Kolyvagin Derivatives of Modular Points on Elliptic Curves
Let E/Q and A/Q be elliptic curves. We can construct modular points derived from A via the modular parametrisation of E. With certain assumptions we can show that these points are of infinite order and are not divisible by a given prime p. We can also show that the group generated by these points is...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2020
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| Online Access: | https://eprints.nottingham.ac.uk/63443/ |
| _version_ | 1848800019467993088 |
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| author | Hatton, Richard |
| author_facet | Hatton, Richard |
| author_sort | Hatton, Richard |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Let E/Q and A/Q be elliptic curves. We can construct modular points derived from A via the modular parametrisation of E. With certain assumptions we can show that these points are of infinite order and are not divisible by a given prime p. We can also show that the group generated by these points is isomorphic to a representation of the projective general linear group.
We further investigate this representation, initially looking at the modular representation theory associated to it. We classify all the Fp-submodules and look at their group cohomology with respect to subgroups of the projective general linear group. We then look at the integral representation theory associated to the representation. We look at some of the Zp-lattices and apply the modular representation theory to determine the size of their group cohomologies.
We finally look at applying the representation theory to the group generated by the modular points. In particular, using Kolyvagin’s construction of derivative classes, we find elements in certain Shafarevich-Tate groups of prime power order. |
| first_indexed | 2025-11-14T20:44:54Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-63443 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:44:54Z |
| publishDate | 2020 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-634432025-02-28T15:04:52Z https://eprints.nottingham.ac.uk/63443/ Kolyvagin Derivatives of Modular Points on Elliptic Curves Hatton, Richard Let E/Q and A/Q be elliptic curves. We can construct modular points derived from A via the modular parametrisation of E. With certain assumptions we can show that these points are of infinite order and are not divisible by a given prime p. We can also show that the group generated by these points is isomorphic to a representation of the projective general linear group. We further investigate this representation, initially looking at the modular representation theory associated to it. We classify all the Fp-submodules and look at their group cohomology with respect to subgroups of the projective general linear group. We then look at the integral representation theory associated to the representation. We look at some of the Zp-lattices and apply the modular representation theory to determine the size of their group cohomologies. We finally look at applying the representation theory to the group generated by the modular points. In particular, using Kolyvagin’s construction of derivative classes, we find elements in certain Shafarevich-Tate groups of prime power order. 2020-12-11 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/63443/1/Thesis.pdf Hatton, Richard (2020) Kolyvagin Derivatives of Modular Points on Elliptic Curves. PhD thesis, University of Nottingham. elliptic curves modular points representation of groups |
| spellingShingle | elliptic curves modular points representation of groups Hatton, Richard Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title | Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title_full | Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title_fullStr | Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title_full_unstemmed | Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title_short | Kolyvagin Derivatives of Modular Points on Elliptic Curves |
| title_sort | kolyvagin derivatives of modular points on elliptic curves |
| topic | elliptic curves modular points representation of groups |
| url | https://eprints.nottingham.ac.uk/63443/ |