Normal integral bases and strict ray class groups modulo 4

Let F be a number field. We construct three tamely ramified quadratic extensions which are ramified at most at some given set of finite primes, such that K₃⊂K₁K₂, both K₁/F and K₂/F have normal integral bases, but K₃/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order ?3, then there exist infinitely many triplets (K₁,K₂,K₃) of such fields.     

Logarithmic Frobenius structures and Coxeter discriminants

We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.     

Automorphisms of symmetric groups are toy examples for mapping class groups

We give a new proof establishing the automorphism groups of the symmetric groups inspired by the analogous result of Ivanov for the extended mapping class group. As a key tool, we consider the actions on the Kneser graphs.     

Resultants and discriminants of Chebyshev and related polynomials

We show that the resultants with respect to of certain linear forms in Chebyshev polynomials with argument are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.

     

Presentation and central extensions of mapping class groups

We give a presentation of the mapping class group of a (possibly bounded) surface, considering either all twists or just non-separating twists as generators. We also study certain central extensions of . One of them plays a key role in studying TQFT functors, namely the mapping class group of a -structure surface. We give a presentation of this extension.

     

On the degree of commutativity of p ‐groups of maximal class

Let G be a p ‐group of maximal class of order p^m , p ≠ 2, and c (G) the degree of commutativity of G. Let c₀ be the nonnegative residue of c modulo p – 1. In this paper, by using only Lie algebra techniques, we prove that 2c ≥ m – 2p + c 0 + 1. Also, we give examples of Lie algebras satisfying the following equalities: In addition, there exist examples of p ‐groups of maximal class satisfying 2c = m – 2p + c₀ + 3 for each c₀ ∈ [2, p – 2] (see [6, Theorem 4.5]). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields

For the function field K of hyperelliptic curves over Q we define a subgroup of the ideal class group called the group of Z-primitive ideals. We then show that there are homomorphisms from this subgroup to ideal class groups of certain quadratic number fields.     

Minus class groups of the fields of the -th roots of unity

We show that for any prime number the minus class group of the field of the -th roots of unity admits a finite free resolution of length 1 as a module over the ring . Here denotes complex conjugation in . Moreover, for the primes we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.

     

The Chern-Mather class of the multiview variety

The multiview varietyassociated to a collection of N cameras records which sequences of image points in ?^2N can be obtained by taking pictures of a given world point x∈?³ with the cameras. In order to reconstruct a scene from its picture under the different cameras, it is important to be able to find the critical points of the function which measures the distance between a general point u∈?^2N and the multiview variety. In this paper we calculate a specific degree 3 polynomial that computes the number of critical points as a function of N. In order to do this, we construct a resolution of the multiview variety and use it to compute its Chern-Mather class.     

On upper bounds on stable commutator lengths in mapping class groups

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.     

Some computations of stable twisted homology for mapping class groups

Abstract

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

Communicated by Jason P. Bell     

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 $ {p} $‐groups of maximal class

Let R be the ring $ {\mathbb Z}[x]/\left({{x^p-1}\over{x-1}}\right) = {\mathbb Z}[\bar{x}] $ and let $ \mathfrak {a} $ be the ideal of R generated by $ (\bar{x}-1) $ . In this paper, we discuss the structure of the $ {\mathbb Z}[C_p] $‐module $ (R/\mathfrak {a}^{n-1}) \wedge (R/\mathfrak {a}^{n-1}) $, which plays an important role in the theory of p‐groups of maximal class (see 2 - 5 ). The generators of this module allow us to obtain the defining relations of some important examples of p‐groups of maximal class with Y₁ of class two. In particular we obtain the best possible estimates for the degree of commutativity of p‐groups of maximal class with Y₁ of class two. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A class of periodic groups

We deal with periodic groups saturated with dihedral groups. In particular, it is proved that periodic groups of bounded period, and also periodic Shunkov groups, saturated with dihedral groups, are locally finite.Supported by RFBR grant No. 03-01-00356 and by the Krasnoyarsk Science Foundation, project 11F0202C.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 114–125, January–February, 2005.     

Dihedral side extensions and class groups

We present a non-abelian mirror-type principle relating the p-ranks of class groups of subfields of a dihedral field of degree 2p for an odd prime p with a limited ramification condition.     

A relationship between mixed discriminants and the joint spectrum of a family of commuting operators in finite-dimensional space

We study the properties of the polynomial operator pencil

The largest character degree,conjugacy class size and subgroups of finite groups

Let G be a finite group, let b(G) denote the largest irreducible character degree of the group G and let bcl(G) denote the largest conjugacy class size of the group G. We study the relations between the sizes of the nilpotent and solvable subgroups of G and b(G). We also study the relations between the sizes of the nilpotent and solvable subgroups of G and bcl(G).     

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z ⁿ +t−1)²+4t(witht|z ⁿ −1), which contains the well-known fields withm=4z ⁿ +1 andm=4z ²ⁿ +4 as special cases. Project supported by the National Natural Science Foundation of China.     

Braids,mapping class groups,and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. The first author was supported by Inha University research grant.