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  • 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:2020-07-03
  • 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 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
  • 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
  • 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:2020-06-26
  • 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:2020-06-25
  • 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
  • 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:2020-06-23
  • 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:2020-06-23
  • 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:2020-06-23
  • 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
  • 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:2020-06-23
  • 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
  • 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) [CK11]. We also find congruence relations satisfied by the Fourier coefficients

    更新日期:2020-06-23
  • 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:2020-06-23
  • 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:2020-06-23
  • 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:2020-06-23
  • 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:2020-06-23
  • On subsequence sums of a zero-sum free sequence over finite abelian groups
    J. Number Theory (IF 0.718) Pub Date : 2020-05-26
    Jiangtao Peng; Yuanlin Li; Chao Liu; Meiling Huang

    Text Let G be a finite abelian group and S be a sequence with elements of G. Let Σ(S)⊂G denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S. We call S zero-sum free if 0∉Σ(S). In this paper, we study |Σ(S)| when S is a zero-sum free sequence of elements from G and 〈S〉 is not cyclic. We improve the results of A. Pixton and P. Yuan on this topic. In particular

    更新日期:2020-05-26
  • Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture
    J. Number Theory (IF 0.718) Pub Date : 2020-05-22
    Trajan Hammonds; Seoyoung Kim; Benjamin Logsdon; Álvaro Lozano-Robledo; Steven J. Miller

    Let X:y2=f(x) be a hyperelliptic curve over Q(T) of genus g≥1. Assume that the jacobian of X over Q(T) has no subvariety defined over Q. Denote by Xt the specialization of X to an integer T=t, let aXt(p) be its trace of Frobenius, and let AX,r(p)=1p∑t=1paXt(p)r be its r-th moment. The first moment is related to the rank of the jacobian JX(Q(T)) by a generalization of a conjecture of Nagao:limX→∞⁡1X∑p≤X−AX

    更新日期:2020-05-22
  • Depth reductions for associators
    J. Number Theory (IF 0.718) Pub Date : 2020-05-22
    David Jarossay

    We prove that for any associator, two specific families of coefficients of the associator can be expressed in terms of coefficients of lower depth. Combining these results to our notions of adjoint p-adic multiple zeta values and multiple harmonic values, we obtain a new point of view on the question of relating p-adic and finite multiple zeta values, and a few other application to the study of p-adic

    更新日期:2020-05-22
  • Modular polynomials on Hilbert surfaces
    J. Number Theory (IF 0.718) Pub Date : 2020-05-20
    Enea Milio; Damien Robert

    We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta

    更新日期:2020-05-20
  • Variations of Lehmer's Conjecture for Ramanujan's tau-function
    J. Number Theory (IF 0.718) Pub Date : 2020-05-20
    Jennifer S. Balakrishnan; William Craig; Ken Ono

    We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove thatτ(n)∉{±1,±3,±5,±7,±691}. This result is an example of general theorems (see Theorems 1.2 and 1.3 of [2]) for newforms with trivial mod 2 residual Galois representation. Ramanujan's well-known congruences for τ(n) allow for the simplified proof in these special cases

    更新日期:2020-05-20
  • Galois structure of the holomorphic differentials of curves
    J. Number Theory (IF 0.718) Pub Date : 2020-05-20
    Frauke M. Bleher; Ted Chinburg; Aristides Kontogeorgis

    Let X be a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Suppose G is a finite group acting faithfully on X such that G has non-trivial cyclic Sylow p-subgroups. We show that the decomposition of the space of holomorphic differentials of X into a direct sum of indecomposable k[G]-modules is uniquely determined by the lower ramification groups

    更新日期:2020-05-20
  • On L-functions for U2n+1 × ResE/FGLm (m > n)
    J. Number Theory (IF 0.718) Pub Date : 2020-05-20
    Kazuki Morimoto; David Soudry

    We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U2n+1 and ResE/FGLm, for a quadratic extension of number fields E/F, when m>n.

    更新日期:2020-05-20
  • Hausdorff dimension of the large values of Weyl sums
    J. Number Theory (IF 0.718) Pub Date : 2020-05-20
    Changhao Chen; Igor E. Shparlinski

    The authors have recently obtained a lower bound of the Hausdorff dimension for the sets of vectors (x1,…,xd)∈[0,1)d with large Weyl sums, namely of vectors for which|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩾Nα for infinitely many integers N⩾1. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.

    更新日期:2020-05-20
  • Elliptic curves and lower bounds for class numbers
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Michael Griffin; Ken Ono

    Ideal class pairings map the rational points of rank r≥1 elliptic curves E/Q to the ideal class groups CL(−D) of certain imaginary quadratic fields. These pairings imply thath(−D)≥12(c(E)−ε)(log⁡D)r2 for sufficiently large discriminants −D in certain families, where c(E) is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.

    更新日期:2020-05-19
  • Some results on multiple polylogarithm functions and alternating multiple zeta values
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Ce Xu

    In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst

    更新日期:2020-05-19
  • Counting rational points on biprojective hypersurfaces of bidegree (1,2)
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    L.Q. Hu

    An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biprojective hypersurfaces of bidegree (1,2) in sufficiently many variables. This confirms the Manin conjecture for this variety.

    更新日期:2020-05-19
  • Rankin-Cohen brackets of eigenforms and modular forms
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Jeffrey Beyerl

    We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. This occurs, for example, when the Rankin-Cohen bracket covers the entirety of Sn. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and

    更新日期:2020-05-19
  • Depth-graded motivic Lie algebra
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Jiangtao Li

    In this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part for the neutral Tannakian category mixed Tate motives over Z. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown's matrix conjecture and the nondegeneracy conjecture about depth-graded motivic Lie subalgebra generated by the

    更新日期:2020-05-19
  • A variant of Siegel's theorem for Drinfeld modules
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Simone Coccia; Dragos Ghioca

    We complete the proof of a Siegel type statement for finitely generated Φ-submodules of Ga under the action of a Drinfeld module Φ.

    更新日期:2020-05-19
  • Rademacher's conjecture and expansions at roots of unity of products generating restricted partitions
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Cormac O'Sullivan

    The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic

    更新日期:2020-05-19
  • Twisting moduli for GL(2)
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Benjamin Bedert; George Cooper; Thomas Oliver; Pengcheng Zhang

    We prove various converse theorems for automorphic forms on Γ0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N∈{18,20,24} - these integers are the smallest multiples of 4 or 9 not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ0(N)

    更新日期:2020-05-19
  • Abelian periods of factors of Sturmian words
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Jarkko Peltomäki

    We study the abelian period sets of Sturmian words, which are codings of irrational rotations on a one-dimensional torus. The main result states that the minimum abelian period of a factor of a Sturmian word of angle α with continued fraction expansion [0;a1,a2,…] is either tqk with 1≤t≤ak+1 (a multiple of a denominator qk of a convergent of α) or qk,ℓ (a denominator qk,ℓ of a semiconvergent of α)

    更新日期:2020-05-19
  • Hopf-Galois structures on finite extensions with almost simple Galois group
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Cindy (Sin Yi) Tsang

    In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group G is an almost simple group in which the socle A has prime index p. Each Hopf-Galois structure is associated to a group N of the same order as G. We shall give necessary criteria on these N in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to A×Cp

    更新日期:2020-05-19
  • On perfect powers that are sums of cubes of a seven term arithmetic progression
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Alejandro Argáez-García; Vandita Patel

    We prove that the equation (x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3=yp only has solutions which satisfy xy=0 for 1≤r≤106 and p≥5 prime. This article complements the work on the equations (x−r)3+x3+(x+r)3=yp in [2] and (x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3=yp in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution

    更新日期:2020-05-19
  • On the ideal class group of the normal closure of Q(np)
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    René Schoof

    For a prime number p and an integer n we determine the Galois cohomology groups of the class group of the normal closure of Q(np) to a certain extent and use this information to prove a result about the group structure of the class group.

    更新日期:2020-05-19
  • Polynomial values of figurate numbers
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Lajos Hajdu; Nóra Varga

    There are a lot of effective, ineffective and explicit results concerning power values and polynomial values of binomial coefficients. Also, many papers deal with generalizations of these problems, involving polygonal numbers and pyramidal numbers. In this paper we prove effective and ineffective theorems concerning polynomial values of figurate numbers. Our results yield common extensions and generalizations

    更新日期:2020-05-19
  • On the number of popular differences in Z/pZ
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Mario Huicochea

    In this paper it is shown that there is an absolute constant κ>0 with the following property. For any prime p and nonempty subsets A,B of Z/pZ such that 1<|A|

    更新日期:2020-05-19
  • On the distribution of products of two primes
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Sumaia Saad Eddin; Yuta Suzuki

    For a real parameter r, the RSA integers are integers which can be written as the product of two primes pq with p

    更新日期:2020-05-19
  • Regular ternary triangular forms
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Mingyu Kim; Byeong-Kweon Oh

    An integer of the form Tx=x(x+1)2 for some positive integer x is called a triangular number. A ternary triangular form aTx+bTy+cTz for positive integers a,b and c is called regular if it represents every positive integer that is locally represented. In this article, we prove that there are exactly 49 primitive regular ternary triangular forms.

    更新日期:2020-05-19
  • Drinfeld cusp forms: Oldforms and newforms
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Andrea Bandini; Maria Valentino

    Let p=(P) be any prime of Fq[t], let m be any ideal of Fq[t] not divisible by p and consider the space of Drinfeld cusp forms of level mp, i.e. for the modular group Γ0(mp). Using degeneracy maps, traces and Fricke involutions we offer definitions for p-oldforms and p-newforms which turn out to be subspaces stable with respect to the action of the Atkin operator UP. We provide eigenvalues and/or slopes

    更新日期:2020-05-19
  • Congruences for coefficients of modular functions in levels 3, 5, and 7 with poles at 0
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Paul Jenkins; Ryan Keck

    We give congruences modulo powers of p∈{3,5,7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the authors and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at ∞.

    更新日期:2020-05-19
  • Realizing Artin-Schreier covers of curves with minimal Newton polygons in positive characteristic
    J. Number Theory (IF 0.718) Pub Date : 2020-05-19
    Jeremy Booher; Rachel Pries

    Suppose X is a smooth projective connected curve defined over an algebraically closed field k of characteristic p>0 and B⊂X(k) is a finite, possibly empty, set of points. The Newton polygon of a degree p Galois cover of X with branch locus B depends on the ramification invariants of the cover. When X is ordinary, for every possible set of branch points and ramification invariants, we prove that there

    更新日期:2020-05-19
  • Bielliptic quotient modular curves with N square-free
    J. Number Theory (IF 0.718) Pub Date : 2020-04-30
    Francesc Bars; Josep González; Mohamed Kamel

    Let N≥1 be an square free integer and let WN be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X0(N) such that the modular curve X0(N)/WN has genus at least two. We determine all pairs (N,WN) such that X0(N)/WN is a bielliptic curve and the pairs (N,WN) such that X0(N)/WN has an infinite number of quadratic points over Q.

    更新日期:2020-04-30
  • A single set improvement to the 3k − 4 theorem
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    David J. Grynkiewicz

    The 3k−4 Theorem is a classical result which asserts that if A,B⊆Z are finite, nonempty subsets with(1)|A+B|=|A|+|B|+r≤|A|+|B|+min⁡{|A|,|B|}−3−δ, where δ=1 if A and B are translates of each other, and otherwise δ=0, then there are arithmetic progressions PA and PB of common difference such that A⊆PA, B⊆PB, |B|≤|PB|+r+1 and |PA|≤|A|+r+1. It is one of the few cases in Freiman's Theorem for which exact

    更新日期:2020-04-29
  • Good reduction of affinoids in the Lubin–Tate curve in even equal characteristic, I
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Takahiro Tsushima

    We define affinoids in the Lubin–Tate curve in even equal characteristic, and compute the reductions of them. Each reduction is isomorphic to a smooth affine curve with Artin–Schreier type equation. We expect that the cohomology of the reductions realizes the local Langlands correspondence and local Jacquet–Langlands correspondence for representations of conductor five.

