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Publication# Diagonal Quartic Surfaces And Transcendental Elements Of The Brauer Group

Abstract

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known

Central simple algebra

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (

This work is dedicated to developing algebraic methods for channel coding. Its goal is to show that in different contexts, namely single-antenna Rayleigh fading channels, coherent and non-coherent MIMO channels, algebraic techniques can provide useful tools for building efficient coding schemes. Rotated lattice signal constellations have been proposed as an alternative for transmission over the single-antenna Rayleigh fading channel. It has been shown that the performance of such modulation schemes essentially depends on two design parameters: the modulation diversity and the minimum product distance. Algebraic lattices, i.e., lattices constructed by the canonical embedding of an algebraic number field, or more precisely ideal lattices, provide an efficient tool for designing such codes, since the design criteria are related to properties of the underlying number field: the maximal diversity is guaranteed when using totally real number fields and the minimum product distance is optimized by considering fields with small discriminant. Furthermore, both shaping and labelling constraints are taken care of by constructing Zn-lattices. We present here the construction of several families of such n-dimensional lattices for any n, and compute their performance. We then give an upper bound on their minimal product distance, and show that with respect to this bound, existing lattice codes are optimal in the sense that no further significant coding gain could be reached. Cyclic division algebras have been introduced recently in the context of coherent Space-Time coding. These are non-commutative algebras which naturally yield families of invertible matrices, or in other words, linear codes that fulfill the rank criterion. In this work, we further exploit the algebraic structures of cyclic algebras to build Space-Time Block codes (STBCs) that satisfy the following properties: they have full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We give algebraic constructions of such STBCs for 2, 3, 4 and 6 antennas and show that these are the only cases where they exist. We finally consider the problem of designing Space-Time codes in the noncoherent case. The goal is to construct maximal diversity Space-Time codewords, subject to a fixed constellation constraint. Using an interpretation of the noncoherent coding problem in terms of packing subspaces according to a given metric, we consider the construction of non-intersecting subspaces on finite alphabets. Techniques used here mainly derive from finite projective geometry.

This thesis is concerned with computations of bounds for two different arithmetic invariants. In both cases it is done with the intention of proving some algebraic or arithmetic properties for number fields. The first part is devoted to computations of lower bounds for the Lenstra's constant. For a number field K the Lenstra's constant is denoted Λ(K) and defined as the length of the largest exceptional sequence in K. An exceptional sequence is a set of units in K such that for any two among them their difference is a unit as well. H.W. Lenstra showed that if Λ(K) is large enough – bigger than a constant depending on the degree and the discriminant of K – then the ring of integers of K is Euclidean with respect to the norm. Using computer software PARI/GP and some algorithms from graph theory we construct exceptional sequences in number fields having a small discriminant. These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to 12. The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers. A method due to C. Soulé yields bounds for the torsion of these K-groups depending on an invariant of hermitian lattices over number fields. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups. The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant.