Duality of Selmer groups of an abelian variety over a number field

TL;DR

This paper explores the duality properties of Selmer groups associated with an abelian variety over a number field, linking their sizes to component groups of the dual variety at various primes.

Contribution

It establishes a relationship between the orders of Selmer groups of an abelian variety and its dual, incorporating component groups at primes, which advances understanding of their arithmetic structure.

Findings

Selmer groups of an abelian variety and its dual are related through their orders.

The orders of these Selmer groups are connected to the component groups of the dual abelian variety.

The results provide new insights into the arithmetic duality of abelian varieties over number fields.

Abstract

Let $A$ be an abelian variety defined over a number field $K$ and let $A^{\vee}$ be the dual abelian variety. For an odd prime $p$, we consider two Selmer groups attached to $A[p]$ and relate the orders of these groups along with those of their corresponding duals to the order of the component groups of $A^{\vee}$ at primes $v$.