In this talk, we discuss a general algebraic theory of ‘higher special elements’ that arise from the derived category of certain `admissible complexes’. These elements constitute a natural generalization of the notion of a ‘higher rank Euler system’ introduced by Burns and Sano and we describe how they encode detailed information about the structure of Galois cohomology groups of p-adic representations.

 

As concrete applications of the theory, we address a question of L. Washington and S. Lang regarding the structure of the quotient formed by the group of global units modulo the cyclotomic units, derive an analogue for the values at s = 1 of p-adic L-functions of Brumer's classical conjecture and formulate refined conjectures of Birch and Swinnerton-Dyer type.

 

This talk is based on a joint work with David Burns and Takamichi Sano, and also a subsequent joint work with Daniel Macias Castillo.