Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution is shown to have an affirmative answer.     

Note on divisor function for quaternion algebras

Let a be an integral ideal in a quaternion algebra U over rational numbers Q which ramifies precisely at p and ∞, and d(a) be its divisor function. Recently, Kim and Zhang proved a quaternion analogue of the classical formula , and then established an asymptotic formula for the sum . In this note we improved the upper bound for the error term in the asymptotic formula, and then considered the quaternion analogue of another well-known formula .     

On embedding numbers into quaternion orders

A generalization of the chevalley-Hasse-Noether theorem from maximal orders to arbitrary Eichler orders in quaternion algebras is given. A stability property for the numbers of orbits for unit groups in quaternion orders acting on optimal embeddings of quadratic orders is proved. The results are applied to Siegel's meanvalue of integral representations by genera of integral definite ternary quadratic forms.     

Levels of quaternion algebras

The level of a ring R with 1 ≠ 0 is the smallest positive integer s such that −1 can be written as a sum of s squares in R, provided −1 is a sum of squares at all. D. W. Lewis showed that any value of type 2 ⁿ or 2 ⁿ + 1 can be realized as level of a quaternion algebra, and he asked whether there exist quaternion algebras whose levels are not of that form. Using function fields of quadratic forms, we construct such examples. Received: 23 March 2007, Revised: 30 October 2007     

Heights and Tamagawa numbers of motives

K. Kato has recently defined and studied heights of mixed motives and proposed some interesting questions. In this article, we relate the study of heights to the study of Tamagawa numbers of motives. We also partially answer one of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives with two graded quotients. Finally, we provide a concrete computation with the number of mixed Tate motives with three graded quotients.     

An embedding theorem for Eichler orders

Let B be a quaternion algebra over number field K. Assume that B satisfies the Eichler condition (i.e., there is at least one archimedean place which is unramified in B). Let Ω be an order in a quadratic extension L of K. The Eichler orders of B which admit an embedding of Ω are determined. This is a generalization of Chinburg and Friedman's embedding theorem for maximal orders.     

Mutually orthogonal matrices from division algebras

Matrices A and B in $M_n(\mathbb{C})$ are said to be mutually orthogonal if $A{B}^{?}+B{A}^{?}=0$, where ^? denotes the conjugate transpose. We study cardinalities of certain $\mathbb{R}$-linearly independent families of matrices arising from matrix embeddings of a division algebra of index m with center a number field Z, satisfying the property that matrices from different families are mutually orthogonal. The question is of importance in the context of coding for certain wireless channels, where the cardinalities of such sets is connected to the maximum code rate consistent with low decoding complexity. It follows from our results that the maximum code rate for the codes we consider is severely limited.     

On the distribution of sociable numbers

For a positive integer n, define s(n) as the sum of the proper divisors of n. If s(n)>0, define s₂(n)=s(s(n)), and so on for higher iterates. Sociable numbers are those n with sk(n)=n for some k, the least such k being the order of n. Such numbers have been of interest since antiquity, when order-1 sociables (perfect numbers) and order-2 sociables (amicable numbers) were studied. In this paper we make progress towards the conjecture that the sociable numbers have asymptotic density 0. We show that the number of sociable numbers in [1,x], whose cycle contains at most k numbers greater than x, is o(x) for each fixed k. In particular, the number of sociable numbers whose cycle is contained entirely in [1,x] is o(x), as is the number of sociable numbers in [1,x] with order at most k. We also prove that but for a set of sociable numbers of asymptotic density 0, all sociable numbers are contained within the set of odd abundant numbers, which has asymptotic density about 1/500.     

Traces of Eichler—Brandt matrices and type numbers of quaternion orders

LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.     

Reduction of matrices over orders of imaginary quadratic fields

A special decomposition (called the near standard form) of (1,2)-matrices over a ring is introduced and a method for a reduction of such matrices is explained. This can be applied for a detection of elementary second order matrices among invertible second order matrices. The tool is used in detail over orders of imaginary quadratic fields, where an algorithm, a number of properties and examples are presented.     

On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities

|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N     

On the distribution of square-full and cube-full numbers

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1k+2). At first an asymptotic result is given for the numberN _k,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceN _k,1(x+h)–N_k,1(x) in the casesk=2, 1=4,5,6,7,18 andk=3, 1=5,6,7, 18.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

Variance of the spectral numbers and Newton polygons

Using the theory of the mixed Hodge structure one can define a notion of spectrum of a singularity or of a polynomial. Recently Claus Hertling proposed a conjecture about the variance of the spectrum of a singularity. Alexandru Dimca proposed a similar conjecture on polynomials. Here, we prove these two conjectures in the case of dimension 2 and when the singularity or the polynomial is Newton non-degenerated and commode.     

On arithmetic numbers

An integer n is said to be arithmetic if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, we give an interesting characterization of Mersenne numbers.