高度配对、例外零点和Rubin公式：乘法群

短算术级数中的积性函数

Multiplicative functions in short arithmetic progressions

该研究探讨有界积性函数在短算术级数中的和的方差，改进推广了相关经典定理，给出近乎最优的例外模集合，并将结果应用于三素数乘积的Linnik型问题。

反圆p进L-函数的例外零点公式

Exceptional zero formulas for anticyclotomic p-adic L-functions

本文研究一般数域上椭圆曲线反圆p进L-函数在乘法约化位置处的例外零点现象，利用Fornea-Gehrmann的plectic点构造推广其主结果，给出用plectic点与Tate周期表示p进L-函数r阶导数的p进Gross-Zagier型公式。

On the sumsets of exceptional units in $\mathbb{Z}_n$

Let $R$ be a commutative ring with $1\in R$ and $R^{\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\ast}$. Let $\mathbb{Z}_n$ be the ring of residue classes modulo $n$. In this paper, given an integer $k\ge 2$, we obtain an exact formula for the number of ways to represent each element of $ \mathbb{Z}_n$ as the sum of $k$ exceptional units. This generalizes a recent result of J. W. Sander for the case $k=2$.

具有例外零点的自守形式p进L函数

p-adic L-functions of automorphic forms and exceptional zeros

设F是数域，p是素数.我们构造了$GL_2(A_F)$（在p和$\infty$以上的位置具有一定条件）的自守表示的（多变量）p-adic L-函数，它在中心临界点插值复L-函数.我们使用这种结构来证明，在F上的模椭圆曲线E的p-adic L-函数的消失阶大于或等于素数在P以上的E有分裂乘性约化，预测的例外零猜想。这是Spiesten（arXiv：1207.2289）在全实域上的类似结果的推广。

关于乘法函数中加法条件的探讨

Playing with additivity conditions in multiplicative functions

探讨乘法函数\(f\)，在给定整数\(a\)、\(d\)及条件下，通过适度额外条件，证明对特定例外集外的\(n\)，\(f(n)=n\chi(n)\)，明确了研究乘法函数的一种方法及相关结论。

On a Turán conjecture and random multiplicative functions

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.

Heegner点的例外零点与Gross-Zagier和Kolyvagin定理的p-逆定理

Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin

本文证明了在素数p>3处具有乘性约化的椭圆曲线的Gross-Zagier和Kolyvagin定理的p-逆定理，关键工具是Bertolini-Darmon-Prasanna的p-进公式的推广和Heegner点的例外零点公式。

Exceptional zero formulae and a conjecture of Perrin-Riou

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\…

关于Hilbert模形式的$p$ -adic $L$ -函数的特殊零点

On special zeros of $p$-adic $L$-functions of Hilbert modular forms

设$E$是全实数域$F$上的模椭圆曲线.我们证明了弱例外零猜想，它把附着在$E$上的$p$ -adic $L$ -函数的一个（高）导数与附着在相应Hilbert模形式上的某些$p$ -adic周期联系起来，在$p$之上的地方，$E$有分裂的乘性约化.在对$p$和$E$的导子的适当限制下，我们导出了强形式的例外零猜想（即其中，自守$p$ -adic周期被根据Tate周期定义的$E$的$\cL$ -不变量所取代）。我们的方法的关键是一个新的构造的$p$ -adic $L$ -函数的$E$在本地数据。

Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional z…

长度四指标猜想的乘法傅里叶证明

A Multiplicative Fourier Proof of the Length-Four Index Conjecture

该研究基于乘法傅里叶分析，结合有限群支撑不确定性不等式与Shen-Xia-Li定理的极小模归约，对所有与6互素的n证明了零和理论中的长度四指标猜想。

Intersections of multiplicative translates of $3$-adic Cantor sets II: two infinite families

This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action inter…

关于多项式自变量的刘维尔函数

On the Liouville function at polynomial arguments

本研究针对刘维尔函数在多项式自变量下的无穷次变号问题，证明了四类多项式满足该性质，并推广了相关积性函数的部分和抵消结论，建立了Elliott猜想的99%版本及相关结果的积性函数类比。

Skew-invariant curves and the algebraic independence of Mahler functions

For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ an…