Non-commutative \(L\)-functions for \(p\)-adic representations over totally real fields (Q2319416)

Summary: We prove a unicity result for the non-commutative \(L\)-functions for \(p\)-adic representations over totally real fields-functions appearing in the non-commutative Iwasawa main conjecture over totally real fields. We then consider continuous representations \(\rho\) of the absolute Galois group of a totally real field \(F\) on adic rings in the sense of \textit{T. Fukaya} and \textit{K. Kato} [Transl., Ser. 2, Am. Math. Soc. 219, 1--85 (2006; Zbl 1238.11105)]. Using our unicity result, we show that there exists a unique sensible definition of a non-commutative \(L\)-function for any such \(\rho\) that factors through the Galois group of a possibly infinite totally real extension. We also consider the case of CM-extensions and discuss the relation with the equivariant main conjecture for realisations of abstract 1-motives of \textit{C. Greither} and \textit{C. D. Popescu} [J. Algebr. Geom. 24, No. 4, 629--692 (2015; Zbl 1330.11070)].