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
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
This thesis consists of 8 chapters in which we discuss various aspects of arithmetic. In the first chapter, we give an introduction to the algebraic theory of valued fields. In the second chapter,... Show moreThis thesis consists of 8 chapters in which we discuss various aspects of arithmetic. In the first chapter, we give an introduction to the algebraic theory of valued fields. In the second chapter, we give an introduction to the theory of normal projective curves. In particular, we study curves over finite fields. In the third and fourth chapters, we study images of maps between curves over large fields and over finite fields. In the fifth chapter, we study the subset sum problem for finite abelian groups. In the sixth and seventh chapters, we discuss how to obtain generators for the Picard group of a hyperelliptic curve over a finite field in a deterministic way. In the last chapter, we study automorphism groups of field extensions which are not necessarily algebraic. Show less
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
