Class number relations in almost abelian number fields

We all should learn in our abstract algebra classes that every Euclidean domain R has a minimal Euclidean function, ϕ R . Our short history starts with their introduction via Motzkin’s Lemma and moves onto Lenstra’s categorization of Euclidean functions in imaginary quadratic number fields. We examine computing minimal Euclidean functions in these fields, and apply them in short, easy proofs of standard results. Using the author’s simple formula for ϕ Z [ i ] , the only explicitly computable minimal Euclidean function the author knows for a number field other than Q , we apply the pre-images’ geometry to give a new elementary proof affirming Lenstra’s algebraic description of these sets. Figures illustrate the definitions and arguments.

Let K {\mathbf {K}} be a CM-field that is a quadratic extension of a totally real number field k {\mathbf {k}} . Under a technical assumption, we show that the relative class number of K {\mathbf {K}} is large compared with the absolute value of the discriminant of K {\mathbf {K}} , provided that the Dedekind zeta function of k {\mathbf {k}} has a real zero s s such that 0 > s > 1 0 > s > 1 . This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

In 2000, Paulus and Takagi introduced a public key cryptosystem called NICE that exploits the relationship between maximal and non-maximal orders in imaginary quadratic number fields. Relying on the intractability of integer factorization, NICE provides a similar level of security as RSA, but has faster decryption. This paper presents REAL-NICE, an adaptation of NICE to orders in real quadratic fields. REAL-NICE supports smaller public keys than NICE, and while preliminary computations suggest that it is somewhat slower than NICE, it still significantly outperforms RSA in decryption.KeywordsIdeal ClassQuadratic OrderEncryption TimeIdeal Class GroupEncryption StepThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 ⋅ 163 10921 = 67 \cdot 163 , which is the conductor of the biquadratic number field Q ( − 67 , − 163 ) {\mathbf {Q}}(\sqrt { - 67} ,\sqrt { - 163} ) .

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.

p-adic Eisenstein–Kronecker series and non-critical values of p-adic Hecke L-function of an imaginary quadratic field when the conductor is divisible by p

On a question of Serre

Here we include some classical results from the theory of binary hermitian forms which originate from Hermite (1854). We discuss the reduction theory of binary hermitian forms as described for example in Bianchi (1892). Eventually our considerations lead to Humbert’s computation of the covolume of SL(2, Thuong) where Thuong is the ring of integers in an imaginary quadratic number field. The work of Humbert on hermitian forms is contained in his papers (1915), (1919a)—(1919e). It contains an interesting error, we correct it in Section 9.6. We also develop a theory of representation numbers of binary hermitian forms which is analogous to the theory of binary quadratic forms as in Landau (1927).KeywordsPrime NumberPrime IdealZeta FunctionDirichlet SeriesHermitian FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter is dedicated to a survey of the classification of Picard modular surfaces \(\widehat{mathbb{B}/\Gamma },\Gamma = mathbb{U}\left( {(2,1,{O_K}} \right),K = mathbb{Q}\left( {\sqrt { - d} } \right)\) an imaginary quadratic number field. For the leading examples of Eisenstein and Gauss numbers we classify completely the corresponding surfaces in sections 1 and 2. These examples clarify the way of the Kodaira classification of Picard modular surfaces corresponding to K with higher discriminants. There are some new results for the Picard modular surfaces of Eisenstein numbers which are of number theoretic interest in close connection with 2-dimensional versions of Hilbert’s 12-th (class field) and 7-th (transendence) problem. We refer to the monograph [HO8].

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan's mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay's monstrous Dynkin diagram observation manifesting simultaneously and compatibly.

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

In the p n -th cyclotomic field ℚ p n , p a prime number, n ∈ ℕ , the prime p is totally ramified and the only ideal above p is generated by ω n = ζ p n - 1 , with the primitive p n -th root of unity ζ p n = e 2 π i p n . Moreover these numbers represent a norm coherent set, i.e. N ℚ p n + 1 / ℚ p n ( ω n + 1 ) = ω n . It is the aim of this article to establish a similar result for the ray class field K 𝔭 n of conductor 𝔭 n over an imaginary quadratic number field K where 𝔭 n is the power of a prime ideal in K . Therefore the exponential function has to be replaced by a suitable elliptic function.

The Non-normal Quartic CM-Fields and the Octic Dihedral CM-Fields with Relative Class Number Two