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2.0

Jun 30, 2018
06/18

by
Lior Bary-Soroker; Moshe Jarden; Danny Neftin

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We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.

Topics: Mathematics, Number Theory, Group Theory

Source: http://arxiv.org/abs/1403.3266

4
4.0

Jun 28, 2018
06/18

by
Alex Bartel; Hendrik W. Lenstra

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We develop a theory of commensurability of groups, of rings, and of modules. It allows us, in certain cases, to compare sizes of automorphism groups of modules, even when those are infinite. This work is motivated by the Cohen-Lenstra heuristics on class groups. The number theoretic implications will be addressed in a separate paper.

Topics: Number Theory, Mathematics, Rings and Algebras

Source: http://arxiv.org/abs/1510.02758

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2.0

Jun 30, 2018
06/18

by
Yang Cao; Yongqi Liang; Fei Xu

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We prove that any open subset $U$ of a semi-simple simply connected quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$ over a number field satisfies strong approximation by establishing a fibration of $G$ over a toric variety. We also extend this result to strong approximation with Brauer-Manin obstruction for groupic varieties of which all invertible functions are constant. As a by-product of the fibration method, we provide an example which satisfies strong...

Topics: Algebraic Geometry, Number Theory, Mathematics

Source: http://arxiv.org/abs/1701.07259

4
4.0

Jun 30, 2018
06/18

by
Dzmitry Badziahin

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The $p$-adic Littlewood conjecture (PLC) states that $\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and let $\mathrm{T}$ be the standard left shift map. Assuming that $x$ is a counterexample to PLC we get several restrictions on limit elements of the sequence $\{\mathrm{T}^n w_{CF}(x)\}_{n\in\mathbb{N}}$. As a consequence we show that for any such limit element...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1406.3594

3
3.0

Jun 29, 2018
06/18

by
Franziska Wutz

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We show the existence of a hypersurface that contains a given closed subscheme of a projective space over a finite field and intersects a smooth quasi-projective scheme smoothly, under some condition on the dimension. This generalizes a Bertini theorem by Poonen and is the finite field analogue of a Bertini theorem by Altman and Kleiman. Furthermore, we add the possibility of modifying finitely many local conditions of the hypersurface. We show that the condition on the dimension is fulfilled...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1611.09092

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Jun 28, 2018
06/18

by
Samuele Anni; Pedro Lemos; Samir Siksek

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Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\bar{\rho}_{A,\ell}$ contains a transvection then $\bar{\rho}_{A,\ell}$ is...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1508.00211

3
3.0

Jun 29, 2018
06/18

by
Keenan Kidwell

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We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the $\mathbf{Z}_p$-extension being considered, resulting in the local cohomology groups...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1611.02727

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3.0

Jun 30, 2018
06/18

by
Henrik Ueberschaer; Par Kurlberg

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We consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes ---"old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a...

Topics: Nonlinear Sciences, Chaotic Dynamics, Mathematics, Number Theory, Analysis of PDEs, Mathematical...

Source: http://arxiv.org/abs/1409.6878

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Jun 26, 2018
06/18

by
Felix Breuer; Zafeirakis Zafeirakopoulos

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Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as...

Topics: Symbolic Computation, Number Theory, Combinatorics, Mathematics, Computing Research Repository

Source: http://arxiv.org/abs/1501.07773

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5.0

Jun 28, 2018
06/18

by
Daniel Allcock; Itamar Gal; Alice Mark

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We develop the notational system developed by Conway and Sloane for working with quadratic forms over the 2-adic integers, and prove its validity. Their system is far better for actual calculations than earlier methods, and has been used for many years, but it seems that no proof has been published before now.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.04614

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8.0

Jun 28, 2018
06/18

by
Christoph Aistleitner; Gerhard Larcher

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An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n\geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n \geq1}$ is uniformly distributed modulo one for almost all $\alpha$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy $D_{N}$ of $\left(\left\{a_{n} \alpha \right\}\right)_{n \geq 1}$ for almost all $\alpha$. By a result of R. C. Baker this...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1507.06472

3
3.0

Jun 28, 2018
06/18

by
Naser T Sardari

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For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $00$. Finally assume that an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$ is given. Then we show that there exists an integral solution $x=(x_1,\dots,x_d)$ of $F(x)=N$ such that $x_i\equiv \lambda_i \text{ mod }...