    更新日期:2020-04-29
  • On the error term concerning the number of subgroups of the groups Zm×Zn with mn ≤ x
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Yankun Sui; Dan Liu

    Let s(m,n) and c(m,n) denote the total number of subgroups and cyclic subgroups of Zm×Zn, respectively. For any x≥1, we consider the asymptotic behavior of Ds(x):=∑mn≤xs(m,n) and Dc(x):=∑mn≤xc(m,n) and obtain two asymptotic formulas by using the method of exponential sums. We also study the upper bound of the mean-square estimate of Δs(x) and Δc(x).

    更新日期:2020-04-29
  • Taylor coefficients of the Jacobi θ3(q) function
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Tanay Wakhare; Christophe Vignat

    We extend some results recently obtained by Dan Romik [15] about the Taylor coefficients of the theta function θ3(e−π) to the case θ3(q) of a real valued variable 0

    更新日期:2020-04-29
  • Graded rings of integral Jacobi forms
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Valery Gritsenko; Haowu Wang

    We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of (0,2)-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients

    更新日期:2020-04-29
  • Iteration of polynomials AXd + C over finite fields
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Rufei Ren

    For a polynomial f(X)=AXd+C∈Fp[X] with A≠0 and d≥2, we prove that if d|p−1 and f∘i(0)≠f∘j(0) for 0≤i

    更新日期:2020-04-29
  • Primes with Beatty and Chebotarev conditions
    J. Number Theory (IF 0.718) Pub Date : 2020-04-29
    Caleb Ji; Joshua Kazdan; Vaughan McDonald

    We study the prime numbers that lie in Beatty sequences of the form ⌊αn+β⌋ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence with α of finite type and a Chebotarev class of some Galois extension is precisely the product of the densities α−1⋅|C||G|. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri–Vinogradov

    更新日期:2020-04-29
  • Estimates for representation numbers of binary quadratic forms and Apollonian circle packings
    J. Number Theory (IF 0.718) Pub Date : 2020-04-28
    Radu Toma

    Fix a primitive, positive definite binary quadratic form g with integer coefficients. We prove asymptotic formulas for sums of the form ∑rg(n)β and ∑rg⁎(n)β, where β≥0 and rg(n), resp. rg⁎(n), denote the number of inequivalent representations, resp. proper inequivalent representations, of n by g. These estimates generalize a previous result by Blomer and Granville (2006) by allowing for non-fundamental

    更新日期:2020-04-28
  • Vanishing of the Brauer group of a del Pezzo surface of degree 4
    J. Number Theory (IF 0.718) Pub Date : 2020-04-28
    Manar Riman

    We explicitly construct a del Pezzo surface X of degree 4 over a field k such that H1(k,PicX‾)≃Z/2Z while BrX/Brk is trivial.

    更新日期:2020-04-28
  • Effective bounds for traces of singular moduli
    J. Number Theory (IF 0.718) Pub Date : 2020-01-20
    Havi Ellers; Meagan Kenney; Riad Masri; Wei-Lun Tsai

    We give an asymptotic formula for traces of weak Maass forms at CM points with an effective bound on the error term. Upon specializing to the modular j-function, we deduce such a result for traces of singular moduli. Due to work of Zagier, and Bringmann and Ono, these traces of weak Maass forms at CM points appear as Fourier coefficients of half-integral weight weakly holomorphic modular forms. Hence

    更新日期:2020-04-22
  • Endomorphism rings of reductions of Drinfeld modules
    J. Number Theory (IF 0.718) Pub Date : 2019-03-22
    Sumita Garai; Mihran Papikian

    Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let

    更新日期:2020-04-22
  • Hyperfields, truncated DVRs, and valued fields
    J. Number Theory (IF 0.718) Pub Date : 2019-11-20
    Junguk Lee

    For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th valued hyperfields of K1 and K2 are isomorphic over p for each n≥1, then K1 and K2 are isomorphic. More generally, for n1,n2≥1, if n2 is large enough, then any homomorphism, which is over p, from the n1-th valued hyperfield of K1 to the n2-th valued hyperfield of K2 can

    更新日期:2020-04-22
  • Unramified extensions over low degree number fields
    J. Number Theory (IF 0.718) Pub Date : 2019-11-21
    Joachim König; Danny Neftin; Jack Sonn

    For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply

    更新日期:2020-04-22
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