Arithmetic differential equations and $E$-functions
Said Manjra
设$K$为数字字段。给出了$E$ -函数在$x$中零化的最小次数的$K[x,d/dx]$的微分算子在无穷远处的算术刻划.
Let $K$ be a number field. We give an arithmetic characterization at infinity of the differential operator of $K[x,d/dx]$ with minimal degree in $x$ annihilating a given $E$-function.
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The theory of differential equations has an arithmetic analogue in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear differential equations is the same as the study of the differential cocycle from $GL_n$ into its Lie algebra given by the logarithmic derivative. However we prove here that there are no such cocycles in the context of arithmetic differential equations. In sequels of this paper we will remedy the situation by introducing arithmetic analogues of Lie algebras and a skew v…
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Arithmetic differential operators on Z_p
本文证明了从Z_p到自身的函数f是解析的当且仅当它可以表示为f（x）=F（x，dx，.，其中dx=（x-x^p）/p是Fermat商算子，F是系数在Z_p中的受限幂级数.
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考虑特征p的完备离散赋值域，其中可能存在非完美剩余域.设V是其绝对伽罗瓦群的局部单值有限的秩1连续表示.我们将证明V的算术天鹅导体（在[Kat89]中的Kato之后定义，其适合于[AS02]和[AS06]的更一般的理论）与Kedlaya在[Ked]中定义的相关差分模块$D^†(V)$的差分天鹅导体重合。这种构造是对[Tsu98a]中提出的Fontaine形式主义的非完美剩余情况的推广。我们的证明方法将使我们能够对精炼天鹅导体给出新的解释。
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Arithmetic partial differential operators on ramified extensions of $\ZZ_p$
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Arithmetic partial differential equations, II: modular curves
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Linking numbers and special differential characters in arithmetic hyperbolic manifolds
该研究构造了编码最简型算术双曲流形纽结数的非全纯Siegel模形式，通过深化对Kudla–Millson理论中上迁形式的理解完成证明，并建立了相关结构性质。
In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.
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本文研究了Appell超几何偏微分方程的单值性，得到了与GL（3）群相联系的四个导数.我们的四个衍生物都有显著的性质。我们发现Appell超几何偏微分方程可以通过我们的四个导数化为四个非线性偏微分方程。我们研究这四个非线性偏微分方程。我们对某些特殊情况感兴趣。在这些情况下，线性偏微分方程系统具有显着的性质。我们找到了酉群$U(2, 1)$的$η$ -函数，它具有有趣的性质.我们的$η$ -函数引导我们研究Picard曲线的算术，特别是变换问题（模问题）和相应的模方程。给出了与某些七次代数曲面相关的两个无理积分之间的一个有理变换。我们找到了一个模方程，它给出了两条Picard曲线的模之间的关系.
We obtain asymptotic results for well known summatory arithmetic functions, such as $ψ(x),$ and establish connections to new summatory functions. A new Volterra integral equation is offered, which is solved by summatory arithmetic functions. We conclude with some further integral formulas and provide number theoretic formulas as applications.
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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。
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Arithmetic Laplacians
我们发展了椭圆型偏微分方程的算术模拟。空间坐标的作用是由一族素数扮演的，而沿着各个素数的空间导数的作用是由相应的费马商算子扮演的，这些商算子服从一定的对易关系。这导致算术线性偏微分方程的代数群是类似的某些运营商在分析构造拉普拉斯算子。我们将这类方程分类到一维群上，并分析了它们的解空间。
Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité
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本文在秩条件下，利用光滑四次Weyl和的算术调和分析及第十矩新估计，证明了32个以上变量中三个对角四次方程组的非奇异Hasse原理。
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本研究利用特征零域下算子的单值群，推导微分算子形式形变算术支集的显式公式，并将结果推广至q-差分算子情形，为算术算子理论提供新的显式计算工具。
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory.
Erdős-Moser Equation in Arithmetic Progressions
本文研究算术级数中的Erdős-Moser方程，证明k=2时该级数若有解必含两项或四项，且两种情况均存在无穷多可完全刻画的解。
The average number of solutions of the Diophantine equation U^2 + V^2 = W^3 and related arithmetic functions
本文给出了一类算术函数的Dirichlet求和函数的带锐误差项的渐近性。结果适用于，例如，到r^2（n）和r（n^3）上的和，其中r（m）表示将m写成两个整数平方和的方法的数量。
We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the canonical norm for a relative differential form on an arithmetic surface, which proves a Height Inequality on the arithmetic surface.
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通过算术与几何单双射群的比较及切比察列夫定理，研究在足够大的基域下，构造具有高差分均匀性的有限域上的三项式。
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Algebraic solutions of linear differential equations: an arithmetic approach
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For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.
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We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterise geometrically the arithmetic points, study their distribution, and prove that in many cases the bi-algebraic curves are the linear ones.
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A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to Sylvester's sequence, Giuga numbers, Znám's problem, the inheritance problem, and Curtiss's bound on solutions of a unit fraction equation. Here we show $K \equiv 6 \pmod{6^2}$ if $6\mid K$, and uncover a remarkable $7$-term arithmetic progression of residues modulo $6^2\cdot8$ in the sequence of known PPNs. On that basis, we pose a conjecture which leads to a cond…
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Automorphic Integrals with Rational Period Functions and Arithmetical Identities
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本文提出用算术正则性方法替代傅里叶分析转移原理，以处理稀疏算术集中的组合定理，并给出一个包含线性方程和高次方程系统的正确下界。
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Inspired by the Erdös-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem and Sźmeredis Theorem that the primes and all subsets of positive upper density are AP-sets. We prove that (1, 1,..., 1) is a solution to the equation Mx = 0 where M is an integer matrix whose null space has dimension at least 2, if and only if the equation has infinitely many solutions such that the coordinates of each solution are elements in the same AP. Thi…
Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified and non-logarithmic case, or p=2 and logarithmic case. As an application, we obtain a Hasse-Arf theorem for filtrations on finite flat group schemes over O_K.
We prove that the equation $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime.
In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
This paper has been withdrawn by the author because Conjecture 1 is false. Please see arXiv:0901.2093 for a justification that Conjecture 1 is false. The other main results are also available from the above URL.
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