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  • On the behavior of large empirical autocovariance matrices between the past and the future
    Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-07-28
    P. Loubaton; D. Tieplova

    The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support 𝒮 is characterized. It is also established that the squared singular

    更新日期:2020-09-18
  • Moments of quadratic twists of elliptic curve L-functions over function fields
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Hung M. Bui; Alexandra Florea; Jonathan P. Keating; Edva Roditty-Gershon

    We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over 𝔽q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct

    更新日期:2020-09-18
  • Nonvanishing of hyperelliptic zeta functions over finite fields
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Jordan S. Ellenberg; Wanlin Li; Mark Shusterman

    Fixing t ∈ ℝ and a finite field 𝔽q of odd characteristic, we give an explicit upper bound on the proportion of genus g hyperelliptic curves over 𝔽q whose zeta function vanishes at 1 2 + it. Our upper bound is independent of g and tends to 0 as q grows.

    更新日期:2020-09-18
  • The maximum number of points on a curve of genus eight over the field of four elements
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Everett W. Howe

    The Oesterlé bound shows that a curve of genus 8 over the finite field F4 can have at most 24 rational points, and Niederreiter and Xing used class field theory to show that there exists such a curve with 21 points. We improve both of these results: We show that a genus-8 curve over F4 can have at most 23 rational points, and we provide an example of such a curve with 22 points, namely the curve defined

    更新日期:2020-09-15
  • Sums of higher divisor functions
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Guangwei Hu; Guangshi Lü

    Let dk(n) denote the k-th divisor function. In this paper, we study the asymptotic formula of the sum∑1≤n1,n2,…,nl≤xdk(n12+n22+⋯+nl2), where n1,n2,…,nl∈Z+, k≥4 and l≥3 are integers. Previously only the cases of k=2,3 are studied.

    更新日期:2020-09-15
  • Signatures of Dirichlet characters and elliptic curves
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Andrzej Dąbrowski; Jacek Pomykała

    We prove a lower bound for the number of conductors q≤Q with q≡1(modr) having t prime divisors of Dirichlet characters χ of fixed order r and with given values χ(p) for primes p≤B. We also give an elliptic curve variant of this result.

    更新日期:2020-09-15
  • Up-operators and congruences for Shimura images
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Matthew Boylan

    Recent works [2], [5], [13], [14] studied how Hecke operators map between subspaces of modular forms of the type Ar,s={η(δz)rF(δz):F(z)∈Ms} for suitable δ. For primes p≥5, we give results on how Up maps between subspaces of these spaces with p-integral coefficients modulo powers of p. As a corollary, we prove congruences between Shimura images of a natural family of half-integral weight eigenforms

    更新日期:2020-09-15
  • Obstructions to weak approximation for reductive groups over p-adic function fields
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Yisheng Tian

    We establish arithmetic duality theorems for short complexes of tori associated to reductive groups over p-adic function fields. Using arithmetic dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

    更新日期:2020-09-15
  • Triple Correlation Sums of Coefficients of Cusp Forms
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Thomas A. Hulse; Chan Ieong Kuan; David Lowry-Duda; Alexander Walker

    We produce nontrivial asymptotic estimates for shifted sums of the form ∑a(h)b(m)c(2m−h), in which a(n),b(n),c(n) are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions n−h,n,n+h such that a(n−h)a(n)a(n+h)≠0

    更新日期:2020-09-15
  • On The First Fourier-Jacobi coefficient of Siegel modular forms of degree two
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    M. Manickam

    In this paper we prove that the first Fourier-Jacobi coefficient of a non-zero Siegel cusp form and a Hecke eigenform of degree 2, weight k, for Sp4(Z) is not identically zero.

    更新日期:2020-09-15
  • On uniform distribution of αβ-orbits
    J. Number Theory (IF 0.718) Pub Date : 2020-09-15
    Changhao Chen; Xiaohua Wang; Shengyou Wen

    Let α,β∈(0,1) such that at least one of them is irrational. We take a random walk on the real line such that the choice of α and β has equal probability 1/2. We prove that almost surely the αβ-orbit is uniformly distributed module one, and the exponential sums along its orbit has the square root cancellation. We also show that the exceptional set in the probability space, which does not have the property

    更新日期:2020-09-15
  • Quasimodular forms as solutions of modular differential equations
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-09-14
    Peter J. Grabner

    We study quasimodular forms of depth ≤4 and determine under which conditions they occur as solutions of modular differential equations. Furthermore, we study which modular differential equations have quasimodular solutions. We use these results to investigate extremal quasimodular forms as introduced by M. Kaneko and M. Koike further. Especially, we prove a conjecture stated by these authors concerning

    更新日期:2020-09-15
  • Brauer groups and Néron class groups
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-09-14
    Cristian D. González-Avilés

    Let K be a global field and let S be a finite set of primes of K containing the Archimedean primes. We generalize the duality theorem for the Néron S-class group of an abelian variety A over K established previously by removing the requirement that the Tate–Shafarevich group of A be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper

    更新日期:2020-09-15
  • Ranks of overpartitions modulo 4 and 8
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-09-11
    Su-Ping Cui; Nancy S. S. Gu; Chen-Yang Su

    An overpartition of n is a partition of n in which the first occurrence of a number may be overlined. Then, the rank of an overpartition is defined as its largest part minus its number of parts. Let N¯(s,m,n) be the number of overpartitions of n with rank congruent to s modulo m. In this paper, we study the rank differences of overpartitions N¯(s,m,mn+d)−N¯(t,m,mn+d) for m=4 or 8 and 0≤d,s,t

    更新日期:2020-09-12
  • On norm relations for Asai–Flach classes
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-09-11
    Giada Grossi

    We give a new proof of the norm relations for the Asai–Flach Euler system built by Lei–Loeffler–Zerbes. More precisely, we redefine Asai–Flach classes in the language used by Loeffler–Skinner–Zerbes for Lemma–Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular

    更新日期:2020-09-12
  • Polynomial values of surface point counting polynomials
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-09-11
    L. Hajdu; O. Herendi

    There are many results in the literature concerning power values, equal values or more generally, polynomial values of lattice point counting polynomials. In this paper, we prove various finiteness results for polynomial values of polynomials counting the lattice points on the surface of an n-dimensional cube, pyramid and simplex.

    更新日期:2020-09-12
  • Fluctuations of the spectrum in rotationally invariant random matrix ensembles
    Random Matrices Theory Appl. (IF 1.206) Pub Date : 2020-09-10
    Elizabeth S. Meckes; Mark W. Meckes

    We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of n×n matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for

    更新日期:2020-09-11
  • Product formulas for periods of CM Abelian varieties and the function field analog
    J. Number Theory (IF 0.718) Pub Date : 2020-09-08
    Urs Hartl; Rajneesh Kumar Singh

    We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and A-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and A-motives, including

    更新日期:2020-09-09
  • The local sum conjecture in two dimensions
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-06-26
    Robert Fraser; James Wright

    The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof

    更新日期:2020-09-02
  • The probability that ideals in a number ring are k-wise relatively r-prime
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-07-02
    Brian D. Sittinger; Ryan D. DeMoss

    We say that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime if any k of them are relatively r-prime. In this paper, we provide an exact formula for the probability that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime.

    更新日期:2020-09-02
  • Computing the Galois group of a polynomial over a p-adic field
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-06-26
    Christopher Doris

    We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of ℚ2 of degrees 18, 20 and 22, tables of which are available online.

    更新日期:2020-09-02
  • Some new sums of q-trigonometric and related functions through a theta product of Jacobi
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-06-26
    Mohamed El Bachraoui; József Sándor

    We evaluate some finite and infinite sums involving q-trigonometric and q-digamma functions. Upon letting q approach 1, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper’s q-trigonometric identities.

