Semi-local units modulo Gauss sums

For the p-th cyclotomic field k, Iwasawa proved that p does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related to k. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Received: 2 May 1997 / Revised version: 11 November 1997     

Inequalities for determinants and permanents

Let p be a prime > 3. It is shown that no integral circulant of order p^k exists with determinant p^k+1 . It is also shown that m is the determinant of an integral 9×9 circulant if and only if (m, 3)=l, or m = 0 mod 27. The proof makes use of a criterion which must be satisfied by the difference of two units in the cyclotomic field of level p^k .     

On the Stabilizer of the Automorphism Group of a 4—valent Vertex—transitive Graph with Odd—prime—power Order

Let X be a 4-valent connected vertex-transitive graph with odd-prime-power order p^κ（κ≥1） and let A be the full automorphism group of X.In this paper,we prove that the stabilizer Av of a vertex v in A is a 2-group if p≠5,or a {2,3}-group if p=5.Furthermore,if p=5|Av| is not divisible by 3^2.As a result ,we show that any 4-valent connected vertex-transitive graph with odd-prime-power order p^κ(κ≥1） is at most 1-arc-transitive for p≠5 and 2-arc-transitive for p=5.     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Homology of balanced complexes via the Fourier transform

Let G ₀,…,G _k be finite abelian groups, and let G ₀∗⋯∗G _k be the join of the 0-dimensional complexes G _i . We give a characterization of the integral k-coboundaries of subcomplexes of G ₀∗⋯∗G _k in terms of the Fourier transform on the group G ₀×⋯×G _k . This provides a short proof of an extension of a recent result of Musiker and Reiner on a topological interpretation of the cyclotomic polynomial.     

Estimates on binomial sums of partition functions

Let p(n) denote the partition function and define where p(0)= 1. We prove that p(n,k) is unimodal and satisfies for fixed n≥ 1 and all 1≤k≤n. This result has an interesting application: the minimal dimension of a faithful module for a k-step nilpotent Lie algebra of dimension n is bounded by p(n,k) and hence by , independently of k. So far only the bound n ^{n −1} was known. We will also prove that for n≥ 1 and . Received: 17 December 1999     

The stable free rank of symmetry of products of spheres

A well known conjecture in the theory of transformation groups states that if p is a prime and (ℤ/p) ^r acts freely on a product of k spheres, then r≤k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non-simply connected spaces.     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

Companion forms over totally real fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.     

The class number of cyclotomic function fields

Let k be a rational function field over a finite field. Carlitz and Hayes have described a family of extensions of k which are analogous to the collection of cyclotomic extensions {Q(ζm)| m ≥ 2} of the rational field Q. We investigate arithmetic properties of these “cyclotomic function fields.” We introduce the notion of the maximal real subfield of the cyclotomic function field and develop class number formulas for both the cyclotomic function field and its maximal real subfield. Our principal result is the analogue of a classical theorem of Kummer which for a prime p and positive integer n relates the class number of Q(ζpn + ζpn?1), the maximal real subfield of Q(ζpn), to the index of the group of cyclotomic units in the full unit group of Z[ζpn].     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

Units of modulus 1

If k is an algebraic number field which is normal over the field of rational numbers then it is shown that k has nontrivial units of modulus 1 if and only if the maximal real subfield of k is also a normal extension of the rationals. A characterization of the units is given for fields which satisfy the above conditions. A new proof of Kummer's Theorem on the units of cyclotomic fields is also obtained.     

On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.     

Restriction of characters and products of characters

Let G be a finite p-group, for some prime p, and ψ, θ ∈ Irr(G) be irreducible complex characters of G. It has been proved that if, in addition, ψ and θ are faithful characters, then the product ψθ is a multiple of an irreducible or it is the nontrivial linear combination of at least (p + 1)/2 distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k > 0, we can always find a p-group P and irreducible characters Ψ and Θ of P such that the product ΨΘ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

An Eberhard-Like Theorem for Pentagons and Heptagons

Eberhard proved that for every sequence (p _k ), 3≤k≤r, k≠6, of nonnegative integers satisfying Euler’s formula ∑ _k≥3(6−k)p _k =12, there are infinitely many values p ₆ such that there exists a simple convex polyhedron having precisely p _k faces of size k for every k≥3, where p _k =0 if k>r. In this paper we prove a similar statement when nonnegative integers p _k are given for 3≤k≤r, except for k=5 and k=7 (but including p ₆). We prove that there are infinitely many values p ₅,p ₇ such that there exists a simple convex polyhedron having precisely p _k faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind.     

Examples of λ-invariants

In this paper, we will examine some of the implications of the results in [C1] for the Iwasawa invariants, λ _p , of the cyclotomic fields Q(ζ _n ), wherep + n. In particular, a number of examples, for various primesp, are given.     

Torsion in Mordell-Weil groups of Fermat Jacobians

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups.     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

<Emphasis Type="Italic">K</Emphasis>th Roots of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators are Subscalar Operators of Order 4<Emphasis Type="Italic">k</Emphasis>

In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T^k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T^k is p-hyponormal and σ(T) has nonempty interior in the plane, then T has a nontrivial invariant subspace.