The dual of a constacyclic code,self dual,reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of $\gamma$-constacyclic codes is established and the algebraic characterization of self-dual $\gamma$-constacyclic codes, reversible $\gamma$-constacyclic codes and $\gamma$-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.     

Le nombre des systèmes locaux ℓ-adiques sur une courbe

Let $X_1$ be a projective, smooth, and geometrically connected curve over ${\mathbb{F}}_q$ with $q=p^n$ elements where p is a prime number, and let X be its base change to an algebraic closure of ${\mathbb{F}}_q$. The goal of this article is to announce a formula for the number of irreducible ?-adic local systems ($?\ne p$) with a fixed rank over X fixed by the Frobenius endomorphism. This number behaves like a Grothendieck–Lefschetz fixed point formula for a variety over ${\mathbb{F}}_q$, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. We also sketch our method, which consists in passing to the automorphic side by Langlands correspondence, then calculating all the terms in Arthur's non-invariant trace formula and linking the geometric part of trace formula to the number of ${\mathbb{F}}_q$-points of the moduli space of stable Higgs bundles.     

Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

On finite core-p2p-groups

A p-group G is called a core-$p^2$ group if $|H:H_G|\le p^2$ for every subgroup H of G (where $H_G$ denotes the normal core of H in G). In this paper, it is proved that the nilpotent class of a finite core-$p^2$p-group is at most 5 if $p\ge 3$, which is the best upper bound, and the derived length of a finite core-$p^2$p-group is at most 3 if $p\ge 3$, which is also the best upper bound.     

On exceptional groups of order p5

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient.The smallest examples of exceptional p-groups have order $p^5$. For an odd prime p, we classify all pairs $(G,Q)$ where G has order $p^5$ and Q is a distinguished quotient. (The case $p=2$ has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order $p^5$ with at least one exceptional quotient tends to 1/2.     

Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly $\pi$-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element $a\in R$ is called pseudopolar if there exists $p\in R$ such that $p^2=p\in {\mathrm{comm}}^2(a)\text{,}a+p\in U(R)\mathrm{and}{a}^{k}p\in J(R)$ for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly $\pi$-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

On the gap of finite metric spaces of p-negative type

Let $(X\text{,}d)$ be a metric space of p-negative type. Recently I. Doust and A. Weston introduced a quantification of the p-negative type property, the so called gap $\mathrm{\Gamma }$ of X. This paper gives some formulas for the gap $\mathrm{\Gamma }$ of a finite metric space of strict p-negative type and applies them to evaluate $\mathrm{\Gamma }$ for some concrete finite metric spaces.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

A study of Inkeri's class number formula

Let p be an odd prime and $\mathbb{Q}({\zeta }_p)$ be the cyclotomic field defined by a primitive pth root of unity ${\zeta }_p$. It is well-known that the relative class number $h_p^{-}$ of $\mathbb{Q}({\zeta }_p)$ can be expressed by means of the determinant of Inkeri's matrix. The main purpose of this paper is to study Inkeri's class number formula from the viewpoints of bases of Stickelberger ideals in a certain group ring and some special polynomial values.