Sudhansu Sekhar Rout
We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of multivariate polynomials over the integers, we study the periodic points and the intersection of orbits of such dynamical systems over finite fields. 1
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On the Fixed Points of the Map x -> x^x Modulo a Prime
本文证明了对几乎所有素数p，同余式x^x == x mod p在[2，p-1]中存在整数解x，这些解可解释为映射x -> x^x mod p的不动点，并对与此映射相关的动力系统进行了数值研究和讨论.
We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly …
On the Cycle Structure of Repeated Exponentiation Modulo a Prime Power
本文从算术动力系统的观点出发，得到了模素幂重幂运算的一些结果。特别地，我们将模素数情形下关于周期点和尾部的两个渐近公式推广到模素数幂情形。
We establish the asymptotic density of the Motzkin numbers modulo small primes.
We establish the asymptotic density and some structure results for the Motzkin numbers modulo primes up to 29 using automata.
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g-1$ a square, and that for any prime $p>3$ there is a prime $q<p$ with the Bernoulli number $B_{q…
Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers
本文开始了对模形式和模素幂的模伽罗瓦表示的同余关系的计算研究。算法描述计算最大整数模，其中两个一元互质积分多项式有一个共同的根在某种意义上说，是定义。这些技巧被应用到研究的同余的模形式和模伽罗瓦表示模素幂。最后，一些计算结果的影响（非）提升的模形式模素幂和可能的推广的水平提高。
Automorphic products that are singular modulo primes
本文利用O(n, 2)上的Rankin--Cohen括号，证明了反射Borcherds乘积的傅里叶系数通常满足模某些素数的同余关系。
In this article we prove level raising for cuspidal eigenforms modulo prime powers (for odd primes) of weight $k\geq 2$ and arbitrary character, extending the result in weight two established by the work of Tsaknias and Wiese and generalizing (partially) Diamond-Ribet's celebrated level raising theorems.
In this paper we will consider new bounds on the smallest primitive root modulo a prime. We will make more judicious use of the Pòlya--Vinogradov and Burgess inequalities, and use them to prove that the smallest primitive root is smaller than $p^{0.68}$ for all primes $p$.
Periodicity of traces of Hecke operators modulo prime powers
本文研究了Hecke算子迹在模素数幂下的周期性性质，揭示了其在不同重量下的周期行为。
Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers
本文研究切比雪夫多项式模奇素数幂的不动点与周期结构，给出显式公式，并揭示 p=3 的例外及 p 整除 n 时的周期对应关系。
Distribution of twisted Kloosterman sums modulo prime powers
本文研究了模素幂乘特征标扭曲的Kloosterman和。我们表明，由一个初等计算，这些款项成为equidistributed在真正的线相对于一个适当的措施。
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.
Towards a bifurcation theory for perturbed monomial dynamical systems modulo a prime
研究了$\mathbb{F}_p$上由$x\mapsto x^n+c\bmod{p}$迭代得到的扰动单项动力系统，其中$c\in \mathbb{F}_p$ .我们不是一次研究一个系统，而是同时研究所有系统。周期点的复杂分布被可视化在所谓的周期点图中，它可以被看作是经典分叉图的离散版本。我们还证明了关于周期点分布的一些一般性结果。文章最后提出了一个关于周期点总数的猜想。
Level lowering modulo prime powers and twisted Fermat equations
讨论了权$2$尖点形式的模素幂的一个简洁的水平降低定理.此外，我们说明了如何这可以用来完全解决某些扭曲的费马方程$ax^n+by^n+cz^n=0$。
On the distribution of αp modulo one for primes p of a special form
解析数论中的一个经典问题是研究$αp$模1的分布，其中$α$是无理数，并且$p$在素数集合上运行。我们考虑由素数$p$生成的子序列使得$p+2$是一个几乎素数（存在无穷多个这样的$p$是素数理论中的另一个热门结果），并证明了它的分布具有类似的性质.
Galois Groups of Apéry-like Series Modulo Primes
本文研究了阿佩里型级数模素数的伽罗华群，揭示了其行为由素数的同余条件决定。
In this paper, we will give a series of non-congruent numbers with $\equiv3\pmod8$ prime factors.
Solutions of a polynomial equation modulo a prime power
本文探讨了如何求解多项式方程模整数（特别是素数幂）的整数解问题。
Congruences with intervals and subgroups modulo a prime
我们得到了关于几乎所有模素数$p$的剩余都用一个小整数和$({\mathbb Z}/p{\mathbb Z})^*$的小乘法子群的一个元素的乘积表示的新结果。这些结果是基于J.Cilleruelo和M. Z. Garaev（2014年），
In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.
Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli
模m <= 195的约化剩余系的Dirichlet特征标列表。相关的L-级数被制成表格，m <= 14，小的正整数参数s精确到10^（-50），它们的一阶导数m <= 6。素数上的限制求和只定义了导致欧拉乘积的狄利克雷素数L-函数（素数Zeta模函数）。两者都在相似的模和参数范围内实现。公式和数值技术是众所周知的;目的是提供直接访问参考值。
We give an explicit construction of vector-valued Yoshida lifts and derive a formula of the Bessel periods of Yoshida lifts, by which we prove the non-vanishing modulo a prime of Yoshida lifts attached to a pair of elliptic modular newforms. As a consequence, we obtain a new proof of the non-vanishing of Yoshida lifts.
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.
Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes
最近，Bruinier和Ono分类的尖形$f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{λ+1/2}(Γ_0(N),χ)\cap \mathbb{Z}[[q]]$，不满足一定的分布性质的模奇素数$p$。本文利用Rankin-Cohen定理，将这一结果推广到素数$p \geq 5$的半整权模形式.作为我们的主要定理的应用，我们得到的分布性质，模素数$p\geq5$，奇异模和Hurwitz类数的迹。我们还研究了一个类似的纽曼猜想的超分区。
We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.
Well-distribution modulo one and the primes
本文证明存在一个无理数α，使得素数序列乘以α后模1不具有良分布性质，揭示了素数在模1分布中的一种非均匀现象。
Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.
We study the supersingular curves on Picard modular surfaces modulo a prime $p$ which is inert in the underlying quadratic imaginary field. We analyze the automorphic vector bundles in characteristic $p$, and as an application derive a formula relating the number of irreducible components in the supersingular locus to the second Chern class of the surface.
On the smallest simultaneous power nonresidue modulo a prime
设$p$是素数，$p_1,\ldots, p_r$是$p-1$的不同素除子.本文证明了对于满足$c_r\ge e^{-(1+o(1))r} \; (r\to \infty).$的正$c_r$，最小正整数$n$是一个同时的$p_1,\ldots,p_r$ -幂模$p$非剩余，它满足$$ n<p^{1/4 - c_r+o(1)}\quad(p\to\infty) $$
We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.
On the representation of the number of integral points of an elliptic curve modulo a prime number
在本文中，我们将研究的椭圆曲线的整数点的数目表示的问题，模一个素数p。我们提出了一种方法来表示一个指数和，其中涉及多项式的三次，在显式的非指数项。在这个过程中，我们提出了明确的公式计算的一些系列涉及黎曼ζ函数。
On non-congruent numbers with 1 modulo 4 prime factors
本文利用2-decent方法求出了一系列素因子为$\equiv1\pmod4$的奇非全等数$\equiv1\pmod 8$，使得全等椭圆曲线具有次低Selmer群，其中包括Li和Tian的结果（Li and Tian，2000）作为特例.
Matrix Kloosterman sums modulo prime powers
本文研究素数幂模的矩阵 Kloosterman 和，将既有素数模结果推广到素数幂情形，给出最优界，并指出其与 $\mathrm{GL}_n$ 上 horocyclic 流的联系。
Congruences between modular forms modulo prime powers
给定一个素数$p \ge 5$和一个系数模为$p^n$（对于某些$n \ge 1$）和大图像的抽象奇表示$ρ_n$，我们证明了当局部提升存在时（在某些技术条件下），$ρ_n$提升到特征$0$的存在性.此外，我们可以在除了素数之外的所有素数上选择升力的惯性类型（其中升力是斯坦伯格型）。我们把这个结果应用到模形式的领域，证明了模素幂的一个降阶定理，并给出了升阶的例子。特别地，我们的方法表明，给定一个没有复数乘法或内部扭曲的模本征形$f$，对于所有的素数$p$，但不超过1000，并为所有的正整数$n$，存在另一个本征形$g\neq f$，这是全等的$f$模$p^n$。
For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.
We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.
We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.
Short Cycles in Repeated Exponentiation Modulo a Prime
给定一个素数$p$，我们考虑由模$p$的重复幂运算生成的动力系统，即由映射$u \mapsto f_g(u)$生成的动力系统，其中$f_g(u) \equiv g^u \pmod p$和$0 \le f_g(u) \le p-1$。该映射特别用于密码安全伪随机生成器的许多构造中。我们得到了上述动力系统的不动点和短圈个数的非平凡上界。
Small Doubling and Additive Structure Modulo a Prime
由于定理6的证明错误，本文已被作者撤回。
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.
An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $α\in\mathbb{R}\backslash\mathbb{Q},\,β\in\mathbb{R}$ and $0<θ<10/1561$, there exist infinitely many primes $p$, such that $\|αp^2+β\|<p^{-θ}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.
On the distribution of $αp^2+β$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors
本文针对 α 无理时 αp²+β 模 1 的分布问题，考察 p+2 为殆素数的素数子序列，证明该子序列保持与经典素数序列相似的分布性质。
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is equal to $1/8$, in line with a more general conjecture of Gerth. Vinogradov's method is the key analytic tool for our work.
Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the product of central binomial coefficients modulo $p$. The directed sub-graph of the automata for $C_n \mod p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.
On some arithmetic conditions of recurrent sequences modulo prime p
研究K-斐波那契序列模素数p，估计集合和与积的基数，并提出估计m维递归集合加倍常数的方法。
Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.
Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.
On the distribution of $αp+β$ modulo one for primes of type $p=ar^2+1$
本文研究形如 $p=ar^2+1$ 的素数子序列，证明其 $αp+β$ 模 1 的小数部分分布与全体素数情形具有类似性质。
Binomial Character Sums Modulo Prime Powers
我们表明，二项式和相关的乘法字符和$$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$有一个简单的评价足够大的$m$（$m\geq 2$，如果$p\nmid ABk$）。
On polynomials with roots modulo almost all primes
该研究定义了在几乎所有素数模下有根但无整数根的首一整系数多项式为例外多项式，分类了与不可约首一二次多项式乘积为例外多项式的不可约首一整系数多项式，并构造了指定因式形式的例外多项式。
Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$
本研究拓展前期工作，算出更高素数幂下的威尔逊商及对应阶乘同余，确定费马商幂和模性质，揭示相关p进系数规律并验证原始威尔逊同余的适配性。
Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials
该研究将作者此前用费马商幂和超同余描述威尔逊定理与威尔逊商的成果，转化为幂和与伯努利数的同余式，给出更简洁的证明，还算出了模p⁴的威尔逊商等结果并可推广到更高素数幂。
A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the know…
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