Topics: Combinatorics, Number Theory, Mathematics

Source: http://arxiv.org/abs/1510.00462

3
3.0

Jun 29, 2018
06/18

by
Usha K. Sangale

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Hardy's theorem for the Riemann zeta-function $\zeta(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1606.00680

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7.0

Jun 28, 2018
06/18

by
Felix Breuer; Dennis Eichhorn; Brandt Kronholm

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In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called {\it supercranks} that combinatorially witness every instance of divisibility of $p(n,3)$ by any prime $m \equiv -1 \pmod 6$, where $p(n,3)$ is the number of partitions of $n$ into three parts. A rearrangement of lattice...

Topics: Number Theory, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1508.00397

3
3.0

Jun 28, 2018
06/18

by
Bo He; Ákos Pintér; Alain Togbe; Shichun Yang

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Dujella and Peth\H{o}, generalizing a result of Baker and Davenport, proved that the set $\{1, 3\}$ cannot be extended to a Diophantine quintuple. As a consequence of our main result, it is shown that the Diophantine pair $\{1, b\}$ cannot be extended to a Diophantine quintuple if $b-1$ is a prime.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1510.05579

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9.0

Jun 27, 2018
06/18

by
Marzieh Eini Keleshteri; Nazim I. Mahmudov

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In this paper we aim to specify some characteristics of the so called family of $q$-Appell Polynomials by using $q$-Umbral calculus. Next in our study, we focus on $q$-Genocchi numbers and polynomials as a famous member of this family. To do this, firstly we show that any arbitrary polynomial can be written based on a linear combination of $q$-Genocchi polynomials. Finally, we approach to the point that similar properties can be found for the other members of the class of $q$-Appell polynomials.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1505.05067

3
3.0

Jun 29, 2018
06/18

by
Igor C. Oliveira; Rahul Santhanam

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We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence $\{p_n\}_{n \in \mathbb{N}}$ of increasing primes and a randomized algorithm $A$ running in expected sub-exponential time such that for each $n$, on input $1^{|p_n|}$, $A$ outputs $p_n$ with probability $1$. In other words, our result provides a pseudodeterministic construction of primes in...

Topics: Number Theory, Data Structures and Algorithms, Discrete Mathematics, Computational Complexity,...

Source: http://arxiv.org/abs/1612.01817

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6.0

Jun 28, 2018
06/18

by
Masao Oi

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For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r_x defined by Kato [2]. The main theme of this paper is to give a simple formula to compute r_x' defined in [4], which is equal to r_x for good Artin-Schreier extension. We also prove Kato's...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1507.00097

2
2.0

Jun 30, 2018
06/18

by
Elmar Grosse-Klönne

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We show that the categories of smooth ${\rm SL}_2({\mathbb Q}_p)$-representations (resp. ${\rm GL}_2({\mathbb Q}_p)$-representations) of level $1$ on $p$-torsion modules are equivalent with certain explicitly described equivariant coefficient systems on the Bruhat-Tits tree; the coefficient system assigned to a representation $V$ assigns to an edge $\tau$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $\tau$. The proof relies on computations of the group cohomology of...

Topics: Mathematics, Number Theory, Representation Theory

Source: http://arxiv.org/abs/1408.3367

4
4.0

Jun 30, 2018
06/18

by
Ling Long; Robert Osburn; Holly Swisher

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We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis.

Topics: Mathematics, Number Theory, Combinatorics

Source: http://arxiv.org/abs/1404.4723

3
3.0

Jun 29, 2018
06/18

by
Yuan Ren

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In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second section, we give an example of modular abelian varieties due to Gross and study its properties in some details. In the third section, we define Eisenstein quotients of the modular Jacobians in general and give a criterion of the non-triviality of Heegner points...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1612.08592

4
4.0

Jun 30, 2018
06/18

by
Pierre-Alain Fouque; Mehdi Tibouchi

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In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a

Topics: Cryptography and Security, Mathematics, Computing Research Repository, Number Theory

Source: http://arxiv.org/abs/1406.7078

3
3.0

Jun 30, 2018
06/18

by
Adrian Dudek

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We prove some results concerning the distribution of primes on the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x} \log x,x]$ for all $x \geq 2$; this improves a result of Ramar\'{e} and Saouter. We then show that the constant $4/\pi$ may be reduced to $(1+\epsilon)$ provided that $x$ is taken to be sufficiently large. From this we get an immediate estimate for a well-known theorem of Cram\'{e}r, in that we show the...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1402.6417

4
4.0

Jun 29, 2018
06/18

by
Lynne H. Walling

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We extend some recent work of D. McCarthy, proving relations among some Fourier coefficients of a degree 2 Siegel modular form $F$ with arbitrary level and character, provided there are some primes $q$ so that $F$ is an eigenform for the Hecke operators $T(q)$ and $T_1(q^2)$.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1608.00158

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4.0

Jun 30, 2018
06/18

by
Tamar Ziegler

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We survey some of the ideas behind the recent developments in additive number theory, combinatorics and ergodic theory leading to the proof of Hardy- Littlewood type estimates for the number of prime solutions to systems of linear equations of finite complexity.

