On non-commutative Iwasawa theory and derivatives of Euler systems

We use the theory of reduced determinant functors from [24] to give a new, computationally useful, description of the relative $K_0$-groups of orders in finite dimensional separable algebras that need not be commutative. By combining this approach with a canonical generalization to non-commutative algebras of the notion of `zeta element' introduced by Kato [52], we then formulate, for each odd prime $p$, a natural main conjecture of non-commutative $p$-adic Iwasawa theory for $\mathbb{G}_m$ over arbitrary number fields. This conjecture predicts a simple relation between a canonical Rubin-Stark non-commutative Euler system that we introduce and the compactly supported $p$-adic cohomology of $\mathbb{Z}_p$ and is shown to simultaneously extend both the higher rank (commutative) main conjecture for $\mathbb{G}_m$ formulated by Kurihara and the present authors [19] and the $K$-theoretical formalism of main conjectures in non-commutative Iwasawa theory developed by Ritter and Weiss [73] and by Coates, Fukaya, Kato, Sujatha and Venjakob [27]. In particular, via these links we obtain strong evidence in support of the conjecture in the setting of Galois CM extensions of totally real fields. Our approach also leads to the formulation over arbitrary number fields of a precise conjectural `higher derivative formula' for the Rubin-Stark non-commutative Euler system that is shown to recover upon appropriate specialisation the classical Gross-Stark Conjecture for Deligne-Ribet $p$-adic $L$-functions. We then show that this conjectural derivative formula can be combined with the main conjecture of non-commutative $p$-adic Iwasawa theory to give a strategy for obtaining evidence in support of the equivariant Tamagawa Number Conjecture for $\mathbb{G}_m$ over arbitrary finite Galois extensions of number fields, thereby obtaining a wide-ranging generalization of the main result of [19].