We show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and... Show moreWe show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and consider the ring of algebraic integers as such a lattice. We compute some invariants using capacity theory and obtain a partial solution to the (algorithmic) closest vector problem. We generalize the results on (universally) graded rings by Lenstra and Silverberg. We study the special case of group rings, and show that under similar assumptions rings can be uniquely decomposed into a group ring in a maximal way. We give a functorial algorithm to compute roots of fractional ideals of orders in number rings. Show less
This dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field... Show moreThis dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field extensions generatedby n-th roots of some base field. Its generalization to commutativealgebraic groups involves fields generated by the division points of afixed algebraic group, such as an elliptic curve or a higher dimensionalabelian variety. Of particular interest in this dissertation is the degreeof such field extensions. In the first two chapter, classical results forelliptic curves are improved by providing explicitly computable bounds anduniform and explicit bounds over the field of rational numbers. In thelast two chapters a general framework for the study of similar problemsis developed. Show less
To an algebraic curve C over the complex numbers one can associate a non-negative integer g, the genus, as a measure of its complexity. One can also associate to C, via complex analysis, a g×g... Show moreTo an algebraic curve C over the complex numbers one can associate a non-negative integer g, the genus, as a measure of its complexity. One can also associate to C, via complex analysis, a g×g symmetric matrix Ω called period matrix (or equivalently, its analytic Jacobian). Because of the natural relation between C and Ω, one can obtain information of one by studying the other. In this thesis we consider the inverse problem."Given a matrix Ω, is it the period matrix associated to any curve? If so, give a model of such a curve."We focus on two families of superelliptic curves, i.e., curves of the form y^k = (x -\alpha_1)....(x - \alpha_l): Picard curves, with (k,l) = (3,4) and genus 3, and CPQ curves, with (k,l) = (5,5) and genus 6.In particular, we characterize the period matrices of said families and provide an algorithm to obtain the curve from the period matrix.We also present one application: constructing curves whose Jacobians have complex multiplication. In particular, we determine a complete list of CM-fields whose maximal order occur as the endomorphism ring over the complex numbers of the Jacobian of a CPQ curve defined over the rationals. Show less
This thesis consists of three chapters. Each chapter is on a different subject. However, all three chapters address issues that arise in counting arithmetically interesting objects. Chapter 1 is... Show moreThis thesis consists of three chapters. Each chapter is on a different subject. However, all three chapters address issues that arise in counting arithmetically interesting objects. Chapter 1 is on the unit equation in positive characteristic. Chapter 2 is about the statistical behavior of ray class groups, of fixed integral conductor, in families of imaginary quadratic fields. Chapter 3 concerns the study of the unit group of local fields in the category of filtered groups. Show less
We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number... Show moreWe prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number field of discriminant D. Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a^2+16c^4. Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4. These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion. Show less
Is there a good continued fraction approximation between every two bad ones? What is the entropy of the natural extension for alpha-Rosen fractions? How do you find multi-dimensional continued... Show moreIs there a good continued fraction approximation between every two bad ones? What is the entropy of the natural extension for alpha-Rosen fractions? How do you find multi-dimensional continued fractions with a guaranteed quality in polynomial time? These, and many more, questions are answered in this thesis. Show less