    更新日期:2020-09-02
  • Quadratic residues and quartic residues modulo primes
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-06-26
    Zhi-Wei Sun

    In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let p be an odd prime and let A be any integer. We determine completely the product fp(A):=∏1≤i,j≤(p−1)/2p∤i2−Aij−j2(i2−Aij−j2) modulo p; for example, if p≡1(mod 4) then fp(A)≡−(A2+4)(p−1)/4(mod p)ifA2+4p=1,(−A2−4)(p−1)/4(mod p)ifA2+4p=−1, where (⋅p) denotes the Legendre symbol. We also determine

    更新日期:2020-09-02
  • The polylog quotient and the Goncharov quotient in computational Chabauty–Kim Theory I
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-08-18
    David Corwin; Ishai Dan-Cohen

    Polylogarithms are those multiple polylogarithms that factor through a certain quotient of the de Rham fundamental group of the thrice punctured line known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers, and Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec ℤ. To do

    更新日期:2020-09-02
  • The Spinning Particles – Classical Description
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Cezary J. Walczyk, Zbigniew Hasiewicz

    The classical model of the spinning particle is analyzed in detail in two versions – with single spinor and two spinors put on the trajectory. Equations of motion of the first version are easily solvable. The system with two spinors becomes nonlinear. Nevertheless, the equations of motion are analyzed in detail and solved analytically and numerically. In either case the trajectories are illustrated

    更新日期:2020-09-02
  • Quantum-Mechanical Explicit Solution for the Confined Harmonic Oscillator Model with the Von Roos Kinetic Energy Operator
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    E.I. Jafarov, S.M. Nagiyev, A.M. Jafarova

    Exactly-solvable confined model of the nonrelativistic quantum harmonic oscillator is proposed. Its position-dependent effective mass Hamiltonian is defined via the von Roos kinetic energy operator. The confinement effect to harmonic oscillator potential is included as a result of certain behaviour of the position-dependent effective mass. The corresponding Schrödinger equation in the canonical approach

    更新日期:2020-09-02
  • Order Structures of ( D, ℰ)-Quasi-Bases and Constructing Operators for Generalized Riesz Systems
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Hiroshi Inoue

    The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and ( D, ℰ)-quasi bases for two fixed biorthogonal sequences {ϕn} and {Ψn}. In a previous paper, we have studied the order structure of the set Cϕ of all constructing operators for a generalized Riesz system {ϕn}, and furthermore we have shown that

    更新日期:2020-09-02
  • The Center of a d0-Algebra
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Anna Avallone, Paolo Vitolo

    We define the center C(A) of a d0-algebra A as a set of self-mappings on A. The center can be regarded as the set of all possible d0-subdirect factors of A which are subalgebras of A. we show that C(A) is always a Boolean algebra. we also show that C(A) admits an embedding in the power Aκ, where κ is the cofinality of A. In case κ = 1 (i.e. A is a D-lattice) we reobtain the well-known fact that C(A)

    更新日期:2020-09-02
  • Dirac-Bergmann Procedure Having Regard to Interaction for Light-Front Yukawa Model
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Jan Żochowski

    In this work we applied the Dirac-Bergmann procedure to establish the Dirac brackets, which have regard to interaction for the light-front Yukawa model in D = 1 + 3 dimensions. We made use of a simple matrix equation leading to solution of the set task, wherein the main problem was to calculate the inverse matrix to the array composed of the constraints for enabled interaction. The proposed device

    更新日期:2020-09-02
  • Analytical and Combinatorial Aspects of the Eigenproblem for the Two-Magnon Sector of XXX Heisenberg Rings
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    P. Krasoń, M. Łabuz, J. Milewski

    In this paper we study both analytical and combinatorial properties of solutions of the eigenproblem for the Heisenberg s-1/2 model for two deviations. Our analysis uses Chebyshev polynomials, inverse Bethe Ansatz, winding numbers and rigged string configurations. We show some combinatorial aspects of strings in a geometric way. We discuss some exceptions from the connection between the combinatorial

    更新日期:2020-09-02
  • Quantum Conditional Probability and Measurement Induced Disturbance of a Quantum Channel
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Dariusz Chruściński, Takashi Matsuoka

    In classical information theory the conditional probability uniquely defines a classical channel. Introducing a quantum analogue of conditional probability we analyze the mutual information between input and output states of a given quantum channel when the compound state is fully quantum-quantum and classical-quantum. We introduce an analogue of the quantum discord of the channel which measures the

    更新日期:2020-09-02
  • On a Generalized Wave Equation and its Application
    Rep. Math. Phys. (IF 0.86) Pub Date : 2020-09-02
    Do-Hyung Kim

    Wave equations on flat Minkowski spacetimes are generalized to curved spacetimes. We present two candidates for generalized wave equation, discuss their legitimacy and consider their applications to Klein–Gordon equation.

