Jacobi forms over totally real number fields

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.     

Ideal lattices over totally real number fields and Euclidean minima

No Abstract. . Received: 14 February 2005     

On the Iwasawa invariants of totally real number fields

We shall show two sufficient conditions under which the Iwasawa invariants λ _k and μ _k of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK _n be the composite ofk and thel ⁿ-th cyclotomic extension of the fieldQ of rational numbers. LetC _n be the factor group of thel-class group ofK _n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ₁ be an idempotent of the group ringZ _l[Gal(K ₁/k)] defined in the below. We shall prove λ _k = μ _k =0 if there is a natural numbern such that ε₁ C _n vanishes, under additional conditions concerning ramifications inK _n/k.     

On rotated D n -lattices constructed via totally real number fields

In this paper we construct families of rotated D _n -lattices, which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels via subfields of cyclotomic fields. These constructions exhibit full diversity and good minimum product distance, which are important parameters related to the signal transmission error probability. It is also shown that for some Galois extensions ${\mathbb{K}|\mathbb{Q}}$ , it is impossible to construct rotated D _n -lattices via fractional ideals of ${\mathcal{O}_{\mathbb{K}}}$ .     

About the maximal rank of 3-tensors over the real and the complex number field

Tensor data are becoming important recently in various application fields. In this paper, we consider the maximal rank problem of 3-tensors and extend Atkinson and Stephens’ and Atkinson and Lloyd’s results over the real number field. We also prove the assertion of Atkinson and Stephens: ${{\rm max.rank}_{\mathbb{R}}(m,n,p) \leq m+\lfloor p/2\rfloor n}$ , ${{\rm max.rank}_{\mathbb{R}}(n,n,p) \leq (p+1)n/2}$ if p is even, ${{\rm max.rank}_{\mathbb{F}}(n,n,3)\leq 2n-1}$ if ${\mathbb{F}=\mathbb{C}}$ or n is odd, and ${{\rm max.rank}_{\mathbb{F}}(m,n,3)\leq m+n-1}$ if m < n where ${\mathbb{F}}$ stands for ${\mathbb{R}}$ or ${\mathbb{C}}$ .     

On the maximal conjugate of a totally real algebraic integer

Let α be an algebraic integer of degreed whosed conjugates are all real. We give a lower bound for the absolute value of conjugates of α in terms ofd and of the number of conjugates outside the interval [−2; 2]. Combining this with a lower bound for Mahler's measure of a polynomial, we obtain a lower bound for the maximal conjugate of a totally real algebraic integer. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 1, pp. 18–25, January–March, 1997.     

On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

Theta series of a real algebraic number field

We investigate transformation formulas for theta series with spherical functions on a Hilbert-Siegel space. As an application we show that some of Hilbert-Siegel modular varieties are of general type.     

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

Integer valued polynomials over a number field

A number field is called a Pólya field if the module of integer valued polynomials over that field is generated by (f_i) _i=0 over the ring of integers, with deg(f_i)=i, i=0, 1, 2,... In this paper bounds on the class numbers and on the number of ramified primes in Pólya fields are derived.     

On the number of directions determined by a pair of functions over a prime field

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than pairs with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e.