Isogeny graphs, modular polynomials, and applications

Leiden Repository

Isogeny graphs, modular polynomials, and applications

Type: Doctoral Thesis
Title: Isogeny graphs, modular polynomials, and applications
Author: Martindale, C.R.
Issue Date: 2018-06-14
Keywords: Isogeny graph
Hilbert modular polynomials
Point counting
Genus 2 curves
Real multiplication
Canonical lifts
Ordinarya abelian variety
Cyclic isogeny
Abstract: This thesis has three main parts. The first part gives an algorithm to compute Hilbert modular polynomials for ordinary abelian varieties with maximal real multiplication. Hilbert modular polynomials of a given level b give a way of finding all of the abelian varieties that are b-isogeneous to any given abelian varieties satisfying the right conditions. The second part is the proof of a theorem giving the structure of an isogeny graph of simple ordinary abelian varieties with maximal real multiplication. The third part gives a new polynomial time algorithm to count points on genus 2 curves with maximal real multiplication. This algorithm is the fastest known for curves satisfying the right properties.
Promotor: Supervisor: Stevenhagen P., Enge A. Co-Supervisor: Streng M.
Faculty: Science
University: Leiden

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application/pdf Title Pages_Contents 1.479Mb View/Open
application/pdf Introduction 257.7Kb View/Open
application/pdf Chapter 1 376.1Kb View/Open
application/pdf Chapter 2 402.4Kb View/Open
application/pdf Chapter 3 680.8Kb View/Open
application/pdf Chapter 4 539.6Kb View/Open Full text at publisher site
application/pdf Appendix 296.8Kb View/Open
application/pdf Bibliography_Index 310.4Kb View/Open
application/pdf Summary in English 327.9Kb View/Open
application/pdf Summary in Dutch 328.1Kb View/Open
application/pdf Summary in French 338.3Kb View/Open
application/pdf Acknowledgements_Curriculum Vitae 127.8Kb View/Open
application/pdf Propositions 186.7Kb View/Open

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