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...
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.
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