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Multiple zeta values and iterated log-sine integralsApr 22 2019We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among ... More
Improved algorithms for left factorial residuesApr 19 2019We present improved algorithms for computing the left factorial residues $!p=0!+1!+...+(p-1)! \!\mod p$. We use these algorithms for the calculation of the residues $!p\!\mod p$, for all primes $p$ up to $2^{40}$. Our results confirm that Kurepa's left ... More
An Elementary Approach to a Curious Identity from RamanujanApr 19 2019In his notebooks, Ramanujan wrote the following identity: \begin{equation} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 ... More
Periodicities for Taylor coefficients of half-integral weight modular formsApr 18 2019Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less ... More
On singular moduli that are S-unitsApr 18 2019Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that ... More
An algorithm for determining equations for superelliptic curves with minimal invariantsApr 18 2019Let $k$ be a number field and $\mathcal O_k$ its ring of integers. For any superelliptic curve $\mathcal X$, defined over $\mathcal O_k$, and equation $z^m y^{d-m} = f(x, y)$ we give an algorithm which determines a minimal model over $\mathcal O_k$ in ... More
A $q$-analogue of Wilson's congruenceApr 18 2019Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{\sigma\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,\sigma} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes ... More
Perfect State Transfer on Weighted Graphs of the Johnson SchemeApr 18 2019We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$ and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory results, that $X$ ... More
Critical $L$-values for some quadratic twists of Gross curvesApr 18 2019Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. ... More
A proof of the Hasse-Weil inequality for genus 2 à la ManinApr 18 2019We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.
Approaching Cusick's conjecture on the sum-of-digits functionApr 18 2019Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2. \] We prove ... More
$\wp$-adic continuous families of Drinfeld eigenforms of finite slopeApr 18 2019Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is divisible by $\wp$. Let $k\geq ... More
Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomialsApr 18 2019In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on Zp and generating functions. In addition, we study two variable ... More
Cyclicity of Elliptic Curves Modulo Primes in arithmetic progressionsApr 17 2019We 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 ... More
Poissonian pair correlation on manifolds via the heat kernelApr 17 2019We define a notion of Poissonian pair correlation (PPC) for Riemannian manifolds without boundary and prove that PPC implies uniform distribution in this setting. This extends earlier work by Grepstad and Larcher, Aistleitner, Lachmann, and Pausinger, ... More
Quotients of numerical semigroups generated by two numbersApr 17 2019In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, denoted <a,b> d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like ... More
Subsets of $\mathbb{F}^*_p$ with only small products or ratiosApr 17 2019Let $p$ be a fixed prime. We estimate the number of elements of a set $A \subseteq \mathbb{F}^*_p$ for which $$ s_1s_2 \equiv a \pmod{p} \quad \mbox{for some}\quad a \in [-X,X] \quad \mbox{for all}\quad s_1,s_2 \in A. $$ We also consider variations and ... More
The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan]Apr 17 2019This is the text to accompany my Bourbaki seminar from 30th March 2019, on the maximum size of the Riemann zeta function in "almost all" intervals of length 1 on the critical line. It surveys the conjecture of Fyodorov--Hiary--Keating on the behaviour ... More
Hausdorff and packing dimension of Diophantine setsApr 17 2019Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a result of Jarn\'ik ... More
The Siegel variance formula for quadratic formsApr 17 2019We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum in terms of ... More
The module of vector-valued modular forms is Cohen-MacaulayApr 17 2019Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote ... More
The product of parts or "norm" of a partitionApr 16 2019In this article we study the norm of an integer partition, which is defined as the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current research ... More
Quotients of Hurwitz PrimesApr 16 2019Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger ... More
Distribution of determinant of sum of matricesApr 16 2019Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\det S$ for certain types of subsets $S$ in the ring $M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For $i\in \mathbb{F}_q$, let $D_i$ ... More
Gamma factors for the Asai cube representationApr 16 2019We prove an equality between the gamma factors for the Asai cube representation of ${\rm R}_{E/F}{\rm GL}_2$ defined by the Weil$-$Deligne representations and the local zeta integrals of Ikeda and Piatetski-Shapiro$-$Rallis, where $E$ is an \'etale cubic ... More