    更新日期:2020-09-02
  • Left–right pairs and complex forests of infinite rooted binary trees
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-07-02
    Nina Zubrilina

    Let 𝒟0:={x+iy|x,y∈ℝ,x,y>0}, and let (L,R) be a pair of Möbius transformations corresponding to SL2(ℕ0) matrices such that L(𝒟0) and R(𝒟0) are disjoint. Given such a pair (called a left–right pair), we can construct a directed graph ℱ(L,R) with vertices 𝒟0 and edges {(z,L(z))}z∈𝒟0∪{(z,R(z))}z∈𝒟0, which is a collection of infinite binary trees. We answer two questions of Nathanson by classifying

    更新日期:2020-09-01
  • Convolution of values of the Lerch zeta-function
    J. Number Theory (IF 0.718) Pub Date : 2020-07-23
    M. Ram Murty; Siddhi Pathak

    We investigate generalizations arising from the identityζ2(n−1,1)=n−12ζ(n)−12∑j=2n−2ζ(j)ζ(n−j), where ζ2(k,1) denotes a double zeta value at (k,1), or an Euler-Zagier sum. In particular, we prove analogues of the above identity for Lerch zeta-functions and Dirichlet L-functions. Such an attempt has met with limited success in the past. We highlight that this study naturally leads one into the realm

    更新日期:2020-08-26
  • Non-real poles and irregularity of distribution
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    David Lowry-Duda

    We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical

    更新日期:2020-08-26
  • On a problem related to discrete mean values of Dirichlet L-functions
    J. Number Theory (IF 0.718) Pub Date : 2020-06-20
    Ertan Elma

    Let χ be a nonprincipal Dirichlet character modulo a prime number p⩾3 of order k⩾2. DefineAp(χ):=1p−1∑1⩽N⩽p−1∑1⩽n1,n2⩽Nχ(n1)=χ(n2)1. We prove thatAp(χ)=p(2p−1)6k+(k−1)(p+1)12k+aχp2π2k(p−1)∑j=1k/2|L(1,χ2j−1)|2 where aχ:=(1−χ(−1))/2.

    更新日期:2020-08-26
  • Ramanujan graphs and exponential sums over function fields
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Naser T. Sardari; Masoud Zargar

    We prove that q+1-regular Morgenstern Ramanujan graphs Xq,g (depending on g∈Fq[t]) have diameter at most (43+ε)logq⁡|Xq,g|+Oε(1) (at least for odd q and irreducible g) provided that a twisted Linnik–Selberg conjecture over Fq(t) is true. This would break the 30 year-old upper bound of 2logq⁡|Xq,g|+O(1), a consequence of a well-known upper bound on the diameter of regular Ramanujan graphs proved by

    更新日期:2020-08-26
  • Holomorphic differentials of generalized Fermat curves
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Rubén A. Hidalgo

    A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse

    更新日期:2020-08-26
  • Some properties of Zumkeller numbers and k-layered numbers
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Pankaj Jyoti Mahanta; Manjil P. Saikia; Daniel Yaqubi

    Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer n is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be σ(n)/2. Generalizing even further, we call n a k-layered number if its divisors can be partitioned into k sets with equal sum. In this paper, we completely

    更新日期:2020-08-26
  • On the growth of cuspidal cohomology of GL(2) and GL(3)
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Chaitanya Ambi

    We estimate the growth of cuspidal cohomology of GL2(AQ). Quantitatively, we provide bounds on the total number of normalised eigenforms of Hecke operators which are obtained by automorphic induction from Hecke characters of imaginary quadratic fields grows as level structure varies. We further investigate how much of cuspidal cohomology of GL3(AQ) is obtained by symmetric square transfer from GL2(AQ)

