Special values of triple-product -adic L-functions and non-crystalline diagonal classes

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)$ associated to a triple of modular forms $(f, g, h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(f \otime...

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Autores: Gatti, Francesca, Guitart Morales, Xavier, Masdeu Sabaté, Marc, Rotger, Victor
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2021
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:2445/202122
Acceso en línea:https://hdl.handle.net/2445/202122
Access Level:acceso abierto
Palabra clave:Funcions L
Anàlisi p-àdica
L-functions
p-adic analysis
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spelling Special values of triple-product -adic L-functions and non-crystalline diagonal classesGatti, FrancescaGuitart Morales, XavierMasdeu Sabaté, MarcRotger, VictorFuncions LAnàlisi p-àdicaL-functionsp-adic analysisThe main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)$ associated to a triple of modular forms $(f, g, h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(f \otimes g \otimes h, s)$ (which typically has sign +1$)$ does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E / \mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)(2,1,1)$ is either 0 (when the order of vanishing of the complex $L$-function is $>2$ ) or related to logarithms of global points on $E$ and a certain Gross-Stark unit associated to $g$ (when the order of vanishing is exactly 2). We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $E_p^g(\mathbf{f}, \mathbf{g}, \mathbf{h})(2,1,1)$ in the case where $L(f \otimes g \otimes h, 1) \neq 0$.Société Arithmétique de Bordeaux and Centre Mersenne2023202320212023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion26 p.application/pdfapplication/pdfhttps://hdl.handle.net/2445/202122Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: https://doi.org/10.5802/jtnb.1179Journal de Théorie des Nombres de Bordeaux, 2021, vol. 33, p. 809-834https://doi.org/10.5802/jtnb.1179cc-by-nd (c) Gatti, Francesca et al., 2021https://creativecommons.org/licenses/by-nd/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2445/2021222026-05-29T05:05:01Z
dc.title.none.fl_str_mv Special values of triple-product -adic L-functions and non-crystalline diagonal classes
title Special values of triple-product -adic L-functions and non-crystalline diagonal classes
spellingShingle Special values of triple-product -adic L-functions and non-crystalline diagonal classes
Gatti, Francesca
Funcions L
Anàlisi p-àdica
L-functions
p-adic analysis
title_short Special values of triple-product -adic L-functions and non-crystalline diagonal classes
title_full Special values of triple-product -adic L-functions and non-crystalline diagonal classes
title_fullStr Special values of triple-product -adic L-functions and non-crystalline diagonal classes
title_full_unstemmed Special values of triple-product -adic L-functions and non-crystalline diagonal classes
title_sort Special values of triple-product -adic L-functions and non-crystalline diagonal classes
dc.creator.none.fl_str_mv Gatti, Francesca
Guitart Morales, Xavier
Masdeu Sabaté, Marc
Rotger, Victor
author Gatti, Francesca
author_facet Gatti, Francesca
Guitart Morales, Xavier
Masdeu Sabaté, Marc
Rotger, Victor
author_role author
author2 Guitart Morales, Xavier
Masdeu Sabaté, Marc
Rotger, Victor
author2_role author
author
author
dc.subject.none.fl_str_mv Funcions L
Anàlisi p-àdica
L-functions
p-adic analysis
topic Funcions L
Anàlisi p-àdica
L-functions
p-adic analysis
description The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)$ associated to a triple of modular forms $(f, g, h)$ of weights $(2,1,1)$, in the case where the classical $L$-function $L(f \otimes g \otimes h, s)$ (which typically has sign +1$)$ does not vanish at its central critical point $s=1$. When $f$ corresponds to an elliptic curve $E / \mathbb{Q}$ and the classical $L$-function vanishes, the Elliptic Stark Conjecture of Darmon-Lauder-Rotger predicts that $E_p^g$ (f, $\left.\mathbf{g}, \mathbf{h}\right)(2,1,1)$ is either 0 (when the order of vanishing of the complex $L$-function is $>2$ ) or related to logarithms of global points on $E$ and a certain Gross-Stark unit associated to $g$ (when the order of vanishing is exactly 2). We complete the picture proposed by the Elliptic Stark Conjecture by providing a formula for the value $E_p^g(\mathbf{f}, \mathbf{g}, \mathbf{h})(2,1,1)$ in the case where $L(f \otimes g \otimes h, 1) \neq 0$.
publishDate 2021
dc.date.none.fl_str_mv 2021
2023
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/202122
url https://hdl.handle.net/2445/202122
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.5802/jtnb.1179
Journal de Théorie des Nombres de Bordeaux, 2021, vol. 33, p. 809-834
https://doi.org/10.5802/jtnb.1179
dc.rights.none.fl_str_mv cc-by-nd (c) Gatti, Francesca et al., 2021
https://creativecommons.org/licenses/by-nd/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by-nd (c) Gatti, Francesca et al., 2021
https://creativecommons.org/licenses/by-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 26 p.
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Société Arithmétique de Bordeaux and Centre Mersenne
publisher.none.fl_str_mv Société Arithmétique de Bordeaux and Centre Mersenne
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.811543

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