A note on the Bloch-Tamagawa space and Selmer groups

TL;DR

This paper constructs a new extension of the Tate-Shafarevich group by the Bloch-Tamagawa space for abelian varieties over number fields, providing a novel example of a Zagier sequence for Selmer groups.

Contribution

It introduces a new extension linking the Tate-Shafarevich group and Bloch-Tamagawa space, utilizing recent advances by Lichtenbaum and Flach.

Findings

New extension of Tate-Shafarevich group by Bloch-Tamagawa space

Provides a novel example of a Zagier sequence for Selmer groups

Enhances understanding of Selmer group structures

Abstract

For any abelian variety $A$ over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach. This gives a new example of a Zagier sequence for the Selmer group of $A$.