Controlling $\lambda$-invariants for the double and triple product $p$-adic $L$-functions
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 733-778.

In the late 1990s, Vatsal showed that a congruence modulo ${p}^{\nu }$ between two modular forms implied a congruence between their respective $p$-adic $L$-functions. We prove an analogous statement for both the double product and triple product $p$-adic $L$-functions, ${\mathbf{L}}_{p}\left(\mathbf{f}\otimes \mathbf{g}\right)$ and ${\mathbf{L}}_{p}\left(\mathbf{f}\otimes \mathbf{g}\otimes \mathbf{h}\right)$: the former is cyclotomic in its nature, while the latter is over the weight-space. As a corollary, we derive transition formulae relating analytic $\lambda$-invariants of congruent Galois representations for ${V}_{\mathbf{f}}\otimes {V}_{\mathbf{g}}$, and for ${V}_{\mathbf{f}}\otimes {V}_{\mathbf{g}}\otimes {V}_{\mathbf{h}}$, respectively.

À la fin des années 1990, Vatsal a montré qu’une congruence modulo ${p}^{\nu }$ entre deux formes modulaires implique une congruence entre leurs fonctions $L$ $p$-adiques. Nous prouvons des énoncés analogues pour les fonctions $L$ $p$-adiques ${\mathbf{L}}_{p}\left(\mathbf{f}\otimes \mathbf{g}\right)$ et ${\mathbf{L}}_{p}\left(\mathbf{f}\otimes \mathbf{g}\otimes \mathbf{h}\right)$ associées aux produits double et triple de formes modulaires : la première est de nature cyclotomique, tandis que l’autre est définie sur l’espace des poids.

Comme corollaire, nous obtenons des formules de transition reliant les invariants $\lambda$ analytiques des représentations de Galois congruentes pour ${V}_{\mathbf{f}}\otimes {V}_{\mathbf{g}}$ et ${V}_{\mathbf{f}}\otimes {V}_{\mathbf{g}}\otimes {V}_{\mathbf{h}}$ respectivement.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1177
Classification: 11F33, 11F67, 11G40, 11R23
Keywords: Iwasawa theory, $p$-adic $L$-functions, automorphic forms
Daniel Delbourgo 1; Hamish Gilmore 1

1 Department of Mathematics and Statistics University of Waikato Gate 8, Hillcrest Road Hamilton 3240, New Zealand
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Daniel Delbourgo; Hamish Gilmore. Controlling $\lambda$-invariants for the double and triple product $p$-adic $L$-functions. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 733-778. doi : 10.5802/jtnb.1177. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1177/

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