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...
| Authors: | ; ; ; |
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| Format: | Article |
| Status: | Published version |
| Publication date: | 2021 |
| Country: | España |
| Institution: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repository: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/202122 |
| Online Access: | https://hdl.handle.net/2445/202122 |
| Access Level: | Open access |
| Keyword: | Funcions L Anàlisi p-àdica L-functions p-adic analysis |
| _version_ | 1878735427576266752 |
|---|---|
| author | Gatti, Francesca |
| author2 | Guitart Morales, Xavier Masdeu Sabaté, Marc Rotger, Victor |
| author2_role | author author author |
| author_browse | Gatti, Francesca Guitart Morales, Xavier Masdeu Sabaté, Marc Rotger, Victor |
| author_facet | Gatti, Francesca Guitart Morales, Xavier Masdeu Sabaté, Marc Rotger, Victor |
| author_role | author |
| collection | Recercat. Dipósit de la Recerca de Catalunya |
| 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 $L(f \otimes g \otimes h, 1) \neq 0$. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | ES_7c871c5f2d2f72eb92a50e2ef4d8fee5 |
| institution | 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) |
| network_acronym_str | ES |
| network_name_str | España |
| oai_identifier_str | oai:recercat.cat:2445/202122 |
| publishDate | 2021 |
| publishDateSort | 2021 |
| publisher | Société Arithmétique de Bordeaux and Centre Mersenne |
| publisherStr | Société Arithmétique de Bordeaux and Centre Mersenne |
| reponame_str | Recercat. Dipósit de la Recerca de Catalunya |
| spelling | Special values of triple-product -adic L-functions and non-crystalline diagonal classes Gatti, Francesca Guitart Morales, Xavier Masdeu Sabaté, Marc Rotger, Victor Funcions L Anàlisi p-àdica L-functions p-adic analysis 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 $L(f \otimes g \otimes h, 1) \neq 0$. Société Arithmétique de Bordeaux and Centre Mersenne https://hdl.handle.net/2445/202122 |
| 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 |
| status_str | publishedVersion |
| title | 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_short | 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 |
| topic | Funcions L Anàlisi p-àdica L-functions p-adic analysis |
| topic_facet | Funcions L Anàlisi p-àdica L-functions p-adic analysis |
| url | https://hdl.handle.net/2445/202122 |