Kummer generators and lambda invariants

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.

We show that there are finitely many imaginary quadratic number fields for which the class group has exponent 5. Indeed there are finitely many with exponent at most 6. The proof is based on a method of Pierce [Pierce L. B.: A bound for the 3-part of class numbers of quadratic fields by means of the square sieve. Forum Math. 18 (2006), 677–698]. The problem is reduced to one of counting integral points on a certain affine surface. This is tackled using the author's “square-sieve”, in conjunction with estimates for exponential sums. The latter are derived using the q-analogue of van der Corput's method.

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.

We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E6.

The diophantine equation ax2+by2+cz2 = dxyz with a, b, c, d ∈ ℤ{0} and a, b, c ¦ d has been studied in connection with discrete subgroups of PSL(2, ℝ) ([R2, KR, K, Sch]). Rosenberger and Kern-Isberner have determined the complete set of integral solutions. Further Silverman ([S]) gave a description of the solutions of the equation x2 + y2 + z2 = dxyz, |d| ≥ 3, over orders in quadratic imaginary fields, whereas Bowditch, Maclachlan and Reid studied the equation x2 + y2 + z2 = xyz in order to describe the arithmetic once-punctered torus bundles ([BMR]). In this paper we give a survey of the set of solutions over the ring of integers Ok in quadratic imaginary fields, \(\mathbb{Q}\left( {\sqrt { - k} } \right)\), k > 0 squarefree, of the equation ax2 + by2 + cz2 = dxyz, where the coefficients are chosen as follows: $$\bullet a,b,c,d \in O_k \backslash \{ 0\} ,\left. {a,b,c} \right|d,\left| {\tfrac{d} {{\sqrt {abc} }}} \right| \geqslant 3$$ $$\bullet a,b,c,d \in \mathbb{Z}\backslash \{ 0\} ,\left. {a,b,c} \right|d,1 \leqslant \left| {\tfrac{d} {{\sqrt {abc} }}} \right| < 3, resp. a = b = c = 1, d \in O_k \backslash \{ 0\} ,\left| d \right| < 3.$$

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

In this paper, over imaginary quadratic fields, we consider the family of L L -functions L ( s , f ) L (s, f) for an orthonormal basis of spherical Hecke–Maass forms f f with Archimedean parameter t f t_f . We establish asymptotic formulae for the twisted first and second moments of the central values L ( 1 2 , f ) L\big (\frac 1 2, f\big ) , which can be applied to prove that at least 33 33 % of L ( 1 2 , f ) L\big (\frac 1 2, f\big ) with t f ⩽ T t_f \leqslant T are non-vanishing as T → ∞ T \rightarrow \infty . Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.

Kneser's method of constructing adjacent lattices will be used to determine class numbers of unimodular positive definite hermitian lattices of rank 2 and 3 over rings of integers in some imaginary quadratic fields. The same method will also be applied in order to construct indecomposable unimodular positive definite hermitian lattices of rank 2 and 3 over almost all orders in imaginary quadratic fields, even non-maximal ones. All exceptional cases will be determined explicitly.

In this paper, we give an explicit description of 3-Hilbert class field of certain imaginary quadratic fields.

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields$K$for the quartic del Pezzo surface$S$of singularity type${\boldsymbol{A}}_{3}$with five lines given in${\mathbb{P}}_{K}^{4}$by the equations${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

It is known that infinitely many imaginary quadratic fields allow Hermitian lattices which are generated by minimal vectors but have no basis of minimal vectors. In this article we construct systematically such Hermitian lattices over other imaginary quadratic fields. These lattices are binary and unimodular. This construction requires specific non-principal ideals.

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

Let p be a prime number, F a number field, and H n the set of all unramified cyclic extensions over F of degree p having a relative normal integral basis. When ζp ∈ F×, Childs determined the set H n in terms of Kummer generators. When p=3 and F is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all p ⩾ 3 and F with ζp ∉ F× (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.

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.

We present empirical results that suggest that there are real quadratic fields with properties similar to imaginary quadratic fields in terms of size and structure of the class group. Therefore, these class groups can also be used for encryption schemes such as the ElGamal scheme, where up to now, only class groups of imaginary quadratic fields have been considered. Some security aspects are also addressed.