An equivalence of categories in algebraic geometry and some unlikely intersections in powers of elliptic curves

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dc.contributor.advisorBoxall, Garethen_ZA
dc.contributor.advisorMarques, Sophieen_ZA
dc.contributor.authorSmith, David Jasonen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2024-03-05T10:49:36Z
dc.date.accessioned2024-04-26T10:02:34Z
dc.date.available2024-03-05T10:49:36Z
dc.date.available2024-04-26T10:02:34Z
dc.date.issued2024-03
dc.descriptionThesis (MSc)--Stellenbosch University, 2024. en_ZA
dc.description.abstractENGLISH ABSTRACT: Let E be an elliptic curve defined over a number field and let C₁, C₂ EN C be irreducible closed algebraic curves where N 3. Suppose that C₁ is not contained in a 1-dimensional algebraic subgroup of EN C and C₁ C₂ is not contained in a 2-dimensional algebraic subgroup of EN C . Extending work of Boxall on the multiplicative group to elliptic curves, we prove that, if at least one of C₁ and C₂ is not defined over Q, then there are at most finitely many points x C₁ such that there exists an n N such that nx C₂ and that n C₁ C₂ where n C₁ nx x C₁ . Moreover, we consider a defini- tion of affine varieties and prevarieties, in the classical sense, over an arbitrary field and provide expository development of many well-known properties of these classical affine varieties. Additionally, extending well-known definitions of functors in the algebraically closed field case, we rigorously construct func- tors in both directions, between the category of these prevarieties and the category of reduced schemes of finite type over the same arbitrary field, which we show to be quasi-inverse so that they give rise to an equivalence of cate- gories. Finally, in an appendix, we include the well-known definition and some properties of schemes as well as some other basic topics for convenience.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: Laat E ’n elliptiese kurwe wees wat oor ’n getalveld gedefinieer word en laat C₁, C₂ EN C ‘n onherleibare geslote algebraïese kurwe wees waar N 3. Gestel C₁ is nie vervat in ’n 1-dimensionele algebraïese subgroep van EN C nie en C₁ C₂ is nie vervat in ’n 2-dimensionele algebraïese subgroep van EN C nie. Uitbreiding van werk van Boxall op die vermenigvuldigende groep tot elliptiese kurwes, bewys ons dat, as ten minste een van C₁ en C₂ nie oor Q gedefinieer word nie, is daar hoogstens eindige hoeveelheid punte x C₁, sodanig dat daar ’n n N bestaan sodat nx C₂ en dat n C₁ C₂ waar n C₁ nx x C₁ . Verder beskou ons ’n definisie van verwante variëteite en voorvariëteite, in die klassieke sin, oor ’n arbitrêre veld en bied blootstel- lingsontwikkeling van baie bekende eienskappe van hierdie klassieke verwante variëteite. Daarbenewens, om bekende definisies te verleng van funktore in die algebraïes geslote veldsaak, konstrueer ons funktore streng in beide rigtings, tussen die kategorie van hierdie voorvariëteite en die kategorie van verminderde skemas van eindige tipe oor dieselfde arbitrêre veld, wat ons wys dat dit kwasi- omgekeerd is sodat dit aanleiding gee tot ’n ekwivalensie van kategorieë. Ten slotte, in ’n aanhangsel, sluit ons die bekende definisie en sommige eienskappe van skemas sowel as ander basiese onderwerpe vir gerief.af_ZA
dc.description.versionMastersen_ZA
dc.format.extentviii, 140 pagesen_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/130230
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshArithmetical algebraic geometryen_Z
dc.subject.lcshGeometry, Algebraicen_ZA
dc.subject.lcshCurves, Elliptic -- Mathematical modelsen_ZA
dc.subject.nameUCTDen_ZA
dc.titleAn equivalence of categories in algebraic geometry and some unlikely intersections in powers of elliptic curvesen_ZA
dc.typeThesisen_ZA
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