Topics: Mathematics, Number Theory, Dynamical Systems

Source: http://arxiv.org/abs/1404.0775

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10.0

Jun 27, 2018
06/18

by
Nariya Kawazumi

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We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.

Topics: Quantum Algebra, Number Theory, Mathematics, Geometric Topology

Source: http://arxiv.org/abs/1506.03174

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8.0

Jun 30, 2018
06/18

by
David Dummit; Andrew Granville; Hershy Kisilevsky

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We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1411.4594

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9.0

Jun 30, 2018
06/18

by
David Simmons

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A theorem of Kurzweil ('55) on inhomogeneous Diophantine approximation states that if $\theta$ is an irrational number, then the following are equivalent: (A) for every decreasing positive function $\psi$ such that $\sum_{q = 1}^\infty \psi(q) = \infty$, and for almost every $s\in\mathbb R$, there exist infinitely many $q\in\mathbb N$ such that $\|q\theta - s\| < \psi(q)$, and (B) $\theta$ is badly approximable. This theorem is not true if one adds to condition (A) the hypothesis that the...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1412.5992

9
9.0

Jun 27, 2018
06/18

by
Shabnam Akhtari

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The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue's inequalities. This will provide an upper bound for the number of integer points on a family of elliptic curves with j-invariant equal to 1728.

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1504.00710

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4.0

Jun 29, 2018
06/18

by
David Jarossay

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This is the first part of a work whose goal is study associator equations in a way which is adapted to the framework of crystalline pro-unipotent fundamental groupoids. Our general goal is to reformulate (some natural consequences of) the associator equations as an explicit comparison between the respective modules of coefficients of an associator and its image under a certain automorphism, this comparison being compatible with their respective depth filtrations and defined over a ring of...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1601.01161

3
3.0

Jun 30, 2018
06/18

by
A. Iosevich; B. Murphy; J. Pakianathan

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Let $R$ be a finite ring and define the hyperbola $H=\{(x,y) \in R \times R: xy=1 \}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following "square root law" bound holds with a constant $C>0$ for all non-trivial characters $\chi$ on $R^2$: \[ \left| \sum_{(x,y)\in H}\chi(x,y)\right|\leq C\sqrt{|H|}. \] Then, with a finite number of exceptions, those rings are fields. For rings of even order we show that there are other infinite families...

Topics: Mathematics, Number Theory, Combinatorics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1405.7657

2
2.0

Jun 29, 2018
06/18

by
P. Habegger

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Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structure which give a general framework to work with sets mentioned above. It complements the theorem of Pila-Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1608.04547

3
3.0

Jun 30, 2018
06/18

by
Wee Teck Gan; Shuichiro Takeda

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We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for any nonarchimedean local field of characteristic not 2 and in arbitrary residual characteristic.

Topics: Mathematics, Number Theory, Representation Theory

Source: http://arxiv.org/abs/1405.2626

2
2.0

Jun 30, 2018
06/18

by
Shuo Li

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This article consists to give a necessary and sufficient condition of the meromorphic continuity of Dirichlet series defined as $\sum_{x\in \mathbf{N}^n} \frac{a_{x}}{P(x)^s}$, Where $a_{x}$ is a $q$-automatic sequence of $n$ parameters and $P: \mathbf{C}^n \to \mathbf{C}$ a polynomial, such that $P$ does not have zeros on $\mathbf{Q}^{n}_{+}$. And some specific cases of $n=1$ will also be studied in this article as examples to show the possibility to have an holomorphic continuity on the whole...

Topics: Combinatorics, Number Theory, Mathematics

Source: http://arxiv.org/abs/1701.08603

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4.0

Jun 30, 2018
06/18

by
Adam Harris

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This thesis develops some of the basic model theory of covers of algebraic curves. In particular, an equivalence between the good model-theoretic behaviour of the modular j-function, and the openness of certain Galois representations in the Tate-modules of abelian varieties is described.

