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 |
| Summary: | 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 $L(f \otimes g \otimes h, 1) \neq 0$. |
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