Arithmetic differential geometry in the arithmetic PDE setting, I: connections

Arithmetic properties of the Herglotz function

In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group $\mathrm{PSL}(2,…

算术型完整全纯D-模与中间卷积

Holonomic $\mathcal{D}$-modules of arithmetic type and middle convolution

本文建立将G-联络扩展为完整全纯D-模的框架并证明其导出范畴具有格罗滕迪克六函子形式体系，由此得到中间卷积关于全局逆半径的稳定性，进而证明刚性Fuchs系统上多种算术性质等价，肯定回答了André与Baldassarri的猜想。

Arithmetic analogues of some basic concepts from Riemannian geometry

Following recent work of the author, partly in collaboration with T. Dupuy and M. Barrett, we describe arithmetic analogues of some key concepts from Riemannian geometry such as: metrics, Chern connections, curvature, etc. Theorems are stated to the effect that the spectrum of the integers has a non-vanishing curvature.

$p$-进流形上 Weil 丛的算术方面

Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds

本文建立 $p$-进解析流形上 Weil 丛的系统理论，证明其继承典范解析结构，并给出提升定理、Galois 等变结构及与晶体上同调的比较同构，应用于 $p$-进模形式和椭圆曲线的无穷小形变研究。

Frobenius结构在刚性连接上的应用及算术应用

Frobenius structure on rigid connections and arithmetic applications

本文构建了刚性不规则\check{G}-连接的Frobenius结构，并研究了其在局部monodromy表示和epipelagic Langlands参数预测中的应用。

Sums of singular series along arithmetic progressions and with smooth weights

Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.

The Arithmetic Fourier Transform

The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and can be performed by parallel processing. The AFT has some deep connections with the Prime Number Theorem and its rich history is discussed in this expository article.

Derivatives of Eisenstein series and arithmetic geometry

We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties $M$ associated to rational quadratic forms $(V,Q)$ of signature $(n,2)$. In the case $n=1$, we define generating series $\hatϕ_1(τ)$ for 1-cycles (resp. $\hatϕ_2(τ)$ for 0-cycles) on the arithmetic surface $\Cal M$ associated to a Shimura curve over $\Bbb Q$. These series are related to the second term in the Laurent expansion of an Eisenstein series of weight $\frac32$ and genus 1 (resp. genus 2) at the Siegel--Weil point, and these …

算术函数与几何

Arithmetic Functions and Geometry

本文在统一的代数框架下重新审视Euler函数、除数和函数与Dedekind的$ψ$-函数，揭示其与几何的联系，并为未来类似构造提供背景，同时对Dedekind的$ψ$-函数提出了原创性观察。

微分算子的微解与算术Gevrey级数的值

Microsolutions of differential operators and values of arithmetic Gevrey series

本研究通过Fuchsian微分算子微解从G-算子构造E-算子的解基，给出André定理的构造性证明，还引入算术逆Laplace变换研究相关常数的算术性质，定义算术Gevrey级数特殊值并探讨其与指数周期环的关系。

源自Sun猜想的三角行列式的整性

Integrality of a trigonometric determinant arising from a conjecture of Sun

本文通过Dirichlet L函数和字符间联系，解决Sun提出的三角行列式整性猜想，并揭示其与傅里叶分析和算术不变量的关联。

Semi-galois Categories II: An arithmetic analogue of Christol's theorem

In connection with our previous work on semi-galois categories, this paper proves an arithmetic analogue of Christol's theorem concerning an automata-theoretic characterization of when a formal power series over finite field is algebraic over the polynomial ring. There are by now several variants of Christol's theorem, all of which are concerned with rings of positive characteristic. This paper provides an arithmetic (or F_1) variant of Christol's theorem in the sense that it replaces the polynomial ring with the ring of integers of a number field and the ring of formal power series with the r…

在Tate猜想与算术不变量之间搭建桥梁

Building bridges between Tate conjectures and arithmetic invariants

本文澄清并建立了受Serre和Tate工作启发的多个猜想之间的联系，重点研究了代数圈的Tate猜想及其动形推广，并探讨了与Nagao猜想的关联。

Some results on affine Deligne-Lusztig varieties

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.