例外数域正类的2-群的计算

单值群与例外Hodge类，I：Fermat Jacobians

Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians

本文计算了超椭圆曲线y^2=x^m+1的Jacobian簇J_m的等元分解、端元定义域和连通单值域，并展示了高维退化现象。

某些阿贝尔域的岩泽不变量与例外对

Iwasawa invariants and exceptional pairs of certain abelian fields

本文针对Q(√d,ζₚ)研究岩泽不变量与例外对，确定岩泽模的Λ同构类及例外对结构差异，在指定范围内发现15个新例外对，包括λ不变量为2的特殊情形。

有限域上单 Abel 簇的 Galois 群与例外 Tate 类

Galois groups of simple abelian varieties over finite fields and exceptional Tate classes

本文利用 Frobenius 极小多项式 Galois 群与 Newton 多边形作用的组合判据，研究有限域上几何单 Abel 簇的例外 Tate 类，证明非最大角秩等新情形，并给出 CM 域实现与角秩结果。

On the sums of squares of exceptional units in residue class rings

Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if $\gcd(1-u,n)=1$. We denote by $\mathcal{N}_{k,c,e}(n)$ the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^e+...+x_k^e\equiv c \pmod n$ with all $x_i$ being exceptional units in the ring $\mathbb{Z}_n$. In 2017, Mollahajiaghaei presented a formula for the number of solutions $(x_1,...,x_k)$ of the congruence $x_1^2+...+x_k^2\equiv c\pmod n$ with all $x_i$ be…

高度配对、例外零点和Rubin公式：乘法群

Height pairings, exceptional zeros and Rubin's Formula: The multiplicative group

在本文中，我们证明了一个公式，由于鲁宾，这表示的首项系数的各种p-adic L-函数在一个例外零的存在下Nekovar的p-adic高度配对他的扩展Selmer群的精神。在一个特殊的情况下，我们证明的鲁宾式公式恢复了一个p-adic克罗内克极限公式。在不相交的情况下，我们观察到，对于Kubota-Leopoldt p-adic L-函数的首项系数，我们使用Nekovar高度的计算与Ferrero- Greenberg公式（更一般地，Gross的结构公式）一致（分别为，Deligne-Ribet p-adic L-函数）。

Counting Exceptional Points for Rational Numbers Associated to the Fibonacci Sequence

If $α$ is a non-zero algebraic number, we let $m(α)$ denote the Mahler measure of the minimal polynomial of $α$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(α)$. For fixed $α\in \bar{\mathbb Q}$, the map $t\mapsto m_t(α)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $α$. It remains open to determine whether there is a sequence of elements $α_n\in \bar{\mathbb Q}$ such that …

数域的2-群窄对数divisor类的计算

Computation of 2-groups of narrow logarithmic divisor classes of number fields

给出了数域F的0次窄对数除子类2-群的一个计算算法。作为应用，我们在某些情况下计算野生核WK 2（F）的2-秩。

Explicit Computations of Fundamental Classes

We use the techniques of group cohomology to give explicit computations of the local fundamental class. As an application, we discuss how to compute the Tate canonical class for the extension $\mathbb Q(ζ_{p^ν})/\mathbb Q$, where $p^ν$ is an odd prime power.

单三次函数域与类数计算

Simple cubic function fields and class number computations

本文在函数域的背景下研究了单三次域，并将例外单位集的概念推广到三次函数域，即$k$ -例外单位的概念.本文给出了伽罗瓦单三次函数域是Shanks最简三次数域的直接类比的一个简单证明。除了计算的不变量，包括一个公式的调节器，我们计算类数的伽罗瓦简单的三次函数字段在$\mathbb{F}_{5}$和$\mathbb{F}_{7}$使用截断欧拉产品。最后，作为一个额外的应用，我们确定所有的伽罗瓦单三次函数域的类数1，受到一个温和的限制。

Cyclotomic derivatives of Beilinson--Flach classes and a new proof of a Gross--Stark formula

We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal L$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson--Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be se…

全域上的病态分布集合与丢番图几何中的例外集

Ill-distributed sets over global fields and exceptional sets in Diophantine Geometry

本文运用离散分析技术，证明了Wilkie关于有理点密度的猜想等价于可定义集在剩余类层面的病态分布，并在全域背景下研究了相应的反问题，为证明该猜想提供了新策略。

数域上 K3 曲面约化的 Picard 秩的例外跳跃

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

本文证明了数域上处处具有潜在好约化的 K3 曲面在无限多个素数处几何 Picard 秩发生跳跃，并基于 GSpin Shimura 簇的例外类理论探讨了其在有理曲线与阿贝尔曲面同构中的应用。

Investigating the exceptionality of scattered polynomials

Scattered polynomials over a finite field $\mathbb{F}_{q^n}$ have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-$q^t$-partially scattered and R-$q^t$-partially scattered polynomials, for $t$ a divisor of $n$. Indeed, a polynomial is scattered if and only if it is both L-$q^t$-partially scattered and R-$q^t$-partially scattered. In this pap…