Dimensions of Automorphic Representations, $L$-Functions and LiftingsApr 16 2019There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a priority for ... More
p-adic equidistribution of CM pointsApr 16 2019Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. ... More
Continued fractions, the Chen-Stein method and extreme value theoryApr 16 2019In this paper, we investigate extreme value theory in the context of continued fractions using the Chen-Stein method. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu asymptotics for the exceedances and deduce several consequences. ... More
On the Ramsey number of the Brauer configurationApr 16 2019We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions monochromatic with their common difference. Such a result has been obtained independently and in much greater generality by Sanders. ... More
Dimension bound for doubly badly approximable affine formsApr 16 2019We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension ... More
Uniform bound for the number of rational points on a pencil of curvesApr 15 2019Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number ... More
Modeling rational numbers by Cantor seriesApr 15 2019In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case of an arbitrary ... More
The doubling Archimedean zeta integrals for p-adic interpolationApr 15 2019We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic interpolation conjectured ... More
An infinitesimal variant of Guo-Jacquet trace formula: the case of $(GL_{2n}, GL_n\times GL_n)$Apr 15 2019We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of $(GL_{2n}, GL_n\times GL_n)$ using an analogue of Arthur's truncation procedure. It results from the Poisson summation formula on $(\mathfrak{gl}_n\oplus\mathfrak{gl}_n)(\mathbb ... More
An Asymptotic Form of the Generating Function $\prod_{k=1}^\infty (1+x^k/k)$Apr 15 2019It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function $\prod_{k=1}^\infty (1+x^k/k)$ converges to a limit $C > 0$. $C$ can be expressed as $C = \exp\Bigl(-\sum_{k = 2}^\infty \frac{(-1)^k}{k}\ \zeta(k) ... More
An algorithm for determining torsion growth of elliptic curvesApr 15 2019We present a fast algorithm that takes as input an elliptic curve defined over $\mathbb Q$ and an integer $d$ and returns all the number fields $K$ of degree $d'$ dividing $d$ such that $E(K)_{tors}$ contains $E(F)_{tors}$ as a proper subgroup, for all ... More
Infinite class field towers of number fields of prime power discriminantApr 15 2019For every prime number p, we show the existence of a solvable number field L ramified only at {p and infinity whose p-Hilbert Class field tower is infinite.
Ranks of overpartitions: asymptotics and inequalitiesApr 15 2019In this paper we compute asymptotics for $ \overline{N}(a,c,n), $ the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c. $ As an application we prove, among others, some inequalities conjectured by Ji, Zhang and Zhao (2018), and ... More
Universal norms and Greenberg conjectureApr 15 2019We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.
Mean values of divisors twisted by quadratic charactersApr 15 2019In this paper, we evaluate the sum $\sum_{m,n}\leg {m}{n}d(n)$, where $\leg {m}{n}$ is the Kronecker symbol and $d(n)$ is the divisor function.
An elliptic analogue of Fukuhara's trigonometeric identitiesApr 15 2019We obtain new elliptic function identities, which are an elliptic analogue of Fukuhara's trigonometric identities. We show that the coefficients of Laurent expansions at $z=0$ of our elliptic identities give rise to some reciprocity laws for elliptic ... More
Integral points on twisted Markoff surfacesApr 15 2019We study the integral Hasse principle for affine varieties of the form ax^2+y^2+z^2-xyz=m ,using Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements .We will construct their explicit representatives and compute ... More
Value distribution of derivatives in polynomial dynamicsApr 15 2019For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ ... More
Interesting identities involving weighted representations of integers as sums of arbitrarily many squaresApr 15 2019In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier coefficients of weight ... More
Representation of an integer as the sum of a prime in arithmetic progression and a squarefree integer with certain parityApr 14 2019Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a squarefree integer with an even (or odd) number of prime ... More
A non-commutative differential module approach to Alexander modulesApr 14 2019The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we give examples ... More
Absence of irreducible multiple zeta-values in melon modular graph functionsApr 13 2019The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term ... More
On the largest element in D(n)-quadruplesApr 13 2019Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$ is called a $D(n)$-$m$-tuple. In this paper, we consider the question, for given integer $n$ which is not a ... More
The Massey vanishing conjecture for number fieldsApr 13 2019A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. We establish this conjecture when k is a number field.
