Special values of triple-product p-adic L-functions and non-crystalline diagonal classes
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 809-834.

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 (f,g,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(fgh,s) (which typically has sign +1) does not vanish at its central critical point s=1. When f corresponds to an elliptic curve E/ and the classical L-function vanishes, the Elliptic Stark Conjecture of Darmon–Lauder–Rotger predicts that Ł p g (f,g,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 (f,g,h)(2,1,1) in the case where L(fgh,1)0.

L’objectif principal de cette note est de comprendre l’arithmétique encodée dans la valeur de la fonction L p-adique Ł p g (f,g,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(fgh,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/ et la fonction L classique s’annule, la conjecture elliptique de Stark de Darmon–Lauder–Rotger prédit que soit la valeur Ł p g (f,g,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 (f,g,h)(2,1,1) dans le cas où L(fgh,1)0.

Published online:
DOI: 10.5802/jtnb.1179
Classification: 11G40, 11F85
Keywords: $p$-adic $L$-functions, Selmer groups, elliptic curves
Francesca Gatti 1; Xavier Guitart 2; Marc Masdeu 3; Victor Rotger 1

1 Departament de Matemàtiques Universitat Politècnica de Catalunya Edifici Omega, Campus Nord Carrer de Jordi Girona 1–3 08034 Barcelona, Catalonia
2 Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran via de les Corts Catalanes 585 08007 Barcelona, Catalonia
3 Departament de Matemàtiques Universitat Autònoma de Barcelona Edicifi C, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Special values of triple-product $p$-adic {L-functions} and non-crystalline diagonal classes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Francesca Gatti; Xavier Guitart; Marc Masdeu; Victor Rotger. Special values of triple-product $p$-adic L-functions and non-crystalline diagonal classes. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 809-834. doi : 10.5802/jtnb.1179. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1179/

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