Witte, Malte

Non-Commutative \(L\)-Functions for \(p\)-Adic Representations over Totally Real Fields

Doc. Math. 24, 1413-1511 (2019)
DOI: 10.25537/dm.2019v24.1413-1511
Communicated by Otmar Venjakob


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 Fukaya and Kato. 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 Greither and Popescu.

Mathematics Subject Classification

11R23, 11R42, 19F27


main conjecture, non-commutative Iwasawa theory, totally real fields


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Witte, Malte
Ruprecht-Karls-Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany