Gatti, FrancescaGuitart Morales, XavierMasdeu Sabaté, MarcRotger, Victor2023-09-212023-09-2120211246-7405https://hdl.handle.net/2445/202122The 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 $L(f \otimes g \otimes h, 1) \neq 0$.26 p.application/pdfengcc-by-nd (c) Gatti, Francesca et al., 2021https://creativecommons.org/licenses/by-nd/4.0/Funcions LAnàlisi p-àdicaL-functionsp-adic analysisinfo:eu-repo/semantics/article7208042023-09-21info:eu-repo/semantics/openAccess