相关论文: $g$-invariant on unary Hermitian lattices over ima…

Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over…

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$…

For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of…

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for…

We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the…

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of…

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.

Let $\mathfrak o$ be the ring of integers of a totally real number field. If $f$ is a quadratic form over $\mathfrak o$ and $g$ is another quadratic form over $\mathfrak o$ which represents all proper subforms of $f$, does $g$ represent…

We present an adaptation of Voronoi theory for imaginary quadratic number fields of class number greater than 1. This includes a characterisation of extreme Hermitian forms which is analogous to the classic characterisation of extreme…

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let…

We study the representation ${\cal D}$ of a simple compact Lie algebra $\g$ of rank l constructed with the aid of the hermitian Dirac matrices of a (${\rm dim} \g$)-dimensional euclidean space. The irreducible representations of $\g$…

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

Let $k$ be a positive integer and let $F$ be a finite unramified extension of $\mathbb{Q}_2$ with ring of integers $\mathcal{O}_F$. An integral (resp. classic) quadratic form over $\mathcal{O}_F$ is called $k$-universal (resp. classically…

We study the algebras of hermitian automorphic forms for the lattice $L_n=diag(1,1,\ldots,1,-1)$ and for the field $K=\mathbb{Q}(\sqrt{-d})$ such that $p=2$ is unramified and the ring of integers $\mathcal{O}_K$ is a p.i.d. We prove that…

An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the…

Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational…

We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally…

Let K be an algebraic number field and O_K be its ring of integers. Let S_K be the set of elements in O_K which are sums of squares in O_K and s(O_K) the minimal number of squares necessary to represent -1in O_K. Let g( S_K ) be the…

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra…

In this paper, we will prove that if $d$ is sufficiently large square-free positive rational integer, then there is no integral septenary universal quadratic lattice over $\mathcal{O}_F$ where $F=\mathbb Q(\sqrt{d})$.