Controlling λ-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 ν 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, L p (fg) and L p (fgh): the former is cyclotomic in its nature, while the latter is over the weight-space. As a corollary, we derive transition formulae relating analytic λ-invariants of congruent Galois representations for V f V g , and for V f V g V h , respectively.

À la fin des années 1990, Vatsal a montré qu’une congruence modulo p ν entre deux formes modulaires implique une congruence entre leurs fonctions L p-adiques. Nous prouvons des énoncés analogues pour les fonctions L p-adiques L p (fg) et L p (fgh) 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 λ analytiques des représentations de Galois congruentes pour V f V g et V f V g V h respectivement.

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

1 Department of Mathematics and Statistics University of Waikato Gate 8, Hillcrest Road Hamilton 3240, New Zealand
     author = {Delbourgo, Daniel and Gilmore, Hamish},
     title = {Controlling $\lambda $-invariants for the double and triple product $p$-adic $L$-functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {733--778},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Delbourgo, Daniel; Gilmore, Hamish. 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.

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