Controlling $\lambda$-invariants for the double and triple product $p$-adic $L$-functions

Delbourgo, Daniel; Gilmore, Hamish

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 733-778.

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 ${\mathbf{L}}_p(\mathbf{f}\otimes \mathbf{g})$ et ${\mathbf{L}}_p(\mathbf{f}\otimes \mathbf{g}\otimes \mathbf{h})$ 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.

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, ${\mathbf{L}}_p(\mathbf{f}\otimes \mathbf{g})$ and ${\mathbf{L}}_p(\mathbf{f}\otimes \mathbf{g}\otimes \mathbf{h})$: 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.

Classification : 11F33, 11F67, 11G40, 11R23
Mots clés : Iwasawa theory, $p$-adic $L$-functions, automorphic forms

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