Imaginary Quadratic Number Field Research Articles

Small generators of quadratic fields and reduced elements

Ruppert proved that there is a constant d 2 such that every imaginary quadratic number field with discriminant D K has a generator α which satisfies H ( α ) ⩽ d 2 | D K | , where H ( α ) is the height of α . The constant d 2 in Ruppertʼs result is non-effective. Ruppert conjectured that one can take d 2 = 3.2 . In the first part of this paper, we give an effective version to Ruppertʼs result and deduce Ruppertʼs conjecture in many cases. Ruppert proved some results about the height of the reduced elements in a real quadratic field. In the second part of this paper, among other results, we establish a best possible constant for a result of Ruppert connecting the heights of reduced elements and generators of quadratic fields. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ghF9_nTo3aI . Author Video Watch what authors say about their articles

GLOBAL UNITS MODULO ELLIPTIC UNITS AND IDEAL CLASS GROUPS

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two distinct primes 𝔭 and [Formula: see text]. Let k∞be the unique ℤp-extension of k which is unramified outside of 𝔭, and let K∞be a finite extension of k∞, abelian over k. Following closely the ideas of Belliard in [1], we prove that in K∞, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same μ-invariant and the same λ-invariant. We deduce that a version of the classical main conjecture, which is known to be true for p ∉ {2, 3}, holds also for p ∈ {2, 3} once we neglect the μ-invariants.

Arithmetic properties of shimura sums related to several modular forms

The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained depends on the splitting of primes in certain imaginary quadratic number fields.

ON WELL-ROUNDED IDEAL LATTICES

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

An even unimodular 72-dimensional lattice of minimum 8

An even unimodular 72-dimensional lattice Γ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant −7. The automorphism group of Γ contains the absolutely irreducible rational matrix group (PSL2(7)× SL2(25)) : 2.

Generators for the Euclidean Picard modular groups

The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the relations of the isotropy subgroup by analysis of the combinatorics of the fundamental domain in Heisenberg group.

8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

For imaginary quadratic number fields F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$</TEX>, where <TEX>${\varepsilon}{\in}$</TEX>{-1,-2} and distinct primes <TEX>$p_i{\equiv}1$</TEX> mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = <TEX>$\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$</TEX>, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of <TEX>$p_i$</TEX> modulo <TEX>$2^n$</TEX>, the quartic residue symbol <TEX>$(\frac{p_1}{p_2})4$</TEX> and binary quadratic forms such as <TEX>$p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$</TEX>, where <TEX>$h+(2p_1)$</TEX> is the narrow class number of <TEX>$\mathbb{Q}(\sqrt{2p_1})$</TEX>. The results are also very useful for numerical computations.

On the divisibility of the class number of imaginary quadratic fields

On the divisibility of the class number of imaginary quadratic fields

The integral homology of [formula omitted] of imaginary quadratic integers with nontrivial class group

We show that a cellular complex defined by Flöge allows us to determine the integral homology of the Bianchi groups PS L 2 ( O − m ) , where O − m is the ring of integers of an imaginary quadratic number field Q [ − m ] for a square-free natural number m . In the cases of nontrivial class group, we handle the difficulties arising from the cusps associated to the nontrivial ideal classes of O − m . We use this to compute the integral homology of PS L 2 ( O − m ) in the cases m = 5 , 6 , 10 , 13 and 15 , which previously was known only in the cases m = 1 , 2 , 3 , 7 and 11 with trivial class group.

Irreducibility criteria of Schur-type and Pólya-type

Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k ≥ 1. In 1919 Polya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Polya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Polya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.

Fast ideal cubing in imaginary quadratic number and function fields

We present algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks' NUCOMP algorithm that computes a reduced output and keeps the sizes of the intermediate operands small. Extensive numerical results are included demonstrating that in many cases our formulas, when combined with double base chains using binary and ternary exponents, lead to faster exponentiation.

Improvements in the computation of ideal class groups of imaginary quadratic number fields

We investigate improvements to the algorithm for the computationof ideal class groups described by Jacobson in the imaginary quadratic case.These improvements rely on the large prime strategy and a new method forperforming the linear algebra phase. We achieve a significant speed-up and areable to compute ideal class groups with discriminants of 110 decimal digits inless than a week.

2-universal Hermitian lattices over imaginary quadratic fields

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields.

A Mordell inequality for lattices over maximal orders

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that the 16-dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structures.

Thue's Fundamentaltheorem, I: The general case

In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to previously unconsidered families of algebraic numbers. Furthermore, we extend the method to also cover approximation by algebraic numbers in imaginary quadratic number fields.

Linear Independence of q‐Logarithms over the Eisenstein Integers

For fixed complex q with |q | &gt; 1, the q‐logarithm Lq is the meromorphic continuation of the series ∑n&gt;0zn/(qn − 1), |z | &lt;|q|, into the whole complex plane. If K is an algebraic number field, one may ask if 1, Lq(1), Lq(c) are linearly independent over K for q, c ∈ K× satisfying |q | &gt; 1, c ≠ q, q2, q3, …. In 2004, Tachiya showed that this is true in the Subcase K = ℚ, q ∈ ℤ, c = −1, and the present authors extended this result to arbitrary integer q from an imaginary quadratic number field K, and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if K is the Eisenstein number field , q an integer from K, and c a primitive third root of unity. Under these conditions, the linear independence holds also for 1, Lq(c), Lq(c−1), and both results are quantitative.

Bernoulli–Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant D K ≠ − 3 , − 4 and class number one. Fix a prime p ⩾ 7 which is unramified in K. Given an elliptic curve A / Q with complex multiplication by K, let ρ A ¯ : Gal ( K ¯ / K ( μ p ∞ ) ) → SL ( 2 , Z p ) be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation ρ A : Gal ( K ¯ / K ( μ p ∞ ) ) → SL ( 2 , Z p 〚 X 〛 ) of ρ A ¯ is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL ( 2 , Z p ) . If p splits in K, then the same result holds as long as a certain Bernoulli–Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert–Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.

A Golod–Shafarevich equality and p-tower groups

Text All current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod–Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod–Shafarevich type inequalities, we prove a Golod–Shafarevich equality for analytic pro- p-groups. As an application, we further work of Skopin [V.A. Skopin, Certain finite groups. Modules and homology in group theory and Galois theory, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 31 (1973) 115–139 (in Russian)], showing that groups of the third of the three types mentioned above are necessarily tremendously large. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=13GudVNQUUI.

On the divisibility of the class number of imaginary quadratic number fields

We prove that if at least one of the prime divisors of an odd integer U ≥ 3 U\geq 3 is equal to 3 3 mod 4 4 , then the ideal class group of the imaginary quadratic field Q ( 1 − 4 U n ) \mathbf {Q}(\sqrt {1-4U^n}) contains an element of order n n .

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants Ono K are equal to their class numbers h K . Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if Ono K is large enough compared with h K , then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values d K of their discriminants are less than 2.3 ⋅ 10 9 . We determine all these K's with d K ⩽ 10 6 . There are 76 such K's. We prove that there is at most one such K with d K ⩾ 1.8 ⋅ 10 11 .