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

Gatti, Francesca; Guitart, Xavier; Masdeu, Marc; Rotger, Victor

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 809-834.

L’objectif principal de cette note est de comprendre l’arithmétique encodée dans la valeur de la fonction $L$ $p$-adique ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associée à un triplet de formes modulaires $(f,g,h)$ de poids $(2,1,1)$, dans le cas où la fonction $L$ classique $L(f\otimes g\otimes h,s)$ (qui est généralement de signe $+1$) ne s’annule pas au point central critique $s=1$. Lorsque $f$ correspond à une courbe elliptique $E/\mathbb{Q}$ et la fonction $L$ classique s’annule, la conjecture elliptique de Stark de Darmon–Lauder–Rotger prédit que soit la valeur ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ est $0$ (lorsque l’ordre d’annulation de la fonction $L$ complexe est $>2$), soit elle est liée aux logarithmes des points globaux sur $E$ et à une certaine unité de Gross–Stark associée à $g$ (lorsque l’ordre d’annulation est exactement $2$). Nous complétons la conjecture de Stark elliptique en donnant une formule pour la valeur ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ dans le cas où $L(f\otimes g\otimes h,1)\ne 0$.

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ 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 ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ 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 ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})(2,1,1)$ in the case where $L(f\otimes g\otimes h,1)\ne 0$.

Classification : 11G40, 11F85
Mots-clés : $p$-adic $L$-functions, Selmer groups, elliptic curves

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