Descent methods and torsion on Jacobians of higher genus curves
<p>In this thesis we accomplish four main results related to Jacobians of curves.</p> <p>Firstly, we find a large number of hyperelliptic curves of genus 2, 3 and 4 whose Jacobians have torsion points of large order. The genus 2 case is particularly well-studied in the literature,...
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| Format: | Thesis |
| Language: | English |
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2018
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| _version_ | 1817931756200787968 |
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| author | Nicholls, C |
| author2 | Flynn, V |
| author_facet | Flynn, V Nicholls, C |
| author_sort | Nicholls, C |
| collection | OXFORD |
| description | <p>In this thesis we accomplish four main results related to Jacobians of curves.</p> <p>Firstly, we find a large number of hyperelliptic curves of genus 2, 3 and 4 whose Jacobians have torsion points of large order. The genus 2 case is particularly well-studied in the literature, and we provide a new example of a geometrically simple Jacobian of a genus 2 curve with a point of order 25, an order which was not previously known. For geometrically simple Jacobians of curves of genus 3 and 4, we extend the known orders of points, increasing the largest known order in both cases to 91 and 88, respectively.</p> <p>Secondly, we find an explicit embedding of the Kummer variety of a genus 3 superelliptic curve into projective space. This is a natural extension of the embeddings that are already known for the Kummer varieties of hyperelliptic curves of genus 2 and 3.</p> <p>Thirdly, we classify the genus 2 curves whose Jacobians admit a (4,4)- isogeny. We find an infinite family of genus 2 curves for which the elements of the kernel of the (4,4)-isogeny are defined over the ground field, and make partial progress on classifying the genus 2 curves with this property. We also extend Flynn’s example of a genus 2 curve whose Jacobian admits a (5, 5)-isogeny to infinitely many geometrically nonisomorphic curves.</p> <p>Finally, we extend Schaefer’s algorithm for computing the Selmer group of a Jacobian to carry out a (4, 4)-descent on Jacobians of curves that admit a (4, 4)-isogeny.</p> |
| first_indexed | 2024-03-06T18:16:33Z |
| format | Thesis |
| id | oxford-uuid:04cef70a-2ab9-44c2-8bbe-ca2ac33bfe41 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:27:04Z |
| publishDate | 2018 |
| record_format | dspace |
| spelling | oxford-uuid:04cef70a-2ab9-44c2-8bbe-ca2ac33bfe412024-12-01T09:53:16ZDescent methods and torsion on Jacobians of higher genus curvesThesishttp://purl.org/coar/resource_type/c_db06uuid:04cef70a-2ab9-44c2-8bbe-ca2ac33bfe41Arithmetic geometryEnglishORA Deposit2018Nicholls, CFlynn, V<p>In this thesis we accomplish four main results related to Jacobians of curves.</p> <p>Firstly, we find a large number of hyperelliptic curves of genus 2, 3 and 4 whose Jacobians have torsion points of large order. The genus 2 case is particularly well-studied in the literature, and we provide a new example of a geometrically simple Jacobian of a genus 2 curve with a point of order 25, an order which was not previously known. For geometrically simple Jacobians of curves of genus 3 and 4, we extend the known orders of points, increasing the largest known order in both cases to 91 and 88, respectively.</p> <p>Secondly, we find an explicit embedding of the Kummer variety of a genus 3 superelliptic curve into projective space. This is a natural extension of the embeddings that are already known for the Kummer varieties of hyperelliptic curves of genus 2 and 3.</p> <p>Thirdly, we classify the genus 2 curves whose Jacobians admit a (4,4)- isogeny. We find an infinite family of genus 2 curves for which the elements of the kernel of the (4,4)-isogeny are defined over the ground field, and make partial progress on classifying the genus 2 curves with this property. We also extend Flynn’s example of a genus 2 curve whose Jacobian admits a (5, 5)-isogeny to infinitely many geometrically nonisomorphic curves.</p> <p>Finally, we extend Schaefer’s algorithm for computing the Selmer group of a Jacobian to carry out a (4, 4)-descent on Jacobians of curves that admit a (4, 4)-isogeny.</p> |
| spellingShingle | Arithmetic geometry Nicholls, C Descent methods and torsion on Jacobians of higher genus curves |
| title | Descent methods and torsion on Jacobians of higher genus curves |
| title_full | Descent methods and torsion on Jacobians of higher genus curves |
| title_fullStr | Descent methods and torsion on Jacobians of higher genus curves |
| title_full_unstemmed | Descent methods and torsion on Jacobians of higher genus curves |
| title_short | Descent methods and torsion on Jacobians of higher genus curves |
| title_sort | descent methods and torsion on jacobians of higher genus curves |
| topic | Arithmetic geometry |
| work_keys_str_mv | AT nichollsc descentmethodsandtorsiononjacobiansofhighergenuscurves |