    更新日期:2020-08-26
  • Small doubling in prime-order groups: From 2.4 to 2.6
    J. Number Theory (IF 0.718) Pub Date : 2020-06-25
    Vsevolod F. Lev; Ilya D. Shkredov

    Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A⊆Fp with p prime and |A|<0.0045p, (i) if |A+A|<2.59|A|−3 and |A|>100, then A is contained in an arithmetic progression of size |A+A|−|A|+1, and (ii) if |A−A|<2.6|A|−3, then A is contained in an arithmetic progression of size |A−A|−|A|+1. The improvement comes from using the properties of higher energies

    更新日期:2020-08-26
  • Hilbert cubes meet arithmetic sets
    J. Number Theory (IF 0.718) Pub Date : 2020-06-25
    Norbert Hegyvári; Péter Pál Pach

    We show that an additive Hilbert cube (in prime fields) of sufficiently large dimension always meets certain kinds of arithmetic sets, namely, product sets and reciprocal sets of sumsets satisfying certain technical conditions.

    更新日期:2020-08-26
  • On the degeneracy of integral points and entire curves in the complement of nef effective divisors
    J. Number Theory (IF 0.718) Pub Date : 2020-06-26
    Gordon Heier; Aaron Levin

    As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges

    更新日期:2020-08-26
  • Polynomial analogue of the Smarandache function
    J. Number Theory (IF 0.718) Pub Date : 2020-07-09
    Xiumei Li; Min Sha

    In the integer case, the Smarandache function of a positive integer n is defined to be the smallest positive integer k such that n divides the factorial k!. In this paper, we first define a natural order for polynomials in Fq[t] over a finite field Fq and then define the Smarandache function of a non-zero polynomial f∈Fq[t], denoted by S(f), to be the smallest polynomial g such that f divides the Carlitz

    更新日期:2020-08-26
  • Integral points on varieties defined by matrix factorization into elementary matrices
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Bruce W. Jordan; Yevgeny Zaytman

    Let K be a number field and S be a finite set of valuations of K containing the archimedean valuations. Let O be the ring of S-integers. For A∈SL2(O) and k≥1, we define matrix-factorization varieties Vk(A) over O which parametrize factoring A into a product of k elementary matrices beginning with lower triangular; the equations defining Vk(A) are written in terms of Euler's continuant polynomials.

    更新日期:2020-08-26
  • Congruences for Hecke eigenvalues in minus spaces
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    SoYoung Choi; Chang Heon Kim; Kyung Seung Lee

    The minus space Mk!−(p) is defined to be the subspace of the space Mk!(p) of weakly holomorphic weight k modular forms for Γ0(p) consisting of all eigenforms of the Fricke involution Wp with eigenvalue −1. In this paper, we study congruences for Hecke eigenvalues in minus spaces. This is extended results for Choi and Kim (2011) [5]. We also find congruence relations satisfied by the Fourier coefficients

    更新日期:2020-08-26
  • Strong Selmer companion elliptic curves
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Ching-Heng Chiu

    Let E1 and E2 be elliptic curves defined over a number field K. Suppose that for all but finitely many primes ℓ, and for all finite extension fields L/K,dimFℓ⁡Selℓ(L,E1)=dimFℓ⁡Selℓ(L,E2). We prove that E1 and E2 are isogenous over K.

    更新日期:2020-08-26
  • Linear operators, the Hurwitz zeta function and Dirichlet L-functions
    J. Number Theory (IF 0.718) Pub Date : 2020-07-13
    Bernardo Bianco Prado; Kim Klinger-Logan

    At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity [5]. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies

    更新日期:2020-08-26
  • On a Diophantine equation of Erdős and Graham
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Szabolcs Tengely; Maciej Ulas; Jakub Zygadło

    We study solvability of the Diophantine equationn2n=∑i=1kai2ai, in integers n,k,a1,…,ak satisfying the conditions k≥2 and ai

