Positive definite binary hermitian forms with finitely many exceptions

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

Let $$\ell $$ be a rational prime. Previously, abelian $$\ell $$ -towers of multigraphs were introduced which are analogous to $${\mathbb {Z}}_{\ell }$$ -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the $$\ell $$ -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $${\mathbb {Z}}_{\ell }$$ -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian $$\ell $$ -towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in $${\mathbb {Z}}_{\ell }$$ and then study the special value at $$u=1$$ of the Artin-Ihara L-function $$\ell $$ -adically.

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.

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.

Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Toth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

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 prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x 2+Ny 2 for a squarefree integer N.

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then(ii) for , if m is a square-free positive integer with , then

On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

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.

Units from 5-Torsion on the Jacobian ofy2=x5+1/4 and the Conjectures of Stark and Rubin

By the problem, called a representation of the integers, is meant the encoding of the positive integers without a known upper bound, for which unique decoding can be provided. From the viewpoint of information theory, this problem has been discussed theoretically in its relation to the universality, asymptotic optimality, and computation theory. The result in those approaches has been applied to the data compression scheme for the type of data referring to the past as well as the URR and other floating-pointnumber representations with variable exponential length. This paper discusses representation of the integers problem from the viewpoint of the codeword-length function, which is the function to represent the length of the codeword. As a result, a general construction method is presented which has the desirable property and can uniquely encode all representations. Several methods are given from this approach, which can construct not only the traditional representations of the integers but also the first representations of the integers which has not been considered. It is noted first that two construction methods of codeword length, which are independent of the encoding, are considered. One is based on the combination of the codeword-length functions, and the other is based on the codeword-length distribution. More precisely, it is shown that not only are the traditional representations derived, but also the representations that have not been considered in the past. Using detailed examples, it is shown that the computational complexity of the encoding and decoding depends on the codeword-length function.

In this paper, for any given natural numbersn anda, we can construct explicitly positive definite indecomposable integral Hermitian forms of rankn over\(\mathbb{Q}(\sqrt { - 3} )\) with discriminanta, with the following ten exceptions:n=2,a=1, 2, 4, 10;n=3,a=1, 2, 5;n=4,a=1;n=5,a=1; andn=7,a=1. In the exceptional cases there are no forms with the desired properties. The method given here can be applied to solving the problem of constructing indecomposable positive definite HermitianRm-lattices of any given rankn and discriminanta, whereRm is the ring of algebraic integers in an imaginary quadratic field\(\mathbb{Q}(\sqrt { - m} )\) with class number unity.

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.