Heegner points, stark-Heegner points, and diagonal classes
Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/537064 |
| Acceso en línea: | http://hdl.handle.net/2072/537064 |
| Access Level: | acceso abierto |
| Palabra clave: | Elliptic curves, modular forms, p-adic L-functions, Heegner points, Stark-Heegner points, generalised Kato classes |
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Heegner points, stark-Heegner points, and diagonal classesDarmon, H.Rotger, V.Elliptic curves, modular forms, p-adic L-functions, Heegner points, Stark-Heegner points, generalised Kato classesStark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that remains inert in K, but are conjectured to be rational over ring class fields of K and to satisfy a Shimura reciprocity law describing the action of GK on them. The main conjectures of [Da01] predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character ψ of K should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when L0 (E/K, ψ, 1) 6= 0. Building on the results on families of diagonal classes described in the remaining contributions to this volume, this note explains how such linear combinations arise from global classes in the idoneous pro-p Selmer group, and are non-trivial when the first derivative of a weight-variable p-adic L-function Lp(f/K, ψ) does not vanish at the point associated to (E/K, ψ).The first author was supported by an NSERC Discovery grant. The second author also acknowledges the financial support by ICREA under the ICREA Academia programme. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682152). It is a pleasure to thank M.L. Hsieh and M. Longo for detailed explanations of their respective recent preprints, and M. Bertolini, M. Seveso, and R. Venerucci for their complementary works [BSVa], [BSVb] appearing in this volume.Société Mathematique de France2022info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion28 p.application/pdfhttp://hdl.handle.net/2072/537064RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésAstériqueL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5370642026-05-29T05:05:01Z |
| dc.title.none.fl_str_mv |
Heegner points, stark-Heegner points, and diagonal classes |
| title |
Heegner points, stark-Heegner points, and diagonal classes |
| spellingShingle |
Heegner points, stark-Heegner points, and diagonal classes Darmon, H. Elliptic curves, modular forms, p-adic L-functions, Heegner points, Stark-Heegner points, generalised Kato classes |
| title_short |
Heegner points, stark-Heegner points, and diagonal classes |
| title_full |
Heegner points, stark-Heegner points, and diagonal classes |
| title_fullStr |
Heegner points, stark-Heegner points, and diagonal classes |
| title_full_unstemmed |
Heegner points, stark-Heegner points, and diagonal classes |
| title_sort |
Heegner points, stark-Heegner points, and diagonal classes |
| dc.creator.none.fl_str_mv |
Darmon, H. Rotger, V. |
| author |
Darmon, H. |
| author_facet |
Darmon, H. Rotger, V. |
| author_role |
author |
| author2 |
Rotger, V. |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
Elliptic curves, modular forms, p-adic L-functions, Heegner points, Stark-Heegner points, generalised Kato classes |
| topic |
Elliptic curves, modular forms, p-adic L-functions, Heegner points, Stark-Heegner points, generalised Kato classes |
| description |
Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field K. They are constructed analytically as local points on elliptic curves with multiplicative reduction at a prime p that remains inert in K, but are conjectured to be rational over ring class fields of K and to satisfy a Shimura reciprocity law describing the action of GK on them. The main conjectures of [Da01] predict that any linear combination of Stark-Heegner points weighted by the values of a ring class character ψ of K should belong to the corresponding piece of the Mordell-Weil group over the associated ring class field, and should be non-trivial when L0 (E/K, ψ, 1) 6= 0. Building on the results on families of diagonal classes described in the remaining contributions to this volume, this note explains how such linear combinations arise from global classes in the idoneous pro-p Selmer group, and are non-trivial when the first derivative of a weight-variable p-adic L-function Lp(f/K, ψ) does not vanish at the point associated to (E/K, ψ). |
| publishDate |
2022 |
| dc.date.none.fl_str_mv |
2022 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion |
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article |
| status_str |
acceptedVersion |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/2072/537064 |
| url |
http://hdl.handle.net/2072/537064 |
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Inglés |
| language_invalid_str_mv |
Inglés |
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Astérique |
| dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
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openAccess |
| dc.format.none.fl_str_mv |
28 p. application/pdf |
| dc.publisher.none.fl_str_mv |
Société Mathematique de France |
| publisher.none.fl_str_mv |
Société Mathematique de France |
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RECERCAT (Dipòsit de la Recerca de Catalunya) reponame:Recercat. Dipósit de la Recerca de Catalunya instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Recercat. Dipósit de la Recerca de Catalunya |
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Recercat. Dipósit de la Recerca de Catalunya |
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