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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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.