Root numbers and parity of ranks of elliptic curves

TL;DR

This paper advances the understanding of elliptic curves by proving the parity conjecture assuming the Shafarevich-Tate conjecture, providing formulas for root numbers, and confirming a conjecture of Kramer and Tunnell in characteristic 0.

Contribution

It establishes the parity conjecture for all elliptic curves over number fields assuming the Shafarevich-Tate conjecture and completes the proof of Kramer and Tunnell's conjecture in characteristic 0.

Findings

Shafarevich-Tate conjecture implies the parity conjecture for elliptic curves over number fields

Formulas for local and global root numbers of elliptic curves are provided

Proof of Kramer and Tunnell's conjecture in characteristic 0 is completed

Abstract

The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.