Stark points on elliptic curves via Perrin-Riou's philosophy
In the early 90’s, Perrin-Riou [PR] introduced an important refinement of the Mazur-Swinnerton-Dyer p-adic L-function of an elliptic curve E over Q, taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-fun...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| _version_ | 1797109573059870720 |
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| author | Darmon, H Lauder, AG |
| author_facet | Darmon, H Lauder, AG |
| author_sort | Darmon, H |
| collection | OXFORD |
| description | In the early 90’s, Perrin-Riou [PR] introduced an important refinement of the Mazur-Swinnerton-Dyer p-adic L-function of an elliptic curve E over Q, taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret-Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations %g and %h respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by %g ⊗ %h, in the style of the regulators that arise in [DLR1], and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series. |
| first_indexed | 2024-03-07T07:43:37Z |
| format | Journal article |
| id | oxford-uuid:39dba4ab-3cf1-4dab-8239-f56bf46c9aa7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:43:37Z |
| publishDate | 2021 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:39dba4ab-3cf1-4dab-8239-f56bf46c9aa72023-05-16T07:50:05ZStark points on elliptic curves via Perrin-Riou's philosophyJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:39dba4ab-3cf1-4dab-8239-f56bf46c9aa7EnglishSymplectic ElementsSpringer2021Darmon, HLauder, AGIn the early 90’s, Perrin-Riou [PR] introduced an important refinement of the Mazur-Swinnerton-Dyer p-adic L-function of an elliptic curve E over Q, taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret-Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations %g and %h respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by %g ⊗ %h, in the style of the regulators that arise in [DLR1], and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series. |
| spellingShingle | Darmon, H Lauder, AG Stark points on elliptic curves via Perrin-Riou's philosophy |
| title | Stark points on elliptic curves via Perrin-Riou's philosophy |
| title_full | Stark points on elliptic curves via Perrin-Riou's philosophy |
| title_fullStr | Stark points on elliptic curves via Perrin-Riou's philosophy |
| title_full_unstemmed | Stark points on elliptic curves via Perrin-Riou's philosophy |
| title_short | Stark points on elliptic curves via Perrin-Riou's philosophy |
| title_sort | stark points on elliptic curves via perrin riou s philosophy |
| work_keys_str_mv | AT darmonh starkpointsonellipticcurvesviaperrinriousphilosophy AT lauderag starkpointsonellipticcurvesviaperrinriousphilosophy |