JournalTOCs

Research in Number Theory
Journal Prestige (SJR): 0.754

Number of Followers: 0

Hybrid journal (It can contain Open Access articles)
ISSN (Print) 2522-0160 - ISSN (Online) 2363-9555
Published by SpringerOpen

[228 journals]

A criterion of algebraic independence of values of modular functions and
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  Abstract: Abstract The aim of this paper is to give a criterion of algebraic independence for the values at the same point of two modular functions under certain conditions. As an application, we show that any two infinite products in $$\begin{aligned} \prod _{n=1}^{\infty }\left( 1+\frac{1}{F_n} \right) ,\quad \prod _{n=3}^{\infty }\left( 1-\frac{1}{F_n} \right) ,\quad \prod _{n=1}^{\infty }\left( 1+\frac{1}{L_n} \right) ,\quad \prod _{n=2}^{\infty }\left( 1-\frac{1}{L_n} \right) \end{aligned}$$ are algebraically independent over $\mathbb {Q}$ , where $\{F_n\}$ and $ \{L_n\} $ are the Fibonacci and Lucas sequences, respectively. The proof of our main theorem is based on the properties of the field of all modular functions for the principal congruence subgroup, together with a deep result of Yu. V. Nesterenko on algebraic independence of the values of the Eisenstein series.
  PubDate: 2022-05-10
Special zeta Mahler functions
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  Abstract: Abstract We study the zeta Mahler function (ZMF, also zeta Mahler measure), which is closely related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials $k + (x_1 + x_1^{-1}) \cdots \left( x_r + x_r^{-1}\right) $ , where k is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore connections with the Mahler measure.
  PubDate: 2022-04-24
Totally real bi-quadratic fields with large Pólya groups
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  Abstract: Abstract For an algebraic number field K with ring of integers $\mathcal {O}_{K}$ , an important subgroup of the ideal class group $Cl_{K}$ is the Pólya group, denoted by Po(K), which measures the failure of the $\mathcal {O}_{K}$ -module $Int(\mathcal {O}_{K})$ of integer-valued polynomials on $\mathcal {O}_{K}$ from admitting a regular basis. In this paper, we prove that for any integer $n \ge 2$ , there are infinitely many totally real bi-quadratic fields K with $Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}$ . In fact, we explicitly construct such an infinite family of number fields. This also provides an infinite family of bi-quadratic fields with ideal class groups of 2-ranks at least n.
  PubDate: 2022-04-24
Splitting of primes in number fields generated by points on some modular
curves
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  Abstract: Abstract We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on $X_1(N)$ generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves $X_0(N)$ which are hyperelliptic (except for $N=37$ ) and in cubic fields generated by points on $X_1(2,14)$ .
  PubDate: 2022-04-21
Explicit minimal embedded resolutions of divisors on models of the
projective line
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  Abstract: Abstract Let K be a discretely valued field with ring of integers $\mathcal {O}_K$ with perfect residue field. Let K(x) be the rational function field in one variable. Let ${\mathbb {P}}^1_{\mathcal {O}_K}$ be the standard smooth model of ${\mathbb {P}}^1_K$ with coordinate x. Let $f(x) \in \mathcal {O}_K[x]$ be a squarefree polynomial with corresponding divisor of zeroes ${{\,\mathrm{div}\,}}_0(f)$ on ${\mathbb {P}}^1_{\mathcal {O}_K}$ . We give an explicit description of the minimal embedded resolution $\mathcal {Y}$ of the pair $({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))$ by using Mac Lane’s theory to write down the discrete valuations on K(x) corresponding to the irreducible components of the special fiber of $\mathcal {Y}$ .
  PubDate: 2022-04-13
Bounds on shifted convolution sums for Hecke eigenforms
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  Abstract: Abstract Shifted convolution sums play a prominent rôle in analytic number theory. Here these sums are considered in the context of holomorphic Hecke eigenforms. We investigate pointwise bounds, mean-square bounds consistent with the optimal conjectural bound, and find asymptotics on average for their variance.
  PubDate: 2022-04-08
Two-variable polynomials with dynamical Mahler measure zero
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  Abstract: Abstract We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of the Boyd–Lawton formula, and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker’s lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer’s question is true.
  PubDate: 2022-03-31
Bailey pairs and indefinite quadratic forms, II. False indefinite theta
functions
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  Abstract: Abstract We construct families of Bailey pairs $(\alpha _n,\beta _n)$ where the exponent of q in $\alpha _n$ is an indefinite quadratic form, but where the usual $(-1)^j$ is replaced by a sign function. This leads to identities involving “false” indefinite binary theta series. These closely resemble q-identities for mock theta functions or Maass waveforms, but the sign function prevents them from having the usual modular properties.
  PubDate: 2022-03-23
On the Mordell–Weil lattice of $$y^2 = x^3 + b x + t^{3^n + 1}$$ y 2 = x
3 + b x + t 3 n + 1 in characteristic 3
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  Abstract: Abstract We study the elliptic curves given by $y^2 = x^3 + b x +t^{3^n+1}$ over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve $u^3 + b u = v^{3^n + 1}$ . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension $2 \cdot 3^n$ for every $n \ge 1$ , which improve on the currently best known sphere packing densities in dimensions 162 (case $n=4$ ) and 486 (case $n=5$ ). For $n=3$ , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for $n=1$ it has the same density as the lattice $E_6$ in dimension 6.
