Imaginary Quadratic Fields Research Articles

Codes over $$\mathbb {F}_4$$ and $$\mathbb {F}_2 \times \mathbb {F}_2$$ and theta series of the corresponding lattices in quadratic fields

AbstractUsing codes defined over $$\mathbb {F}_4$$ F 4 and $$\mathbb {F}_2 \times \mathbb {F}_2$$ F 2 × F 2 , we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields $$\mathbb {Q}(\sqrt{d})$$ Q ( d ) with $$d \equiv 1\mod 4$$ d ≡ 1 mod 4 a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of d. For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.

Elliptic curves with complex multiplication and abelian division fields

AbstractLet be an imaginary quadratic field, and let be the order in of conductor . Let be an elliptic curve with complex multiplication by , such that is defined by a model over , where . In this article, we classify the values of and the elliptic curves such that (i) the division field is an abelian extension of , and (ii) the ‐division field coincides with the th cyclotomic extension of the base field.

Integral points on a del Pezzo surface over imaginary quadratic fields

Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. In this paper, we generalise the torsor method for integral points from Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{A}}_3$$\\end{document} over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.

ON abelian extensions and Kronecker’s jugendtraum in algebraic number theory

Abstract Each subdivision within the cyclotomic domain constitutes an Abelian numerical field, and Kronecker established the reverse of this statement. His aspirations extended beyond the realm of rational numbers, delving into the Abelian extensions of imaginary quadratic numerical fields and introducing the renowned jugendtraum. It wasn’t until the emergence of class field theory in the 20th century that this youthful dream was confirmed as valid. This concept also maintains a close connection to Hilbert’s 12th problem, which remains unresolved to this day.

On the torsion part in the K-theory of imaginary quadratic fields

We obtain upper bounds for the torsion in the K-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.

Non-cyclic torsion of elliptic curves over imaginary quadratic fields of class number 1

Let [Formula: see text] be a non-cyclotomic imaginary quadratic field with class number 1 and [Formula: see text] an elliptic curve with [Formula: see text] In this paper, we determine the torsion groups that can arise as [Formula: see text] where [Formula: see text] is any quadratic extension of [Formula: see text]

Class group and factorization in orders of a PID

In this paper, we study properties of factorization in orders of a PID via the computation of algebraic invariants that measure the failure of unique factorization. The focus is on the numerical semigroup rings over a finite field and the orders of imaginary quadratic fields with class number 1. We also give a complete description of the class group structure of those rings.

ℓ-adic digits and class number of imaginary quadratic fields

Motivated by the work of [K. Girstmair, A “popular” class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of [Formula: see text] in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of [Formula: see text] and digits of [Formula: see text], Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] whenever [Formula: see text] is a product of two distinct primes or a prime power. More explicitly, if [Formula: see text] is an integer such that [Formula: see text] and suppose that [Formula: see text] is the [Formula: see text]-adic expansion of [Formula: see text] then we establish the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] in terms of [Formula: see text] and the “trace” of generalized Bernoulli numbers [Formula: see text] where [Formula: see text]’s are odd Dirichlet characters modulo [Formula: see text] As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7).

Imaginary Quadratic Fields With ℓ-Torsion-Free Class Groups and Specified Split Primes

Abstract Given an odd prime $\ell $ and finite set of odd primes $S_{+}$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell $ and which splits at every prime in $S_{+}$. Notably, we do not require that $p \not \equiv -1 \,\;(\mathrm{mod}\, \ell )$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.

Class numbers, Ono invariants and some interesting primes

Our aim is to find all the prime numbers p such that p+x2 has at most two different prime factors, for all the odd integers x such that x2≤p. We solve entirely the cases p≡1,3,5(mod8), using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case p≡7(mod8) is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value p≡7(mod8) besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers n>1 such that n+x2 has at most two different prime factors, for all the odd integers x such that x2≤n.

On Some Multipliers Related to Discrete Fractional Integrals

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓp→ℓq bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators.

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

On the solutions of some Lebesgue–Ramanujan–Nagell type equations

Denote by [Formula: see text] the class number of the imaginary quadratic field [Formula: see text] with [Formula: see text] prime. It is well known that [Formula: see text] for [Formula: see text]. Recently, all the solutions of the Diophantine equation [Formula: see text] with [Formula: see text] were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation [Formula: see text] in unknown integers [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.

A criterion for nondensity of integral points

AbstractWe give a general criterion for Zariski degeneration of integral points in the complement of a divisor with components in a variety of dimension defined over or over a quadratic imaginary field. The key condition is that the intersection of the components of is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest.

On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting

Abstract We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.

Tempered perfect lattices in the binary case

A new algorithm for computing Hecke operators for SLn was introduced in [14]. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL2(Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.

Hilbert’s tenth problem in anticyclotomic towers of number fields

Let K K be an imaginary quadratic field and p p be an odd prime which splits in K K . Let E 1 E_1 and E 2 E_2 be elliptic curves over K K such that the Gal ⁡ ( K ¯ / K ) \operatorname {Gal}(\bar {K}/K) -modules E 1 [ p ] E_1[p] and E 2 [ p ] E_2[p] are isomorphic. We show that under certain explicit additional conditions on E 1 E_1 and E 2 E_2 , the anticyclotomic Z p \mathbb {Z}_p -extension K anti K_{\operatorname {anti}} of K K is integrally diophantine over K K . When such conditions are satisfied, we deduce new cases of Hilbert’s tenth problem. In greater detail, the conditions imply that Hilbert’s tenth problem is unsolvable for all number fields that are contained in K anti K_{\operatorname {anti}} . We illustrate our results by constructing an explicit example for p = 3 p=3 and K = Q ( − 5 ) K=\mathbb {Q}(\sqrt {-5}) .

Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences.

Attractor-repeller construction of Shintani domains for totally complex quartic fields

The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields.To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields.

On -unramified extensions over imaginary quadratic fields

AbstractLet $n$ be an integer congruent to $0$ or $3$ modulo $4$ . Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$ . The same result is obtained unconditionally in special cases.