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An introduction to p-adic period ringsAug 22 2019This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{\'e}tale and crystalline or de Rham cohomologies ... More

Remarks on generating series for special cyclesAug 22 2019In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F:\Q|=d, of signature ((m,2)^{d_+},(m+2,0)^{d-d_+}), 1\le d_+<d. For each n, 1\le n\le m, there are special cycles ... More

Étale cohomology of rank one $\ell$-adic local systems in positive characteristicAug 22 2019We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem.

Motivic sheaves revisitedAug 22 2019In earlier work (arXiv:0801.0261), we gave a definition of an abelian category of motivic (constructible) sheaves over a base in characteristic zero using Nori's method. This category has Hodge and etale realizations, and is stable under inverse and direct ... More

The modularity of special cycles on orthogonal Shimura varieties over totally real fields under the Beilinson-Bloch conjectureAug 21 2019We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree $d$ associated with a quadratic form in $n+2$ variables whose signature is $(n,2)$ at $e$ real places and $(n+2,0)$ at the remaining $d-e$ real places ... More

Monogenic trinomials with non-squarefree discriminantAug 21 2019For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific ... More

Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representationsAug 21 2019In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric ... More

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groupsAug 21 2019In this paper we analyze Fourier coefficients of automorphic forms on adelic split simply-laced reductive groups $G(\mathbb{A})$. Let $\pi$ be a minimal or next-to-minimal automorphic representation of $G(\mathbb{A})$. We prove that any $\eta\in \pi$ ... More

Torsion groups of Mordell curves over cubic and sextic fieldsAug 21 2019In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce ... More

Generating Functions and Congruences for Some Partition Functions Related to Mock Theta FunctionsAug 21 2019Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition functions as ... More

Some results on vanishing coefficients in infinite product expansionsAug 21 2019Recently, M. D. Hirschhorn proved that, if $\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$ and $\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. ... More

Tensor Product $L$-Functions On Metaplectic Covering Groups of $GL_r$Aug 21 2019In this note we compute some local unramified integrals defined on metaplectic covering groups of $GL$. These local integrals which were introduced by Suzuki, represent the standard tensor product $L$ function $L(\pi^{(n)}\times \tau^{(n)},s)$ and extend ... More

Quadratic residues in $\mathbb{F}_{p^2}$ and related permutations involving primitive rootsAug 20 2019Let $p=2n+1$ be an odd prime, and let $\zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[\zeta_{p^2-1}]$ be a primitive root modulo $p\mathbb{Z}_p[\zeta_{p^2-1}]$ ... More

Algebraic integer totally in a compactAug 20 2019An algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients. An algebraic integer is said to be totally in a compact of the complex plan if all its conjugates are in the same compact as well. Given a compact, ... More

Gamma functions, monodromy and Apéry constantsAug 20 2019In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as an Ap\'ery series associated to an ordinary differential operator L with a point of maximal unipotent monodromy (MUM) at 0, a conifold singularity ... More

PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion ManuscriptsAug 20 2019In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever ... More

Torsors on loop groups and the Hitchin fibrationAug 20 2019In his proof of the fundamental lemma, Ng\^o established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant ... More

A note on Misiurewicz polynomialsAug 20 2019Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz points $c_0,c_1\in ... More

The mean square of real character sumsAug 20 2019In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symbol and $\sideset{}{^*}\sum$ ... More

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbersAug 20 2019The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. ... More

Iwasawa theory for $\mathrm{U}(r,s)$, Bloch-Kato conjecture and Functional EquationAug 20 2019In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic ... More

Moduli stacks of étale (phi,Gamma)-modules and the existence of crystalline liftsAug 20 2019We construct stacks which algebraize Mazur's formal deformation rings of local Galois representations. More precisely, we construct Noetherian formal algebraic stacks over Spf Zp which parameterize \'etale (phi,Gamma)-modules; the formal completions of ... More

On the distribution of primitive subgroups of $\mathbb{Z}^{d}$ of large covolumeAug 20 2019We prove existence and compute the limiting distribution of the image of rank-$\left(d-1\right)$ primitive subgroups of $\mathbb{Z}^{d}$ of large covolume in the space $X_{d-1,d}$ of homothety classes of rank-$\left(d-1\right)$ discrete subgroups of $\mathbb{R}^{d}$. ... More

Nonuniform Distributions of Residues of Prime Sequences in Prime ModuliAug 19 2019For 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 $\varphi(q)$ reduced residue classes ... More

Local $\mathbb F_l$-monodromy and level fixingAug 19 2019We tackle three related problems. The first deals with freeness of localized cohomology groups of Harris-Taylor perverse sheaves, defined on the special fiber of some Kottwitz-Harris-Taylor Shimura variety. We then study the nilpotent monodromy operator ... More

Moduli stacks of two-dimensional Galois representationsAug 19 2019We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).

Linnik's large sieve and the $L^{1}$ norm of exponential sumsAug 19 2019The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the lower bound due ... More

Weil descent and cryptographic trilinear mapsAug 19 2019It has recently been shown that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop a method for constructing such maps on the Weil descent (restriction) of abelian varieties over finite ... More

Weil descent and cryptographic trilinear mapsAug 19 2019Aug 21 2019It has recently been shown that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop a method for constructing such maps on the Weil descent (restriction) of abelian varieties over finite ... More

Fast multi-precision computation of some Euler productsAug 19 2019For every modulus $q\ge3$, we define a family of subsets $\mathcal{A}$ of the multiplicative group $(\mathbb{Z}/{q}\mathbb{Z})^\times$ for which the Euler product $\prod_{p\text{mod}q\in\mathcal{A}}(1-p^{-s})$ can be computed in double exponential time, ... More

Combinatorial Proof of the Minimal Excludant TheoremAug 19 2019The maximal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is defined to be the least gap of $\lambda$. For each positive integer $n$, the function $ \sigma\, \rm{mex}(n)$ is defined to be the sum of the least gaps in all partitions of $n$. ... More

Optimal Lifting for the Projective Action of $SL_3(Z)$Aug 19 2019Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon}$, whose projection ... More

Some new congruences for $(7,t)$-regular bipartitions modulo $t$Aug 19 2019In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ ... More

On the remainder term of the Weyl law for congruence subgroups of Chevalley groupsAug 19 2019Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen ... More

Some identities of type 2 Degenerate Bernoulli polynomials of the second kindAug 19 2019I recent years, many mathematicians studied various degenerate version of some spcial polynomials of which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and ... More

On the Chow group of the self-product of a CM elliptic curve defined over a number fieldAug 18 2019In this note we study the cokernel of the restriction map from the Chow group of codimension 2 cycles on the spread of the self product of a CM-elliptic curve over the ring of integers of a number field to the codimension 2 cycles on the self of product ... More

The rational cuspidal divisor class group of $X_0(N)$Aug 18 2019For any positive integer $N$, we completely determine the structure of the rational cuspidal divisor class group $\mathcal{C}(N)$ of $X_0(N)$, which is conjecturally equal to the group of rational torsion points on $J_0(N)$. More specifically, let $\ell$ ... More

The crystalline comparison of Ainf-cohomology: the case of good reductionAug 18 2019We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.

A Dedekind's Criterion over Valued FieldsAug 18 2019Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient conditions for ... More

Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fieldsAug 17 2019We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level ... More

On the number of gaps of sequences with Poissonian Pair CorrelationsAug 17 2019A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\# \{1\le i\neq j\le N: |x_i-x_j| \le s/N\}=2sN(1+o(1))$ for all reals $s>0$, as $N\to \infty$. It is known that, if $(x_n)$ has Poissonian pair correlations, then the number ... More

Algebraic independence of certain entire functions of two variables generated by linear recurrencesAug 17 2019In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear ... More

Uniform Bounds for Periods of Endomorphisms of VarietiesAug 17 2019Suppose $X$ is a projective variety defined over a finite extension $K$ of $\mathbb{Q}_p$ and suppose $X$ admits a model $\mathcal{X}$ defined over the ring of integers $R$ of $K$. Let $f:{X}\rightarrow {X}$ be an endomorphism of $X$ defined over $K$ ... More

On the modularity of 2-adic potentially semi-stable deformation ringsAug 16 2019Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_2)$ and an ordinary $R = \mathbb{T}$ theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable ... More

Symmetric primes revisitedAug 16 2019A pair of odd primes is said to be symmetric if they are 1 modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the current author gets an upper bound for the distribution of primes that belong to a symmetric pair. In this paper that ... More

The Minkowski chain and Diophantine approximationAug 16 2019The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the ... More

Lower bounds in the polynomial Szemerédi theoremAug 16 2019We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's ... More

Congruences in Hermitian Jacobi and Hermitian modular formsAug 16 2019In this paper we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian Jacobi forms to characterize ... More

$\boldsymbol{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometryAug 16 2019A Sturmian sequence is an infinite string over two letters with low subword complexity. The history of the research surveyed in the present chapter starts with two papers written by Morse and Hedlund as well as Coven and Hedlund in 1940 and 1973, respectively. ... More

