Many sources suggest a folklore procedure to determine if a smooth curve of genus 1 has a rational point. This procedure terminates conditionally on the Tate-Shafarevich conjecture. In this thesis, we provide an exposition for this procedure, making several steps explicit. In some instances, we also provide MAGMA implementations of the subroutines. In particular, we give an algorithm to determine if a smooth, genus 1 curve of arbitrary degree is locally soluble, we compute its Jacobian, and we give an exposition for descent in our context. Additionally, we prove there exists an algorithm to decide if smooth, genus 1 curve has a rational point if and only if there exists an algorithm to compute the Mordeil-Weil group of an elliptic curve. / Master of Science / It is unknown whether an algorithm can determine if an equation with rational coefficients has a solution in the rational numbers. This thesis examines the simplest class of such equations: those representing so called smooth curves of genus 1. We demonstrate that an algorithm can decide if these equations have a rational solution if and only if there is an algorithm that can compute all rational solutions given a single rational solution. A procedure exists for the latter, but its success relies on a conjecture. Assuming this conjecture, we explicitly construct the corresponding algorithm to decide if an equation representing a smooth curve of genus 1 has a rational solution.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/119085 |
| Date | 23 May 2024 |
| Creators | Swanson, Nicolas J. Brennan |
| Contributors | Mathematics, Morrison, Travis William, Sikora, Jamie, Matthews, Gretchen L., LeGrow, Jason Travis |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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