On the dihedral Euler characteristics of Selmer groups of abelian varieties

TL;DR

This paper explores the use of Euler characteristic formulas to analyze Selmer groups of abelian varieties in dihedral and anticyclotomic extensions, connecting to the refined Birch and Swinnerton-Dyer conjecture within Iwasawa theory.

Contribution

It introduces a framework to verify the p-part of the refined BSD conjecture for Selmer groups in specific extensions, including cases where the groups are not cotorsion.

Findings

Verifies the p-part of the refined BSD conjecture when Selmer groups are cotorsion.

Provides a conjectural description of Euler characteristics for non-cotorsion Selmer groups.

Deduces inequalities from main conjecture divisibilities by Perrin-Riou and Howard.

Abstract

This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule, and explain how to deduce inequalities from the associated main conjecture divisibilities of Perrin-Riou and Howard.