Geometric local $\varepsilon$-factors in higher dimensionsAug 16 2019We use former results on geometric local $\varepsilon$-factors over curves in order to prove a factorization result for the determinant of the cohomology of an $\ell$-adic sheaf over an arbitrary proper scheme over a perfect field of positive characteristic ... More

On the $p$-adic properties of Stirling numbers of the first kindAug 15 2019Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti ... More

Singer difference sets and the projective norm graphAug 15 2019We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets as a subset ... More

A New Class of Irreducible PolynomialsAug 15 2019In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside the closed unit disc centered at the origin in the complex plane and deduce the irreducibility over the ring of integers. ... More

Dessins d'enfants and Brauer configuration algebrasAug 15 2019In this paper we associate to a dessin d'enfant an associative algebra, called a Brauer configuration algebra. This is an algebra given by quiver and relations induced by the monodromy of the dessin d'enfant. We show that the dimension of the Brauer configuration ... More

Factorization of Dickson polynomials over Finite FieldsAug 15 2019Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these polynomials in the case ... More

Tensor Operations on Group SchemesAug 15 2019In this paper we study multilinear morphisms between commutative group schemes and the associated tensor constructions. We will also do some explicit calculations and give examples that show that this theory behaves in a way that one would naturally expect. ... More

A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number PuzzleAug 15 2019In this article, we give a particular recreational application of two integer sequences. These sequences are respectively the sequence A000533 and sequence A261544 in "The On-line Encyclopedia of Integer Sequences" (OEIS). The recreational application ... More

An analogue of a formula for Chebotarev DensitiesAug 15 2019In this short note, we show an analogue of Dawsey's formula on Chebotarev densities for finite Galois extensions of $\mathbb{Q}$ with respect to the Riemann zeta function $\zeta(ms), m\geq2$. Her formula may be viewed as the limit version of our formula ... More

Epsilon factors of symplectic type characters in the wild caseAug 14 2019By John Tate we can associate the epsilon factor with every multiplicative character of a local field. In this paper we determine the explicit signs of the epsilon factors for symplectic type characters of $K^\times$, where $K/F$ is a wildly ramified ... More

Maximally additively reducible subsets of the integersAug 14 2019Let $A, B \subseteq \mathbb{N}$ be two finite sets of natural numbers. We say that $B$ is an additive divisor for $A$ if there exists some $C \subseteq \mathbb{N}$ with $A = B+C$. We prove that among those subsets of $\{0, 1, \ldots, k\}$ which have $0$ ... More

Bilateral Ramanujan-like series for $1/π^k$ and their congruencesAug 14 2019We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In ... More

Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta valuesAug 14 2019We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every ... More

On the divisor problem with congruence conditionsAug 14 2019Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\geq (q_1q_2)^{1+\varepsilon}, ... More

A Constructive Proof of Masser's TheoremAug 14 2019The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+\epsilon}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $ where ... More

On odd deficient-perfect numbers with four distinct prime divisorsAug 14 2019For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number ... More

Pair correlation for Dedekind zeta functions of abelian extensionsAug 13 2019Here we study problems related to the proportions of zeros, especially simple and distinct zeros on the critical line, of Dedekind zeta functions. We obtain new bounds on a counting function that measures the discrepancy of the zeta functions from having ... More

Twelfth moment of Dirichlet L-functions to prime power moduliAug 13 2019We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement ... More

$p$-adic Integral GeometryAug 13 2019We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving ... More

On functional equations for Nielsen polylogarithmsAug 13 2019We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also ... More

On Arthur's unitarity conjecture for split real groupsAug 12 2019Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the "Langlands element" (i.e., the one specified by Arthur) of all unipotent Arthur packets ... More

Moment estimates for the exponential sum with higher divisor functionsAug 12 2019We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for ... More

Algorithms for the Multiplication Table ProblemAug 12 2019We present several algorithms for computing $M(n)$, the function that counts the number of distinct products in an $n\times n$ multiplication table. In particular, we consider their run-times and space bounds for single evaluation, tabulation, and Monte-Carlo ... More

Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus ThreeAug 12 2019We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double ... More

Subspaces of tensors with high analytic rankAug 12 2019It is shown that for any subspace $V\subseteq \mathbb{F}_p^{n\times\cdots\times n}$ of $d$-tensors and integer $1\leq r\leq n$, there is subspace $W\subseteq V$ of dimension $\Omega_d(\dim(V)/rn^{d-1})$ in which every nonzero element has analytic rank ... More

