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Promotie

Inverse Jacobian and related topics for certain superelliptic curves

Datum
donderdag 28 maart 2019
Tijd
Locatie
Academiegebouw
Rapenburg 73
2311 GJ Leiden

Samenvatting

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. Therefore, it makes sense to consider the inverse problem.

Inverse Jacobian problem: "Given a matrix Ω, is it the period matrix associated to any curve? If so, give a model of such a curve."

In this thesis we treat these two problems for two families of superelliptic curves, that is, curves of the form y^k = (x - \alpha_1)....(x - \alpha_l).
We focus on the family of Picard curves, with (k,l) = (3,4) and genus 3, and the family of cyclic plane quintic curves (CPQ curves), with (k,l) = (5,5) and genus 6. We solve the inverse Jacobian problem from a computational point of view, that is, we provide an algorithm to obtain a model for the curve from the period matrix.

We also present one application for the above algorithms: constructing curves such that their Jacobians have complex multiplication. In particular, we determine a complete list of CM-fields whose ring of integers occur as the endomorphism ring over the complex numbers of the Jacobian of a CPQ curve defined over the rational numbers.

Extended summary (pdf)

Promotoren

  • Prof.dr. P. Stevenhagen
  • Prof.dr..J.-C. Lario Loyo (Universitat Politècnica de Catalunya)

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Persvragen

Maarten Muns, adviseur wetenschapscommunicatie Universiteit Leiden
m.a.muns@bb.leidenuniv.nl
071 527 3282

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