Noncommutative Iwasawa Main Conjectures over Totally Real Fields

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The algebraic techniques developed by Kakde are poised to significantly advance the study of congruences between automorphic forms and the main conjectures of non-commutative Iwasawa theory for various motives. This field has seen remarkable progress in the last decade, highlighted by the recent proof of the non-commutative main conjecture for the Tate motive over a totally real p-adic Lie extension of a number field, achieved independently by Ritter and Weiss, as well as Kakde. The foundational ideas for the non-commutative main conjecture were initially proposed by Venjakob and further explored in subsequent works by Coates-Fukaya-Kato-Sujatha-Venjakob and Fukaya-Kato, alongside related contributions from Burns and Flach regarding the equivariant Tamagawa number conjecture. Kato's pivotal insight into studying the K_1 groups of non-abelian Iwasawa algebras through their abelian quotients has been beautifully developed by Kakde, effectively reducing the study of non-abelian main conjectures to abelian cases. In contrast, Ritter and Weiss employ a more classical approach, drawing inspiration from Frohlich and Taylor's techniques. This volume aims to provide a self-contained exposition of the key themes underlying these developments, making it a valuable resource for researchers in both Iwasawa theory and automorphic forms.