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, 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 ${Ł}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ 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 $L(f\otimes g\otimes h,1)\ne 0$.

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 $L(f\otimes g\otimes h,1)\ne 0$.

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

[1] Belaïche, Joël An introduction to the conjecture of Bloch and Kato (Two lectures at the Clay Mathematical Institute Summer School, Honolulu, Hawaii, 2009, http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf)

[2] Bertolini, Massimo; Darmon, Henri A Rigid Analytic Gross–Zagier Formula and Arithmetic Applications, Ann. Math., Volume 146 (1997) no. 1, pp. 111-147 | DOI | MR | Zbl

[3] Bloch, Spencer; Kato, Kazuya $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 333-400 | MR | Zbl

[4] Darmon, Henri; Lauder, Alan; Rotger, Victor Stark points and $p$-adic iterated integrals attached to modular forms of weight one, Forum Math. Pi, Volume 3 (2015), e8, 95 pages | DOI | MR | Zbl

[5] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems I: A $p$-adic Gross–Zagier formula, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 4, pp. 779-832 | DOI | MR | Zbl

[6] Darmon, Henri; Rotger, Victor Elliptic curves of rank two and generalised Kato classes, Res. Math. Sci., Volume 3 (2016) no. 1, p. 27 | DOI | MR | Zbl

[7] Darmon, Henri; Rotger, Victor Diagonal cycles and Euler systems II: The Birch and Swinnerton–Dyer conjecture for Hasse–Weil–Artin $L$-functions, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 601-672 | DOI | MR | Zbl

[8] Darmon, Henri; Rotger, Victor Stark–Heegner points and generalised Kato classes (2019) (preprint)

[9] Gatti, Francesca; Rotger, Victor Kolyvagin classes versus non-cristalline diagonal classes (2021) (https://arxiv.org/abs/2103.11492)

[10] Hsieh, Ming-Lun Hida families and $p$-adic triple product $L$-functions, Am. J. Math., Volume 143 (2021) no. 2, pp. 411-532 | DOI | MR | Zbl

[11] Lauder, Alan Efficient computation of Rankin $p$-adic $L$-functions, Computations with modular forms (Contributions in Mathematical and Computational Sciences), Volume 6, Springer, 2014, pp. 181-200 | DOI | MR | Zbl

[12] Mazur, Barry; Rubin, Karl Kolyvagin systems, Mem. Am. Math. Soc., Volume 168 (2004) no. 799, p. viii+96 | DOI | MR | Zbl

[13] Ohta, Masami On the $p$-adic Eichler–Shimura isomorphism for $\Lambda$-adic cusp forms, J. Reine Angew. Math., Volume 463 (1995), pp. 49-98 | DOI | MR | Zbl

[14] The Sage Developers SageMath, the Sage Mathematics Software System (Version 9.1) (2020) (http://www.sagemath.org)