A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.

Eisenstein-素数 障碍 筛选 用于 单元 生成 性

Eisenstein-prime Obstruction Sieve for Monogenicity

本文 证明 在 纯 场 家族 中， 单元 生成 性 与 $α$--单 元 生成 性 具有 相同 的 自然 密度， 并 提出 了 Eisenstein-素数 障碍 筛选 机理。

纯三次域中素数幂指标的单生成序的比例

The proportion of monogenic orders of prime power indices of the pure cubic field

本文证明纯三次域中指标为固定素数（非2或3）幂的序中单生成序的比例为零，通过计数满秩子模并利用Thue-Mahler方程本原解有限性得到上界。

Frobenius Finds Non-monogenic Division Fields of Abelian Varieties

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $π\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}π$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When $\text{End}_k(A)=\mathbb{Z}[π,v]$, we construct a matrix which describes the action of $π$ on the prime-to-$p$-torsion points of $A$. We employ this matrix in an algorithm that detects when $p$ is an obstruction to the monogeneity of division fields of certain abelian varieties.

Lecacheux五次域参数族中Pólya群的秩

Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields

研究Lecacheux五次域族中Pólya群的结构，证明其可任意大的初等阿贝尔5-群，并给出正密度下5-秩的下界，进而得到无穷5-类域塔的存在性。

A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends …