K1-congruences between L-values of elliptic curves
We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q. First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which int...
| Main Author: | |
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2009
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| Online Access: | https://eprints.nottingham.ac.uk/10766/ |
| _version_ | 1848791126449848320 |
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| author | Ward, Thomas |
| author_facet | Ward, Thomas |
| author_sort | Ward, Thomas |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q.
First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which interpolates the values the L-values of an Artin twist of E, at a family of finite-order character twists. To do this, we exploit the fact that such an L-value may be written as the Rankin convolution of two Hilbert modular forms, when the representation factors through the false Tate curve extension. Recent developments in non-abelian Iwasawa theory predict certain strong congruences between these p-adic L-functions, and we shall establish weakened versions of these congruences.
Next, we prove analogous results for an elliptic curve with complex multiplication; we do this using work of Hida and Tilouine on the p-adic interpolation of Hecke L-functions over a CM-field. We go on to investigate the ratio of the automorphic and motivic periods associated to E in this setting. We describe how the p-valuation of this ratio may be explicitly calculated, and use the computer package MAGMA to produce some numerical examples. We end by proving a formula for the growth of this quantity in terms of the Iwasawa invariants associated to the two-variable extension of the CM-field. |
| first_indexed | 2025-11-14T18:23:33Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-10766 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:23:33Z |
| publishDate | 2009 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-107662025-02-28T11:09:30Z https://eprints.nottingham.ac.uk/10766/ K1-congruences between L-values of elliptic curves Ward, Thomas We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q. First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which interpolates the values the L-values of an Artin twist of E, at a family of finite-order character twists. To do this, we exploit the fact that such an L-value may be written as the Rankin convolution of two Hilbert modular forms, when the representation factors through the false Tate curve extension. Recent developments in non-abelian Iwasawa theory predict certain strong congruences between these p-adic L-functions, and we shall establish weakened versions of these congruences. Next, we prove analogous results for an elliptic curve with complex multiplication; we do this using work of Hida and Tilouine on the p-adic interpolation of Hecke L-functions over a CM-field. We go on to investigate the ratio of the automorphic and motivic periods associated to E in this setting. We describe how the p-valuation of this ratio may be explicitly calculated, and use the computer package MAGMA to produce some numerical examples. We end by proving a formula for the growth of this quantity in terms of the Iwasawa invariants associated to the two-variable extension of the CM-field. 2009-07-22 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/10766/1/Thomas_Ward-_Thesis.pdf Ward, Thomas (2009) K1-congruences between L-values of elliptic curves. PhD thesis, University of Nottingham. |
| spellingShingle | Ward, Thomas K1-congruences between L-values of elliptic curves |
| title | K1-congruences between L-values of elliptic curves |
| title_full | K1-congruences between L-values of elliptic curves |
| title_fullStr | K1-congruences between L-values of elliptic curves |
| title_full_unstemmed | K1-congruences between L-values of elliptic curves |
| title_short | K1-congruences between L-values of elliptic curves |
| title_sort | k1-congruences between l-values of elliptic curves |
| url | https://eprints.nottingham.ac.uk/10766/ |