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  • On the nonnegative integer solutions to the equation Fn ± Fm = ya
    J. Number Theory (IF 0.718) Pub Date : 2020-09-22
    Salima Kebli; Omar Kihel; Jesse Larone; Florian Luca

    In this paper, we study the solutions to the titular Diophantine equation in integers n≥m≥0, y≥2 and a≥2. We show that there are only finitely many of them for a fixed y, and we provide a bound on the largest such solution. As an application, we find all the solutions when y∈[2,1000]. We also show that the abc-conjecture implies that there are only finitely many integer solutions (n,m,y,a) with min⁡{y

    更新日期:2020-09-22
  • 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
  • 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 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
  • On Fourth and Higher Moments of Short Exponential Sums Related to Cusp Forms
    J. Number Theory (IF 0.718) Pub Date : 2020-08-14
    Anne-Maria Ernvall-Hytönen; Esa V. Vesalainen

    We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators. We obtain the conjectured best possible bound in the case of the fourth moment.

    更新日期:2020-08-14
  • On the compactification of the Drinfeld modular curve of level Γ1Δ(n)
    J. Number Theory (IF 0.718) Pub Date : 2020-08-14
    Shin Hattori

    Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in Fq[t] which has a prime factor of degree prime to q−1. In this paper, we define a Drinfeld modular curve Y1Δ(n) over A[1/n] and study the structure around cusps of its compactification X1Δ(n), in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X1Δ(n)

    更新日期:2020-08-14
  • On the Hasse invariants of the Tate normal forms E5 and E7
    J. Number Theory (IF 0.718) Pub Date : 2020-08-14
    Patrick Morton

    A formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic field K=Q(−l), proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term −1, and a theorem is stated for the number

    更新日期:2020-08-14
  • Sturm-type bounds for modular forms over function fields
    J. Number Theory (IF 0.718) Pub Date : 2020-08-14
    Cécile Armana; Fu-Tsun Wei

    In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic

    更新日期:2020-08-14
  • Comments on Deuring's zero-spacing phenomenon
    J. Number Theory (IF 0.718) Pub Date : 2020-08-14
    Mark Watkins

    We codify some aspects of Deuring's zero-spacing phenomenon, namely that in the presence of an exceptional Landau/Siegel zero, various product L-functions have their low-height zeros lie in a very regular pattern, nearly an arithmetic progression on the central line. This piece of “analytic number theory folklore” seems to have had its actual proof escape the literature in generality, yet we still

    更新日期:2020-08-14
  • On a problem of Nathanson on minimal asymptotic bases
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Cui-Fang Sun

    Let N denote the set of all nonnegative integers and A be a subset of N. Let h be an integer with h≥2. Let n∈N and rh(A,n)=♯{(a1,…,ah)∈Ah:a1+⋯+ah=n}. The set A is called an asymptotic basis of order h if rh(A,n)≥1 for all sufficiently large integer n. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. In 1988, Nathanson posed a problem on minimal

    更新日期:2020-08-14
  • On the exceptional set of transcendental functions with integer coefficients in a prescribed set: The Problems A and C of Mahler
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Diego Marques; Carlos Gustavo Moreira

    In 1976, Mahler posed the question about the existence of a transcendental function f∈Z{z} with bounded coefficients and such that f(Q‾∩B(0,1))⊆Q‾. In this paper, we prove, in particular, the existence of such a function but with the weaker requirement that the coefficients have only 2 and 3 as prime factors. More generally, we shall prove that any subset of Q‾∩B(0,1), which is closed under complex

    更新日期:2020-08-14
  • The integrated fourth moment of Dirichlet L-functions over rational function fields
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Goran Djanković; Dragan Đokić; Nikola Lelas; Ilija Vrećica

    We consider the hybrid fourth shifted moment of Dirichlet L-functions over rational function fields, where the moment average is taken over all odd primitive characters of modulus Q∈Fq[t] and over the critical circle, which is the symmetry line of the corresponding functional equation. We obtain an asymptotic formula for this moment with the full main term for arbitrary modulus Q, as deg⁡Q→∞ and q

    更新日期:2020-08-14
  • Classification of standard modules with linear periods
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Miyu Suzuki

    Suppose that F is a non-Archimedean local field of characteristic not 2 and D is a central division algebra over F. Let n be a positive integer. We show a classification modulo essentially square-integrable representations of standard modules of GLn(D) which have non-zero linear periods. By this classification, the conjecture of Prasad and Takloo-Bighash is reduced to the case of essentially square-integrable

    更新日期:2020-08-14
  • A generic effective Oppenheim theorem for systems of forms
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Prasuna Bandi; Anish Ghosh; Jiyoung Han

    We prove a uniform effective density theorem as well as an effective counting result for a generic system comprising a polynomial with a mild homogeneous condition and several linear forms using Rogers' second moment formula for the Siegel transform on the space of unimodular lattices.

    更新日期:2020-08-14
  • On the values of representation functions II
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Xing-Wang Jiang; Csaba Sándor; Quan-Hui Yang

    For a set A of nonnegative integers, let R2(A,n) and R3(A,n) denote the number of solutions to n=a+a′ with a,a′∈A, a

    更新日期:2020-08-14
  • Moderate Expanders Over Rings
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Dao Nguyen Van Anh; Le Quang Ham; Doowon Koh; Mozhgan Mirzaei; Hossein Mojarrad; Thang Pham

    In this note, we provide a large class of moderate expanders with the exponents 38 and 513 over arbitrary finite fields and prime fields, respectively. Our main ingredients are an energy result due to the third, fourth, sixth listed authors and Shen (2019) and a theorem on two-variable expanding functions given by Hegyvári and Hennecart (2009). Using the same approach, we derive similar results in

    更新日期:2020-08-14
  • A genus formula for the positive étale wild kernel
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Hassan Asensouyis; Jilali Assim; Youness Mazigh

    Let F be a number field and let i≥2 be an integer. In this paper, we study the positive étale wild kernel WK2i−2ét,+F, which is the twisted analogue of the 2-primary part of the narrow class group. If E/F is a Galois extension of number fields with Galois group G, we prove a genus formula relating the order of the groups (WK2i−2ét,+E)G and WK2i−2ét,+F.

    更新日期:2020-08-14
  • On the sum of squares of the middle-third Cantor set
    J. Number Theory (IF 0.718) Pub Date : 2020-08-13
    Zhiqiang Wang; Kan Jiang; Wenxia Li; Bing Zhao

    Let C be the middle-third Cantor set. In this paper, we show that for every x∈[0,4], there exist x1,x2,x3,x4∈C such that x=x12+x22+x32+x42, which was conjectured in [Athreya, J. S.; Reznick, B.; Tyson, J. T. Cantor set arithmetic. Amer. Math. Monthly 126 (2019), no. 1, 4–17].

    更新日期:2020-08-14
  • Absolute series for higher Euler constants
    J. Number Theory (IF 0.718) Pub Date : 2020-08-05
    Paul Thomas Young

    We give families of series for generalized Euler constants of multiple Hurwitz zeta functions in terms of absolute zeta functions of algebraic groups, extending recent work of Kurokawa and Tanaka. Our series also converge p-adically to give p-adic limit formulas for higher p-adic Euler constants.

    更新日期:2020-08-05
  • 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:2020-07-23
  • From Pólya fields to Pólya groups (II) Non-Galois number fields
    J. Number Theory (IF 0.718) Pub Date : 2020-07-22
    Jean-Luc Chabert; Emmanuel Halberstadt

    The Pólya group of a number field K is the subgroup of the class group of K generated by the classes of the products of the maximal ideals with same norm. A Pólya field is a number field whose Pólya group is trivial. Our purpose is to start with known assertions about Pólya fields to find results concerning Pólya groups. In this second paper we describe the Pólya group of some non-Galois extensions

    更新日期:2020-07-22
  • A modular interpretation of BBGS towers
    J. Number Theory (IF 0.718) Pub Date : 2020-07-22
    Rui Chen; Zhuo Chen; Chuangqiang Hu

    In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined

    更新日期:2020-07-22
  • Category of C-motives over finite fields
    J. Number Theory (IF 0.718) Pub Date : 2020-07-22
    Esmail Arasteh Rad; Urs Hartl

    In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure L of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and

    更新日期:2020-07-22
  • Fermat's polygonal number theorem for repeated generalized polygonal numbers
    J. Number Theory (IF 0.718) Pub Date : 2020-07-20
    Soumyarup Banerjee; Manav Batavia; Ben Kane; Muratzhan Kyranbay; Dayoon Park; Sagnik Saha; Hiu Chun So; Piyush Varyani

    Text In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized m-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized m-gonal numbers are repeated

    更新日期:2020-07-20
  • Surjectivity of the adelic Galois representation associated to a Drinfeld module of rank 3
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Chien-Hua Chen

    In this paper, we prove that the adelic Galois representationρφ:Gal(Fq(T)sep/Fq(T))⟶lim←aAut(φ[a])≅GL3(Aˆ) associated to the Drinfeld module φ over Fq(T) of rank 3, φ defined by φT=T+τ2+Tq−1τ3, is surjective.

    更新日期:2020-07-17
  • Ramification in the Inverse Galois Problem
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Benjamin Pollak

    This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite

    更新日期:2020-07-17
  • Integral points on twisted Markoff surfaces
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Sheng Chen

    We study the integral Hasse principle for affine varieties of the shapeax2+y2+z2−xyz=m, using the Brauer–Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements. We describe these elements explicitly, and in some cases, we show that there is no Brauer–Manin obstruction to the integral Hasse principle for them.

    更新日期:2020-07-17
  • Elliptic surfaces of rank one and the topology of cubic-line arrangements
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Shinzo Bannai; Hiro-o Tokunaga

    Let φ:S→C be an elliptic surface over a smooth curve C with a section O. We denote its generic fiber by ES which can be considered as an elliptic curve over C(C). For a divisor D on S, not contained in fibers of φ, we canonically associate a C(C)-rational point PD of ES. In this note, we give a description of PD, when the rank of the group of C(C)-rational points of ES is one. We apply our description

    更新日期:2020-07-17
  • Modular hyperbolas and bilinear forms of Kloosterman sums
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    I.D. Shkredov

    In this paper we study incidences for hyperbolas in finite field with p elements and show how linear sum–product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums.

    更新日期:2020-07-17
  • A local to global principle for higher zero-cycles
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Johann Haas; Morten Lüders

    We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Thélène for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our approach also allows to reprove the ramified global class field theory of Kato and Saito. Finally, we apply the Kato conjectures to study the p-adic cycle class map over

    更新日期:2020-07-17
  • On Galois extensions with prescribed decomposition groups
    J. Number Theory (IF 0.718) Pub Date : 2020-07-17
    Kwang-Seob Kim; Joachim König

    We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of Q all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups

    更新日期:2020-07-17
  • 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:2020-07-17
  • Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers II
    J. Number Theory (IF 0.718) Pub Date : 2020-07-16
    Daniel A. Goldston; Sidney W. Graham; Apoorva Panidapu; Janos Pintz; Jordan Schettler; Cem Y. Yıldırım

    We show that for any positive integer n, there is some fixed A such that d(x)=d(x+n)=A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x+n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x>1 is the multiset of nonzero exponents which appear in the prime factorization of x.

    更新日期:2020-07-16
  • Epsilon factors of representations of finite general linear groups
    J. Number Theory (IF 0.718) Pub Date : 2020-07-16
    Rongqing Ye; Elad Zelingher

    We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums.

    更新日期:2020-07-16
  • Analogues of Alladi's formula
    J. Number Theory (IF 0.718) Pub Date : 2020-07-15
    Biao Wang

    In this note, we mainly show the analogue of one of Alladi's formulas over Q with respect to the Dirichlet convolutions involving the Möbius function μ(n), which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if (k,ℓ)=1, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where

    更新日期:2020-07-15
  • Integral points on the modular curves X0(p)
    J. Number Theory (IF 0.718) Pub Date : 2020-07-15
    Yulin Cai

    In this paper, we give an explicit bound for the height of integral points on X0(p) by using a very explicit version of the Chevalley-Weil principle. We improve the bound given by Sha in [12].

    更新日期:2020-07-15
  • 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:2020-07-13
  • 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:2020-07-09
  • On the structure of locally potentially equivalent Galois representations
    J. Number Theory (IF 0.718) Pub Date : 2020-06-29
    Vijay M. Patankar; C.S. Rajan

    Suppose ρ1,ρ2 are two ℓ-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair

    更新日期:2020-06-29
  • On the equation ∑j=1kjFjp=Fnq
    J. Number Theory (IF 0.718) Pub Date : 2020-06-29
    Alaa Altassan; Florian Luca

    In this paper, we show that the title equation, where Fm is the mth Fibonacci number, in positive integers (k,n,p,q) with k>1 entails max⁡{k,n,p,q}≤102500.

    更新日期:2020-06-29
  • A probabilistic model for the distribution of ranks of elliptic curves over Q
    J. Number Theory (IF 0.718) Pub Date : 2020-06-25
    Álvaro Lozano-Robledo

    In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions of our model with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and

    更新日期:2020-06-26
  • 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:2020-06-26
  • 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:2020-06-25
  • 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:2020-06-25
  • Slow Recurrences
    J. Number Theory (IF 0.718) Pub Date : 2020-06-23
    Sam Spiro

    For positive integers α and β, we define an (α,β)-walk to be any sequence of positive integers satisfying wk+2=αwk+1+βwk. We say that an (α,β)-walk is n-slow if ws=n with s as large as possible. Slow (1,1)-walks have been investigated by several authors. In this paper we consider (α,β)-walks for arbitrary positive α,β. We derive a characterization theorem for these walks, and with this we prove several

    更新日期:2020-06-23
  • Asai cube L-functions and the local Langlands correspondence
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Guy Henniart; Luis Lomelí

    Let F be a non-archimedean locally compact field. We study a class of Langlands-Shahidi pairs (H,L), consisting of a quasi-split connected reductive group H over F and a Levi subgroup L which is closely related to a product of restriction of scalars of GL1's or GL2's. We prove the compatibility of the resulting local factors with the Langlands correspondence. In particular, let E be a cubic separable

    更新日期:2020-06-23
  • L-series and isomorphisms of number fields
    J. Number Theory (IF 0.718) Pub Date : 2020-06-22
    Harry Smit

    Two number fields with equal Dedekind zeta function are not necessarily isomorphic. However, if the number fields have equal sets of Dirichlet L-series then they are isomorphic. We extend this result by showing that the isomorphisms between the number fields are in bijection with L-series preserving isomorphisms between the character groups.

    更新日期:2020-06-23
  • 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:2020-06-23
  • 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:2020-06-22
  • 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:2020-06-22
  • 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:2020-06-22
  • 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:2020-06-22
  • 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:2020-06-22
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