Browsing by Author "Smith, David Jason"

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    An equivalence of categories in algebraic geometry and some unlikely intersections in powers of elliptic curves
    (Stellenbosch : Stellenbosch University, 2024-03) Smith, David Jason; Boxall, Gareth; Marques, Sophie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH 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.