Notes on 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.

minor corrections, to appear in a CRM Advanced Courses volume "Elliptic curves, Hilbert modular forms and Galois deformations"; 43 pages

References in corpus (6)

• Regulator constants and the parity conjecture
• Root numbers and parity of ranks of elliptic curves
• Parity of ranks for elliptic curves with a cyclic isogeny
• Self-duality of Selmer groups
• Root numbers of elliptic curves in residue characteristic 2
• A note on the Mordell-Weil rank modulo n

Cited by in corpus (3)

• Generalized Heegner cycles at Eisenstein primes and the Katz $p$-adic $L$-function
• Root numbers and parity of local Iwasawa invariants
• Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture