Inverse Jacobian and related topics for certain superelliptic curves

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Inverse Jacobian and related topics for certain superelliptic curves

Type: Doctoral Thesis
Title: Inverse Jacobian and related topics for certain superelliptic curves
Author: Somoza, Henares A.
Issue Date: 2019-03-28
Keywords: Abelian variety
Algorithm
Complex multiplication
Curve
Geometry
Hermitian lattice
Inverse Jacobian problem
Jacobian
Number theory
Riemann-Schottky problem
Abstract: 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 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.
Promotor: Supervisor: Lario J., Stevenhagen P. Co-Supervisor: Streng M.
Faculty: Faculty of Science
University: Leiden University
Handle: http://hdl.handle.net/1887/70564
 

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application/pdf Introduction 281.2Kb View/Open
application/pdf Chapter 1 500.0Kb View/Open
application/pdf Chapter 2 424.7Kb View/Open
application/pdf Chapter 3 515.6Kb View/Open
application/pdf Chapter 4 464.7Kb View/Open
application/pdf Bibliography 268.7Kb View/Open
application/pdf Index 286.1Kb View/Open
application/pdf Summary in Dutch 281.6Kb View/Open
application/pdf Summary in Spanish 281.1Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 144.0Kb View/Open
application/pdf Propositions 203.0Kb View/Open

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