Controlling \(\lambda\)-invariants for the double and triple product \(p\)-adic \(L\)-functions
type: Item
instance of: scholarly article
title: Controlling \(\lambda\)-invariants for the double and triple product \(p\)-adic \(L\)-functions
author: Daniel Delbourgo
author: Hamish Gilmore
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2022-02-03T00:00:00Z
review text: This nice and interesting paper answers a very natural and interesting question in the theory of \(p\)-adic \(L\)-functions attached to modular forms. The question, first addressed by \textit{V. Vatsal} [Duke Math. J. 98, No. 2, 397--419 (1999; Zbl 0979.11027)] is the following: given two elliptic modular forms whose Galois representations are congruent modulo a prime number \(p\), show that, after an appropriate choice of complex periods (canonical, up to \(p\)-adic units), the associated cyclotomic \(p\)-adic \(L\)-functions are also congruent modulo \(p\). This result has important applications to the variation of \(\mu\) and \(\lambda\)-invariants of \(p\)-adic \(L\)-functions along congruences. Such a results has been generalised to Hida families of modular forms by \textit{M. Emerton} et al. [Invent. Math. 163, No. 3, 523--580 (2006; Zbl 1093.11065)] and by \textit{F. Castella} et al. in a different setting [Algebra Number Theory 11, No. 10, 2339--2368 (2017; Zbl 1404.11126)]. The goal of the paper under review is to develop similar ideas in the context of double and triple product \(p\)-adic \(L\)-functions for families of modular forms. Given the emerging importance of these objects in modern number theory, the questions addressed in this paper is very natural and quite central in developing a good Iwasawa theory in this setting. The results are of the following form: if the residual Galois representations attached to the relevant Hida families of modular forms are congruent, then the triple and the double product \(p\)-adic \(L\)-functions are also congruent. There are some technical (and natural) conditions to obtain the proof of these results, which are partly different in the two cases. Also, the consequences for the \(\lambda\)-invariants are stated and proved (or conjectured).
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 7469014
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: automorphic forms
MaRDI profile type: Publication
full work available at URL: jtnb.1177
OpenAlex ID: W4210321226
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DOI: 10.5802/JTNB.1177
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description: scientific article; zbMATH DE number 7469014