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
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
