Pär Kurlberg, Florian Luca, Igor Shparlinski
In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.
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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
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 …
We obtain some results about the repeated exponentiation modulo a prime power from the viewpoint of arithmetic dynamical systems. Especially, we extend two asymptotic formulas about periodic points and tails in the case of modulo a prime to the case of modulo a prime power.
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…
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.
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。
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
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.
We investigate perturbed monomial dynamical system over $\mathbb{F}_p$ given by iterations of $x\mapsto x^n+c\bmod{p}$, where $c\in \mathbb{F}_p$. Instead of study the systems one at a time we study all of them at the same time. The complex distibution of periodic points is visualized in the so called Periodic Point Diagram, which can be seen as a discrete version of the classical Bifurcation Diagram. We also prove some general results about the distribution of periodic points. We end the article with a conjecture about the total number of periodic points.
We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.
A classical problem in analytic number theory is to study the distribution of $αp$ modulo 1, where $α$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is an almost-prime (the existence of infinitely many such $p$ is another topical result in prime number theory) and prove that its distribution has a similar property.
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。
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),
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.
The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.
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.
Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty} a_f(n)q ^n \in S_{λ+1/2}(Γ_0(N),χ)\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.
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.
Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 - c_r+o(1)}\quad(p\to\infty) $$ for some positive $c_r$ satisfying $c_r\ge e^{-(1+o(1))r} \; (r\to \infty).$
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.
In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third degree, in explicit non-exponential terms. In the process, we present explicit formulas for the calculation of some series involving the Riemann Zeta function.
In this paper, we use the 2-decent method to find a series of odd non-congruent numbers $\equiv1\pmod 8$ whose prime factors are $\equiv1\pmod4$ such that the congruent elliptic curves have second lowest Selmer groups, which includes Li and Tian's result (Li and Tian, 2000) as special cases.
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。
Given a prime $p \ge 5$ and an abstract odd representation $ρ_n$ with coefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove the existence of a lift of $ρ_n$ to characteristic $0$ whenever local lifts exist (under some technical conditions). Moreover, we can chose the inertial type of our lift at all primes but finitely many (where the lift is of Steinberg type). We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. In particular, our method shows that given a modular eigenform $f$ witho…
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 $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.
Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.
This paper has been withdrawn by the author due to an error in the proof of Theorem 6.
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。
We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).
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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