On conjectures of Sharifi

Abstract

(1) This talk is about Iwasawa theory. In non-commutative geometry, the field F1, the field of one element (which is still an imaginary existence), is regarded as an important object. The spirit of Iwasawa theory is near to the idea of F1. In the analogy between Z and Fq[T ] (Fq is a finite field), we look for an analogue of F̄q[T ] = Fq[T ]⊗Fq F̄q on the Z-side. The analogue should be Z ⊗F1 F̄1, but this is an imaginary existence. Iwasawa used ∪r≥1Z[ζpr ] (p is a fixed prime number and ζpr is a primitive p-th root of unity) as an analogue of F̄q[T ] on the Z-side.