On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

The hyperring of adèle classes

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We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK=AK/K^× of a global field K. After promoting F₁ to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R^× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G∪{0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3LSKD_PfJyc.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Let K be a finite extension of Qp, let L/K be a finite abelian Galois extension of odd degree and let OL be the valuation ring of L. We define AL/K to be the unique fractional OL-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Qp contained in certain cyclotomic extensions, Erez has described integral normal bases for AL/Qp that are self-dual with respect to the trace form. Assuming K/Qp to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.     

On products of two nilpotent subgroups of a finite group

LetG be a finite group with an abelian Sylow 2-subgroup. LetA be a nilpotent subgroup ofG of maximal order satisfying class (A)≦k, wherek is a fixed integer larger than 1. Suppose thatA normalizes a nilpotent subgroupB ofG of odd order. ThenAB is nilpotent. Consequently, ifF(G) is of odd order andA is a nilpotent subgroup ofG of maximal order, thenF(G)?A.     

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).     

Quadratic function fields with invariant class group

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

The classification of finite groups by using iteration digraphs

A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = x^kfor all x ∈ G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed.     

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.     

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.     

Restricted Modules and Conjectures for Modules of Constant Jordan Type

We introduce the class of restricted k[A]-modules and p ^t -Jordan types for a finite abelian p-group A of exponent at least p ^t and a field k of characteristic p. For these modules, we generalize several theorems by Benson, verify a generalization of conjectures stated by Suslin and Rickard giving constraints on Jordan types for modules of constant Jordan type when t is 1. We state conjectures giving constraints on p ^t -Jordan types and show that many p ^t -Jordan types are realizable.     

Exterior algebras and two conjectures on finite abelian groups

Let G be a finite abelian group with |G| > 1. Let a ₁, …, a _k be k distinct elements of G and let b ₁, …, b _k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation π on {1, …,k} such that a ₁ b _π(1), …, a _k b _π(k) are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Károlyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily’s conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters.     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v^′=A(t)v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v^′=A(t)v+f(t,v) under sufficiently small perturbations f. This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation v^′=A(t)v is Lyapunov regular if for every k the limit of Γ(t)^1/t as t→∞ exists, where Γ(t) is any k-volume defined by solutions v₁(t),…,v_k(t). We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.     

On the chromaticity of complete multipartite graphs with certain edges added

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t−1)×p,p+k) by adding a set S of s edges to the partite set of size p+k such that 〈S〉 is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K⁺((t−1)×p,p+k). In this paper, we shall prove that for k=0,1 and p+k≥s+2, each graph G∈K(s,t,p,k) is chromatically unique if and only if 〈S〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K⁺((t−1)×p,p+k) is chromatically unique for k=0,1 and p+k≥3.     

Potentially good reduction of Barsotti-Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K^′ of K. We prove that there exists a constant c?2 which depends on the absolute ramification index e(K^′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron-Ogg-Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge-Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.