On non-abelian higher special elements of $p$-adic representations

TL;DR

This paper develops a non-abelian theory of higher special elements in Galois cohomology of p-adic representations, linking it to Galois module structures and refined BSD conjectures.

Contribution

It introduces a new non-abelian framework for higher special elements, connecting them to organizing matrices and Galois module structures of Selmer groups.

Findings

Establishes a relation between non-abelian special elements and Galois module structures.

Provides a new perspective on refined BSD conjectures.

Links the theory to the structure of Tate-Shafarevich and Selmer groups.

Abstract

We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.