An effective analytic formula for the number of distinct irreducible factors of a polynomial

关于函数域中具有给定重数的不可约因子的数量

On the number of irreducible factors with a given multiplicity in function fields

该研究针对函数域中具有给定重数的首一不可约因子计数函数$\omega_k(f)$，推导了其一二阶矩的渐近估计，证明$\omega_1(f)$的正规阶为$\log(\deg(f))$且满足Erdős-Kac定理，而$k\geq2$时$\omega_k(f)$无正规阶。

A Theorem on Analytic Strong Multiplicity One

Let $K$ be an algebraic number field, and $π=\otimesπ_{v}$ an irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$ with analytic conductor $C(π)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $π$ can be decided completely by its local components $π_{v}$ with norm $N(v)<c\cdot C(π)^{2m+\varepsilon}.$

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements $a\in\mathbb F_{q^n}$ for which Trace$(a)=0$ and Trace$(a^{-1})=0$. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.

On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in the obvious way, is uniformly distributed modulo one. It is the goal of this paper to show that if f(x) and g(x) are a pair of primitive irreducible polynomials of degree greater than one, not necessarily distinct, then the sequence (u/n,v/n), with f(u) = 0(n) and g(v) = 0(n), ordered in the obvious way, is uniformly distributed modulo one in the unit torus.

Locally analytic vectors of some crystabelian representations of GL_2(Qp)

For V a 2-dimensional p-adic representation of G_Qp, we denote by B(V) the admissible unitary representation of GL_2(Qp) attached to V under the p-adic local Langlands correspondence of GL_2(Qp) initiated by Breuil. In this article, building on the works of Berger-Breuil and Colmez, we determine the locally analytic vectors B(V)an of B(V) when V is irreducible, crystabelian and Frobenius semi-simple with Hodge-Tate weights (0,k-1) for some integer k>=2; this proves a conjecture of Breuil. Using this result, we verify Emerton's conjecture that dim Ref^{η\otimesψ}(V)=dim Exp^{η|\cdot|\otimes xψ}…

Monodromy of four dimensional irreducible compatible systems of Q

Let $F$ be a totally real field and $n\leq 4$ a natural number. We study the monodromy groups of any $n$-dimensional strictly compatible system $\{ρ_λ\}_λ$ of $λ$-adic representations of $F$ with distinct Hodge-Tate numbers such that $ρ_{λ_0}$ is irreducible for some $λ_0$. When $F=\mathbb{Q}$, $n=4$, and $ρ_{λ_0}$ is fully symplectic, the following assertions are obtained. (i) The representation $ρ_λ$ is fully symplectic for almost all $λ$. (ii) If in addition the similitude character $μ_{λ_0}$ of $ρ_{λ_0}$ is odd, then the system $\{ρ_λ\}_λ$ is potentially automorphic and the residual image …

On the Extensions of a Discrete Valuation in a Number Field

Let $K$ be a number field defined by a monic irreducible polynomial $F(X) \in \mathbb{Z}[X]$, $p$ a fixed rational prime, and $ν_p$ the discrete valuation associated to $p$. Assume that $\overline{F}(X)$ factors modulo $p$ into the product of powers of $r$ distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly $r$ valuations of $K$ extending $ν_p$. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful…

The Jordan-Hölder series of the locally analytic Steinberg representation

We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed by Orlik and Strauch.

Langlands 参数模空间的不可约分支

Irreducible components of the moduli space of Langlands parameters

本文研究 p 进域上 Langlands 参数模空间的几何结构，确定其不可约分支，刻画驯分歧正则点附近的局部结构及其与 Gelfand–Graev 表示自同态环的联系，并给出 Levi 子群情形下自然模叠映射光滑的开稠密集合。

具有大典范维数的四元数代数在$\mathbb{Q}_p$上的简单Iwasawa模

Simple Iwasawa modules with large canonical dimension of quaternion algebra over $\mathbb{Q}_p$

构造四元数代数上大典范维数的绝对不可约Banach表示，作为Dospinescu-Schraen猜想的反例。

Analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$

We relate the analytic conductor of a generic irreducible representation of $\mathrm{GL}_n(\mathbb{R})$ to the invariance properties of vectors in that representation. The relationship is an analytic archimedean analogue of some aspects of the classical non-archimedean newvector theory of Casselman and Jacquet--Piatetski-Shapiro--Shalika. We illustrate how this relationship may be applied in trace formulas to majorize sums over automorphic forms on $\mathrm{PGL}_n(\mathbb{Z}) \backslash \mathrm{PGL}_n(\mathbb{R})$ ordered by analytic conductor.

关于阿贝尔幺半群中h-free和h-full元素的不同生成元数量分布的研究

On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid

本文首次深入研究阿贝尔幺半群中的h-free与h-full元素，通过分析生成素元数量ω(m)的各阶矩确立其正常阶，并将该统一方法成功应用于数域、全局函数域及几何不可约射影簇。