On triangular numbers, forms of mixed type and their representation numbersApr 12 2019In \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of determining formulas for the number of representations of a natural number $n$ by a sum of $k$ triangular numbers and derived many applications, including the one connecting these ... More
Elekes-Rónyai Theorem revisitedApr 12 2019In this paper it is proven that for any $f\in\mathbb{R}(x_1,x_2)$ and $A_1,A_2$ nonempty finite subsets of $\mathbb{R}$ such that $|A_1|=|A_2|$ and $f$ is defined in $A_1\times A_2$, we have that \begin{equation*} |f(A_1,A_2)|=\Omega\left(|A_1|^{\frac{4}{3}}\right) ... More
Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjectureApr 12 2019The concept of Artin transfer pattern $((\ker(T_{K,N_i}))_i,(\mathrm{Cl}_p(N_i))_i)$ for homogeneous multiplets $(N_1,\ldots,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N_i)$ ... More
Applications of Siegel's Lemma to best approximations for a linear formApr 12 2019Consider a real vector $(1,\zeta_{1},\ldots,\zeta_{n})$. The problem of making linear forms $p_{0}+p_{1}\zeta_{1}+\cdots+p_{n}\zeta_{n}$ for integers $p_{j}$ small naturally induces a sequence of integer vectors called best approximations or minimal points. ... More
On some determinants involving cyclotomic unitsApr 12 2019For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix ... More
A note on small gaps between zeros of the Riemann zeta-functionApr 12 2019Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we generalize a set of ... More
Deformation theory of the trivial mod $p$ Galois representation for $\mathrm{GL}_n$Apr 12 2019We study the rigid generic fiber $\mathcal{X}^\square_{\overline{\rho}}$ of the framed deformation space of the trivial representation $\overline\rho: G_K \to \mathrm{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute ... More
Freeness of $p$-adically Completed Modular Jacobians over a Hecke AlgebraApr 11 2019We construct a Taylor-Wiles system using a family of $p$-adically completed modular Jacobians over suitable Hecke algebras and prove that certain $p$-adically completed Mordell-Weil groups of these Jacobians is free of finite rank over a Hecke algebra. ... More
On sum-product basesApr 11 2019Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.
Non-vanishing theorems for central $L$-values of some elliptic curves with complex multiplication IIApr 11 2019Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence ... More
Zeta-polynomials, Hilbert polynomials, and the Eichler-Shimura identitiesApr 11 2019In 2017, Ono, Rolen, and Sprung [ORS17] answered problems of Manin [Man16] by defining zeta-polynomials $Z_f(s)$ for even weight newforms $f\in S_k(\Gamma_0(N)$; these polynomials can be defined by applying the "Rodriguez-Villegas transform" to the period ... More
Generic cuspidal representations of $U(2,1)$Apr 11 2019Let $F$ be any non-Archimedean local field with a Galois involution $\sigma$ and $F_0$ be the fixed field for the action of $\sigma$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical ... More
Generic cuspidal representations of $U(2,1)$Apr 11 2019Apr 12 2019Let $F$ be any non-Archimedean local field with a Galois involution $\sigma$ and $F_0$ be the fixed field for the action of $\sigma$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical ... More
Arakelov geometry, heights, equidistribution, and the Bogomolov conjectureApr 11 2019This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, ... More
New Identities for Padovan NumbersApr 11 2019In \cite{Choi-Jo}, the $am+b$ ($0\leq b<a$) subscripted Tribonacci numbers are studied. This work is devoted to study a new generalization of Fibonacci numbers called Padovan numbers. In particular, the $am+b$ subscripted Padovan numbers will be expressed ... More
A Framework for Modular Properties of False Theta FunctionsApr 10 2019False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives ... More
On the Optimality of Gauss's Algorithm over Euclidean Imaginary Quadratic FieldsApr 10 2019In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the algebraicvariant of Gauss ... More
On the Random Wave Conjecture for Dihedral Maaß FormsApr 10 2019We prove two results on arithmetic quantum chaos for dihedral Maass forms, both of which are manifestations of Berry's random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level $1$ forms, these ... More
Obstructions to weak approximation for reductive groups over $p$-adic function fieldsApr 10 2019We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for quasi-split reductive groups. Finally, we give an application ... More
Algorithm for studying polynomial maps and reductions modulo prime numberApr 10 2019In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. Also the class of Pascal finite polynomial automorphisms was introduced. Pascal finite polynomial maps constitute a generalization of exponential automorphisms ... More
Value patterns of multiplicative functions and related sequencesApr 10 2019We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short intervals in a ... More
Equivariant splitting of the Hodge--de Rham exact sequenceApr 10 2019Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes ... More
Generating functions of planar polygons from homological mirror symmetry of elliptic curvesApr 10 2019We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers' mock theta function. ... More
Density results for specialization sets of Galois coversApr 10 2019We provide evidence for this conclusion: given a finite Galois cover $f: X \rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this ... More
