Notes on the Parity Conjecture

TL;DR

This paper provides an expository overview connecting the finiteness of the Tate-Shafarevich group to the parity conjecture for elliptic curves over number fields, reviewing root numbers and their implications.

Contribution

It offers a self-contained proof linking Tate-Shafarevich group finiteness to the parity conjecture, clarifying key concepts and consequences.

Findings

Finiteness of Tate-Shafarevich group implies the parity conjecture.

Classification of local and global root numbers of elliptic curves.

Discussion of consequences of the parity conjecture.

Abstract

This is an expository article, based on a lecture course given at CRM Barcelona in December 2009. The purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate-Shafarevich group implies the parity conjecture for elliptic curves over number fields. Along the way, we review local and global root numbers of elliptic curves and their classification, and discuss some peculiar consequences of the parity conjecture.