The theory of complex multiplication makes it possible to construct certain class fields and abelian varieties. The main theme of this thesis is making these constructions explicit for the case... Show moreThe theory of complex multiplication makes it possible to construct certain class fields and abelian varieties. The main theme of this thesis is making these constructions explicit for the case where the abelian varieties have dimension 2. Chapter I is an introduction to complex multiplication, and shows that a general result of Shimura can be improved for degree-4 CM-fields. Chapter II gives an algorithm for computing class polynomials for quartic CM-fields, based on an algorithm of Spallek. We make the algorithm more explicit, and use Goren and Lauter___s recent bounds on the denominators of the coefficients, which yields the first running time bound and proof of correctness of an algorithm computing these polynomials. Chapter III studies and computes the irreducible components of the modular variety of abelian surfaces with CM by a given primitive quartic CM-field. We adapt the algorithm of Chapter II to compute these components. Chapters IV and V construct certain `Weil numbers'. They have properties that are number theoretic in nature and are motivated by cryptography. Chapter IV is joint work with David Freeman and Peter Stevenhagen. Chapter V is joint work with Laura Hitt O'Connor, Gary McGuire, and Michael Naehrig. Show less
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 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
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 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
This thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are... Show moreThis thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are examples of arithmetic surfaces. Therefore, Arakelov theory (intersection theory on arithmetic surfaces) occupies a prominent place in this thesis. Apart from this, a substantial part of it is devoted to studying modular curves over finite fields, and their Jacobian varieties, from an algorithmic viewpoint. The end product of this thesis is an algorithm for computing modular Galois representations. These are certain two-dimensional representations of the absolute Galois group of the rational numbers that are attached to Hecke eigenforms over finite fields. The running time of our algorithm is (under minor restrictions) polynomial in the length of the input. This main result is a generalisation of that of work of Jean-Marc Couveignes, Bas Edixhoven et al. Several intermediate results are developed in sufficient generality to make them of interest to the study of modular curves and modular forms in a wider sense. 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
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
Factorization methods, such as the quadratic sieve and the number field sieve, spend a lot of time on the sieving step, in which the necessary relations are collected for factoring the given number... Show moreFactorization methods, such as the quadratic sieve and the number field sieve, spend a lot of time on the sieving step, in which the necessary relations are collected for factoring the given number N. Relations are smooth or k-semismooth numbers (numbers with either all prime factors below some bound or all with the exception of at most k prime factors that do not exceed a second bound) or pairs of these type of numbers. In this thesis, we predict the amount of k-semismooth numbers needed to factor N, based on asymptotic approximation formulas (these formulas generalize the published results), and compare them with the amount of k-semismooth numbers found during the factorization of N. Furthermore, for the number field sieve we propose a method for predicting the number of necessary relations for factoring N with given parameters, and the corresponding sieving time. The basic idea is to do a small but representative amount of sieving and analyze the relations in this sample. We randomly generate relations according to the relevant distribution as observed in the sample and process these relations. Experiments show that our predictions of the number of necessary relations are within 2% of the number of relations needed in the real factorization. Show less
A common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence... Show moreA common theme in the research on rational points on varieties is: investigating under which conditions rational points are dense with respect to a chosen topology. We prove several existence results concerning K3 surfaces defined over the rational numbers with a dense set of rational points with respect to the p-adic topology, for a prime number p, and product topologies arising from these Show less
Barry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher... Show moreBarry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher degree number fields. Merel proved that for all integers d one has that the number of isomorphsim classes of torsion groups of elliptic curves over number fields of degree d is finite. This thesis consists of 4 chapters, the first is introductory and the other tree are research articles. Chapter two deals with the computation of gonalities of modular curves, and the application of these computations to the question which cyclic subgroups can occur as the torsion subgroup of infinitely many non-isomorphic elliptic curves over number fields of degree <7. In the second chapter a general theory for finding rational points on symmetric powers of curves is developed that is similar to symmetric power Chabauty. Application of this theory to symmetric powers of modular curves allows us to determine which primes can divide the order of the torsion subgroup of an elliptic curve over a number field of degree <7. The last chapter studies elliptic curve with a point of order 17 over a number field of degree 4. Show less
