Real Quadratic Number Fields Research Articles

An application of Birch–Tate formula to tame kernels of real quadratic number fields

An application of Birch–Tate formula to tame kernels of real quadratic number fields

Cyclicity of the 2-decomposed unramified Iwasawa module

Let k be a real quadratic number field, and k∞ its cyclotomic Z2-extension. We study the cyclicity of the Galois group X∞′ over k∞ of the maximal abelian unramified 2-extension, in which all 2-adic primes of k∞ split completely. As consequence, we determine the complete list of real quadratic number fields for which X∞′ is cyclic.When X∞′ is cyclic non-trivial, we give a new infinite family of real quadratic number fields, for which Greenberg's conjecture is valid.

The sum of four squares over real quadratic number fields

Well-known results of Lagrange and Jacobi prove that every [Formula: see text] can be expressed as a sum of four integer squares, and the number [Formula: see text] of such representations can be given by an explicit formula in [Formula: see text]. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is [Formula: see text]. We provide explicit formulas for [Formula: see text] for [Formula: see text] and [Formula: see text]. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bounds for the Eisenstein coefficients. Last, we include examples of the complete theta series decomposition of the sum of four squares over [Formula: see text], [Formula: see text] and [Formula: see text].

Idempotent factorization on some matrices over quadratic integer rings

In 2020, Cossu and Zanardo raised a problem on the idempotent factorization of singular matrices in the form ( p z z ¯ ‖ z ‖ / p ) , where p is a prime integer which is irreducible but not prime in the ring of integers O K , with K a real quadratic number field, and z ∈ O K is such that 〈 p , z 〉 is a non-principal ideal. In this paper, we provide some classes of matrices that factor in a prescribed way and some other classes that do not. We further show that there are matrices in the above form that cannot be written as a product of two idempotent matrices.

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I.M. Vinogradov proved that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4 using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. Recently, in [3], the authors established an analogue of Matomäki's result on imaginary quadratic extensions of the function field k=Fq(T). In this paper, we consider the case of real quadratic extensions of k of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent θ=1/4). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper [3] on the imaginary quadratic case. We also simplify arguments in the paper [1] on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the form y 2 =x(x 2 +b), where b is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.

Cyclicity of the 2-class group of the first Hilbert 2-class field of some number fields

Let $\mathds{k}$ be a real quadratic number field. Denote by $\mathrm{Cl}_2(\mathds{k})$ its $2$-class group and by $\mathds{k}_2^{(1)}$ (resp. $\mathds{k}_2^{(2)}$) its first (resp. second) Hilbert $2$-class field. The aim of this paper is to study, for a real quadratic number field whose discriminant is divisible by one prime number congruent to $3$ modulo 4, the metacyclicity of $G=\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k})$ and the cyclicity of $\mathrm{Gal}(\mathds{k}_2^{(2)}/\mathds{k}_2^{(1)})$ whenever the rank of $\mathrm{Cl}_2(\mathds{k})$ is $2$, and the $4$-rank of $\mathrm{Cl}_2(\mathds{k})$ is $1$.

Lattice minima and units in real quadratic number fields

Lattice minima and units in real quadratic number fields

Measured Foliations and Hilbert 12th Problem

Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves.

On the Distribution of αp Modulo One in Quadratic Number Fields

Abstract We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.

Diophantine approximation with prime restriction in real quadratic number fields

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$\alpha $$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $$\nu >0$$ one can establish the infinitude of primes p satisfying $$||\alpha p||\le p^{-\nu }$$ . The latest record in this regard is Kaisa Matomäki’s landmark result $$\nu =1/3-\varepsilon $$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $$\mathbb {Q}(i)$$ of his result in the context of $$\mathbb {Q}$$ , which yields an exponent of $$\nu =7/22$$ . Marc Technau and the first-named author produced an analogue of Bob Vaughan’s result $$\nu =1/4-\varepsilon $$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman’s sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.

Number-theoretic positive entropy shifts with small centralizer and large normalizer

AbstractHigher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

Algebraic units, anti-unitary symmetries, and a small catalogue of SICs

In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form n^2+3, the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.

The narrow 2-class field tower of some real quadratic number fields

The narrow 2-class field tower of some real quadratic number fields

Indecomposable integers in real quadratic fields

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Abstract Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Associations between personality traits, substance use, and depression in Romanian medical students

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real number field

We show that if K is a monogenic, primitive, totally real number field, that contains units of every signature, then there exists a lower bound for the rank of integer universal quadratic forms defined over K . In particular, we extend the work of Blomer and Kala, to show that there exist infinitely many totally real cubic number fields that do not have a universal quadratic form of a given rank defined over them. For the real quadratic number fields with a unit of negative norm, we show that the minimal rank of a universal quadratic form goes to infinity as the discriminant of the number field grows. These results follow from the study of interlacing polynomials. Specifically, we show that there are only finitely many irreducible monic polynomials related to primitive number fields of a given degree, that have a bounded number of interlacing polynomials.

Torsion of Elliptic Curves Over Real Quadratic Fields of Smallest Discriminant

In [9] and [10], Filip Najman examined the torsion of elliptic curves over the number fields $$\mathbb{Q}\left( {\sqrt { - 1} } \right)$$ and $$\mathbb{Q}\left( {\sqrt { - 3} } \right)$$ . In this paper, we study the torsion structures of elliptic curves over the real quadratic number fields $$\mathbb{Q}\left( {\sqrt 2 } \right)$$ and $$\mathbb{Q}\left( {\sqrt 5 } \right)$$ , which have the smallest discriminants among all real quadratic fields $$\mathbb{Q}\left( {\sqrt d } \right)$$ with d ≢ 1 mod 4 and d ≡ 1 mod 4 respectively.

TWISTED TRIPLE PRODUCT -ADIC L-FUNCTIONS AND HIRZEBRUCH–ZAGIER CYCLES

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross–Zagier formula relating the values of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel–Jacobi image of generalized Hirzebruch–Zagier cycles.