Square-free Research Articles

Tame torsion and the tame inverse Galois problem

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve C/{mathbb {Q}} such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Abstract Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r 2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), then we will deduce forms of d satisfying r 2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

Expansions of abelian square-free groups

We investigate finitary functions from [Formula: see text] to [Formula: see text] for a square-free number [Formula: see text]. We show that the lattice of all clones on the square-free set [Formula: see text] which contain the addition of [Formula: see text] is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all [Formula: see text]-linearly closed clonoids, [Formula: see text], to the [Formula: see text] power, where [Formula: see text]. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most [Formula: see text].

Explicit integral basis of pure sextic fields

Let K=ℚ(𝜃) be an algebraic number field with 𝜃 satisfying an irreducible polynomial x6−m having integer coefficient m. We explicitly provide an integral basis of K along with some examples. As an application, we show that when m is a squarefree integer such that m≡2,3 mod4 and m≢±1 mod9, then K has a power integral basis.

A conjecture of Watkins for quadratic twists

Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.

2-Class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let [Formula: see text] be an odd positive square-free integer. In this paper, we shall investigate the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of the imaginary biquadratic number field [Formula: see text] if [Formula: see text] is of specifiic form. Furthermore, we deduce the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of [Formula: see text].

On the x-coordinates of Pell equations that are sums of two Padovan numbers

Let (P_{n})_{nge 0} be the sequence of Padovan numbers defined by P_0=0 , P_1 = P_2=1, and P_{n+3}= P_{n+1} +P_n for all nge 0 . In this paper, we find all positive square-free integers d such that the Pell equations x^2-dy^2 = N with Nin {pm 1, pm 4} , have at least two positive integer solutions (x, y) and (x^{prime }, y^{prime }) such that both x and x^{prime } are sums of two Padovan numbers.

On the variance of squarefree integers in short intervals and arithmetic progressions

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length H < x^{6/11 - varepsilon } and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with q > x^{5/11 + varepsilon }. On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively H < x^{2/3 - varepsilon } and q > x^{1/3 + varepsilon }. Furthermore we show that obtaining a bound sharp up to factors of H^{varepsilon } in the full range H < x^{1 - varepsilon } is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

Shimura lifts of certain classes of modular forms of half-integral weight

Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map [Formula: see text]) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra’s work (on the image of a class of modular forms under the first Shimura map [Formula: see text]) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the [Formula: see text]th Shimura lifts [Formula: see text], for any positive square-free integer [Formula: see text].

On the rank of the 2-class group of some imaginary triquadratic number fields

Let d be an odd square-free integer and \(\zeta _8\) a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields \(L_d={\mathbb {Q}}(\zeta _8,\sqrt{d})\).

The Eisenstein ideal with squarefree level

We use deformation theory of pseudorepresentations to study the analogue of Mazur's Eisenstein ideal with squarefree level. Given a prime number p>3 and a squarefree number N satisfying certain conditions, we study the Eisenstein part of the p-adic Hecke algebra for Γ0(N), and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that, in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that “multiplicity one” fails for the modular Jacobian J0(N) in these cases. In a particular case, this proves a conjecture of Ribet.

Unit groups of some multiquadratic number fields of degree 16

Let p and q be two primes and $$\ell$$ be an odd positive square-free integer such that $$\ell \not \in \{1,p, q, pq\}$$ . In this paper we give the unit group of some number fields of the form $${\mathbb {Q}}(\sqrt{2},\sqrt{p}, \sqrt{q}, \sqrt{-\ell } )$$ .

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Abstract Let $\varepsilon&gt; 0$ be sufficiently small and let $0 &lt; \eta &lt; 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

Riemann Hypothesis, Catholic Information and Potential of Events with New Techniques for Financial and Other Applications

In this research we are going to define two new concepts: a) “The Potential of Events” (EP) and b) “The Catholic Information” (CI). The term CI derives from the ancient Greek language and declares all the Catholic (general) Logical Propositions () which will true for every element of a set A. We will study the Riemann Hypothesis in two stages: a) By using the EP we will prove that the distribution of events e (even) and o (odd) of Square Free Numbers (SFN) on the axis Ax(N) of naturals is Heads-Tails (H-T) type. b) By using the CI we will explain the way that the distribution of prime numbers can be correlated with the non-trivial zeros of the function ζ(s) of Riemann. The Introduction and the Chapter 2 are necessary for understanding the solution. In the Chapter 3 we will present a simple method of forecasting in many very useful applications (e.g. financial, technological, medical, social, etc) developing a generalization of this new, proven here, theory which we finally apply to the solution of RH. The following Introduction as well the Results with the Discussion at the end shed light about the possibility of the proof of all the above. The article consists of 9 chapters that are numbered by 1, 2, …, 9.

Random multiplicative functions: the Selberg-Delange class

Let 1∕2≤β<1, p be a generic prime number and fβ be a random multiplicative function supported on the squarefree integers such that (fβ(p))p is an i.i.d. sequence of random variables with distribution P(f(p)=−1)=β=1−P(f(p)=+1). Let Fβ be the Dirichlet series of fβ. We prove a formula involving measure-preserving transformations that relates the Riemann ζ function with the Dirichlet series of Fβ, for certain values of β, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sum of fβ.

Applications of Bombieri–Vinogradov type theorems to power-free integers

Studying a variant of a classical result of Walfisz on the number of representations of an integer as the sum of a prime number and a square-free integer with an extra additive constraint on the prime summand, we obtain an asymptotic formula for the numbe

On the $2$-class group of some number fields with large degree

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5\pmod 8$.

О слабой теореме универсальности

Данная работа посвящена вопросам приближения квадратичной алгебраической решётки целочисленной решёткой. В ней вычисляются расстояния между квадратичной алгебраической решёткой и целочисленной решёткой, когда они заданы числителем и знаменателем подходящей дроби к корню квадратному из дискриминанта d — свободного от квадратов натурального числа.Результаты данной работы позволяют изучать вопросы о наилучших приближенияхквадратичных алгебраических решёток целочисленными решётками.

The Prime state and its quantum relatives

The Prime state of n qubits, |Pn⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

Prime Suspects in a Quantum Ladder.

In this Letter we set up a suggestive number theory interpretation of a quantum ladder system made of N coupled chains of spin 1/2. Using the hard-core boson representation and a leg-Hamiltonian made of a magnetic field and a hopping term, we can associate to the spins σ_{a} the prime numbers p_{a} so that the chains become quantum registers for square-free integers. The rung Hamiltonian involves permutation terms between next-neighbor chains and a coprime repulsive interaction. The system has various phases; in particular, there is one whose ground state is a coherent superposition of the first N prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics.