Controlling \(\lambda\)-invariants for the double and triple product \(p\)-adic \(L\)-functions
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).
scientific article; zbMATH DE number 7469014
scientific article