The First Moment of $L(\frac{1}{2},χ)$ for Real Quadratic Function FieldsAug 12 2019In this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of even degree at ... More

The shapes of Galois quartic fieldsAug 11 2019We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of ... More

Counting pattern-avoiding integer partitionsAug 11 2019A partition $\alpha$ is said to contain another partition (or pattern) $\mu$ if the Ferrers board for $\mu$ is attainable from $\alpha$ under removal of rows and columns. We say $\alpha$ avoids $\mu$ if it does not contain $\mu$. In this paper we count ... More

Galois Cohomology For Lubin-tate (phi_q, Gamma_{LT})-modules Over Coefficient RingsAug 11 2019The classification of local Galois representations using (phi, Gamma)-modules by Fontaine has been generalized by Kisin and Ren [8] over Lubin-Tate extensions of local fields using the theory of (phi_q, Gamma_{LT})-modules. We extend the work of Herr ... More

Galois Cohomology For Lubin-Tate (phi_q, Gamma_{LT})-modules Over Coefficient RingsAug 11 2019Aug 13 2019The classification of local Galois representations using (phi, Gamma)-modules by Fontaine has been generalized by Kisin and Ren [8] over Lubin-Tate extensions of local fields using the theory of (phi_q, Gamma_{LT})-modules. We extend the work of Herr ... More

Congruences in fractional partition functionsAug 11 2019Aug 14 2019The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved congruences of the ... More

Congruences in fractional partition functionsAug 11 2019The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved congruences of the ... More

Arithmetic of weighted Catalan numbersAug 11 2019In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the $2$-adic valuations of the weighted Catalan numbers are equal to the $2$-adic valutations of the Catalan numbers. ... More

New Transcendental Numbers from Certain SequencesAug 11 2019We construct three infinite decimals from certain digits of $n^n, n^m$, and $(n!)^m$ (for any fixed $m$) and show that all three are transcendental. In particular, while previous work looked at the last non-zero digit of $n^n$, we look at the digit right ... More

Distribution of Eigenvalues of Random Real Symmetric Block MatricesAug 11 2019Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions. Many statistics in one can be interpreted in terms of quantities of the other; for example, zeros of $L$-functions correspond ... More

A note on degenerate Euler and Bernoulli polynomials of complex variableAug 10 2019In this paper, we study the degenerate version of the new type Euler polynomials, namely degenerate cosine-Euler polynomials and sime-Euler polynomials and also corresponding ones for Bernoulli polynomials, namely degenerate cosine Bernoulli polynomials ... More

Factoring Catalan numbersAug 10 2019The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes of Segment are ... More

Polynomial analogues of restricted multicolor b-ary partition functionsAug 10 2019Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\geq ... More

Discrete Measures and the Extended Riemann HypothesisAug 10 2019In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined arithmetically ... More

All modular forms of weight 2 can be expressed by Eisenstein seriesAug 09 2019We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values ... More

Heights and isogenies of Drinfeld modulesAug 09 2019We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials ... More

Typical representations for level zero Bernstein components of ${\rm GL}_n(F)$Aug 09 2019Let $F$ be a non-discrete non-Archimedean locally compact field. In this article for a level zero Bernstein component $s$, we classify those irreducible smooth representations of ${\rm GL}_n{\integers{F}}$ (called typical representations) whose appearance ... More

Fallings Serre method on three dimensional selfdual representationsAug 09 2019We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre ... More

Faltings Serre method on three dimensional selfdual representationsAug 09 2019Aug 12 2019We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre ... More

Gaussian Integers, Rings, Finite Fields, and the Magic Square of SquaresAug 08 2019We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed to be the Gaussian ... More

Gaussian Integers, Rings, Finite Fields, and the Magic Square of SquaresAug 08 2019Aug 12 2019We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed to be the Gaussian ... More

Bioperational Multisets in Various Semi-ringsAug 08 2019One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or semi-rings. We ... More

On reconstructing subvarieties from their periodsAug 08 2019Let X be a smooth hypersurface of dimension n and Y a subvariety of dimension n/2. We give an algorithm which takes the periods of Y and returns an ideal. If Y is a complete intersection in the ambient space of X then we show that low degree equations ... More

Multipliers and invariants of endomorphisms of projective space in dimension greater than 1Aug 08 2019There is a natural conjugation action on the set of endomorphism of $\P^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $\P^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this ... More

On primeness of the Selberg zeta-functionAug 08 2019In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.

On Euler-Kronecker constants and the generalized Brauer-Siegel conjectureAug 08 2019As a natural generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $\gamma_K$ in a tower of number fields ... More