  PubDate: 2022-03-17
Correction to: A transcendental Brauer–Manin obstruction to weak
approximation on a Calabi–Yau threefold
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  Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s40993-021-00307-4
  PubDate: 2022-03-07
Oriented pro- $$\ell $$ ℓ groups with the
Bogomolov–Positselski property
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  Abstract: Abstract For a prime number $\ell $ we say that an oriented pro- $\ell $ group $(G,\theta )$ has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal $\theta $ -abelian quotient $\pi ^{\mathrm {ab}}_{G,\theta }:G\rightarrow G(\theta )$ is a free pro- $\ell $ group contained in the Frattini subgroup of G. We show that oriented pro- $\ell $ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro- $\ell $ Galois groups of a field $\mathbb {K}$ in case that $\mathbb {K}^\times /(\mathbb {K}^\times )^\ell $ is finite. Secondly, it is shown that for an $H^\bullet $ -quadratic oriented pro- $\ell $ group $(G,\theta )$ the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map $d_2^{2,1}$ in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).
  PubDate: 2022-03-02
Discrete mean square of the coefficients of symmetric square L-functions
on certain sequence of positive numbers
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  Abstract: Abstract In this paper, we will be concerned with the average behavior of the $n\mathrm{th}$ normalized Fourier coefficients of symmetric square L-function (i.e., $L(s,sym^{2}f)$ ) over certain sequence of positive integers. Precisely, we prove an asymptotic formula for $$\begin{aligned} \mathop {\sum }\limits _{\begin{array}{c} a^{2}+b^{2}+c^{2}+d^{2}\le {x} \\ (a,b,c,d)\in {\mathbb {Z}}^{4} \end{array}}\uplambda ^{2}_{sym^{2}f}(a^{2}+b^{2}+c^{2}+d^{2}), \end{aligned}$$ where $x\ge {x_{0}}$ (sufficiently large), and $$\begin{aligned} L(s,sym^{2}f):= \mathop {\sum }\limits _{n=1}^{\infty }\dfrac{\uplambda _{sym^{2}f}(n)}{n^{s}}. \end{aligned}$$
  PubDate: 2022-02-24
Exponential sums in prime fields for modular forms
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  Abstract: Abstract The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski’s bound for exponential sums attached to certain linear recurrence sequences over finite fields.
  PubDate: 2022-02-22
The distribution of rational points for a class of quartic hypersurfaces
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  Abstract: Abstract We give an asymptotic formula for the number of rational points of bounded height of quartic hypersurfaces $Q_n:x^4=(y_1^2+\cdots +y_n^2)z^2$ by following the approach and method presented by Liu, Wu, and Zhao for cubic hypersurfaces. Also, we give an observation for much higher dimension hypersurfaces of similar form.
  PubDate: 2022-02-19
Isolated points on $$X_1(\ell ^n)$$ X 1 ( ℓ n ) with rational
j-invariant
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  Abstract: Abstract Let $\ell $ be a prime and let $n\ge 1$ . In this note we show that if there is a non-cuspidal, non-CM isolated point x with a rational j-invariant on the modular curve $X_1(\ell ^n)$ , then $\ell =37$ and the j-invariant of x is either $7\cdot 11^3$ or $-7.137^3\cdot 2083^3$ . The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.
  PubDate: 2022-02-11
Kummer-faithful fields which are not sub-p-adic
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  Abstract: Abstract A Kummer-faithful field k is a perfect field such that the Kummer map associated to G is injective for every finite extension K of k and every semi-abelian variety G over K. A typical example of a Kummer-faithful field of characteristic zero is a sub-p-adic field for some prime number p. In particular, every number field is Kummer-faithful. In this paper, we give a construction of a Kummer-faithful field which is an infinite algebraic extension of the field of rational numbers. Moreover, we give some examples of Kummer-faithful fields which are not sub-p-adic by using the constructions.
  PubDate: 2022-02-03
On Farey sequence and quadratic Farey sums
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  Abstract: Abstract Let ${{\mathcal {F}}}_n$ be the Farey series of order $n\ge 1$ . We obtain sharp effective estimates of the Farey sums $$\begin{aligned} F_n(f)= \sum _{\frac{{\kappa }}{{\lambda }}\in {{\mathcal {F}}}_n} f\Big (\frac{{\kappa }}{{\lambda }}\Big ),{\qquad }F_{n,{\sigma }}(f)= \sum _{\frac{{\kappa }}{{\lambda }}\in {{\mathcal {F}}}_n} \frac{1}{{\kappa }^{\sigma }{\lambda }^{\sigma }}f\Big (\frac{{\kappa }}{{\lambda }}\Big ),{\qquad }0< {\sigma }\le 1, \end{aligned}$$ for 1-periodic functions f satisfying weak local regularity assumption of Dini’s type at rational points of ]0, 1[. We also prove an unconditional lower bound for Farey sums.
  PubDate: 2022-01-21
Correction to: Twist-minimal trace formula for holomorphic cusp forms
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  Abstract: A Correction to this paper has been published: 10.1007/s40993-021-00302-9
  PubDate: 2022-01-20
A transcendental Brauer–Manin obstruction to weak approximation on a
Calabi–Yau threefold
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  Abstract: Abstract In this paper we investigate the $\mathbb {Q}$ -rational points of a class of simply connected Calabi–Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer–Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb {C}$ each of these Calabi–Yau threefolds Y is derived equivalent to a Reye congruence Calabi–Yau threefold X. We show that these derived equivalences may also be constructed over $\mathbb {Q}$ , and we give sufficient conditions for X to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi–Yau variety over $\mathbb {C}$ .
  PubDate: 2022-01-20
Twist-minimal trace formula for holomorphic cusp forms
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  Abstract: Abstract We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg.
  PubDate: 2021-12-24