Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers O_k, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers O_N of N determines a class in the locally free class group Cl(O_k[Γ]). We show that the set of classes in Cl(O_k[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O_k[Γ]) to the ideal class group Cl(O_k), provided that the ray class group of O_k for the modulus 4O_k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

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.     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Let F be a non-archimedean local field of odd residual characteristic. Let (J,τ) be a maximal simple type in GLN(F) for the inertial class [GLN(F),π]_GLN(F) of a self-contragredient supercuspidal irreducible representation π of GLN(F). Identify GLN(F) to the standard Siegel Levi subgroup in Sp2N(F). We construct, in Sp2N(F), a type for the inertial class [GLN(F),π]_Sp2N(F), as a Sp2N(F)-cover of (J,τ), strongly related to the GL2N(F)-cover of (J×J,τ⊗τ) in GL2N(F) constructed by Bushnell and Kutzko and which induces to a simple type in GL2N(F). In the process, we show that if τ has positive level, then the maximal simple type (J,τ) may be attached to a simple stratum [A,n,0,β] where the field F[β] is a quadratic extension of F[β²], and to a simple character θ in C(A,0,β) Galois conjugate of its inverse.     

On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Let K be a complete discrete valued field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G such that the induced extension of residue fields kL/kK is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group {H¹(G,Wn(OL))}_n∈N is zero, where OL is the ring of integers of L and W(OL) is the ring of Witt vectors in OL w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt?s conjecture for all Galois extensions.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

The Coxeter polynomial for a one point extension algebra

Let Λ be a finite-dimensional algebra over an algebraically closed field k of finite global dimension. Let M be a finitely generated Λ-module and let $\Gamma =\Lambda [M]$ be the one point extension algebra. We show how to compute the Coxeter polynomial for Γ from the Coxeter polynomial of Λ and homological invariants of M.     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Lattice paths with given number of turns and semimodules over numerical semigroups

Let Γ=〈α,β〉 be a numerical semigroup. In this article we consider several relations between the so-called Γ-semimodules and lattice paths from (0,α) to (β,0): we investigate isomorphism classes of Γ-semimodules as well as certain subsets of the set of gaps of Γ, and finally syzygies of Γ-semimodules. In particular we compute the number of Γ-semimodules which are isomorphic with their k-th syzygy for some k.     

Invariant boundary distributions for finite graphs

Let Γ be the fundamental group of a finite connected graph G. Let M be an abelian group. A distribution on the boundary ∂Δ of the universal covering tree Δ is an M-valued measure defined on clopen sets. If M has no χ(G)-torsion, then the group of Γ-invariant distributions on ∂Δ is isomorphic to H₁(G,M).     

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)_k?1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)_k?1 of E(M), we consider the Rademacher averages k_∑εk⊗xk as elements of the noncommutative function space and study estimates for their norms ‖k_∑εk⊗xkE_‖ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖k_∑εk⊗xkE_‖ is equivalent to the infimum of over all yk, zk in E(M) such that xk=yk+zk for any k?1. Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖k_∑εk⊗xkE_‖ is equivalent to . We also study Rademacher averages _∑i,jεi⊗εj⊗xij for doubly indexed families (xij)_i,j of E(M).     

The tame infinitesimal groups of odd characteristic

Given an infinitesimal group G over an algebraically closed field k of characteristic p?3, we provide criteria for the principal block B₀(G) of its algebra of distributions to be of tame representation type. These are employed in conjunction with Galois coverings to determine the structure of G modulo its multiplicative center as well as the quiver and the relations of the algebra B₀(G).     

A corollary of the b-function lemma

Let X be an algebraic variety, f a regular function, ${j:U\hookrightarrow X}$ the complement to the locus of vanishing of f, and M a holonomic D-module on U. Consider the D _U [s]-module ${M\otimes ``{f^{s}}''}$ . The goal of this note is to describe all D <sub>X</sub> [s] submodules ${N\hookrightarrow j_*(M\otimes ``{f^{s}}'')}$ such that ${j^*(N)\simeq M\otimes ``{f^{s}}''}$ .     

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.