Dynamically affine maps in positive characteristicApr 09 2019We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function ... More
Borel's stable range for the cohomology of arithmetic groupsApr 09 2019In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical ... More
The structure and number of Erdős covering systemsApr 09 2019Introduced by Erd\H{o}s in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several ... More
Perfect points on genus one curves and consequences for supersingular K3 surfacesApr 09 2019Apr 10 2019We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any ... More
Perfect points on genus one curves and consequences for supersingular K3 surfacesApr 09 2019We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any ... More
A K-theoretic Selberg trace formulaApr 09 2019Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of operators on the Hilbert space L^2(G/H) associated to compactly supported smooth functions ... More
Quadratic Chabauty for (bi)elliptic curves and Kim's conjectureApr 09 2019We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing ... More
New bounds on even cycle creating Hamiltonian paths using expander graphsApr 09 2019We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved ... More
The Fourth Moment of Derivatives of Dirichlet $L$-functions in Function FieldsApr 09 2019We obtain the asymptotic main term of moments of arbitrary derivatives of $L$-functions in the function field setting. Specifically, the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $Q ... More
Determinant groups of hermitian lattices over local fieldsApr 09 2019We describe the determinants of the automorphism groups of hermitian lattices over local fields. Using a result of G. Shimura, this yields an explicit method to compute the special genera in a given genus of hermitian lattices over a number field.
Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant θ_3Apr 09 2019Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant \theta_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, ... More
Exponential Riesz bases, multi-tiling and condition numbers in finite abelian groupsApr 09 2019Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We show that that every subset of a finite abelian group has such as basis, removing ... More
Hausdorff dimension of the large values of Weyl sumsApr 09 2019The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d n^{d})) \right| ... More
Unit Signatures in Real Biquadratic and Multiquadratic Number FieldsApr 09 2019We consider the signature rank of the units in real multiquadratic fields. When the three quadratic subfields of a real biquadratic field $K$ either (a) all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all have signature rank ... More
A note on multiplicative automatic sequencesApr 08 2019We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin ... More
Values of L-Series of Hecke EigenformsApr 08 2019We determine a formula for the average values of L-series associated to eigenforms at complex values.
A genus formula for the positive étale wild kernelApr 08 2019Let $F$ be a number field and let $i\geq 2$ be an integer. In this paper, we study the positive \'{e}tale wild kernel $\mathrm{WK}^{\mbox{\'{e}t},+}_{2i-2}F$, which is the twisted analogue of the $2$-primary part of the narrow class group. If $E/F$ is ... More
Group Operation on Nodal CurvesApr 08 2019In this work, we present an efficient method for computing in the Generalized Jacobian of special singular curves. The efficiency of the operation is due to representation of an element in the Jacobian group by a single polynomial.
A canonical form of the hypergeometric equation and transformation formulasApr 08 2019We give a new method to prove in a uniform and easy way various transformation formulas for Gauss' hypergeometric functions. The key is a certain canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also ... More
A canonical form of the hypergeometric equation and transformation formulasApr 08 2019Apr 11 2019We give a new method to prove in a uniform and easy way various transformation formulas for Gauss' hypergeometric functions. The key is a certain canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also ... More
A canonical form of the hypergeometric equation and transformation formulasApr 08 2019Apr 15 2019We give a new method to prove in a uniform and easy way various transformation formulas for Gauss' hypergeometric functions. The key is a certain canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric functions is also ... More
On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3Apr 07 2019Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\Z_{\ell}(2))$ ... More
SanD primes and numbersApr 07 2019We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD {\em primes}. ... More
From Ramanujan Graphs to Ramanujan ComplexesApr 06 2019Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. ... More
Integral points on Hilbert moduli schemesApr 06 2019We use the method of Faltings (Arakelov, Par\v{s}in, Szpiro) in order to explicitly study integral points on a class of varieties over $\mathbb Z$ called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are classical ... More
On the cuspidal divisor group and Eisenstein ideal of Drinfeld modular varietiesApr 06 2019Let $A=\mathbb{F}_q[T]$ be the ring of polynomials in $T$ with coefficients in a finite field with $q$ elements. Let $\mathfrak{p}\lhd A$ be a maximal ideal, and denote $|\mathfrak{p}|=\# A/\mathfrak{p}$. Let $Y_0^r(\mathfrak{p})$ be the modular variety ... More
A p-adic analogue of Siegel's Theorem on sums of squaresApr 06 2019Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and ... More