Notes on the Parity Conjecture

2010/09/30 by Tim Dokchitser
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT #msc:11G05 #msc:11G07 #msc:11G40

paper · pdf · doi:10.1007/978-3-0348-0618-3_5

published as Elliptic Curves, Hilbert Modular Forms and Galois Deformations, Advanced Courses in Mathematics - CRM Barcelona, Springer Basel, 2013 · minor corrections, to appear in a CRM Advanced Courses volume "Elliptic curves, Hilbert modular forms and Galois deformations"; 43 pages

arxiv created 2012/01/17 · openalex publication_date 2013/01/01 · arxiv updated 2013/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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.

Citations

• Conjectures on elliptic curves over quadratic fields
• On the behaviour of root numbers in families of elliptic curves
• Good Reduction of Abelian Varieties
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Nonvanishing Theorems for Automorphic L-Functions on GL(2)