    更新日期:2020-08-26
  • Zeroes of quaternionic modular forms and central L-values
    J. Number Theory (IF 0.718) Pub Date : 2020-06-23
    Kimball Martin; Jordan Wiebe

    Values of quaternionic modular forms are related to twisted central L-values via periods and a theorem of Waldspurger. In particular, certain twisted L-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture

    更新日期:2020-08-26
  • Dedekind sums arising from newform Eisenstein series
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-08-21
    T. Stucker; A. Vennos; M. P. Young

    For primitive nontrivial Dirichlet characters χ1 and χ2, we study the weight zero newform Eisenstein series Eχ1,χ2(z,s) at s=1. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of H1(Γ0(N),ℂ). We also give a short proof of the reciprocity formula for

    更新日期:2020-08-22
  • On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-08-21
    Yining Hu; Guoniu Wei-Han

    Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen

    更新日期:2020-08-22
  • Explicit Kummer theory for the rational numbers
    Int. J. Number Theory (IF 0.606) Pub Date : 2020-08-21
    Antonella Perucca; Pietro Sgobba; Sebastiano Tronto

    Let G be a finitely generated multiplicative subgroup of ℚ× having rank r. The ratio between nr and the Kummer degree [ℚ(ζm,Gn):ℚ(ζm)], where n divides m, is bounded independently of n and m. We prove that there exist integers m0,n0 such that the above ratio depends only on G, gcd(m,m0), and gcd(n,n0). Our results are very explicit and they yield an algorithm that provides formulas for all the above

    更新日期:2020-08-22
  • The Riemann Hypothesis for period polynomials of Hilbert modular forms
    J. Number Theory (IF 0.718) Pub Date : 2020-08-19
    Angelica Babei; Larry Rolen; Ian Wagner

    There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a “Riemann Hypothesis” in both classical settings and for cohomological versions extending the classical setting to the case of higher derivatives of L-functions. There thus appears to be a general phenomenon behind these phenomena. In this

    更新日期:2020-08-20
  • p-adic Asai L-functions of Bianchi modular forms
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    David Loeffler; Chris Williams

    The Asai (or twisted tensor) L-function of a Bianchi modular form Ψ is the L-function attached to the tensor induction to ℚ of its associated Galois representation. When Ψ is ordinary at p we construct a p-adic analogue of this L-function: that is, a p-adic measure on ℤp× that interpolates the critical values of the Asai L-function twisted by Dirichlet characters of p-power conductor. The construction

    更新日期:2020-08-20
  • Pro-unipotent harmonic actions and dynamical properties of p-adic cyclotomic multiple zeta values
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    David Jarossay

    p-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of ℙ1 ∖{0,μN,∞}. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of p-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new

    更新日期:2020-08-20
  • Nouvelles cohomologies de Weil en caractéristique positive
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Joseph Ayoub

    Soit K un corps valué de hauteur 1 et d’inégales caractéristiques (0,p), et soit k son corps résiduel. Dans cet article, nous construisons une nouvelle cohomologie de Weil pour les k-schémas de type fini à valeurs dans les AK-modules, avec AK une K-algèbre de « périodes abstraites p-adiques » qui admet une description explicite par générateurs et relations. Nous démontrons des théorèmes de comparaison

    更新日期:2020-08-20
  • Elliptic curves over totally real cubic fields are modular
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Maarten Derickx; Filip Najman; Samir Siksek

    We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.

    更新日期:2020-08-20
  • Motivic Gauss–Bonnet formulas
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Marc Levine; Arpon Raksit

    The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck–Witt ring of the base field. Previous work of the first author and recent work of Déglise, Jin and Khan established a motivic Gauss–Bonnet formula relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. We apply

    更新日期:2020-08-20
  • Burgess bounds for short character sums evaluated at forms
    Algebra Number Theory (IF 0.92) Pub Date : 2020-08-18
    Lillian B. Pierce; Junyan Xu

    We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This n-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least qβ, with β > 1∕2 − 1∕(2(n + 1)). This is the first Burgess bound that applies in all dimensions to

    更新日期:2020-08-20
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