents in $\text{M(K}\text{G}\text{K}\text{)}$ are determined.     

The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Let G be a finite Abelian group acting by tensor-product automorphisms on a UHF-C¹-algebra $\text{D}$. Extending a result of A. Kishimoto it is shown that the number of extremal traces on the fixed-point algebra $\text{D}$^G equals the cardinality of the subgroup K of automorphisms in G which are weakly inner in the trace representation of $\text{D}$.     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

Sur la structure galoisienne du groupe des unités d'un corps abélien réel de type (p, p)

Let p be a rational prime. We classify those $\text{Z}$[($\text{Z}$/p$\text{Z}$)²]-modules arising as submodules of the units (mod. torsion) of a real abelian field K with Galois group ($\text{Z}$/p$\text{Z}$)², up to isomorphism and up to genus. Explicit results are given when p is 2 or 3. We apply our classification to discuss the existence of a Minkowski unit in K for arbitrary p.     

Galois Structure of K-Groups of Rings of Integers

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts, to the root number class of Fröhlich and Cassou-Noguès.     

Algebraic function fields with small class number

It is proved that there is no congruence function field of genus 4 over GF(2) which has no prime of degree less than 4 and precisely one prime of degree 4. This shows the nonexistence of function fields of genus 4 with class number one and gives an example of an isogeny class of abelian varieties which contains no jacobian. It is shown that, up to isomorphism, there are two congruence function fields of genus 3 with class number one. It follows that there are seven nonisomorphic function fields of genus different from zero with class number one. Congruence function fields with class number 2 are fully classified. Finally, it is proved that there are eight imaginary quadratic function fields $\text{F}\text{K(x)}$ for which the integral closure of K[x] in F has class number 2.     

Neighbourhoods of transitive graphs and GRR's

Let X be a vertex-transitive graph with complement $\text{X}$. We show that if both N, the neighbourhood of a vertex in X, and $\text{N}$, the neighbourhood of a vertex in $\text{X}$, are disconnected, then either X is isomorphic to K₃ × K₃ or both N and $\text{N}$ contain isolated vertices. We characterize the graphs which satisfy this last condition and show in consequence that they admit automorphisms of the form (12)(34). It follows that if X is a GRR for some graph G then at least one of N and $\text{N}$ is connected. (X is said to be a graphical regular representation, or GRR, for G if its automorphism group is isomorphic to G and acts regularly on its vertices.) Using this result we determine those groups generated by their involutions which do not have a GRR. The largest such group has order 18. As a corollary we conclude that all non-abelian simple groups have GRR's.     

On subgraph number independence in trees

For finite graphs F and G, let N_F(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If $\text{F}$ and $\text{G}$ are families of graphs, it is natural to ask then whether or not the quantities N_F(G), F∈$\text{F}$, are linearly independent when G is restricted to $\text{G}$. For example, if $\text{F}$ = {K₁, K₂} (where K_n denotes the complete graph on n vertices) and $\text{F}$ is the family of all (finite) trees, then of course N_K1(T) ? N_K2(T) = 1 for all T∈$\text{F}$. Slightly less trivially, if $\text{F}$ = {S_n: n = 1, 2, 3,…} (where S_n denotes the star on n edges) and $\text{G}$ again is the family of all trees, then Σ_n=1^∞(?1)ⁿ⁺¹N_Sn(T)=1 for all T∈$\text{G}$. It is proved that such a linear dependence can never occur if $\text{F}$ is finite, no F∈$\text{F}$ has an isolated point, and $\text{G}$ contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear).     

On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let $L/K$ be a finite separable extension of fields whose Galois closure $E/K$ has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on $L/K$ has the form $E{[N]}^G$ for some group N of order $[L:K]$. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree $p^n$, for p an odd prime number.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

On the failure of spectral synthesis for compact semisimple Lie groups

Let G be a compact connected semisimple Lie group with Lie algebra $\text{g}$. We show that the conjugacy class of a regular element of G is not a set of synthesis for the Fourier algebra of G. Similarly, the Ad(G)-orbit of a regular element of $\text{g}$ is not a set of synthesis for the algebra of Fourier transforms on $\text{g}$. In proving this latter result we demonstrate a regularity property of Ad-invariant Fourier transforms, analogous to the differentiability of radial Fourier transforms.     

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.