Browsing by Subject "Arithmetical algebraic geometry"
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Item Hypergeometric functions in arithmetic geometry(2009-05) Salerno, Adriana Julia, 1979-; Villegas, Fernando RodriguezHypergeometric functions seem to be ubiquitous in mathematics. In this document, we present a couple of ways in which hypergeometric functions appear in arithmetic geometry. First, we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces, [mathematical symbol] ([lamda]), is a linear combination of hypergeometric functions. We use results by Koblitz and Gross to find explicit relationships, which could be useful for computing Zeta functions in the future. We then study more geometric aspects of the same families. A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection. This connection gives rise to a differential equation which is known to be hypergeometric. We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces.Item The intersection of closure of global points of a semi-abelian variety with a product of local points of its subvarieties(2011-05) Sun, Chia-Liang; Voloch, José Felipe; Vaaler, Je ffrey D.; Rodriguez-Villegas, Fernando; Helm, David F.; Tan, Ki-SengThis thesis consists of three chapters. Chapter 1 explains how the research problems considered in this thesis fit into the investigation of local-global principle in the diophantine geometry, as well as gives a unified sketch of the proofs of the two main results in this thesis. Chapter 2 establishes a similar conclusion to Theorem B of a paper by Poonen and Voloch in another settings. Chapter 3 relates to the object considered in the main result of Chapter 2 to an old conjecture proposed by Skolem and solves some cases of its analog.