Topics: Mathematics, Logic, Number Theory

Source: http://arxiv.org/abs/1412.3484

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6.0

Jun 28, 2018
06/18

by
Patrick Meisner

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We determine in this paper the distribution of the number of points on the cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r = F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is fixed and the genus, $g$, tends to infinity. This generalize the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over $\mathbb{F}_q$. In all cases, the distribution is given by a sum of random variables.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.07814

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4.0

Jun 29, 2018
06/18

by
Andrew R. Booker; Carl Pomerance

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We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1607.01557

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2.0

Jun 30, 2018
06/18

by
Jae-Hyun Yang

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In this paper, we obtain some behaviours of theta sums of higher index for the Schroedinger-Weil representation of the Jacobi group associated with a positive definite symmetric real matrix of degree m.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1702.08667

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2.0

Jun 28, 2018
06/18

by
Bouikhalene Belaid; Elqorachi Elhoucien

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In \cite{St3} H. Stetk\ae r obtained the solutions of Van Vleck's functional equation for the sine $$f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y),\; x,y\in G,$$ where $G$ is a semigroup, $\tau$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine $$\mu(y)f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y), \;x,y\in G,$$ where $\mu$ : $G\longrightarrow...

Topics: Number Theory, Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1512.06753

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2.0

Jun 29, 2018
06/18

by
Wataru Kai; Hiroyasu Miyazaki

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The moving lemma of Suslin states that a cycle on $X\times \mathbb{A} ^n$ meeting all faces properly can be moved so that it becomes equidimensional over $\mathbb{A}^n$. This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups. In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting.

Topics: Number Theory, Algebraic Geometry, K-Theory and Homology, Mathematics

Source: http://arxiv.org/abs/1604.04356

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2.0

Jun 30, 2018
06/18

by
F. E. Brochero Martínez; C. R. Giraldo Vergara; L. Batista de Oliveira

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Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into irreducible factors is given. Finally, weakening one of our hypothesis, we also obtain factors of the form $x^{2t}-ax^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is given.

Topics: Mathematics, Computing Research Repository, Information Theory, Number Theory

Source: http://arxiv.org/abs/1404.6281

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3.0

Jun 29, 2018
06/18

by
Kei Yuen Chan; Gordan Savin

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Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $\psi$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}\psi)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed...

Topics: Number Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1605.05130

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5.0

Jun 30, 2018
06/18

by
Enric Nart

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Types over a discrete valued field $(K,v)$ are computational objects that parameterize certain families of monic irreducible polynomials in $K_v[x]$, where $K_v$ is the completion of $K$ at $v$. Two types are considered to be equivalent if they encode the same family of prime polynomials. In this paper, we characterize the equivalence of types in terms of certain data supported by them.

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1409.4345

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2.0

Jun 28, 2018
06/18

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Gerhard Larcher; Florian Puchhammer

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It is known that there is a constant $c>0$ such that for every sequence $x_1, x_2,\ldots$ in $[0,1)$ we have for the star discrepancy $D^{*}_N$ of the first $N$ elements of the sequence that $N D^{*}_N\geq c\cdot \log N$ holds for infinitely many $N$. Let $c^{*}$ be the supremum of all such $c$ with this property. We show $c^{*}>0.065664679\ldots$, thereby slightly improving the estimates known until now.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1511.03869

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Jun 29, 2018
06/18

by
Natalie M. Paquette; Daniel Persson; Roberto Volpato

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We provide a physics derivation of Monstrous moonshine. We show that the McKay-Thompson series $T_g$, $g\in \mathbb{M}$, can be interpreted as supersymmetric indices counting spacetime BPS-states in certain heterotic string models. The invariance groups of these series arise naturally as spacetime T-duality groups and their genus zero property descends from the behaviour of these heterotic models in suitable decompactification limits. We also show that the space of BPS-states forms a module for...

Topics: High Energy Physics - Theory, Number Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1601.05412

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Jun 30, 2018
06/18

by
Salim Ali Altug; Arul Shankar; Ila Varma; Kevin H. Wilson

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We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood....

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1704.01729

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Jun 30, 2018
06/18

by
Fabien Cléry; Gerard van der Geer

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This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also shows the strong relations between these modular forms of different genera. We illustrate this by a number of examples.

Topics: Mathematics, Number Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1409.7176

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Jun 30, 2018
06/18

by
Colin Defant

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Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, the three imaginary...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1412.3072

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Jun 30, 2018
06/18

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Michiel Kosters

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In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is needed.

Topics: Mathematics, Number Theory, Commutative Algebra

Source: http://arxiv.org/abs/1404.3916

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Jun 30, 2018
06/18

by
Stephen Kudla

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We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of...

Topics: Mathematics, Number Theory, Algebraic Geometry

Source: http://arxiv.org/abs/1408.1907